Quantum Field Theory of Neutrino Oscillations

ISSN 1063-7796, Physics of Particles and Nuclei, 2020, Vol. 51, No. 1, pp. 1–106. © Pleiades Publishing, Ltd., 2020.Russian Text © The Author(s), 2020, published in Fizika Elementarnykh Chastits i Atomnogo Yadra, 2020, Vol. 51, No. 1.

Quantum Field Theory of Neutrino OscillationsD. V. Naumova, * and V. A. Naumova, **

aJoint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia*e-mail: [email protected]

**e-mail: [email protected] July 10, 2019; revised August 14, 2019; accepted August 14, 2019

Abstract—The theory of neutrino oscillations in the framework of the quantum field perturbative theory withrelativistic wave packets as asymptotically free in- and out-states is expounded. A covariant wave packet for-malism is developed. This formalism is used to calculate the probability of the interaction of wave packetsscattered off each other with a nonzero impact parameter. A geometric suppression of the probability of inter-action of wave packets for noncollinear collisions is calculated. Feynman rules for the scattering of wave pack-ets are formulated, and a diagram of a sufficiently general form with macroscopically spaced vertices(a “source” and a “detector”) is calculated. Charged leptons ( in the source and in the detector) are pro-duced in the space-time regions around these vertices. A neutrino is regarded as a virtual particle (propagator)connecting the macrodiagram vertices. An appropriate method of macroscopic averaging is developed andused to derive a formula for the number of events corresponding to the macroscopic Feynman diagram. Thestandard quantum-mechanical probability of f lavor transitions is generalized by considering the longitudinaldispersion of an effective neutrino wave packet and finite time intervals of activity of the “source” and the“detector”. A number of novel and potentially observable effects in neutrino oscillations is predicted.

DOI: 10.1134/S1063779620010050

CONTENTS1. INTRODUCTION 22. QUANTUM-MECHANICAL THEORY OF NEUTRINO OSCILLATIONS IN THE PLANE-WAVE APPROXIMATION 5

2.1. Quantum States 52.2. Plane-Wave Theory of Neutrino Oscillations in Vacuum 52.3. Incompleteness and Paradoxes of the Plane-Wave Approximation: Review of the Proposed Solutions 6

3. QUANTUM-MECHANICAL THEORY OF NEUTRINO OSCILLATIONS IN THE MODEL WITH A WAVE PACKET 8

3.1. General Properties of a Wave Packet 83.2. Mean Trajectory of a Wave Packet 83.3. Spreading of a Wave Packet in the Configuration Space 83.4. Transverse Spreading of a Wave Packet Leads to an Inverse Square Law 93.5. Noncovariant Gaussian Wave Packet 103.6. Theory of Vacuum Neutrino Oscillations in the Wave Packet Model 11

3.6.1. Neutrino state in the Wave Packet model and the amplitude of the transition from the source to the detector 11

3.7. Macroscopic Averaging of the Transition Probability 123.8. Qualitative Discussion of the Formula for the Oscillation Probability in the Wave Packet Model 13

4. RELATIVISTIC WAVE PACKET 154.1. Definitions 154.2. Mean Four-Momentum 184.3. Wave Packet for a Fermion 184.4. Commutator Function 204.5. Multipacket States 21

5. RELATIVISTIC GAUSSIAN PACKETS 235.1. Function 235.2. Mean and Root-Mean-Square Velocities 245.3. Wave Function 255.4. Commutator Function 275.5. Approximation of Nonspreading Packets 285.6. Effective Size and Mass of a Packet 315.7. Energy and Momentum Uncertainties of a Packet 315.8. SRGP Applicability Domain 32

6. SCATTERING OF WAVE PACKETS 346.1. Scattering Amplitude 35

±α β

∓

φ ,( )G k p

ψ ,( )G xp, ;$ ( )G xp q

1

2 D. V. NAUMOV, V. A. NAUMOV

6.2. Number of Interactions for Noncollinear Collisions of Wave Packets 366.3. Relativistic Invariance of the Impact Parameter Squared 37

7. MACROSCOPIC DIAGRAMS 387.1. Macroscopic Diagram of the General Form 387.2. Examples of Macrodiagrams 387.3. Feynman Rules 417.4. Overlap Integrals 427.5. Plane-Wave Limit 427.6. Overlap Tensors 427.7. Factors Governing the Energy-Momentum Balance 437.8. Impact Points, Geometric Suppression Factors, Symmetries, and All That 43

7.8.1. Translation group 447.8.2. Group of uniform rectilinear motions 447.8.3. Impact vectors 46

7.9. Asymptotic Conditions 477.10. Phase Factors 507.11. Overlap Volumes 50

8. AMPLITUDE OF THE PROCESS WITH THE PRODUCTION OF CHARGED LEPTONS IN THE SOURCE AND THE DETECTOR AND A VIRTUAL NEUTRINO 51

8.1. Amplitude Asymptotics at Large 538.2. Integration over 54

8.2.1. Ultrarelativistic case 548.2.2. Nonrelativistic case 56

8.3. Resulting Formula for the Amplitude 578.4. Effective Wave Packet of an Ultrarelativistic Neutrino 59

9. MICROSCOPIC PROBABILITY OF MACROSCOPICALLY SEPARATED EVENTS 6210. PROBABILITY AND COUNT RATE 62

10.1. Macroscopic Averaging 6311. DECOHERENCE FUNCTION 67

11.1. Synchronized Measurements 6711.2. Diagonal Decoherence Function 6811.3. Nondiagonal Decoherence Function 7011.4. Additional Methodological Remarks 73

12. DISCUSSION 7412.1. How a Virtual Neutrino Becomes Real 7412.2. Neutrino Wave Function and the Process Amplitude 7412.3. Ensemble Averaging and Role of Relativistic Covariance 75

L

0q

PHYSICS O

12.4. Probability of Flavor Transitions 7512.4.1. Synchronized and nonsynchronized measurements 75

12.5. Wave Packet Observability 7713. CONCLUSIONS 78APPENDIX A. PROPERTIES OF OVERLAP TENSORS 81

A.1. General Formulas for and 81

A.2. Inverse Overlap Tensors 81A.3. Two-Particle Decay in the Source 82

A.3.1. Formulas for arbitrary momenta 83A.3.2. PW limit 84

A.4. Quasi-elastic Scattering in the Detector 86

A.4.1. Formulas for arbitrary momenta 86A.4.2. PW limit 87A.4.3. Low-energy limits of functions and 89A.4.4. High-energy asymptotics of functions and 89

A.5. Three-Particle Decay in the Source 92APPENDIX B. MULTIDIMENSIONAL GAUSSIAN QUADRATURES 93APPENDIX C. FACTORIZATION OF HADRON BLOCKS 94APPENDIX D. STATIONARY POINT FOR AN ARBITRARY CONFIGURATION OF MOMENTA OF EXTERNAL WAVE PACKETS 95APPENDIX E. SPREADING OF A NEUTRINO WAVE PACKET AT EXTREMELY LONG DISTANCES 97APPENDIX F. SPATIAL AVERAGING 100APPENDIX G. COMPLEX ERROR FUNCTION AND ASSOCIATED FORMULAS 102

1. INTRODUCTIONA neutrino is a light fermion with zero electric

charge that participates in weak interactions. This par-ticle was proposed by Pauli in 1930 as a means to pre-serve the energy conservation law and explain the“incorrect” statistics in nuclear -decays and was usedshortly after by Fermi in the development of the firstquantitative theory of -decay [1]. Fermi had rightlyassumed that a neutrino is produced in the decay of anucleus together with a beta particle1, while Paulibelieved neutrinos to be constituents of nuclei. Three

1 Pauli’s famous letter was published in [2] with comments andinsightful historical remarks. The preliminary report of Fermiwas published in 1933 [3].

μν,ℜs d

μν,ℜ s d

μν,ℜ s d

0

0

Fd nd

Fd nd

β

β

F PARTICLES AND NUCLEI Vol. 51 No. 1 2020

QUANTUM FIELD THEORY OF NEUTRINO OSCILLATIONS 3

neutrino f lavors have been discoveredsince then. They are associated with the correspond-ing charged leptons : the production oflepton in weak interactions is always accompaniedby the production of neutrino of the same flavor .This may be formulated as the law of conservation oflepton number .

By analogy with oscillations of neutral kaons,, Bruno Pontecorvo hypothesized the exis-

tence of neutrino f lavor oscillations [4, 5] in the late1950s. The validity of this hypothesis was verified inexperiments with solar [6–8], atmospheric [9, 10],accelerator [10, 11], and reactor [12–15] neutrinos andantineutrinos. Neutrino oscillations (or f lavor transi-tions) are a quantum effect of quasi-periodic varia-tions of neutrino f lavor in time. This effect,which violates the conservation of lepton numbers, isattributable to the nonequivalence of eigenstates ofneutrinos with a definite f lavor and states withdefinite masses the nonzero differ-ence in masses of the latter, and the hypothesis that aneutrino takes part in weak interactions as a coherentsuperposition of the mass eigenstates

(1)

Here, is an element of a unitary vacuum mixingmatrix, which is also called the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix [5, 16]. The tem-poral evolution of state (1) leads to an oscillatorydependence of the probability of detection of a neu-trino with a certain f lavor in time after the produc-tion of state The simplest quantum-mechanicaltheory of oscillations based on the plane-wave approx-imation was formulated by Gribov and Pontecorvo[17]. This theory had a great influence on research inneutrino physics and was developed further in subse-quent years by other physicists (see, e.g., [18–20] andreviews [21–31], where the advances in experimentalstudies and phenomenology of neutrino oscillations invacuum and matter as well as various mechanisms ofneutrino mass generation and mixing, which are notaddressed in the present paper, were discussed indetail).

Although the plane-wave approximation is veryefficient in characterizing the results of experiments, itis not self-consistent and leads to a number of para-doxes, which were discussed widely in literature. Forthose interested, we have compiled a list of studies[32–65] directly related to the subject of the presentpaper. Although this list is extensive, it is not exhaus-tive by any means. A systematic formulation of thetheory itself, critical comments, generalizations, andreferences for further study are given, e.g., in [66].

α μ τν = ν ,ν ,ν( )e

α = ,μ, τ ( )e+α

αν α

αL

↔0 0K K

α βν → ν

ανν ,k = , ,( 1 2 3),k

α α=

∗ν = ν .3

1k k

k

V

αkV

β tαν .

PHYSICS OF PARTICLES AND NUCLEI Vol. 51 No.

The need to describe neutrino oscillations in amodel with neutrino wave packets was noted in [67–72](see also [63, 66, 73–79]). A wave packet (WP) is acoherent superposition of waves with momenta dis-tributed about the most probable value with a certain“width” (dispersion). A WP is localized in themomentum and configuration spaces. The theory ofneutrino oscillations in a model with WPs provides anopportunity to resolve some paradoxes of the plane-wave approximation and predicts several novel phe-nomena. Following the publication of pioneeringstudies [67–72], two distinct paths towards the devel-opment of this theory of oscillations have emerged. Inthe first approach, the formalism was developed withinrelativistic quantum mechanics, and the neutrino wavefunction was postulated (see, e.g., [45, 66, 80]). In thesecond approach, the theoretical apparatus of quan-tum field theory (QFT) with wave packets for all par-ticles involved in the processes of production anddetection of neutrinos (except for neutrinos them-selves, which are regarded as virtual particles) was used[32–48, 51–58, 60–65]. The form of the neutrinowave packet follows from the formalism in this para-digm. Both approaches predicted novel potentiallyobservable effects, which were related to the loss ofcoherence. The interest in decoherence in neutrinooscillations has been on the rise lately, since severalrecent (and future) experiments are potentially sensi-tive to these effects [81–95].

The aim of this work is to examine the theory ofvacuum neutrino oscillations thoroughly and system-atically within the QFT perturbative approach withrelativistic WPs serving as external in- and out-states.This theory has been outlined earlier in [57, 58, 96].We start with a brief discussion of the standard quan-tum-mechanical theory of neutrino oscillations in theplane-wave approximation (see Section 2). As wasalready noted, the quantum-mechanical theory ofneutrino oscillations in the plane-wave approximationis incomplete and contradictory. The underlying fac-tors are discussed in Section 2.3.

The model with a wave packet (see Section 3)solves some of the problems inherent in the theorywith plane waves. This model introduces WP spread-ing and the procedure of macroscopic averaging,which allows one to derive a correct formula for theoscillation probability.

The focus of the paper is on the construction of thetheory of neutrino oscillations within QFT. Conceptu-ally, the mechanism of neutrino oscillations within QFTis the result of interference of diagrams with mass eigen-fields ( ) in the intermediate states. Let ushighlight the key stages and features of our approach.

(1) The problem is formulated as follows. Two spa-tial regions with a macroscopically large intervalbetween them are considered. Charged leptons (and ) are produced in these regions (the “source”

νi = , ,1 2 3i

+α

−β

1 2020


and “detector,” respectively). The source and thedetector remain active within certain time intervals.This corresponds to the most general setup of an oscil-lation experiment. One needs to calculate the numberof such events ( ) within QFT.

(2) Naturally, number of events in the detectoris independent of the reference frame (i.e., is a relativ-istic invariant). Therefore, quantum states corre-sponding to external particles should transform covar-iantly under Lorentz transformations. This impliesthat a WP corresponding to any external particle isrequired to be covariant. The theory of a covariant WPis constructed in Section 4; its properties are alsoexamined in detail. Section 4.5 is focused on multi-packet states for fermions and bosons. In particular, itis demonstrated that if WPs corresponding to identicalparticles are separated by sufficiently large space-timeintervals, the system lacks the Bose–Einstein conden-sation for bosons and the Pauli Exclusion Principle isnot valid for fermions. This intuitively obvious result isimpossible to infer in the model with plane waves.

(3) The model of a relativistic Gaussian WP (RGP)is developed as a working model for further calcula-tions in Section 5. It is demonstrated that a traditional

Gaussian WP of the form is a nonrelativ-istic approximation of an RGP. This is the key resultpresented in Section 5.

(4) The above formalism is applied in solving theproblem of scattering of two WPs in Section 6. Thegeneral case of WP scattering with a nonzero impactparameter is considered. It is demonstrated that thenumber of interactions of two WPs may be presentedas a product of three factors: the cross section (withthe dimension of area), the luminosity (with thedimension of reciprocal area), and a dimensionlessfactor that suppresses interactions at large impactparameters. This result is intuitively expected andillustrates the predictive power of the theory.

(5) A “macroscopic” diagram with vertices (theneutrino source and the detector) separated by a mac-roscopic distance is examined in Sections 7 and 8. Theinitial and final particles are characterized by the con-structed covariant WPs; and neutrinos are describedby a propagator. Charged leptons and are pro-duced in the source and the detector, respectively. TheFeynman rules for such diagrams are formulated, andthe amplitude of the process of interest is calculated.No form of the virtual neutrino WP is assumed; acausal fermion propagator, which emerges automati-cally in QFT, is used instead. It is demonstrated belowthat the asymptotic behavior of the propagator at largedistances between vertices of the diagram reproducesthe properties of the wave function of a free neutrino;in other words, a free neutrino at large distances fromthe source behaves, in a first approximation, as a freewave packet of a particle on the mass shell.

the

αβN

αβN

− − / σ∝2 2( ) 4 pe k p

+α

−β

PHYSICS O

(6) Microscopic probability is calculated inSection 9. Once each external (in or out) WP is char-acterized by mean coordinate at time , mean4-momentum , and momentum dispersion ,

depends parametrically on the set of all theseparameters .

Macroscopic averaging over is performedin Section 10. This provides an opportunity to deter-mine number of events with in the source and

in the detector. The terms “source” and “detector”now acquire their meaning as space-time regionsaround the so-called impact points and definedby parameters at the vertices, where leptons

and are produced. The conditions under which may be presented as an integral over the source

and detector volumes of the product of three factors are examined.

Here, is the f lux of massless neutrinos fromthe source to the detector (it is demonstrated that itbehaves as ); is the cross section of scatter-ing of a massless neutrino in the detector; and ,which depends on flavor indices and , the neutrinoenergy, the effective distance between the source andthe detector, and on neutrino energy-momentum dis-persion , is a generalization of the quantum-mechanical probability of neutrino oscillations. Theobtained “probability” of oscillations is then exam-ined in different experimental settings. Under certainconditions, the obtained formula for the oscillationprobability is numerically close to that from the plane-wave model. However, the new formula predicts anumber of novel effects such as the loss of coherence,the dependence on the temporal windows of activity ofthe source and the detector, and many others.

The decoherence function, which suppresses inter-ference contributions to the oscillation probability, isdiscussed in detail in Section 11.

(7) The main results presented in this article arediscussed in Section 12, and conclusions are drawn inSection 13.

For convenience, the details of cumbersome calcu-lations are given in the appendices. A natural system ofunits, where , is used. Abbreviations of theform of instead of in Minkowski spaceintegrals (the Feynman metric is used throughout thepaper) and instead of in three-dimen-sional Euclidean space integrals are used in thosecases where this does not lead to misunderstanding.Although the number of active f lavors is not essentialto our analysis, we assume that this number is three inthe particular examples considered below.

αβA2

κx κ0x

κp κσ

αβA2

κ κ κσ , , x p

κ κ, x p

αβN +α

−β

sX dXκ κ κ, ,σ x p

+α

−β

αβN

ν ν αβ ν νΦ , × , ; × σ3 D( ) ( ) ( )E L E L E

ν νΦ ,( )E L

21 L νσ( )Eαβ3

α β

D

= = 1c

…dx …

4d x

…dx …

3d x



2. QUANTUM-MECHANICAL THEORY OF NEUTRINO OSCILLATIONS

IN THE PLANE-WAVE APPROXIMATION2.1. Quantum States

Let us first recall the key concepts of quantummechanics essential to the quantum-mechanical the-ory of neutrino oscillations and introduce the defini-tions that will be used in subsequent analysis. We startwith the one-dimensional case and a free theory.

A quantum system in the Hilbert space is charac-terized by abstract time-dependent state . Itsevolution in time is given by the Schrödinger equation

(2)

where is the Hamiltonian. Let be an eigen-state of the free Hamiltonian with eigenvalue :

(3)

Let be an eigenstate of coordinate operator with eigenvalue :

(4)The norms of these states are as follows:

where isthe common Dirac delta function. Quantum state

may be characterized either by coordinate distri-bution in the coordinate representation or bymomentum distribution in the momentumrepresentation for a free or a full Hamiltonian:

(5)

States and are related in the following way:

(6)

In view of (5) and (6), it is evident that

(7)

The formal solution of (2)

(8)by using representation (5) yields:

(9)

where

(10)

ψ( )t

∂ ψ = ψ ,∂

( ) ˆ ( )ti H tt

= 0ˆ ˆH H p

0H pE

= .0ˆ pp E pH

x Xx

= .X x x x

= πδ −2 ( ),q p p q = δ −( ),y x x y δ −( )x y

ψ( )tψ ,( )x t x

ψ ,( )p t p

ψ = ψ , = ψ ; .π ( ) ( ) ( )

2x pdpt dx t x x t p p

x p

− += , = .π 2

ipx ipxdpx e p p dxe x

−

ψ , = ψ , ,π

ψ , = ψ ,

( ) ( )2

( ) ( ).

ipxx p

ipxp x

dpt x e t p

t p dxe t x

−ψ = ψˆ

( ) (0)iHtt e

−ψ = ψ , ,π

= ψ , ; ,

( ) (0 ) 02

(0 )

piE tp

x

dpt e p p

dx x t x

−; = .0îH tt x e x


Expressions (6) and (7) were used to derive (9):

(11)

If where is the momen-tum that can be considered as a parameter of function

, solution (9) takes the well-known form

(12)

that is, the temporal evolution of a state with a strictlydefined momentum is characterized by factor multiplied by a state with a definite momentum inthe momentum representation or by a superposition ofstates with a definite coordinate with weight factor

. This solution is known as a plane wave.

2.2. Plane-Wave Theory of Neutrino Oscillations in Vacuum

Let us define a f lavor state as a coherent super-position of mass states :

(13)

where each mass state is an eigenstate of a free Hamil-tonian with eigenvalue . The evolution of state (13) isgiven by

(14)

where . Let us assume that a coherentsuperposition of states with strictly definedmomenta (13) was produced in a certain weak pro-cess. The projection of this state onto a state with arbi-trary f lavor is not reduced to :

(15)

The corresponding probability of f lavor transition is

(16)

Using ultrarelativistic approximations and putting , where is

− −

− +

ψ , = ψ ,π π

= ψ ,π

= ψ , , .

0

0

ˆ

ˆ

(0 ) (0 )2 2

(0 )2

(0 )

piE t iH tp p

iH t ipxp

x

dp dpe p p e p p

dpdxe e p x

dx x t x

ψ , = πδ −(0 ) 2 ( ),p p p k k

ψ ,(0 )p p

− − +ψ = = ,( ) k kiE t iE t ikxt e k dxe x

− kiE tek

x− +kiE t ikxe

ανν ( )i p

α α∗ν = ν , ( )i i

i

V p

p

− −α α α

∗ν = ν = ν ,ˆ

( ) ( )iiHt iE ti i

i

t e V e p

= +2 2i iE p m

βν αβδ

−β α α β

∗ν ν = .( ) iiE ti i

i

t V V e

α βν → ν

− −αβ β α α α β β

∗∗= ν ν = .( )2( ) ( ) i ji E E t

i j i jij

P t t V V V V e

− −

2 2( ) 2i j i jE E m m p =t L L

1 2020


the distance travelled by a neutrino in the time , weobtain the well-known formula

(17)

where

The expanded form of (17) is

(18)

In the two-neutrino case, probability is a peri-odic function of distance with a period equal to length

. The distance dependence is what justifies the term “neutrino oscillations.” Inthe three-neutrino case, this term is not entirely cor-rect, since oscillation probabilities may have no peri-odicity (except special cases with peculiar relationsbetween the oscillation lengths , , ). In thegeneral case, it would be better to call neutrino oscil-lations as quasi-periodic.

The unitary property of the vacuum mixing matrixgives rise to the expected probability conservation law:

(19)

Note that although the energy of the state withdefinite f lavor is not defined, its mean energy

(20)is a well-defined and conserved quantity. Indeed,

The mean energy of an arbitrary entangled statecharacterized by a certain density matrix is alsoconserved. Indeed, let the initial state have the form

(21)

t

− Δ /αβ α α β β

− π /α α β β

∗ ∗=

∗ ∗=

2

osc

2

2

( )

,

ij

ij

i m L pi j i j

ij

i L Li j i j

ij

P L V V V V e

V V V V e

Δ = − = πΔ

2 2 2 osc2

2and 2 .ij i j ijij

pm m m Lm

( )

( )

αβ α β

α β α β>

α β α β

=

π∗ ∗+

π∗ ∗+ .

2 2

osc

osc

( )

22 Re cos

2Im sin

j jj

j j i ij i ji

j j i iji

P L V V

LV V V VL

LV V V VL

ναβ2 ( )P L

Losc12L ν ν

αβ αβ+ =2 osc 212( ) ( )P L L P L

osc12L osc

23L osc13L

αβ αββ α

= = . 1P P

αν ( )t

α α α= ν νˆ( ) ( ) ( )E t t H t

α α α

α α

α α

αα

∗= ν ν

∗= ν ν

= + ,

= + .

22 2

2

ˆ( ) ( ) ( )

( ) ( )

2

32

i j i jij

i j i i jij

ii i i

i i

ii

i i

E t V V p H p

V V p E p

mV E p Vp

mE E pp

ρ( )t

α α αα

ρ = ν ν ,ˆ(0) (0) (0)w

PHYSICS O

where is the probability of finding pure state in this (mixed) state ( ). The mean energy ofthe mixed state at arbitrary time instant is thenwritten as

Naturally, in the special case of apure initial state If all pure states in initial mix-ture (21) were equally probable, the mean energy

is independent of the mixing parameters.

2.3. Incompleteness and Paradoxes of the Plane-Wave Approximation: Review of the Proposed Solutions

The following crucial assumptions and approxima-tions were used in derivation of the formula character-izing the probability of oscillations:

(i) A coherent superposition of states with definitemasses ( ), which is often called a state with

definite f lavor , where are ele-ments of the unitary vacuum mixing matrix, interactsin the processes of neutrino production and detection.

(ii) States have definite momenta .(iii) All momenta are the same ( ).

(iv) Neutrinos are ultrarelativistic ( ).(v) The neutrino propagation time can be replaced

by the distance travelled ( ).Let us scrutinize all these presumptions. At first

glance, assumption (i), which states that a neutrinointeracts as state in the production and detectionprocesses, appears to be fairly natural. Moreover, inliterature one can often find a mystic statement thatneutrinos interact with matter as f lavor eigenstates,

but between the interaction actsthey evolve in space as mass eigenstates. Note, however,that the SM Lagrangian of interactions of massive neutri-nos with charged leptons can be writteneither in therms of fields and ,α(x) or να(x) and

,i(x) = since the charged lepton current

can be represented in two fully

equivalent forms: and

It is then instructive to ponder

αw αν (0)

αα= 1w

ρ( )t t

( ) ( )−

− −∗α α α

α

α α α αα α

= ρ = ρ

= ν ν

= = .

ˆ ˆ

( )

2

ˆ ˆˆ ˆ( ) Tr ( ) Tr (0)

Tr ( ) ( )

( )

i j

iHt iHt

i E E ti j i i j

ij

i ii

E t H t He e

w V V e E p p

w V E w E t

α=( ) ( )E t E tαν (0) .

αα

= = + ,

21 1( ) ( )3 3 6

ii

i i

mE t E t E pp

im = , ,1 2 3i

α α∗ν = ν i ii

V αiV

νi ip

ip =ip p

@2 2max( )imp

=t L

αν

α α∗ν = ν ,i ii

V

α α = μ τ( , , )l eν ( )i x

α αα∗ ( ),iV x,

μ μα α= ν( ) ( ) ( )j x x O x, ,

μα αα∗ν ( ) ( )i ii

V x O x,

μα αα∗ν ( ) ( ).i ii

V x O x,



the question whether it is possible for a coherentsuperposition of states to be pro-duced and interact and whether oscillations of chargedleptons (similar to neutrino oscillations) are observ-able. Since there is no experimental evidence of suchoscillations, one has to admit that charged leptons areproduced (and interact) incoherently. Therefore,assumption (i) is not valid for charged leptons and, inthe general case, requires a quantitative justification.

Let us now turn to assumption (ii) and discuss the decay as an example. Since the state

does not have definite energy, the energy of at leastone of the two final-state particles is also undefined.Since the pion and muon have definite masses, themomentum of one of them (or both particles) is alsoundefined. Therefore, the momentum of the final-state neutrino should be undefined, but this contra-dicts assumption (ii). Another argument against thisassumption consists in the fact that the spatial coordi-nates of the regions of neutrino production andabsorption (the source and the detector) and, conse-quently, the distance between these regions, involvedin the formula for the f lavor transition probability, arecompletely undefined if the momentum is completelydefined.

It is worth noting here that an unstable particlecannot have definite 4-momentum only due to thefact that the spatial delocalization of such a particle islimited by its decay path length. In other words, theindeterminacy of the 4-momentum of a particle can-not be smaller than its decay width. Therefore, themomenta of neutrinos produced in particle decaysshould also be undefined; i.e., assumption (ii) is amere approximation, and a more general theory isneeded to verify its validity.

Assumption (iii), as well as the similar assumptionof equality of energies found in certain studies, is atodds with relativistic invariance. Let us assume that(iii) holds true in laboratory frame . Both energies

and momenta are then not equal in frame moving with velocity relative to , since it followsfrom the Lorentz transformations that

(22)

It is easy to see that

at . Ifthe velocity of frame satisfies condition

, neutrinos remain ultrarelativisticin ; i.e., condition (iv) is satisfied, but condition (iii)is violated. Naturally, this violation is infinitesimal ifvelocity is low (more precisely, if ). In the

α αα∗= i iV

+ +μπ → μ ν μν

K

'iE 'ip 'K

v K

− = Γ − ≠ ,

− = Γ − ≠ , Γ = .− 2

' ' ( ) 01'' ( ) 0

1

i j i j

i j j i

E E E E

E E

v

v vp p vv

− ≥ − ≈ −2 2' ' (2 )i j i j i jE E m mp p p ≥ 1 2v'K

Γ ! 'min( )i iE mv

'K

v !vp p


general case, regarding the Lorentz transformation asan active one, we conclude that condition (iii) cannotbe simultaneously satisfied for two neutrino beamsemitted by identical sources moving with differentvelocities.

Condition (iv) states that (a) the masses of knownneutrinos are infinitesimal (much below 1–2 eV) and(b) modern current detection techniques are not sen-sitive to neutrinos with energies below ~100 keV; thus,only ultrarelativistic neutrinos are probed in experi-ments. However, this condition is not satisfied for relicneutrinos.

Finally, let us turn to condition (v). This conditionis not equivalent to and does not follow from (iv),which allows for expansion ,since the second expansion term (and, in certain stud-ies, the third) is preserved in (iv) together with the firstterm . By contrast, condition (v) utilizes onlythe first expansion term . The elementaryexpression reproducing the exact relation betweentime and the distances travelled by states and

is

which allows one to rewrite oscillation phase in the following form:

where approximation was used to obtainthe second-to-last expression. The resulting oscilla-tion phase is two times larger compared to (17). Thisdisconcerting fact is the result of voluntary treatment

of orders of smallness of , which is inherent in

both the above “alternative derivation” and condition(v) itself. One may ask oneself, why the differencebetween the times of neutrino detection and productionis equated to distance ? The time of production of theneutrino is unknown in the majority of experiments;therefore, one should integrate over it. Having performedthis integration, we find that all interference terms van-ish, since ; i.e.,

(23)

The theory of neutrino oscillations is based onassumptions (i)–(v). Only condition (iv) is satisfied incurrent experiments (although it is violated for relic

+ +

2 2i iE p m p …

=E p=t L

t νk

ν j

= = ,v v

jk

k j

LLt

−( )k jE E t

− = −

− Δ= − ≈ = ,

v v

2 2 2 22

( )

( )

j jk kk j

k j

j j k j kjk k

E LE LE E t

E L E E L mE Lp p p p

≈ ≈k jL L L

Δ 2

2kjm

p

L

[ ]− − ∝ δ − exp ( ) ( )k j k jdt i E E t E E

αβ α β∝ . 2 2( ) k ktk

P t V V

1 2020


neutrinos). Conditions (ii) and (iii) are evidently non-physical. Condition (i) requires a quantitative justifi-cation. Condition (v) is unjustified. Small correctionsto it induce significant changes in the resultant for-mula for the probability of neutrino oscillations.

3. QUANTUM-MECHANICAL THEORY OF NEUTRINO OSCILLATIONS

IN THE MODEL WITH A WAVE PACKET3.1. General Properties of a Wave Packet

Let us consider a WP at :

(24)

where The evolution of statein (24) with time is given by

(25)

which corresponds to the following wave function inthe coordinate space:

(26)

Let us normalize the state in (25) to unity:

(27)

The mean energy, momentum, and velocity of apacket are defined as follows:

(28)

where the one-particle velocity operator is definedusing

(29)

A quantum WP is similar in some respects to a clas-sical object; in particular, its energy and momentumare conserved, and it moves (on average) along a clas-sical trajectory. Let us establish the latter assertion byperforming explicit calculations.

3.2. Mean Trajectory of a Wave Packet

By definition, the mean WP coordinate is

(30)

= 0t

ψ = ψ ,π p3(0) ( )

(2 )dp p p

= π δ −3 3(2 ) ( ).q p q p

−−ψ = ψ = ψ ,π0

ˆp3( ) (0) ( )

(2 )iE tiH t dt e e pp p p

− +ψ , = ψπx p3( ) ( ).

(2 )iE t idt e p pxpx p

ψ ψ = ψ =π

2p3( ) ( ) ( ) 1.

(2 )dt t p p

= ψ ψ = ψ ,π

= ψ ψ = ψ ,π

= ψ ψ = ψπ

20 p3

2p3

2p3

ˆ( ) ( ) ( )(2 )

ˆ( ) ( ) ( )(2 )

ˆ( ) ( ) ( ) ,(2 )

dE t H t E

dt t

dt t

p

p

p p

pP P p p

pv V p v

= .π 3

ˆ(2 )

dp

pV v p p

= ψ ψ .ˆ( ) ( ) ( )t t tx X

PHYSICS O

Expressing in accordance with (25) andmaking use of relations and

, we determine the right-hand part of (30)by explicit calculations:

(31)

where was used. The integral may beremoved:

(32)

Then,

(33)

where is given by the third expression from (28).Integration by parts was performed in the transitionfrom the first line in (33) to the second. Thus, it wasproven that, on average, the WP coordinate follows aclassical trajectory:

(34)

It was explicitly assumed in this proof that themean WP coordinate is zero at .

3.3. Spreading of a Wave Packet in the Configuration Space

Let us demonstrate by direct calculations that a WPspreads with time. By definition, the spatial coordi-nate dispersion is

(35)

The mean value of the coordinate squared is

(36)

ψ( )t+=

id e pxp x x

=X x x x

− −

+

− − + −

∗ψ ψ = ψ ψπ

×

∗= ψ ψπ

( )p p6

( ) ( )p p6

ˆ( ) ( ) ( ) ( )(2 )

( ) ( ) ,(2 )

i E E t

ikx

i E E t i x

d dt t e

d e

d d e d e

k q

k q k q

k qX q k

q xx x

k q q k xx

−= ie qxq x dx

− − ∂ ∂= − ∂ ∂

∂ ∂= − π δ − ∂ ∂

( ) ( )

3 3

12

1 (2 ) ( ).2

i id e d ei

i

k q x k q xxx xk q

k qk q

( )

− −

− −

∗= ψ ψπ

∂ ∂ × − δ − = − δ − ∂ ∂ π

∂ ∂ ∗× − ψ ψ∂ ∂

= ψ =π

( )p p3

3 33

( )p p

2p3

1( ) ( ) ( )2 (2 )

1( ) ( )2 (2 )

( ) ( )

( ) ,(2 )

i E E t

i E E t

d dt ei

d di

e

d t t

k q

k q

k

k qx q k

k qk q k qk q

q kk q

k k v v

v

= .( )t tx v

= 0t

σ = − .22 2( ) ( ) ( )x t t tx x

− − + −

≡ ψ ψ∗= ψ ψ .

π

2 2

( ) ( )2p p6

ˆ( ) ( ) ( )

( ) ( )(2 )

i E E t i

t t td dd e k q k q x

x Xk qxx q k



Using the approach similar to (32)

and, following (33), integrating by parts in (36), wearrive at

Thus,

Setting , we find that

is the coordinate dispersion at the initial time. Thus,

(37)

where is the WP velocity dispersion.This is the general dispersion law of a nonrelativisticWP, which has already been obtained earlier in [97].

Let us examine how the WP spreading rate dependson relativistic effects. We write in the followingform:

(38)

where and are the mean squares of longitu-dinal and transverse projections of the WP velocitywith respect to mean velocity vector , respectively.Let us rewrite and in terms of vari-ables in the intrinsic frame of reference of the WP

(39)

(40)

(41)

+ − + − ∂ ∂= − + ∂ ∂

∂ ∂= − + π δ − ∂ ∂

2 2

2 ( ) ( )2 2

2 23 3

2 2

12

1 (2 ) ( ),2

i id e d ek q x k q xxx xk q

k qk q

− −

∂ ∂= − δ − + π ∂ ∂

∗× ψ ψ∗ψ ∂ ψ

= − .π ∂

2 22 3

3 2 2

( )p p

22 2 p p

3 2

1 ( )2 (2 )

( ) ( )

( ) ( )Re

(2 )

i E E t

d d

e

dt

k q

k qx k qk q

q k

k kkvk

( )∗ψ ∂ ψ

σ = − − .π ∂

222 2 2 p p

3 2

( ) ( )( ) Re

(2 )x

dt tk kkv v

k

= 0t

∗ψ ∂ ψσ = −

π ∂2

2 p p3 2

( ) ( )(0) Re

(2 )x

d k kkk

( )σ = σ + − = σ + σ ,22 2 2 2 2 2 2( ) (0) (0)x x xt t tvv v

σ = − 22 2v v v

σ2v

σ = − = + −= + − ,

2 22 2 2 2

2 2( )T L

T L

v v v v v v

v v v

2Tv 2

Lv

v2Tv − 2( )Lv v

∗∗= Γ + ,( )LE Ek k v k

∗ ∗= Γ + , = ,*( )L L T TEkk k v k k

∗ += , Γ ≡ Γ =∗ −+ 2

111

LL

L

kk v

k

v vvvv v


in the following way:

(42)

and

(43)

Comparing (42) and (43), one may note that thefollowing is true in the first order for WPs that are suf-ficiently narrow in the momentum space:

Thus, using (37) and considering that

in the intrinsic frame of theWP, we obtain the following in the full spreadingregime:

(44)

The rate of longitudinal WP spreading is timeslower than the rate of transverse spreading. It is evi-dent that

in the intrinsic frame of reference of the WP.

3.4. Transverse Spreading of a Wave Packet Leads to an Inverse Square Law

Any WP spreads with time. In other words, the spa-tial dispersion of a WP increases with time, while itsamplitude decreases. This well-known and inherentproperty of WPs is often considered to be their “draw-back.” This stimulated the search for temporally stablesolutions needed to associate a WP with a particle. Inparticular, although a consistent scattering theory isknown to be impossible to construct without WPs[98, 99], it is not uncommon to come across suchassertions that the decay of the amplitude with timemakes WPs not fully adequate objects to describe theinitial and final states, which are defined formally at

and , respectively. However, as will beshown below , and this is a new result, WP spreading isnot critical for the -matrix formalism, since it has aclear interpretation. This is a novel result. Just as in anensemble of classical particles, WP spreading results insuppression (proportional to ) of the time-inte-

= ψπ

∗= ψ

∗Γ π +

22 2p3

22

p2 3 2

( )(2 )

1 * ( *) ,(2 ) (1 )

T T

T

L

d

d

k

k

k

kv k v

vk kv v

( )

− = ψ −π

∗= ψΓ π ∗+

22 2p3

22

p4 3 2

( ) ( ) ( )(2 )

1 * ( *) .(2 ) 1

L L

L

L

d

d

k

k

k

kv v k v v

vk kv v

∗ ∗= Γ , − = Γ .2 2 2 2 2 4( )T T L Lv v v v v

∗ ∗ ∗= =2 2 21 12 3L Tv v v

∗ ∗σ = Γ , σ = Γ ,

σ = σ Γ .

2 2 2 4 2 2 2 2

2 2 2

1 2( ) ( )3 3

1( ) ( )2

xL xT

xL xT

t t t t

t t

v v

Γ

∗σ = σ + σ = .2 2 2 2 2( ) ( ) ( )x xL xTt t t tv

= −∞t = +∞t

S

21 x

1 2020


grated probability of finding a particle at distance from the point of its production. Thus, the result leadsto simple normalization factors for the initial and finalstates taken at sufficiently long (but finite) times.

The assertion regarding the nature of the inversesquare law was proven in [96] in three different ways:

(i) For two explicit examples of : a Gaussianfunction that is invariant or not invariant with respectto relativistic transformations.

(ii) For a WP of an arbitrary form.(iii) Based on the continuity equation.In the present study, we utilize the simplest and

fairly general approach based on the continuity equa-tion, which agrees with both relativistic and nonrela-tivistic equations of motion.

Let us write the continuity equation for an arbitraryquantum state with scalar function , which isthe probability density of finding a particle at point at time , and vector function , which is the f luxdensity:

(45)

This equation may be rewritten in equivalent form

(46)

where integration is bounded by sphere with radius If this radius is sufficiently large and the time is too

short for a WP to spread (both conditions are satisfiedif ), the integral is expected to be equal tounity based on the normalization condition

therefore, the time derivative of the integral at the left-hand side of this equality is negligible, and the right-hand side of (46) is also negligible. This implies thatthe f lux density should decrease faster than Theexample of a Gaussian WP considered below is inagreement with this assertion.

Let us now integrate over time the right- and left-hand sides of (46) from zero to infinity:

(47)

where is the time integral of theflux density. By definition,

(48)

is the probability of finding a particle inside a spherewith radius at the time instant . Naturally, owing to

x

ψp( )p

ρ ,( ')t x'x

t ,( ')tj x

∂ρ ,+ , = .

∂( ')

( ') 0t

ttx

j x∇

| |≤| |

∂ ρ , = − , ,∂

'

' ( ') ( ')S

d t d tt x x

x x Sj x

S.x

σ@ ( )x tx

| |≤| |

ρ , ≈'

' ( ') 1;d tx x

x x

21 .x

| |≤| | | |≤| |

ρ ∞, − ρ , = − , ' '

' ( ') ' (0 ') ( ')S

d d dx x x x

x x x x S xΦ

∞= ,0( ') ( ')dt tx j xΦ

| |≤| |

, ≡ ρ ,'

( ) ' ( ')P t d tx x

x x x

x t

PHYSICS O

the WP dispersion, the probability of finding a particlewithin an arbitrary finite volume tends to zero as

, since the WP leaves this volume. On the otherhand, if is much larger than the WP size, isvery close to unity, since the WP initially was localizedalmost completely within a sufficiently large volume.Under these conditions, (47) turns into

(49)

The modulus of vector in the intrinsic frameof reference of the WP is independent of the directionof vector . Therefore,

in this frame.

Thus, inverse square law is true for any solu-tions of the Klein–Gordon and/or Dirac equationswith finite normalizations but is not valid for planewaves with a singular normalization. At any giventime, f lux density decreases faster than

3.5. Noncovariant Gaussian Wave PacketLet us examine the useful example of a well-known

noncovariant Gaussian WP with a wave function ofthe following form in the momentum space:

(50)

where is the constant dispersion of the Gaussiandistribution, or the “width” of the momentum distri-bution in a WP. The wave function in the coordinaterepresentation may be obtained by assuming that dis-persion is sufficiently small to calculate accuratelythe integral with expansion

(51)

where . Integration reduces to a Gaussianquadrature, which provides an opportunity to calculate

(52)

where , ,

(53)

→ ∞tx ,(0 )P x

| | ≤| |

ρ , = .

'

' (0 ') ( ') 1S

d dx x

x x S xΦ

( ')xΦ

'x

π 2

1( )4

xx

Φ

21 x

,( ')tj x∇21 .x

( )−σ

−/π ψ = , σ

2

24 p3 4

p 2p

2( ) ek p

k

σp

σp

( )= + −

×+ − + + ,22

23 3

( )

( )2 2

E E

m …E E

k p p

p p

v k p

p kk p

= Epv p

( )

/

ψ , −− − −

σ + τ σ + τ = ,πσ + τ + τ

x2 2

2 2x

2 3 4x

( )

exp4 (1 ) 4 (1 )

(2 ) (1 )(1 )

L T

x L T

L T

t

tipx

it it

it it

x

x v x

= ,( )p Ep p σ = σ2 2x p1 4

τ = Γ τ τ = Γ τ τ = σ Γ =3 2x, , 2 , .L T

Em

mp

p p p



Here, xL and xT are the components of vector that are parallel and perpendicular to mean velocityvector , respectively. Expression (52) shows clearlythat the WP spreads with time. For greater clarity, wewrite down the modulus of function (52):

Thus, a Gaussian WP characterized at by awave function in the configuration space

spreads with time in the longitudinal and transversedirections. The squares of longitudinal andtransverse dispersions are

(54)

where and given by (53) are related as Parameters and have the physical meaning oflongitudinal and transverse spreading times, respec-tively. Parameter is the WP dispersion time in theintrinsic frame, where

Note that the spreading rate of a noncovariantGaussian packet in the longitudinal direction is times lower than in the transverse one. A Gaussian WPhas two spreading regimes: transverse ( ) andlongitudinal ( ). In the case of full spreading inall directions,

Comparing the obtained rates of longitudinal andtransverse spreading with the results of calculationsthat make use of relativistic relations (44), we findthat the spreading rate of a noncovariant Gaussianpacket is times lower. The covariant Gaussianpacket considered in the next section agrees withgeneral formula (44).

Let us verify that the Gaussian WP spreading leadsto in suppression of the time-integrated f lux.

x

v

( )

( ) //

−− − σ + τ σ + τ ψ , = .

πσ + τ + τ

2 2

2 2 2 2 2 2x x T

x 1 42 3 4 2 2 2 2x

exp4 (1 ) 4 (1 )

( )(2 ) 1 1

L T

L

L T

t

t tt

t t

x v x

x

= 0t

/ ψ , = − πσ σ

2

x 2 3 4 2x x

1(0 ) exp(2 ) 4

i xx px

σ2 ( )xL t

σ2 ( )xT t

σ = σ + τ ,σ = σ + τ ,

2 2 2 2

2 2 2 2

( ) (1 )

( ) (1 )xL x L

xT x T

t t

t t

τL τT τ = Γ τ2 .L Tp

τL τT

ττ = τ = τ.L T

Γ2p

τ@ Ttτ@ Lt

σ = σ , τ ,τΓ

σ = σ , τ .τΓ

@

@

31( )

1( )

xL x L

xT x T

tt t

tt t

p

p

Γp

21 x


The flux density is calculated in accordance with thestandard formula

Let us perform calculations in the intrinsic frame ofreference of the WP. The f lux density is then

(55)

The time integral may be calculated exactly:

(56)

The following formula is then obtained at distancesmuch larger than the packet size, :

(57)

3.6. Theory of Vacuum Neutrino Oscillations in the Wave Packet Model

Let us apply the above WP theory to neutrino oscil-lations. It was demonstrated in the previous sectionthat the transverse WP spreading introduces suppres-sion factor to the probability of finding a parti-

cle at distance from the point of its production ininfinite observation time. Let us assume that the trans-verse degrees of freedom do not exert any significantinfluence on the oscillation pattern (with the excep-tion of the above suppression factor). In what follows,we will dispense with this hypothesis and perform acomplete three-dimensional calculation, which willverify the validity of the assumption.

3.6.1. Neutrino state in the wave packet model andamplitude of the transition from the source to the detec-tor. Let us consider a one-dimensional neutrino WPwith mean coordinate at time :

(58)

where the state with the normal-ization in accordance with ,and

(59)

The state norm in (59) is The detectedstate should also be characterized by a WP with mean

, = −

∗ ∗× ψ , ψ , − ψ , ψ , .

x x x x

( )2

( ) ( ) ( ) ( )

itm

t t t t

j x

x x x x∇ ∇

//

τ −

σ + τ , = π σ + τ

2

2 2

5 23 2 5 2

( )exp2 (1 ( ) )

( ) .(2 ) 2 1 ( )

x

x

tt

tm t

xx

j x

∞

− σ/

= , =

σπ

− .π σ

2 2

2

3 20

223 2

( ) ( ) erf24

(2 )x

x

x

dt t

e x

x xx j xx

xx

Φ

σ@2 2

xx

.π

3( )4

xxx

Φ

π 21

4 xx

sx st− +

α α∗ν = ψ ν

π p( ) ( ) ,2

i s ss s iE t ikxi i

k

dkV k e k

= +2 2,i iE k m ν ( )i kν ν = πδ −( ) ( ) 2 ( )j iq k q k

−−σπ ψ = σ

2

2( )1

4 4p 2

2( ) .s

s

k ps

s

k e

α αν ν = 1.s s

1 2020


coordinate xd at the time instant td. In principle, thestate in the detector may differ in f lavor:

(60)

where is defined as in (59) with obvious substi-

tutions . The projection of state onto yields the transition amplitude:

(61)

where is the distance between the meanpositions of neutrino WPs in the source and the detec-tor and The product of Gaussian formfactors in (61) may be presented as a Gaussian func-tion in (59) with momentum

(62)

and dispersion defined by

(63)

The amplitude in (61) takes the following form:

(64)

The momentum integral may be calculated in theapproximation of a small width of the effective WP2

( ) by considering of the one-dimensionalequivalent of the expansion in (51)

(65)

where is the group velocity ofthe WP state with mass . This allows one to calculatethe amplitude:

(66)

2 A WP is called “effective,” since it incorporates the processes ofneutrino production and detection.

− +β β

∗ν = ψ ν ,π p( ) ( )

2i d dd d iE t ikx

i ii

dkV k e k

ψp( )d k

→s d ανs

βνd

αβ β α+∞

− +∗α β

−∞

, ≡ ν ν

∗= ψ ψ ,π p p

( )

( ) ( )2

j

d s

iE t ikLs dj j

j

A t L

dkV V k k e

= −d sL x x

= −( ).d st t t

σ + σ= = σ + σ + σ σ σ

2 22

2 2 2 2 ,s d d s s dp

s d s d

p p p pp

σp

= + .σ σ σ2 2 21 1 1

p s d

+∞

αβ α β−∞

π∗, =σ σ π

−−× − + − − . σ σ + σ

22

2 2 2

2( )2

( )( )exp4 4( )

i ij s d

s dj

p s d

dkA t L V V

p pk piE t ikL

σ !p p

( ) ( )+ − + − , v

22

32j

kj pj pjpj

mE E k p k p

E

= =v pj j pjdE dp p Ejm

( )( )

/

αβ α β/

/

πδ − ∗, =πσ

−− −

σ + τ × ,+ τ

v

1 2

1 42

2

2

1 2

2 ( )( )

2

exp4 (1 )

(1 )

s dj j

jx

pjpj

x Lj

Lj

p pA t L V V

L tiE t

it

it

PHYSICS O

where

(67)

is the “spread” equivalent of the func-tion, which exercises the momentum conservationlaw, is the characteristic time of lon-gitudinal spreading of a neutrino WP with in strictaccordance with (53), and

3.7. Macroscopic Averaging of the Transition Probability

How does one determine the probability of neu-trino oscillations in the model with a WP? The proba-bility density corresponding to the amplitude from (64) depends both on distance andtime difference and on the mean neutrinoWP momenta in the source and the detector. Impor-tantly, all these parameters are completely arbitrary;i.e., we do not impose constraints such as and

. The macroscopic number of “events” aver-aged over nonobservables (production time andmean momentum at production) is the observablequantity:

(68)

where is the number density of neu-trinos at the source in unit time and unit momen-tum . Brackets denote averaging over time andmomentum within the intervals where

decreases rapidly. The probabilityof f lavor transition averaged over time andmomentum at the source is

(69)

Using (67), one may easily integrate over momen-tum . In order to perform integration over time, oneshould consider that terms of the form are vary-ing slowly compared to the exponential suppressionoccurring at times , since relation

(70)

holds true in a wide range of neutrino masses mj 1 eVand spatial dispersions . Thus, the integral over time

reduces to a Gaussian one. This provides an oppor-

/ −ππδ − = − σ + σ σ + σ

1 2 2

2 2 2 2( )22 ( ) exp2( )

s ds d

s d s d

p pp p

πδ −2 ( )s dp p

τ = σ3 2 22Lj pj j pE m

jmσ = σ1 2 .x p

αβ ,( )A t L= −d sL x x

= −( )d st t t

=t L=s dp p

stsp

ναβ

ναβ

− , ,

,

22

2

( )

( )

s s d s s ds s

s s

d ndt dp A t t p pdt dp

d n P Ldt dp

ν ,2 ( )s s s sd n t p dt dpsdt

sdp …

αβ − , − 2( )s s dA t t p pα βν → ν

αβ αβ= − ; , .2( ) ( )s s s s dP L dt dp A t t p p

spτLjt

− σv *2 2( ) 4 1j xL t

− στ σ

& !

2

2 2

( 2 )1x j

Lj pj x

L mtp E

&

σx

st



tunity determine the oscillation probability in theWP model:

(71)

where

(72)

The probability of oscillations in (71) depends onthe following parameters with a dimension of length:

(73a)

(73b)

(73c)

(73d)

where dimensionless parameter Phase depends linearly on distance :

and is inversely proportional to oscillation length ,which turned out to be the same as in the plane-wavecase. However, the following addition emerged in theWP model in the standard phase:

(74)

Let us discuss the obtained results qualitatively.

3.8. Qualitative Discussion of the Formula for the Oscillation Probability

in the Wave Packet ModelIt follows from (71) that neutrino oscillations in the

WP model depend on several parameters with adimension of length:

(i) Oscillation length .

(ii) Coherence length .

to

( )β α α β

αβ,

∗ ∗=

+ × − ϕ + ϕ − − ,

! @

2d4

d 2 2

( )1

exp (

i i j j

i jij

ij ij ij ij

V V V VP L

L L

i

( )

= +

πσ= .

!

@

22

2 cohd

22

osc

1

1

2and

ijijij

xij

ij

LLL L

L

= πΔ

osc2

22 ,ijij

pLm

= =πσσ Δ

2coh osc

2relp

2 2 1 ,2ij ij

ij

pL Lm

= =σσ Δ

3d coh

2 2relp

1 ,2 2ij ij

ij

pL Lm

σ =σ1 ,

2xp

σ ≡ σrel .p pϕ ( )ij L L

ϕ = π osc( ) 2 ,ijij

LLL

oscijL

( )

ϕ = − + . +

2d

2 coh d dd1 1( ) arctan

21ij

ij ij ijij

L LLLL L LL L

oscijLcohijL


(iii) Dispersion length .

(iv) Effective spatial size of a neutrino WP,which may differ by many orders of magnitude. Let usdiscuss the physical meaning of these scales. Coher-ence length corresponds to the distance at whichtwo WPs with masses and travelling with differ-

ent group velocities and

cease to overlap spatially. This occursat time , when , where is the spatialsize of the neutrino WP. The estimate of time

agrees to within a numerical factor

with the definition of coherence length from(73b). Note that the coherence length is proportionalto oscillation length and is inversely proportionalto relative dispersion of the WP momentum.Therefore, the lower the relative dispersion of the WPmomentum, the longer is the time interval of preserva-

tion of the WP coherence. Factor in the expo-

nential function in (71) represents the suppressionrelated to the coherence length.

Dispersion length emerges as a result of expan-sion of energy up to the second order (see (65)). Thiscorresponds to WP spreading in the configurationspace as demonstrated in Section 3.5. Note that isinversely proportional to squared. The spatial sizeof WPs increases after spreading, which compensatespartially the loss of overlap of WPs and due to thedifference in their group velocities. The characteristictime of longitudinal spreading of a WP of a neutrinowith mass (parameter ) turned outto be the same as in (53). The time derivative of spatialWP dispersion parameter defined in accordancewith (54) may be interpreted as the spreading rate:

(75)

It can be seen that grows linearly withtime until it reaches the asymptotic value of spreadingrate . The spreading rate is much lower than thevelocity of neutrinos but is not necessarily lower than thedifference between their group velocities . Indeed,

(76)

dijL

σx

cohijL

im jm

−v2 21 2i im p

−v2 21 2j jm p

t − σv v( )i j xt σx

σ Δ

2 22 x ijt p mcohijL

oscijLσrel

2

cohij

LL

dijL

dijL

σrel

νi ν j

jm τ = σ3 2 22Lj pj j pE m

σ ( )x t

,σ σ τ=

+ τσ τ , τ ,= σσ τ = , τ .

!

! @

2

2

2

2

2

( ) ( )

1 ( )

( )

1

x j x Lj

Lj

x Lj Lj

j px Lj Lj

pp

d t tdt t

t t

mt

EE

,σ ( )x jd t dt

σ τx Lj

−v vi j

,Δ Δ σ− =

σv v

2 2

2 2max

( ),

2 2ij ij p x j

i jpp

m m E d tdtE m

1 2020


where mmax is the maximum neutrino mass. With thecosmological constraint (mmax 0.2 eV) factored in

and taking eV as an estimate, wefind that is comparable to WP spreading rate

if The spreading rate forpackets with much smaller spatial dispersions is muchlower than . At much higher values of (stillsatisfying the condition), the spreading ofpackets will compensate (partially) the loss of theiroverlap more efficiently. For example, at distances

from the source, where it is reasonable toinstall detectors to maximize the sensitivity to neu-trino oscillations, the maximum suppression of inter-ference terms in (71) is at .

In the limit,

(77)

and spreading does not compensate completely theloss of coherence in (71). Thus, oscillations arestrongly suppressed at distances that are much greaterthan the coherence and dispersion lengths. Appar-ently, one may state that neutrinos from astrophysicaland cosmological sources are noncoherent and do notoscillate.

It may seem surprising that the partial restorationof coherence due to WP spreading is sensitive to thedifference between spreading rates rather than to theirsum, which is to be expected for WPs with their spa-tial widths increasing with time. In fact, it is indeedeasy to demonstrate that the coherence loss compen-sation for real Gaussian widths depends on the sumof spreading rates. The explanation is that the spread-ing effect manifests itself as complex-valued function

, which is characterized both by its abso-lute value and its phase. Therefore, just as the oscilla-tion length depends on energy difference ,the WP spreading effect, which depends on ,should depend on phase difference in the interference terms of the amplitude modulussquared in (66).

Since the WP spatial width is complex-valuedfunction , the spreading of WPs bothaffects their overlap and produces a correction to theinterference phase. Therefore, the oscillation phaseacquires addition defined in (74).

Additional suppression factor defined in (72) isindependent of distance. This suppression is substan-tial if the spatial width of a neutrino WP is comparablein order of magnitude to oscillation length (or exceeds it). In the mathematical sense, this corre-

&

−Δ = . ×2 32 45 10ijm 2

−v vi j

,σ ( )x jd t dt σ . 0 03 .p pE

−v vi j σrel

σ !rel 1

= osc 2ijL L

−π/8e σ = π .rel 1 2 0 4→ ∞L

( )→∞

= = , σ+

@

2 2d 2

2 coh coh 2d

1lim 181

ij

Lij ij pij

L pLL LL L

σ + τ2(1 )x Ljit

−( )i jE E tτLjit

∝ τ − τ(1 1 )Lj Liit

σ + τ2(1 )x Ljit

ϕd( )ij L

@2ij

= oscijL L

PHYSICS O

sponds to the plane-wave limit, wherein the spatialwidth of a neutrino packet tends to infinity. This has atrivial interpretation: the interference terms are aver-aged over distances of the order of the packet size. For-mula (71) may be rewritten in the following form:

(78)

which allows one to interpret the suppression of inter-ference of states and if , where

may be interpreted formally as theuncertainty of the neutrino mass squared [68]. This isthe factor that explains why charged leptons do notoscillate. Actually, the theory developed above mayformally be applied to the oscillations of charged lep-tons by substituting with under theassumption that all charged leptons are ultrarelativis-tic. However, all interference terms are suppressedvery strongly in this case, since quantities

are much larger than any realistic

value of due to the factor . Note thatthe interference terms in (71) are also suppressed bynumerator

(79)

Interference vanishes at and in (71), and the oscillation probability becomes a non-coherent sum

(80)

which is independent of energy, distance, and neu-trino masses.

Note finally that the limit, which corre-sponds formally to the plane-wave limit, predicts, incontrast to the plane-wave model itself, the lack ofneutrino oscillations. Although this result is obviouspost factum, it may seem paradoxical at first glance,since WPs turn into states with definite momentum at

, which is a throwback to the model with planewaves. However, one more important step is made inthe WP model: integration over the neutrino emissiontime is performed instead of the unjustified use ofequality in the plane-wave model. If one is to actconsistently in the plane-wave model, one should alsoperform integration over time (as a nonobservable).The plane-wave model then also yields a probability inthe form of a noncoherent sum in (80). Luckily, thisstep was omitted in the pioneering works on oscilla-tions. Thus, it is evident that the “obvious” substitu-tion is far from being innocuous.

Let us now examine whether the assumptions (dis-cussed in Section 2.3) made in the derivation of for-

Δ= , σ

@2

222 1

4kj

kj

m

m

νi ν j Δ σ@ 22ij mm

σ = σ2 2 2 pm p

ν ,ν ,ν1 2 3 ,μ, τe

μ μΔ ≡ −2 2 2e em m m

σp μ − 2exp 1e@ !

( )= + .2d4 1kj kjq L L

σ → 0p σ → ∞p

αβ α β= , 2 2k k

k

P V V

σ → 0p

σ → 0p

=t L

=t L



mula (17) for the probability of f lavor transitions arevalid.

Assumption (i), which states that a coherent super-position of states with masses ( )

interacts in the processes of neu-trino production and detection, may be erroneous if

. In particular, this hypothesis is defi-nitely not valid for charged leptons, which is why theiroscillations are infeasible.

Condition (ii) that states have definitemomenta is violated, since it corresponds to aninfinite spatial width which yields

. This leads to (80).

Hypothesis (III) states that all momenta are thesame ( ). This contradicts relativistic invariance.The noncovariant theory of neutrino oscillations inthe WP model, which was discussed above, did notaddress this deficiency.

Approximation (iv) of ultrarelativistic neutrinos( ) was also used in the theory of neu-trino oscillations in the WP model.

Hypothesis (v), which states that neutrino propa-gation time , is not valid. In fact, a consistent cal-culation requires integrating over time, which wasdone in (68). The largest contribution to the time inte-gral is produced by times of the order of where

which corresponds to the time instant of the maxi-mum overlap of two WPs representing states and .

4. RELATIVISTIC WAVE PACKET

4.1. Definitions

Let be the eigenstate of the operator of4-momentum of a particle with mass :

(81)

transforms state to . States arenormalized in a relativistically invariant way:

. In what follows, The completeness relation takes the form

im = , ,1 2 3i

α α∗ν = ν i ii

V

− @ &2exp 1ij

νi

ip−σ = σ 1(2 ) ,x p

− → @2exp 0ij

ip=ip p

@2 2max( )imp

=t L

v ,ijL

+= ,

+2 21 i j

ij i j

v v

v v v

νi ν j

k= ,0ˆ ˆ ˆ( )P P P m

μ μ= μ = , , , .ˆ ( 0 1 2 3)P kk k

= Λ 'k k k k 'k k=q k

π δ −3 3(2 ) 2 ( )Ek q k = =0k Ek +2 2.mk

= .π 3 1

(2 ) 2d

Ek

k k k


An arbitrary “single-particle” spin-free state |almay be presented as an expansion in the full set ofstates (i.e., as a wave packet):

(82)

State may be expanded in eigenstates of anyother self-adjoint operator (e.g., 3-dimensional coor-dinate operator ( ,

)). Choosing the norm for, which gives , we write

(83)

Since operator acting on state is writtenas ,

Taking into consideration the chosen norms, wefind that . Therefore,

(84)

Thus, functions and are the Fouriertransforms of each other:

(85)

The following relations will also be of use in subse-quent analysis:

(86)

The norm of state takes the form

(87)

If is a real function, WP has mean coordi-nate at . A WP with three-dimensionalcoordinate , which is, in the general case, nonzero,at time is needed for further calculations. To obtainit, one just needs to apply translation operator tostate :

(88)

k

= ψ , ψ = .π p p3 ( ) ( )

2(2 ) 2d aa

EE kk

k kk k k

a

= , ,1 2 3ˆ ˆ ˆ ˆ( )X X XX =ˆ

i iX xx x= , ,1 2 3i = δ −( )y x y x

x = 1dx x x

= ψ , ψ = . x x( ) ( )a d ax x x x x

P x−i x∇

= = − .ˆ i xk x k x P k x k∇

= 2 iE e kxkx k

−+= , = .

π 3 2(2 ) 2

iid e E d e

E

kxkx

kk

kx k k x x

ψp( )k ψx( )x

+

−

ψ = ψ ,π

ψ = ψ

x p3

p x

( ) ( )(2 )

( ) ( ).

i

i

d e

d e

kx

kx

kx k

k x x

−

+

∂ = − ,∂ π

∂ = + .∂

3(2 ) 2

22

i

i

d eiE

i E d eE

kx

k

kxkk

k

x k k kxvk k x x x

k

a

= ψ = ψ .π

2 2p x3 ( ) ( )

(2 )da a dk k x x

ψx( )x a= 0ax = 0t

axat

âiPxe

a

→ , =

= ψ , ,π

ˆ

p3 ( )(2 ) 2

a

a

iPxa a

ikxa

a x e ad e

Ek

pk k p k

1 2020


where wave function depends both on a variable(vector ) and on a parameter (mean momentum vector

). Function is also characterized by disper-sion parameter in the momentum space. For conve-nience, we require that the WP evolve into a state withdefinite momentum at : .This implies that factor is present in wave func-tion to compensate factor . In what follows,the presence of this factor is assumed for a uniquetransition to the plane-wave limit to exist. However,since this factor is constant, it exerts no influence onthe predicted number of interactions. Therefore, toavoid overcomplicating the formulas, we omit suchfactors.

It is convenient to rewrite wave function in the form to determine itstransformation properties. Using the transformationproperties of the state in responce ofthe Lorentz transformation

(89)

where is a unitary operator, one readily sees thatfunction is a relativistic invariant:

(90)

The state in (88) may be expressed in terms ofLorentz-invariant function (form factor):

(91)

The wave function in the configuration space cor-responding to the state in (88) is

(92)

We prefer not to equate the norm of state

(93)

to unity in the present study, although this appears tobe natural. The theoretical model with a WP in quan-tum theory, where a limiting transition to the ordi-nary state with definite momentum and norm

exists, is considered here.This norm is often substituted in handbooks with

, where is a “certain infinitely large spatial vol-ume.” In order to obtain a correct limiting transitionof our formalism to the common plane-wave one, wedefine the Lorentz-invariant norm of state in (93) as

ψ ,p( )ak pk

ap ψ ,p( )ak pσ

σ → 0 σ→ , =0lim a a axp p− a aip xe

ψp( )k aikxe

ψ ,p( )ak pψ , = φ ,p( ) ( ) 2a a Ekk p k p

= †2 0E ak kk

→ Λ = Λ , → Λ ,( )U k kk k k

Λ( )Uφ ,( )ak p

φ , = φ , = Λ , = Λ .''( ' ) ( ) ( ' )a a a ak k p pk p k p

φ ,( )ak p

−, = φ , .π 3 ( )

(2 ) 2aikx

a a adx e

Ek

kp k p k

, − + −

ψ ≡ ψ , ,

= ψ , .π

0 0

x x

( )p3

( ) ( )

( )(2 )

a a

a a a

i k xa

x

d ek x x

x x p

k k p

≡ , , = ψ ,π

= φ ,π

2p3

23

( )(2 )

( )(2 ) 2

a a a a a

a

da a x x

dEk

kp p k p

k k p

= π δ −3 3(2 ) 2 ( )Ekq k q k

2E Vk V

PHYSICS O

, where is a finite quantity with adimension of volume, which is defined uniquely by theWP form factor. Its explicit form is derived in Section 4.2.

Note on “hidden variables.” If WP is treated asa physical quantum state produced in collisions ordecays of other particles, function is expected todepend parametrically on 4-momenta of the pri-mary and, possibly, secondary particles involved in theprocess of WP production3. In the most general case,the set of parent and accompanying daughter particlesincludes particles from the entire chain of reactionsand decays leading to the production of state (seeFig. 1). “Hidden variables” may enter scalar func-tion only in the form of scalar products ,

, and . If one considers the expected prop-erties of function , it should be positivelydefined around and satisfy the following condi-tions:

The last equations are satisfied identically only in thenonphysical case when the most probable velocities of allpackets are the same ( ). In view of thisfact and the arbitrariness of configurations of 4-momenta

, we conclude that In much

the same fashion, we find that Thus, the dependence of on scalar products

and should vanish at least near the maxi-mum of function . Postulating analyticity offunction in the vicinity of its maximum ( ),we find that in the neighborhood of this pointmay depend on 4-momenta only via invariants

(94)

where is the metric tensor and , , G4, etc. aretensors constructed from the components of 4-vectors

. Since we consider only very narrow packets(“smoothed” -functions), the behavior of inthe vicinity of the maximum is of interest. Thus, thisfunction should include only “building blocks” (94).

3 The packet evolution under the influence of external fields isincorporated into this representation, since any interaction inthe -matrix QFT formalism should be regarded as a localinteraction of real or virtual fields.

= w2 Va a m wV

, xp

φ ,( )k pQ

û

S

, xpQ

û

φ ,( )k p ( )Q kû

( )Q pû '( )Q Q

û û

φ ,( )k p=k p

= =

=

=

∂φ , ∂φ ,∂ −= ∂ ∂ ∂ −

∂ ∂φ ,+ ∂ ∂

∂φ ,= − = = , , . ∂

2

2

00 0

( ) ( )( )( )

( ) ( )( )

( ) 0 ( 1 2 3)( )

l l

l

l l

k pk k k p

Q kk Q k

p QQ lQ kp Q

k p k p

k p

k p

k p k p

k p

k p

û

û û

û

û ûû

û =0Q EpQ pû û

Qû [ ] = =∂φ , ∂ 0.( ) ( )Q k

k pk p

û

[ ] = =∂φ , ∂ 0.( ) ( )Q pp k

k pû

φ ,( )k p( )Q k

û( )Q p

û

φ ,( )k pφ ,( )k p =k p

φ ,( )k pQ

û

μν μνμ ν μ ν

μνλμ ν λ

− − , − − ,− − − , ,

2

3

( ) ( ) ( ) ( )

( ) ( ) ( )

g k p k p G k p k p

G k p k p k p …

g 2G 3G

Qû

δ φ ,( )k p



Fig. 1. Schematic illustration of chains of WP production processes.

Q1

Q2

Q2Q3

Q3Q4 Q4

Q5

Q5

Q6

Q6

Q7 Q7

Q8

Q9

Q1

p

p

The remaining scalar products may be incor-porated into the definition of scalar parameters ,which determine the explicit form of ; in otherwords, these parameters, in the general case, may benot just constants, but scalar functions of 4-momentaof particles involved in the processes forming theWP. Therefore, WPs produced in different reactionsand decays (or chains of reactions and decays) andconstructed from identical single-particle states are, strictly speaking, not identical. As a result, thequantum statistics of such “packets with memory”may differ greatly from the statistics of their elemen-tary components (states with definite momenta).

To avoid unnecessary (though by no means fatal)overcomplication of the formalism, we sacrifice gen-erality and assume that parameters are effectiveconstants. In addition, we limit ourselves to the sim-plest “memoryless”4 WPs, which are independent oftensors and, consequently, of hidden variables.Undoubtedly, the issue outlined here is not specific toour approach, but is often disregarded or intentionallyignored in postulating the WP properties.

Correspondence principle and normalization of awave packet. We require that the state in (91) turn into

in the plane-wave limit. This condition is calledhere the correspondence principle:

Note that the dimensionless Lorentz-invariantintegral

is independent of and, consequently, may dependonly on parameter and mass . However, in accor-dance with the condition in (4.3), the considered inte-

4 A more complex WP model was examined in [100].

'( )Q Qû û

σiφ ,( )k p

û

p

σi

iG

p

( )σ→φ , = π δ − .3 3

0lim (2 ) 2 ( )Epk p k p

φ ,φ , =π π 3 3

'

' ( ' ')( )(2 ) 2 (2 ) 2

ddE Ek k

k k pk k p

pσ m


gral tends to unity in the limit. Therefore, it isalso natural to set it equal to unity at a finite small :

(95)

Although this technical condition is not required, itprovides a practically convenient norm of form factors

Certain general properties of a wave packet. Thefollowing are the key properties of a WP:

(1) Since the invariance of with respect to

rotations , is a specialcase of (90), functions and arealso rotationally invariant.

(2) The modulus of function is alsoinvariant with respect to spatial translations but is notinvariant with respect to temporal ones.

(3) It follows from (92) that at

; i.e., a WP spreads with time in the configu-ration space.

(4) Since is independent of , onecan conclude that is the center of spherical symme-try of a packet. In what follows, we call this point thepacket center.

(5) It follows from (90) that function maydepend only on Lorentz-invariant quantity . There-fore, function is symmetric with respect to per-mutation of its arguments; i.e., .

(6) It is also evident that ,where is the rotationally invariant function of asingle variable .

(7) Another simple and important corollary of (90)is the independence of norm (93) of both space-timecoordinate and momentum .

σ = 0σ

φ , φ ,= = .π π 3 3

( ) ( 0) 1(2 ) 2 (2 ) 2d d

E Ek k

k k p k k

φ ,( ).k p

φ ,( )ak p

= 'k k Ok = 'a a ap p Opψ ,p( )ak p ψ , ,x( )a axx p

ψ , ,x( )a axx p

ψ , , →x( ) 0a axx p

→ ∞0ax

ψ , ,x( )a axx p axax

φ ,( )k pkp

φ ,( )k pφ , = φ ,( ) ( )k p p k

φ , = φ , = φ 0( 0) (0 ) ( )kk kφ 0( )k

=0k Ek

ax ap

1 2020


4.2. Mean Four-MomentumLet us now find the mean values of components of

4-momentum of a packet, which aredefined by the standard quantum-mechanical rule

Inserting (88) into this equation, we find

(96)

where denominator is equal to the norm of state:

(97)

Here and elsewhere, index of variables and is omitted for brevity (if this does not lead to misun-derstanding). Formulas (96) and (97) demonstratethat mean 4-momentum is an integral of motion,and quantity , which has the dimension of spatialvolume, is completely independent of the momentumand the space-time coordinate.

In view of the function’s parity, it followsfrom (96) that ; in other words, the meanmomentum turns to zero in the intrinsic frame of ref-erence of the packet5, which is defined by condition

. Therefore, the mean (effective) mass of apacket, , is equal to its mean energy (the zerothcomponent of 4-vector in this frame):

(98)

Using (97) and writing (88) as

(99)

we find that with equality attainedonly in the plane-wave limit; i.e., WP is“heavier” than its component states with definitemomenta. This effect is of a general nature and is a

5 This also follows from simple dimensional considerations: since3-vector depends on just a single vector quantity , .

( )= ,0P P P

μμ μ

, ,= = .

, ,

ˆ( ) a a a a

aa a a a

x P xP P

x xp p

pp p

μ μφ ,= ,π

w

2

3( )1( )

2 V (2 ) 2dP k

m Ek

k k pp

w2 Vm,a axp

∞

φ , φ ,= =π π

= − φ .π

w

2 2

3 3

22 20 0 02

( ) ( 0)2 V

(2 ) 2 (2 ) 2

1 ( )4 m

d dm

E E

dk k m k

k k

k k p k k

a ap ax

PwV

φ ,( 0)k=(0) 0P

P p ∝P p

= 0pm

P

∞

φ ,= =π

= − φ .π

w

w

2

0 3

22 20 0 0 02

( 0)(0)4(2 ) V

1 ( )8 V m

dm Pm

dk k k m km

k k

∞

∞

− φ= ,

− φ

22 20 0 0 0

22 20 0 0

( )

( )

m

m

dk k k m kmm

dk m k m k

≥m m =m m, xp

PHYSICS O

manifestation of mass nonadditivity. In the case underconsideration, it is attributable to the fact that thecomponents of momenta of Fock states transverseto vector , which are constituent parts of the WP, donot contribute to the mean momentum value( ), but produce a nonnegative contributionto the mean packet energy6. Note that the mean valueof the mass squared is ; thus, .

Let us demonstrate that the packet is, on the aver-age, on the mass shell. It follows from the dimensionalconsiderations mentioned above that and

, where and are certain dimensionlessquantities independent of . We then obtain from (96)the following relation:

(100)

Since the left-hand and the right-hand expressionsin (100) are Lorentz-invariant, . Setting in(100) and using the definition in (98) we find

. This proves that

It follows from these relations that group velocity of the packet is the same as its most

probable velocity

4.3. Wave Packet for a FermionWave packets of fermions with single-particle Fock

states of the form with Lorentz-invariant norm = areneeded for further calculations within the -matrixQFT formalism. Operators and are standardcreation and annihilation operators for a particle withmass , 3-momentum , and spin projection .They satisfy the following anticommutation relations:

Let us construct a relativistic fermion WP, which isa state localized in the configuration and momentumspaces that features transformation properties similarto those of Fock states and evolves into a Fock state inthe plane-wave limit. Similar to a scalar particle, a WP

6 This is similar in certain respects to the following well-knownrelativistic effect: the mass of gas in a vessel increases in the pro-cess of its (uniform) heating. The vessel acquires no additionalmomentum, but the internal energy and, consequently, the massof gas increase.

kp

× = 0p P

2m ≥ =2 2 2P P m

= κ0P Ep

= κ'P p κ κ'p

μμ = κ − κ

φ ,= κ + κ − κ = .π

w

2 2

22 2

3

'

( ) ( )1( ')2 V (2 ) 2

p P E

d pkmm E

p

k

p

k k pp

κ = κ' = 0p

κ = ≥ 1m m

μμ

= , = ,

= = .0

2 2

( ) ( )( )

and

P P m m E

P P P mpP p

=0 0P PP P∇

= .Ep pv p

, = †2 0ss E ak kk, ,r sq k ( )π δ δ −3(2 ) 2 srEk k q

S†sak sak

> 0m k s

( )= =

= π δ δ −

† †

† 3

, , 0,

, (2 ) .r s r s

r s sr

a a a a

a aq k q k

q k k q



is characterized here by the most probable 3-momen-tum and spin projection . The most general con-struction of this type at a fixed point in time takesthe form

(101)

where functions depend both on momen-tum and discrete spin variables and on a set (finite orinfinite) parameters , which are inde-pendent of these variables and specify the properties ofthe packet. By definition, momenta and lie on themass shell; i.e., = . We requirethat state (101) evolve into in the plane-wavelimit. Since one can always define parameters insuch a way that they satisfy this limit at ( ),we may formulate a simple correspondence principle:

Since the right-hand side of this equation is a rela-tivistic scalar, it is natural to assume that functions are also scalars and at . Assumingthat parameters are sufficiently small, one maysummarize these requirements using the followingsimple ansatz:

where is a spin-independent function suchthat

and

(102)

Thus, if at least some parameters in set arenonzero (and all of them are sufficiently small), func-tion is (to within a factor) a “smeared”

-function in the momentum space. More precisely,as in the quantum-mechanical case, we assume that

has a peak at and decreases rapidlyoutside the small neighborhood of this point. Inorder to simplify notation, we do not specify theexplicit dependence of on parameter set insubsequent analysis and write expression (101) in theform

(103)

The WP in (103) characterizes a state with spin pro-jection , the most probable momentum , and amean three-dimensional coordinate of the packet cen-ter that is equal to zero at time . To set the initialstate of a WP, one just needs to define its mean coor-dinate at an arbitrary fixed time point. It makes no dif-

p s

( )Φ , ;σ , ,π 3(2 ) 2

srr

d rEk

k k p k

( )Φ , ;σsr k p

= σ σ …1 2 , , σ

p k

−2 2Ep p − =2 2 2E mk k, sp

σi

σ → 0i ∀i

( )σ→Φ , ;σ = π δ δ − .3

0lim (2 ) 2 ( )sr srEpk p k p

Φsr

Φ Φsr ss! ≠r sσi

( )Φ , ; = δ φ , ; ,( )sr srk p k pσ σ

φ , ;( )k p σ

φ , ; = φ , ; = Λ , = Λ( ' ' ) ( ) ( ' ' ),p p k kk p k pσ σ

σ→φ , ; = π δ − .3

0lim ( ) (2 ) 2 ( )Epk p k pσ

σi σ

φ , ;( )k p σ

δ

φ , ;( )k p σ =k p

φ , ;( )k p σ σ

φ , , .π 3 ( )

(2 ) 2d s

Ek

k k p k

s p

= 0t


ference whether this point is located in the future orpast relative to , since the temporal evolution of aWP allows one to reconstruct its trajectory at any timepoint. For visual convenience (more specifically, forobjects emerging in the formalism to have an explicitlytranslation-invariant form [ rather than

]), we set the initial condition for a WP withmean three-dimensional coordinate ( ) at a certaintime point ( ) located in a sufficiently distant past;i.e., we use transformation , whichyields the following expression for a WP at an arbitraryspace-time point :

(104)

Coordinate representation. Let us consider a free

fermion field with spin and a field operator of thefollowing form:

State is often called the state of a fermionwith definite coordinate. This is suggested by the fol-lowing elementary calculation:

(105)

yielding a plane wave multiplied by spinor .In much the same fashion, we define coordinate wavefunction

(106)

where the approximate equality corresponds to narrowpackets. The condition of applicability of this approx-imation may be written as

(107)

Lorentz-invariant function is defined as follows:

(108)

It is evident that the function in (108) satisfies(at any ) the Klein–Gordon equation

i.e., it is a relativistic WP in terms of the standard QFTscattering theory. It is also clear that, under theassumptions made above, function dependsonly on two independent scalar variables and .

= 0t

ψ −( )x yψ +( )x y

−x− 0x

+, ,

îPxs e sp p

x

+, , = φ , , .π 3 ( )

(2 ) 2ikxds x e s

Ek

kp k p k

12

[ ]−Ψ = + .π v

†3( ) ( ) ( )

(2 ) 2iky iky

s s s ss

dy a u e b eE

k kk

k k k

Ψ0 ( )y

−Ψ , = ,0 ( ) ( ) ipysy s u ep p

−ipye ( )su p

− −Ψ , , =π

× φ , ψ , − ,

( )30 ( ) ( )

(2 ) 2( ) ( ) ( )

ik y xs

s

dy s x u eE

u y xk

kp k

k p p p

∇ ψ , − + .ln ( ) 2i x y Ey pp p !

ψ

−ψ , ≡ ψ , , = φ , .π0 3( ) ( ) ( )

(2 ) 2ikxdx x e

Ek

kp p x k p

p

∂ + ψ , = ,2 2( ) ( ) 0m xp

ψ ,( )xp2x ( )px

1 2020


Scalar function remains unchanged after thetransition to the intrinsic frame of reference of apacket ( ); i.e.,

(109)

where 4-vector is related to by the Lorentz transformation along the direction(from the laboratory frame to the intrinsic frame),

(110a)

and is the Lorentz factor ofthe packet. Since is an even function of vari-able , it can depend only on variables and ,which are expressed in terms of invariants and in the following way:

(110b)

Using (110a) and (110b), one can easily verify that

(111)

Effective volume of a wave packet. According to theadopted WP normalization in (95), dimensionlessfunction is not a normalized wave function.This allows one to interpret

(112)

as a three-dimensional volume in space occupied bythe WP. The narrower the WP in the momentumspace, the more accurate the above interpretation. Thestate norm in this approximation may be written as

(113)

Note that in the general case.

ψ

=w 0p

−ψ , = ϕ , = ψ , ,π w

w3( ) ( 0) (0 )(2 ) 2

ikxdx e xEk

kp k

= ,w w w

0( )x x x = ,0( )x x x−p

( )= Γ − ,Γ = + Γ − , Γ +

w

w

0 0

0( )

1

x x

x

p p

p pp p

p

v xv x

x x v

Γ = = − 21 1E mp p pvψ , w(0 )x

wx w

0x wx

( )px 2x

[ ]

= , = −

= − .

w w w

20 0 2 2

2 2 22

( ) ( )

1 ( )

pxx x xm

px m xm

x

( ) ( )( ) ( )

= Γ − − ×

= − − × .

w 220

2 20

1x

E xm

p p p

p

x x v v x

x p p x

ψ ,( )xp

φ ,= ψ , = =Γπ

2def 2

3 2( ) V(0)V( ) ( )

(2 ) (2 )dd x

E pk

k pkp x p

φ ,, , =π

φ , = .π

2

3

2

3 2

( )(2 ) 2

( )2 2 V( )

(2 ) (2 )

dp x p x

E

dE E

E

k

p pk

k k p

k k pp

≠wV V(0)

PHYSICS O

4.4. Commutator FunctionIt is convenient to introduce an auxiliary WP cre-

ation operator:

(114)

which provides an opportunity to rewrite the statein (103) in the form similar to the plane-wave limit,

(115)

Naturally, turns into in the limit.The following (anti)commutation relations are easy

to derive from the definition of operators and

:

(116a)

(116b)

(116c)

Lorentz- and translation-invariant commutatorfunction

(117)

was introduced here. It follows from (104) and (116c)that

(118)Certain properties of the commutator function

in (117) become especially clear in the center-of-iner-tia frame of two packets. We denote quantities in thisframe by an asterisk subscript or (when it cannot bemistaken for the complex conjugation sign) super-script. Since in the center-of-inertiaframe, we have

(119)

The characteristic behavior of form factor inthe vicinity of suggests that the modulus of func-tion has a sharp maximum at

and vanishes rapidly at larger values, sincethe maxima of factors and in theexpression under the integral sign in (119) are widelyseparated in this case; therefore, their product is smallfor any values of integration variable . This is illus-trated schematically in Fig. 2. It is easy to see that theintegral in (119) becomes vanishingly small if points and are widely separated in space (i.e., when thevalue of is large). This is attributable to rapidlyoscillating factor in the integrand in (119).

φ ,= ,π

† †3( )( )

(2 ) 2 2

ikx

s rd eA x a

E Ep k

k p

k k p

, , = .†2 ( ) 0ss x E A xp pp

† ( )sA xp†sap σ → 0

( )sA xp† ( )sA xp

( )− /, = δ φ , ,1 2† ( ) 4 ( )iqxr s sra A x E E eq p q p q p

, = , = ,† †( ) ( ) ( ) ( ) 0r s r sA y A x A y A xq p q p

( )− /, = δ , ; − .1 2†( ) ( ) 4 ( )r s srA y A x E E x yq p q p p q$

−, ; − = φ , φ ,π

( )3( ) ( ) *( ) ,

(2 ) 2ik x ydx y e

Ek

kp q k p k q$

, , , , = δ , ; − .( )srr y s x x yq p p q$

+ = 0* *p q

−

, − ; −

= φ , φ , − .π

( )* *3

( )* * * *( ) *( )* *(2 ) 2

ik x y

x yd e

Ek

p pk k p k p

$

φ ,( )k p=k p

, − ; −( )* * * *x yp p$

= 0*p *pφ ,( )*k p φ , −( )*k p

k

*x*y

−* *x y− −( )* *ie k x y



Fig. 2. Schematic illustration of the smallness of the integrand in (119) at large values of momentum in the center-of-inertiaof wave packets.

0

|φ(k, –p )|

|φ(k, p )|

ky

kx

*

*

*p

In order to return to the laboratory (or any other)frame, one needs to express variables in the center-of-inertia frame in terms of variables in the frame of inter-est. The corresponding Lorentz transformation takesthe form

(the same goes for and ), where

denote the velocity and the Lorentz factor of the cen-ter-of-inertia frame in the laboratory frame; theenergy and the momentum in the center-of-inertiaframe are given by

The latter equality demonstrates that decreasesat and increases as grows larger. There-fore, it follows from the above considerations thatfunction reaches its maximum at and decreases at large values of .

To conclude this section, we write kinematicidentities

(120a)

(120b)

( )= Γ − ,∗Γ = + Γ −

Γ +

0 0

0( )* 1

x x

x

vx

x x vx v

*y −* *x y

++= Γ = =+ +− 2

* *

1and1

E EE E E E

p q

p q p q

p qvv

= = + ≡ ,

= = − − ≡ .

2* *

2

1 ( ) *21 ( )* * *2

E E p q E

p q P

p q

p q

*P→p q −p q

, ; −( )x yp q$ =p q−p q

( )= + , = − ,∗

= − ,

0

2 0

2 ( ) 2 ( )* * *

*

E x p q x q p x

x

p x

vx vx v

[ ] ( )+= − = − − × ,∗ +

22 22 2 2 0

2

( )( )p q x

x xp q

x x v v x


which will be used often in subsequent analysis. It fol-lows from the last identity that

4.5. Multipacket StatesMultipacket states are crucial for the theory of neu-

trino oscillations in matter when one considers theeffects of coherent neutrino scattering off matter parti-cles. We assume that identical WPs, which correspondto states of one and the same quantum field but havedifferent momenta and spin projection, are character-ized by identical sets of parameters .Therefore, the dependence on is not stated explicitly.Ket state of identical WPs is defined as

(121)

The corresponding bra state is derived from (121)using Hermitian conjugation. It is evident thatstate (121) is completely symmetric (antisymmetric)for bosons (fermions) with respect to permutations

for any pair of indices ( , . The general symmetry relationtakes the form

Here and elsewhere, the upper and the lower signscorrespond to bosons and fermions, respectively, and

is the parity of permutation

= ⇔ = .00* xx x v

= σ ,σ ,1 2 …σ

σn

=

, , ; , , ; ; , ,

= . ∏

1 1 2 2

1 1 1 2 2 2

† † †1 2

12 ( ) ( ) ( ) 0

i n n

n n nn

s s s ni

s x s x … s x

E A x A x A xp p p p

p p p

, , ↔ , ,( ) ( )i i i j j js x s xp p ,i j≤ , ≤1 i j n ≠ )i j

η

, , ; , , ; ; , ,= ± , , ; , , ; ; , , .

1 1 1 2 2 2

1 1 1 2 2 2

( 1) P

n n n

n n n

i i i i i i i i i

s x s x … s x

s x s x … s x

p p p

p p p

ηP

= . 1 2

1 2

n

… ni i … i

3

1 2020


In order to find the norm of states (121), we intro-duce an matrix

Let us prove that

(122)

This relation is trivial if . It is easy to verify bydirect calculations that

Therefore, equality (122) also holds true at .Let us now consider matrix element at .According to (121) and (116b),

Following successive permutations of operator with operators , , with

the use of (122) for -packet matrix elements and with(anti) commutation relations (116c) factored in, wethan find

The right-hand side of this relation may be rear-ranged in compact form

where is the nth-order minor of determinant arising from elimination of the th row and

the jth column from . The sum over of the latterexpression is nothing but the minor expansion ofdeterminant over the -th row. Therefore,

, which completes the inductive proof.Thus, according to (122), -boson ( -fermion)

matrix element is proportional to the matrix permanent (determi-

×n n

( )+

≡ ,, , , ,

= δ , ; − .∓

D D

( 1) ( )i j

n n ni j

s r i j i j

r y s x

x y

q p

p q$

≡ , , ; , , ; ; , ,, , ; , , ; ; , , = .D

1 1 1 2 2 2

1 1 1 2 2 2

n n n

n n n n n

M r y r y … ry s x s x … s x

q q qp p p

= 1n

[]

≡ , , ; , , , , ; , ,= δ δ , ; − , ; −

± δ δ , ; − , ; − = .D

1 1 2 2

1 2 2 1

2 1 1 1 2 2 2 1 1 1 2 2 2

1 1 1 1 2 2 2 2

1 2 1 2 2 1 2 1 2

( ) ( )( ) ( )

s r s r

s r s r

M r y r y s x s xx y x y

x y x y

q q p pp q p q

p q p q$ $

$ $

= 2n+1nM ≥ 2n

+ +

+ +

/+

+=

+

+

=

×× .

∏

1 1 1 1

1 1 1 1

1 21

11

1 1† † †

1 1

4

0 ( ) ( ) ( )

( ) ( ) ( ) 0

i i

n n n n

n n n n

n

ni

r r n r n

s n s n s

M E E

A y A y A y

A x A x A x

q p

q q q

p p p

+ + +1 1

†1( )

n ns nA xp ( )n nr nA yq …

1 1 1( )rA yqn

( ) ( )

+

+

− − + +

+ +

/+ ++ +

+==

− /+ +

− +

+

= ± δ

× , ; −×

.

∏

1

1

1 1 1 1 1 1

1 1 1 1

1 21 11

111

1 21 1

1 1 1

† †1 1

4 ( 1)

40 ( ) ( ) ( )

( ) ( ) ( ) ( ) 0

i i n j

j n

j j j j

n n n n n n

n nn j

n s rji

n j n j

r r j r j

r n r n s n s

M E E

E E x yA y A y A y

A y A y A x A x

q p

q p

q q q

q q p p

p q$

( )+

++ +

+ + +=± δ , ; − , D

1

11 ( )

1 1 11

( 1)n j

nn j j

s r n j n j nj

x yp q$

+D( )

1j

n

+D 1n +( 1)n+D 1n j

+D 1n +( 1)n+ += D1 1n nM

n nnM

δ , ; −( )i js r i j i jx yp q$

PHYSICS O

nant). Let us now turn to the normalization problem.Since

matrix is Hermitian. Therefore,-packet matrix element

(123)

is a real polynomial combination of order of com-mutator functions with different arguments. Forexample, the following is obtained for n = 1, 2, and 3:

(124)

The properties of the commutator function, dis-cussed in Section 4.4, provide an opportunity to ana-lyze these results in two simple limiting cases. If allspace-time points ( ) are sufficientlyseparated and/or 3-momenta differstrongly in amplitude or direction (nonintersectionmode), it follows from (122) that

In the opposite case, when packets with identicalspin projections intersect strongly both in the momen-tum space and in the configuration space,

A matrix element is independent of momenta,coordinates, and spin projections in both limitingcases. The behavior of an -packet matrix element inthe intersection regime is just the manifestation ofBose attraction and Pauli blocking for identical bosonsand fermions, respectively. Much less trivial is the factthat the WP formalism validates the intuitive expecta-tions that free identical bosons (fermions) with thesame momenta and spin projections do not condense(may coexist freely) if they are separated by sufficientlylarge space-time intervals. This physically under-standable result is incomprehensible within the plane-wave QFT formalism. In fact, the Pauli principle issometimes formulated in the following categoricalform: “the probability of finding two identical fermi-ons with the same momenta and spin projections iszero.” We see that, strictly speaking, this is not true,

, ; − = , ; − ,( ) *( )x y y xp q q p$ $

( ),, , , ,D n ns x s xp pn

, , ; , , ; ; , , , , ;, , ; ; , ,≡ , , , ,

1 1 1 2 2 2 1 1 1

2 2 2

n n n

n n n

n n

s x s x … s x s xs x … s x

s x s x

p p p pp p

p p

n$

[ ]≤ < ≤

≤ < ≤

, , , , = ,, , , ,

= ± δ , ; − ,, , , , = ±

× δ , ; −

+ δ , ; − .

∏

1 2

1 1

2 222

1 2 1 23

3 32

1 3

1 3

2 V

(2 V ) ( )

(2 V ) 2 V

( )

2Re ( )

i j

i j

s s

s s i j i ji j

s s i j i ji j

s x s x ms x s x

m x x

s x s x m m

x x

x x

p pp p

p p

p p

p p

p p

$

$

$

w

w

w w

, , ,1 2 nx x … x ≥ 2n, , ,1 2 n…p p p

, , , , ≈ = , , , , .w(2 V ) nnn ns x s x m s x s xp p p p

! ,, , , , ≈

w for (2 bosonV )0

sfor fermions.

n

n nn m

s x s xp p

n



unless particles with definite momenta (idealizedmathematical objects existing in the entire infinitespace-time) are considered instead of WPs, which arelocalized in finite space-time regions and character-ized by the most probable momentum values. We willrevisit this conceptually important issue in Section 5.5and clarify the meaning of words “sufficiently largespace-time interval” using a simple model of a relativ-istic WP.

5. RELATIVISTIC GAUSSIAN PACKETS

5.1. Function

This section is focused on the simple (but import-ant for the considered formalism) model of a relativis-tic Gaussian packet that is applicable in characterizingsufficiently narrow (in the momentum space) wavepackets that are spherically symmetric in the intrinsicframe of reference and have a single maximum7. Sincefunction for such packets is essentially (towithin a known positively defined factor) a smeared

-function, we may assume that withoutloss of generality. It is also assumed that form factor

depends on a single scalar variable . It iseasy to see that . Since , we obtainthe following in the intrinsic frame of reference of thepacket ( ):

Note also that at. Since function has its maximum at point, one can apply a natural additional constraint

Function may then be approximated in asufficiently small neighborhood of the maximum bythe following expression:

(125)

7 The RGP model was proposed in [57]. It was thoroughly ana-lyzed mathematically and generalized in [101]. Intriguing appli-cations of the RGP (apart from neutrino physics) were discussedin [102]. Another class of models (covariant asymmetric WP,AWP) was proposed in [100]. It was demonstrated that an AWPis never the same as an RGP, although it features all the proper-ties needed for a covariant description of localized “particle-like” quantum states. It was also demonstrated that the WP con-sidered below is the only relativistic “memoryless” packet (i.e., apacket independent of 4-momenta of “parent” states (particles)involved in the reactions or decays producing the WP; see Sec-tion 4.1). We follow the logic of [57], which is sufficient for ourpurpose.

φ ,( )G k p

φ ,( )k p

δ φ , >( ) 0k p

φ ,( )k p − 2( )k p− ≤2( ) 0k p ≥

wE mk

=w 0p

( )− = − = − − ≤ .ww w

2 2( ) ( ) 2 0k p k p m E mk

[ ]− ≈ − − + − 22 2( ) ( ) ( )k p pk p v k p∼k p φ ,( )k p=k p

=

φ , ≡ > , σ = .− σ2 2

ln ( ) 1 0 const( ) 4

dd k p k p

k p

φ ,( )k p

σ −φ , ≈ = φ , . σ

2def

2( )( ) exp ( )

4G

k pNk p k p


We call the states with such form factors RGPs byanalogy with noncovariant (nonrelativistic) Gaussianpackets in (50) (NGPs) with form factor

which is normalized by condition

It will be demonstrated below that the properties ofRGPs differ considerably from those of NGPs. Nor-malization constant in (125) may be derived fromcondition (95), which states that

Thus, we obtain

(126)

where

denotes the modified Bessel function of the third kindof order 1. We can now rewrite (125) in the followingform:

(127)

Imposing the natural (assumed by default in whatfollows) requirement that

(128)

and making use of the known asymptotic expansion offunction , which is true at large [103],

(129)

we can rewrite expression (127) as follows:

(130)

/ −π ϕ − = − , σ σ

3 4 2

2 2( )2( ) exp

4G

k pk p

ϕ = .π

23 ( ) 1

(2 )G

dk k

σN

( )

∞

σ∞

−= π σ

= Γ Γ − − Γ − . π σ

22

2 20

2 22

21

1 exp(2 ) 2

1exp ( 1)2 2

m mEdN E

m md

k

k

k k

( )σ

π − σ= φ , = ,

σ σ

2 2 2

2 2 21

2 exp 2( )

( 2 )G

mN

K mp p

( )∞

− π= − <2

11

( ) 1 arg2

ztK z z dte t z

− πφ , = − . σ σ σ

2

2 2 2 21

2( ) exp( 2 ) 2

GE E

K mk p kp

k p

σ ,2 2m!

1( )K z z

( )− π + + +

π< ,

∼1 2 33 15 1( ) 1

2 8 2(8 )3arg2

zK z ez z z z

z

2

/ −πφ , = σσ σ

σ σ× + + .

23 2

2 2

2 4

2 4

( )2( ) exp4

314

Gk pm

m m

k p

2

1 2020


The following is obtained from (130) in the nonrel-ativistic case ( ):

(131)

This agrees, to within a normalization, with NGPform factor . However, the differencebetween and may be arbitrarily large(irrespective of the difference in normalizations) atrelativistic momenta. For example, function (130)behaves in the following way in the ultrarelativisticlimit ( , ):

where is the angle between vectors and . In par-ticular, we obtain the following at and :

In the first example, the relativistic effect is limitedto packet broadening (compared to the nonrelativisticcase) in the momentum space ( ). Thiseffect is significant for all processes of neutrino pro-duction and absorption (scattering) involving relativis-tic particles.

If one needs to illustrate the significance of cor-rect normalization, it is instructive to verify explicitlythat the limit at is indeed given byequality (4.3).To this end, it is sufficient to prove that

(132)

for any smooth function .The left-hand side of (132) may be rearranged in

the following way:

( )+ 2 24mk p !

( )/ −πφ , ≈ − ≡ φ , σσ σ

23 2NR

2 22( ) exp ( ).

4G G

m k pk p k p

( )ϕ −G k pφ ,( )G k p ϕ −( )G k p

2 2mp @2 2mk @

( ) ( )

/πφ , ≈σσ

− − ϑ× − − ≡ φ , , σ σ

3 2

2

22UR

2 2

2( )

1 cosexp ( )

4 2

G

G

m

m

k p

k p k pk p

k p

ϑ k pϑ = 0 π 2

( )ϑ=

ϑ=π/

−φ ∝ − σ Γ Γ

φ ∝ − . σ

2UR

0 2

UR2 2

exp4

and exp2

G

G

k p

k p

k p

σ σ Γ Γ k p

φ ,( )G k p σ → 0

σ→

ϕ , =π 30

( 0)lim ( ) (0),(2 ) 2

Gd F FEk

k k k

( )F k

( )

( )

∞

/σ→

∞/

σ→

Ωπ σ × − − σ

= Ω πσ

× − + + , σ

2

3 2 300

2

2

3 22

200

2

2

lim(4 )

exp 12

lim4

exp ( 2) ( 2)2

m d dE

Em Fm

m d dt

m t t t F m t t

kk

k

n

kk

k

n

PHYSICS O

where . The obtained integral over may beestimated using the well-known asymptotic formula(see, e.g., [104])

(133)

which holds true for an arbitrary continuous function . Since

in the case under consideration, the equality in (132)becomes evident.

Thus, we see that function is the simplestmodel of a form factor that satisfies all the require-ments imposed on function of the general form.Let us derive explicit formulas for the mean and root-mean-square velocities of an RGP to understand itsproperties better.

5.2. Mean and Root-Mean-Square Velocities

Let us start with definitions

(134)

Evidently, ; therefore, it is sufficient todetermine the projection of the mean velocity onto themomentum direction ( ). Integrating inangular variables, we find

( ); it is assumed here that becausethe case is trivial. The saddle-point method is

=n k k t

[ ]

( )

∞− −ν −ν Γ +

> ,ν → ∞ ,

∼

1

0

( ) ( ) (0) 1 (1)

0

a t adtt e f t a f o

a

( ),f t ∈ ,∞[0 )t

( )= , ν =

σ= + +

2

232 2

and ( ) 2 ( 2)

ma

f t t F m t t n

φ ,( )G k p

φ ,( )k p

φ ,= ,πφ ,= .π

w

w

2

3 2

222

3 3

( )12 V (2 ) 2

( )12 V (2 ) 2

dm E

dm E

k

k

kk k pv

kk k pv

∝ ∝pv v p

=pn p p

( )

∞

−∞

−∞

π λ=λ

λ × θ θ −

| = λ − λ λ

λ − × − ,

w

32 2

5 2 21 0

1

21

2

2 2 21

2

4 V ( 2)

( )cos cos exp

12 ( )

exp

dm K E

pkdm

mE mK

E E

m

pk

k

p k

k kv n

k k p kp

p k

λ = σ2 2m > 0p= 0p



used in subsequent calculations; we utilize the well-known theorem of asymptotic integral expansion

(135)

(see, e.g., [104], p. 41), which states that

(136)

where

is the only stationary point of function ( ), , and in theneighborhood of . All these conditions are evidentlysatisfied in our special case8. The result is

(137)

As expected, the mean velocity is always lower thanthe most probable. In the nonrelativistic case,

; in the ultrarelativistic one,

. Similar calculations for the root-mean-square velocity and its longitudinal and trans-verse components yield the following:

(138a)

(138b)

(138c)

In the nonrelativistic case, ; i.e., jit-ters are observed in the WP, although the jitters arenegligible. In the ultrarelativistic case, they vanish

8 The linear dependence of function on parameter does notpreclude one from applying the theorem.

[ ]∞

−∞

λ = −λ ,

, ∈ −∞,∞

C

( ) ( )exp ( )

( ) ( ) ( )

F dtf t S t

f t S t

[ ] ( )∞

=+ /

λ −λ Γ +

λ× , λ → ∞, ! λ

∼ 00

1 2

2

0

1( ) exp ( )2

( ) 2(2 ) "( )

nn

n

F S t n

bn S t

[ ]

( )=

λ = λ , ,

, = − + − ,0

0

20 0 0 0

( ) ( )exp ( )

1( ) ( ) ( ) "( )2

n

n nt t

db f t h t tdt

h t t S t S t t t S t

0t ( )S t−∞ < < ∞0t >0"( ) 0S t ∞, ∈C( ) ( )f t S t

0t

( )f t λ

− −= − + + λ .

Γ λ Γ λ

23

2 2 2

7 611 ( )2

pp

p p

vv v 2

− ≈ λp pv v v

( )− ≈ Γ λ2p p pv v n

−−= + − + λ ,

Γ λ Γ λ

22 2 3

4 4 2

3(7 10 )3 ( )2

pp

p p

vv v 2

−

−= = +

Γ λ− +

− + λ ,Γ λ

22 2 2

2

2 43

2 2

1 3( )

7 39 30( )

2

Lp

p pp

p p

p

vv vn v

v v2

−−= × = − + λ .

Γ λ Γ λ

22 2 3

2 2 2

7 62( ) ( )Tp

pp p

vv v n 2

− ≈ λ2 2 3pv v


completely, and a WP behaves as a classical particle.The following is obtained from (137) and (138a):

(139)

Thus, difference (139) assumes the value of in

the nonrelativistic limit and tends to zero as inthe ultrarelativistic limit. The RGP model allows oneto study the properties of the wave and commutatorfunctions in detail.

5.3. Wave Function Inserting (127) into (109), integrating over the

direction of vector , and performing simple transfor-mations, we obtain the following:

This integral is calculated using the known repre-sentation for function

(140)

which holds true for and (see, e.g.,[105, (2.5.42.3), p. 460]). Thus, we arrive at a compactexpression

(141)

which features a dimensionless Lorentz-invariantvariable

(142)

Here and elsewhere, the square root sign is under-stood in the sense of its principal value, and

.It is instructive to verify that function (141) satisfies

the Klein–Gordon equation. Considering that

and , we find

−

−

= − − − + + λ λΓ λ

22

2 2 42

2

35 163 11 7 ( ) .3 3 3 2

p p p

p

v v

v v v2

λ3

Γ λ22 ( )p

ψ ,( )G xp ψG

k

( )

∞ ψ , = − + σ

−× .σ σ

w w

w

w

20

2

2 2

2 2 21

(0 ) exp2

sin2 ( 2 )

m

E mx dE imxm

E mK m

xx

1K

( )

( )

∞− −

= + ,+

2 2

2 212 2

sinbt

a

dte c t a

ac K a b cb c

> 0a >Re Imb c

ζ σψ , = = ψ , ,

ζ σ

2 2def1

2 21

( 2 )( ) ( )

( 2 )G

K mx x

K mp p

[ ]ϕ σζ = ζ = − σ −

σ = − σ − .

w w

22 2 0

2

22 2

2

41

41 ( )

ie x imxm

x i pxm

= −w w w

2 0 2 2( )x x x

= −0 1( ) ( )K z K z = − −1 0 1'( ) ( ) ( )K z K z K z z

μμ

∂ ∂ = − + , 1

0 1( ) 2( ) ( )

zK z K z K zz z z

1 2020


Fig. 3. Three-dimensional plots of functions and ϕ in variables and .

11.0

8.5

6.0

3.5

1.0–5.0

–2.50

2.55.0 0

12

34

5

|ζ|

σ2x0/mw

σ2|x |/mw

σ2x0/mw

ϕ

–5.0–2.5

02.5

5.0 01

23

45

π/2

π/4

0

–π/4

–π/2σ2|x |/m

w

ζ σ w

2 0x m σ w

2 mx

where . Therefore,

(143)

Since

it follows that

Inserting these identities into (143) and using (141)we make sure that

The modulus and the phase of variable (142) aregiven by

(144)

(145)

= ζ σ2 2(2 )z m

μμ

= − +

∂ ∂ − + + .

hh

10 1

0 12

( ) 2( ) ( )

( )( )3 6( ) 1 ( )

zK z K z K zz z z

z zK z K z

z zz

( )μ μ μ∂ = − σ − ,σ

22

2 2m iz x p

z

μ μμ μ= −∂ ∂ = , ∂ ∂ = − .h

223 ( )( )mz z z z m

z

∂ + ψ = .2 2( ) 0Gm

[ ]

σ σζ = − +

σσ= + + + ,

w w

w

w w

2 24 2 2 04

2

24 240 2 2

2 2

4 41

481 ( )

x xmm

xxm m

x

σ − σϕ = , ϕ = .ζ ζ

w w

2 0 2 4 2

2 224 4sin 2 cos 2x m xm m

PHYSICS O

It follows from the last equality in (144) that, forany ,

(146)with equality holding true only at the packet center( , ). Relying on (145) and analyticityconsiderations, one may easily demonstrate that

(147)Indeed, it follows from (145) that at

. Since this is the only zero of phase at, the continuity of implies that at

Figure 3 illustrates the behavior of and as afunction of two independent dimensionless variables

and It can be demonstrated thatfunction has no zeros at and tendsto unity (zero) at ( ) in sector (147). It fol-lows from the properties of parameter that

if (at fixed ) or (at fixed ).

The behavior of wave function dependsstrongly on the value of ratio . Figure 4 presentsthe profile of the modulus of function (leftpanel) and an enlarged fragment of this function in theneighborhood of its maximum (right panel) as exam-ples. Calculations were performed with in

terms of dimensionless variables and

, and the reference frame was chosen so thatvector is directed along the third axis. Note that the

*xζ ≥ ,1

=w

0 0x =w 0x

ϕ < π .2ϕ → ∓0

→ ±w

0 0x ϕζ < ∞ ϕ ϕ → π∓2→ ±∞w

0 .xζ ϕ

σ w

2 0x m σ w

2 .mxψ ,( )G xp ≤ ζ < ∞1

ζ → 1 ζ → ∞ζ

ψ , →w( ) 0G x0 → ∞w

0x wx → ∞wx

w

0xψ , w( )G x0

σ mψ , w( )G x0

σ = .0 1m

σ w

2 0x m

σ w

2 3x mwx



Fig. 4. Three-dimensional plot of function in variables and (left) and its zoomed fragment in the

neighborhood of the maximum (right). It is assumed that ). Calculations were performed for .

σ2x0/mw

σ2x3/mw

1.00.90.80.70.60.50.40.30.20.10

3 2 1 00

0.51.0

–1

–1.0–0.5

–2 –3

|ψG|

σ2x0/mw

σ2x3/mw

|ψG |

1.0

0.9

0.8

0.7

0.6

0.50.09

0.090.04 0.04–0.01

–0.01

–0.06

–0.06

–0.11

–0.11

ψ , w( )G x0 σ w

2 0x m σ w

2 3x m

= , ,w w

3(0 0 )xx σ = .0 1m

value of σ/m is entirely unrealistic and was chosen onlyfor illustrative purposes, since the fine features ofbehavior of would be indiscernible at

. Naturally, is an even function ofboth variables; this explains the symmetry of the

profile in Fig. 4. In a sufficiently smallneighborhood of the maximum and as the

value decreases, the profile of function changes relatively slightly along the time

axis, but f lattens rapidly along the space axis. Figure 4provides an insight as to why the effective packet vol-ume defined in accordance with (97) is independent oftime, although wave function itself tendsasymptotically to zero at . The reason is thatthe packet spreads with time in such a manner that thisspreading compensates accurately the reduction. This compensation preserves the norm ofthe packet. Thus, contrary to the claims made in cer-tain textbooks on quantum mechanics, the spreadingof a relativistic WP does not imply that it vanishes.

5.4. Commutator Function

Inserting (127) into the definition of commutatorfunction (117) written in the intrinsic frame of refer-ence of the packet and integrating in angular variables,we obtain

ψ , w( )G x0σ .0 1m ! ψ , w( )G x0

ψ , w( )G x0ψ , w( )G x0

σ / mψ , w( )G x0

ψ , w( )G x0→ ∞w

0x

ψ , w( )G x0

, ;( )G xp q$

( )( )

∞

∗, − ; = − +

σ

−× .σ σ

w

02

2 2

2 2 21

( )* * * *exp2 V

sin *( )*

m

x E mEdE imxm m

E mK m

p p

xx

$


Expression (140) allows one to calculate theremaining integral over easily; we find [cf. (141)]

(148)

Invariant dimensionless variable is defined as

(149)

The relations between kinematic variables in theintrinsic frame of reference and the laboratory frame(see the end of Section 4.4) were used in deriving thelast equality. The modulus and the phase of variable are given by

Using these relations, one can prove that

(150)The obtained formulas for the wave and commuta-

tor functions provide an opportunity to examine theproperties of WPs and multipacket states in detail.However, the calculation of macroscopic Feynmandiagrams of interest to us is a rather technically com-plicated task even in such a simple model as the RGPbecause one needs to calculate multidimensional inte-grals of products of Bessel functions of complex vari-

E

σ, ; = = , ; .

σw

2 2def1

2 21

( )( ) 2 V ( )

( )G

K zmx m x

zK mp q p q$ $

z

σ = − σ − ∗∗ ∗

= + − σ σ − + .

22 2 0

2

2 2 2 2

* 1 2 *

1 ( ) 4 ( )2

Ez x iE xm E

p q x i p q xm

z

− σ σ += + + ,

σ += − .

2 24 2 24

2 2

2

22

2 ( )1 14

( )1arg arcsin2

pq x p q xzm m

p q xzm z

≥ ≥ < π .1 and arg 2*z E m z

1 2020


ables. In addition, for most practically important sce-narios, the regime is interestig in which the WPspreading can be neglected. This regime is typical,e.g., of particles of a not too rarefied gas if the meantime between two successive collisions of a particle ismuch shorter than the effective spreading time of thepacket that characterizes its state between the colli-sions. The case is similar for unstable particles withtheir lifetimes being short relative to the packet spread-ing time. The corresponding approximation in theRGP model is considered below.

5.5. Approximation of Nonspreading Packets

Let us determine the physical conditions underwhich the RGP spreading may be neglected9.

Owing to inequalities (128), (146), and (147), onecan use asymptotic equation (129), which yields

(151)

This formula holds true for any and , but it isstill too complicated for our purpose. Imposing addi-tional constraints, one may simplify expression (151)considerably by expanding variable in powers ofsmall parameter . In the intrinsic frame of refer-ence of a packet,

(152)

Elementary analysis suggests that one may limitthis asymptotic series to the leading correctionsif the following (necessary and sufficient) conditionsare satisfied:

(153)

It is evident that the space-time region defined bythese conditions becomes arbitrarily wide at . If(153) is satisfied, formula (152) evolves into a simpleand physically transparent expression:

(154)

9 It is clear that nonspreading WPs are easier to associate with(quasi) stable particles (i.e., localized objects) and to use asasymptotically free states of in- and out-fields in the -matrixQFT formalism instead of common plane waves, which occupythe entire spacetime and are thus unable to serve as an adequatemathematical model of particles.

S

/ − ζ σψ , = − − ζζ σ

σ σ+ − + + ζ ζ

2 2

3 2 2 2

4 6

4 6

(1 )1 3 1( ) exp 1 12 4

3 1 51 11 .32

Gmx

m

m m

p

2

p x

ζσ2 2m

[ ]

( )[ ]

ψ , σ σ= − + − −

σ σ− σ − + + .

2 40 2 2

2 4

4 62 2 0 2 2

2 6

ln ( )

3 31 22

3 ( )

G x

imx mm m

xm m

0

x

x x

w

w w

w w w 2

σ2 2m

σ σ σ σ .w w

2 0 2 2 2 2 2 2 2( ) andx m mx! !

σ → 0

( )ψ , = − − σ .w w

20 2( ) expG x imx0 x

PHYSICS O

It is evident that, in the intrinsic frame of referenceof a packet,

(i) behaves as a plane wave at (i.e., in the vicinity of the packet center).

(ii) is independent of time variable (as required, RGP (154) does not spread).

(iii) decreases in accordance with theGaussian law at large distances from the packet center,

; note that the last inequality does not con-tradict the second one from (153).

Using kinematic relations (110a) and (111), wewrite function in the laboratory frame:

(155a)

(155b)

where and are the components of vector ( ) that are parallel and perpendicular tovelocity vector , respectively. It follows from (155b)that an RGP undergoes relativistic compression in thedirection of velocity in the configuration space. Thisagrees with the abovementioned relativistic packetbroadening in the momentum space for Fock compo-nents with momenta codirectional with vector (seeSection 5.1).

It follows from (155a) that function isinvariant with respect to a one-parameter group of trans-formations with (or, in the com-ponent-wise notation, ). Itfollows that = . It is alsoevident that and along classical world line , but for any deviation from it. The probability of quantumdeviation from classical motion is sup-pressed by factor

It can be seen that transverse (with respect to veloc-ity vector ) deviations are less suppressed than longi-tudinal ones. Moreover, the ultrarelativistic packetscan deviate from classical trajectories exclusively inthe transverse direction; i.e., the quantum motion ofpackets is confined to classical light cones.

The nonrelativistic limit corresponding to approxi-mation (131) for function is given by

ψ , w( )G x0 σw

2 21x !

ψ , w( )G x0 w

0x

ψ , w( )G x0

σw 1x *

ψG

( )( ) ( )

ψ , = − − − σ Γ − ×−

0222 2

0

( ) expG x iE x

xp

p pp

p v x

v xx v

( )[( ) ⊥

= − −− σ Γ − − σ ,

022 2 2 2

0

exp iE x

xp

p p

v x

x v x

x ⊥x x

⊥= +x x x

pv

p

ψ ,( )G xp

+ θx x p Ep θ < ∞+ θ0 0 ,x x + θ px x v

ψ , ,0( )G xp x ψ , , − 0( 0 )G xpp x vψ , =( ) 1G xp [ ]ψ , = Γ0arg ( )G x mx pp

= 0xpx v ψ , <( ) 1G xp

δ = − 0xpx x v

( ) − σ δ + Γ δ . 22 2 2exp 2 ( ) p px v x

pv

φ ,( )G k p

( ) ψ , ≈ − − − σ −

≡ ψ , , .

220 0

2NR

( ) exp

( ) 1G

G

x im x x

xp p

p

p v x x v

p v !



The phase of the nonrelativistic wave function is independent of velocity ( at

and ).

Using relation (110b), one may rewrite inequalities(153) and expression (154) in an explicitly relativisti-cally invariant form:

(156)

(157)

Condition follows from (156). Inwhat follows, we call the considered approximation astable relativistic Gaussian packet (SRGP). Applyingthe Klein–Gordon operator to (157), we obtain

Thus, function approximately satisfies the Klein–Gordon equation (i.e., if conditions (128) and(158), which define the domain of applicability of theSRGP approximation, are fulfilled. It will be demon-strated below that this region is sufficiently wide forour purpose.

Note that the inclusion of contributions into (152) results in an even more rapid reduction in

with distance at and reduction in

with time at . One of the techni-cal corollaries of this behavior, which is useful inamplitude calculations, consists in the fact that theSRGP approximation may be extended to the entirespace-time integration region in integrals of the fol-lowing type:

where functions correspond to relativistic WPscharacterizing the states of particles , and is anarbitrary smooth function of without sharp extrema.Integrals of this type are found in formulas for themacroscopic amplitudes of scattering and decay of rel-ativistic WPs.

Direct (and rather cumbersome) calculations ofmean packet position with the use of (157) demon-strate that, as expected, a WP follows a classical trajec-tory. Thus, an SRGP reproduces completely the gen-eral properties of a relativistic WP.

The SRGP model allows one to verify the fulfill-ment of condition (107) needed for approximate for-

ψNRG

− = Γ ≈20 0 0x x xpv x

= 0xpx v 2 1pv !

σ − σ ,2 4 4 2 2 2 4 4( ) and ( )px m px m x m! !

[ ] σψ , = − − − .2

2 2 22( ) exp ( ) ( )G x i px px m x

mp

σ σ2 2 2 2x m!

[ ]

∂ + − Δ ψ , = ,

σ σΔ = − − .

2 2

2 42 2 2

2 4

(1 ) ( ) 0

46 ( )

Gm x

px m xm m

p

ψG

Δ 1)!

( )σ4 4m2

ψG σ σw

2 2 2 2mx *

ψG σ σw

2 0 2 2 2( )x m*

ψG

ψ , − ,∏ ( ) ( )dxf x x xpû û û

û

ψû

û ( )f xx

x


mula (106), which a significant element of the for-malism, to be applicable. It follows from (157) that

Therefore, condition (107) may be rewritten as

(158)

where . Elementaryalgebra gives

Thus, we have proven that

Therefore, inequality (158) is not an independentcondition and follows from the second constraintfrom (156), which defines the domain of applicabilityof the SRGP approximation.

Let us now examine the properties of commutatorfunction (148) in the SRGP approximation. Usinginequalities (150) and condition (128), we may writean asymptotic expansion for it [cf. (151)]:

(159)

For self-consistency, one should rewrite this for-mula in an approximation satisfying the SRGPapproximation conditions for function . Weuse the truncated series for for this purpose:

Inserting it into (159) and expanding the logarithmof ratio (written in the center-of-inertia

is

σ ∇ ψ , = − − .

22

2ln ( ) 2 ( )Gi x i px mm

x p p p x

( )− σ ,2 2 2( )pX m m Epp X !

= , = − , −0 0 0( ) ( )X X y xX y x

( )

−

= − + = + + −

≤ + + − .

22

2 2 2 4 2

2 2 2 2 2 2 2 20

2 2 2 2 2 2 2 2 20

( )

( ) 2 ( )( )

( ) ( )

( )

pX m

pX m pX m

pX E m m E X

pX E m m E Xp p

p p

p X

p pX X

pX X

p X X

− ≤ − .2 2 2 2( ) ( )pX m E pX m Xpp X

/ −, ; = σ

σ σ × − − +

σ × − + +

w

2

3 2 2

2 4

2 4

6

6

2 V (1 )( ) exp

3 1 31 18 128

1 51 11 .

Gm m zxz

zm m

z z m

p q$

2

ψ ,( )G xpz

σ ∗ = + ∗

σ σ σ∗ ∗ + − + . ∗

4 2

2

2 0 4 2 8

2 8

* 12

12

Ezm E

xi

m E m

x

x2

w(2 V )G m$

1 2020


frame) in powers of small parameter we arriveat the following relation:

where . It can be seen that, if conditions

(160)

are satisfied, it is sufficient to use just four principalterms of the expansion in . The resultant for-mula in this approximation is

(161)

where one should set (see the nextsection).

As expected, decreases rapidly if or (or both quantities) are sufficiently large. Theformal conditions for this, examined qualitatively inSection 4.4 for a WP of the general form, now becomeevident. In addition, commutator function (161) hascertain properties that were not obvious in advance. Itcan be seen, e.g., that the dependence of on vari-ables and vanishes on classical trajectories, whileratio is exponentially small even at sub-relativistic energies ( ) and tends to zero atultrarelativistic energies ( ) almost inde-pendently of . It is easy to demonstrate that

at any .Transforming (161) to the laboratory frame and

switching to nonrelativistic energies, we obtain

(162)

σ2 2,m

( )

, − ; σ= − + ∗

σ× + − ∗

σΓ − σ Γ −∗− − − +Γσ

σ σ∗− + − + , ∗ ∗ ∗

20

22 2

2 22 2

2

24 62 0 2 2

2 2 2 6

( ) 3* * *ln 12 V 2 *

41 14 3*

( 1) 3 ( 1)3 * *ln*2 2 8* *3 ( )

4 4

GD x imxm mE

mmE

mmE

Pm x

m E m m

p p

x

x

x

w

2

Γ =* *E m

σ σ σ σ ,∗ ∗ ∗ ∗2 0 2 2 2 2 2 2 2( ) andx E Ex! !

σ2 2m

( )

/, − ; =Γ∗

σΓ − ∗× − − − , ∗ Γσ

w

3 2

2 220

2

2 V( )* * *

1*exp2 *

Gmx

mimx

p p

x

$

/= π σw

2 3 2V [ (2 )]

, − ;( )* * *G xp p$ *p

*x

G$

0x xw(2 V )GD m

Γ − ∼1 1*Γ 1* @

*x

( ), − ; ≤ − − Γ w

( ) 1* * * exp 2 1 ,*2 V *G x m

mp p x$

σ

( ), ; ≈ − − − σσ× − − − .

w

2

0 2

22 2

0

( ) 2 V exp8

2

Gmx m im x

xp q

p q vx

v v x v

$

PHYSICS O

Here, and are the group veloc-ities of packets in the laboratory frame,

, and it is assumed that and

. Note that the contribution in (162)may be large if and parameter is sufficientlysmall.

The transition to the plane-wave limit in (161) isnot entirely trivial. In order to find this limit, we useasymptotic formula (133) (see Section 5.1), which pro-vides an opportunity to prove that the following rela-tion holds true for an arbitrary smooth function :

Therefore,

Using kinematic relations (120a), one may rewriteinequalities (160) in an explicitly relativistically invari-ant (although somewhat less clear) form:

(163)

It is easy to demonstrate that conditions (163) agreecompletely with conditions (156), which define thedomain of SRGP applicability. In much the same way,one may also rewrite expression (161) in an explicitlyLorentz-invariant form. We do not cite the corre-sponding formula, since it is rather cumbersome andnot very useful in practice. It is often easier to operatein the center-of-inertia frame of two packets and boostto the laboratory frame only in the resultant expres-sions.

Let us examine -packet matrix elements (123)with equal momenta ( , ) as an importantapplication of formula (161). The general case, whichwas discussed brief ly in Section 4.5, is illustrated wellby the simplest examples with and 3. Rewritingrelations (124) in the center-of-inertia frame, which isthe same as the intrinsic frame of reference (the samefor all one-packet “substates”) in this case, and insert-ing (161), we find:

= mpv p = mqv q

( )= +12 p qv v v 2 1pv !

2 1qv ! ∝ σ2 2( )m≠p qv v σ

( )F p

−

σ→, − ; = .

π0*

30*

1*lim ( ) ( ) (0)* * * * 8(2 ) 2imx

Gd x F e F

Ep

p p p p$

−

σ→−

, ; = π δ

= π δ − .

03 *

03

lim ( ) (2 ) 2 (2 )* *(2 ) 2 ( )

imxG

ipx

x E e

E ep

p q p

p q

$

[ ] + + σ ,

+ + σ .

22 2 2

22 2 2 2

( ) ( )

( ) ( )

p q x p q

p q x p q

!

!

n=ip p ∀i

= 2n

( )

( )<

<

, , , , = × ± δ −σ − ,

, , , , =× ± δ −σ −

σ+ δ δ δ − −

w

w w

w

w w

w w

1 2

1 2 2 3 3 1

22 2

221 2

33 3

22

2 2

(2 V )

1 exp

(2 V )

1 exp

2 exp ,2

i j

s s

s s i ji j

s s s s s s i ji j

s x s x m

s x s x m

p p

x x

p p

x x

x x



Fig. 5. SRGP in the configuration space (left) and themomentum space (right). Arrows denote the direction ofthe packet momentum.

where plus and minus signs correspond to bosons andfermions, respectively. It can be seen that the Bose–Einstein attraction and the Fermi repulsion, which arefound at for any pair of particles , are sig-nificant only at short distances between the centers ofpackets and satisfying condition

(164)

In other words, both effects become significantwhen the spatial distance between identical packets(measured in their common rest frame) is comparableto the size of packets (i.e., precisely when it becomesnecessary to consider the dynamic interaction of pack-ets). At sufficiently large distances between packets,the difference in quantum statistics is irrelevant. Thisconclusion remains valid for states with an arbitrarynumber of noninteracting identical packets and is cru-cial to constructing the quantum WP statistics.

5.6. Effective Size and Mass of a Packet

It is easy to find an explicit form of volume ,which is defined by (97), in the RGP approximation:

(165)

The explicit formula for mean mass may also bederived as an asymptotic expansion in parameter .It is convenient to use Watson’s lemma, which statesthat the term-wise Laplace transform for the powerexpansion of function is the expansion of theLaplace transform of this function, for this purpose. Inthe case under consideration,

Applying this asymptotic formula to the numeratorand the denominator in (99) and inserting the expan-sion for the normalization constant from (126)

we find the ratio of the mean packet mass to the fieldmass:

Thus, the mean (effective) mass of a WP exceedsthe mass of its constituent Fock states by

.

=i js s ,( )i j

i j

σ − .w w22 1i jx x &

wV

[ ]/π λ λ π = =

σλ

× − + + , λ = . λ λ λ σ

3 221

2 231

2

2 3 2

( )V22 ( 2)

9 249 118 128

Km K

m

w

2

mλ1

( )f t

( )

∞ ∞− −λ

+=

Γ +!λ

> ,λ .

∼

( )1

00

( ) (0)( )

0 1

na t

n an

n a fdtt e f tn

a @

/

σπλ = − + + , λ λ λ

3 2

2 2 32( ) 3 33 11

4 32N

m2

= + − + . λ λ λ2 33 3 11

2 8mm

2

δ ≈ . σ21 5m m


Let us now turn to the SRGP model:

(166)

One may define (with the same accuracy) theeffective WP size in the intrinsic frame of reference asdiameter of a sphere with volume ; i.e.,

(167)

Evidently, the effective size of a packet in the labo-ratory frame in the direction of its momentum is

. An oblate spheroid with a cross diameterof ~ and an angular eccentricity of may serve as an illustration of a packet in the configu-ration space (see Fig. 5, left panel). Volumetric density

of a packet at its center exceeds the density atthe (nominal) boundary by a factor of more thanthree. To be more precise, this density ratio in the intrin-sic frame is

5.7. Energy and Momentum Uncertainties of a Packet

Let us examine the relation between parameter and quantum uncertainties of the WP energy andmomentum in the SRGP approximation. Using themean-value theorem and conditions (95), we obtainthe following identity:

where is the vector to be determined with a smallabsolute value. This vector has a clear physical mean-ing: the WP momentum uncertainty. It is natural to

/π . ≈ ≈ . σ σ

w

3 2

2 31 969V

2

wd wV

( )/ /π . = ≈ ≈ . π σ σ w

w

1 3 1 66V 9 1 1 5552

d

p= Γ wd d p

. σ1 555 Γarccos( )p

ρ ,( )xp

( ) ( ) / . σ / = π ≈ ≈ . . w

1 32 2 1 21exp 2 exp 9 16 3 35d e

σ

= ϕ + δ , φ ,π

= φ + δ , ,

w 32 V ( ) ( )(2 ) 2

( )

G G

G

dmEk

kp p p k p

p p p

δp

1 2020


assume that 10. This condition allows one toperform the expansion

which yields

where and are the longitudinal and transverse(with respect to momentum vector ) components ofsought-for vector . In accordance with (130),we have

(168)

where denotes vector written in the intrinsicframe of reference of the packet. Comparing rela-tions (168) and (166) (derived in the same approxima-tion), we find

(169)Formulas (167) and (166) yield “uncertainty relation”

(170)

To illustrate this, we consider a hadron WP. Itseffective size obviously cannot be smaller than the nat-ural size of the hadron itself (i.e., fm). It thenfollows from (169) and (170) that MeV;therefore, σ ≈ MeV. Thus, 300 MeV isthe maximum possible value of for a hadron WP. Wewill discuss the extreme allowed values in the SRGPmodel in Section 5.8.

Owing to the spherical packet symmetry in theintrinsic frame of reference,

(171a)

from which it follows that

(171b)

Relations (171) allow one to estimate the momen-tum uncertainty and the effective WP (half) width inthe momentum space. We can see that the relative

10 This assumption is verified a posteriori by relation (171).

δ Epp !

( )+δ

δ ⋅δ ⋅ δ≈ + + − ,

22

2 2 412 2

E EE E E

p p pp p p

p pp p p

( )

( )

+δ

⊥

δ− + δ ≈ +

δδ ⋅ δ− = + + ,

22

22 22

2 2

2

22 2

E E m

mE

p p p

p p

pp p p

pp p p

δ p ⊥δpp

δp

/⊥

/

δ + Γ δπ ≈ − σ σ Γ

δπ = − , σ σ

w

w

2 223 2

2 2 2

3 2 2

2 2

V exp4

exp4

p

p

p p

p

δ wp δp

δ ≈ σ ≈ . σ .w

2 2 26 ln 2 4 159p

/δ ≈ π ≈ . .w w

1 33 (4 3) ln 2 3 171d p

1*d *

δ 600*p &

δ 300*0.5 p &

σσ

⊥δ = δ ≈ σ

δ = δ Γ ≈ σ Γ ,

w

w

2 2 2

2 2 2 2 2

2 4 ln 231and 2 ln 23 p p

p p

p p

( ) ⊥δ ≈ Γ + σ, δ δ ≈ Γ .

22 ln 2 2 2p pp p p

PHYSICS O

momentum uncertainty, , is small for ultrarela-tivistic momenta, but becomes arbitrarily large as

:

The corresponding energy uncertainty is

Therefore, the relative energy uncertainty is alwayssmall, and the ultrarelativistic asymptotics,

, defines the upperlimit of this ratio.

5.8. SRGP Applicability Domain

Depending on the WP production mode (i.e., thechain of reactions producing it), the effective WP sizesmay vary from microscopically small to macroscopi-cally large. The latter possibility should come as nosurprise if one recalls that the “size” of a plane isinfinite. Let us stress once again that the effective WPsize is a characteristic of the quantum state of a fieldrather than an internal property of this field. This sizenaturally depends on the field properties, but notexclusively on them. The “natural” and, probably,practically unreachable upper limit, , for theeffective size of a WP characterizing an unstable parti-cle should be of the order of its own lifetime (macro-scopic for long-lived particles such as a neutron or amuon), while the “lower limit”, , is set by theinherent particle size, which is on the order of 1 fm forhadrons (compound particles) and on the order of theCompton wavelength (~1/m) for leptons and gaugebosons (structureless particles). Whatever the case,owing to the general restriction, the packetsize allowed by the formalism should be much largerthan .

In the case of unstable particles, conditions (153) ofapplicability of the SRGP approximation impose anadditional tight constraint on the value of parameter

. For these conditions to be satisfied at (i.e., to avoid substantial WP spreading within the statelifetime), the inequality should hold true.Therefore, sets the absolute upper limit11

for values of allowed in the SRGP model. Thus,

11In the physical (rather than in the formally mathematical)sense.

δp p

→ 0p

σ → ∞,δ → σ → .

2 ln 2 for

6 ln 2 for 0

m ppp p p

δ ≈ δ ⋅ ≈ δ ≈ σ . 2 ln 2E E mp p pp p p v p

δ ≈ δ ≈ σ2 ln 2E E mp p p p

w

maxd

τ

w

mind

σ2 2m!

w

mind

σ ≤ τw

00 x &

σ τ4 2 2m!

σ = τmax mσ

( ) ( ) ( )/ / /π π τ= =σw

1 6 1 6 1 2min

max

9 1 92 2

dm



Table 1. The maximum allowed value of parameter σ ( ) and ratio , and the minimum allowed

effective size of wave packets ( ) for some unstable particles in the SRGP approximation

Particle , eV , cm

σ = Γmax m Γ σ = Γmax m

≈ Γmin 1.55*d m

σmax Γ σmax w

mind

±μ −. × 11 78 10 −. × 91 68 10 −. × 41 72 10±τ . × 32 01 10 −. × 61 13 10 −. × 81 53 10±π .1 88 −. × 81 35 10 −. × 51 63 10

π0 . × 43 25 10 −. × 42 41 10 −. × 90 94 10±K .5 12 −. × 81 04 10 −. × 65 99 100SK . × 16 05 10 −. × 71 22 10 −. × 75 07 100LK .2 53 −. × 95 08 10 −. × 51 21 10±D . × 31 09 10 −. × 75 82 10 −. × 82 82 100D . × 31 73 10 −. × 79 28 10 −. × 81 77 10±sD . × 31 61 10 −. × 78 18 10 −. × 81 91 10±B . × 31 46 10 −. × 72 76 10 −. × 82 11 100B . × 31 51 10 −. × 72 86 10 −. × 82 03 100sB . × 31 55 10 −. × 72 89 10 −. × 81 98 10

n −. × 52 64 10 −. × 142 81 10 .1 16

Λ . × 15 28 10 −. × 74 74 10 −. × 75 81 10±Λc . × 32 74 10 −. × 61 87 10 −. × 81 12 10

sets the lower limit for the effective spatial size aWP characterizing an unstable particle in the SRGPapproximation.

Since WPs are not related in any way to naturaldecay width in the considered formalism, it isapplicable only to states with (more precisely,

), that is, to packets with spatial sizes. The lifetime of a particle needs to be

sufficiently large to reconcile this constraint withconditions (153). Since for all known long-lived elementary particles and atomic nuclei,

and, naturally, .Therefore, the SRGP approximation is expected to beapplicable both to stable and unstable long-lived par-ticles (neutrons, muons, -leptons, charged pions,etc.) and nuclei, but inapplicable to broad hadronicresonances, gauge bosons, and other particles that donot satisfy the condition. Note also that themaximum value of allowed in the SRGPmodel is . Therefore, the“weighting effeсt” is substantial for the nucleon andbaryon resonances, while its impact to physicalobservables needs dedicated investigation.

of

Γ = τ1σ Γ@

σ Γ2 2@

τ∼w w

maxd d!

Γ m!

σ = τ = Γ Γmax m m @ σmax m!

τ

σ Γmax @

δ = −m m mδ ≈ . σ = . Γ2

max max1 5 1 5m m


The values of , , and for a numberof unstable elementary particles are presented for illus-trative purposes in Table 1. The masses and lifetimes ofparticles used in the calculation were taken from [106].The particles to be listed in Table 1 were chosen for thefollowing reasons: the decays of , , , and arethe dominant sources of atmospheric and acceleratorneutrinos and antineutrinos with energies TeV;the decays of short-lived , , , , and mesonsand , hyperons become the dominant sources of(anti) neutrinos at TeV (see, e.g., [107]); lep-tons, which emerge as rare events in interactions of(anti) neutrino beams with matter, serve an importantpurpose in oscillation experiments as “indicators” offlavor transitions . A neutral pion and a neu-tron are listed to illustrate two extreme cases (verysmall and very large values of , respectively). Inaddition, the decays of free neutrons are sources oflow-energy astrophysical electron antineutrinos [108].

Numerous -active radionuclides, which act asdecay sources of and in reactor experiments andfuture experiments with beams [109–111]; heavy

σmax Γ σmax w

mind

±π ±K 0LK ±μ

ν 1E &

0SK D sD B sB

Λ Λc

ν 1E @ τ

μ τν → ν

σmax

±βνe νe

β

1 2020


ions, which emit in the process of capture of orbitalelectrons from low-lying electron shells and regardedas candidate sources of monochromatic neutrinobeams [112]; as well as excited ions moving along circu-lar orbits and coherently emitting pairs of all flavors[113, 114] are not listed in Table 1. The value of forall such ions and nuclei is of the order of either thenucleus size or the -orbit diameter, and their half-lifeperiod is long. Therefore, ;i.e., the SRGP approximation conditions are defi-nitely fulfilled.

It follows from Table 1 that the typical valuesare “mesoscopic” (i.e., span the range from an ang-strom to a micrometer) and are much smaller than thewidths of tracks of charged particles in detectors.Ratios are no greater than for all thementioned particles (except for a neutral pion), thusallowing for a fairly wide “spectrum” of values satis-fying the SRGP applicability conditions:

It should be noted that hadronic resonances andother short-lived particles are also of interest in thecontext of neutrino oscillations, since neutrinos andantineutrinos may be produced in their decays and,what is probably even more important, may produceshort-lived particles in interactions with matter. Theinapplicability of the SRGP approximation to short-lived particles does not imply that the processesinvolving them cannot be considered within the -matrix formalism, since resonances may be regardedas virtual particles (i.e., internal lines of Feynman dia-grams). It is then not necessary to construct WPs forresonances. The same approach is actually applicableto processes involving neutral pions with a relativelylarge ratio, since the events of -meson pro-duction are identified in neutrino experiments basedon gamma quanta produced in their decays12. Thatsaid, the generalization of this formalism to particleswith arbitrary decay widths is of a methodologicalinterest, if for no other reason than because a moregeneral approach would allow one to define moreaccurately the domain of applicability of the theoryoutlined here and provide a way to simplify the calcu-lations for specific processes with the use of knownphenomenological models for the resonance neutrinoproduction of mesons.

12It is worth noting here that the states of gamma quanta emerg-ing in processes involving (anti) neutrinos (real or virtual) maybe considered in the plane-wave limit, although a rigorous sub-stantiation of this statement is outside the bounds of the presentstudy. See [115, 116] for a thorough discussion of the theory ofphoton WPs.

νe

νν

w

mind

K

/1 2T /Γ σ ∼ w

minmax 1 2 1d T

w

mind

Γ σmax−≈ × 62 10

σ

Γ σ σ .2 2 2 2max m! ! !

S

Γ σmax π0

PHYSICS O

6. SCATTERING OF WAVE PACKETSLet us apply the above formalism in calculating the

probability of interaction of covariant WPs. Althoughthe results are similar in part to those discussed byPeskin and Schroeder [117], some of them are noveland intriguing. For example, the explicit form of sup-pression of the interaction probability of WPs (in theSRGP model) scattering with a nonzero impactparameter was calculated. This result is conceptuallyand practically important.

The cross section for certain processes (e.g.,) calculated in the plane-wave approxi-

mation does indeed contradict experimental data.This has been observed for the first time at the VEPP-4 collider in Novosibirsk in the MD-1 detector data[118]. The measured cross section turned out to be~30% lower than the calculated one at low photonenergies. It was demonstrated in [119] that impactparameters up to 5 cm produce a considerable con-tribution to the cross section of the pro-cess calculated in the plane-wave approximation,while transverse size a of the colliding beams did notexceed ≈10–3 cm. If the impact parameters are limitedas , the observed number of photons decreases.The theory in which the finite size of colliding beamsare considered is developed in [120].

The results of our calculation are similar to theones obtained in pioneering study [120], although theformalism developed here differs considerably fromthe formalism used in [120]. Let us discuss these dif-ferences. The definition (Eq. (4.1)) and the normal-ization condition (Eq. (4.2)) of WPs used in [120]demonstrate that they are noncovariant. Statisticalaveraging over the states of particles (Eq. (4.7)) wasperformed to pass to the density matrix in the momen-tum representation, which is a Fourier transform ofWigner function (a fine introduction to thetheory of Wigner functions and Weyl transformationsmay be found in [121]). Integrals =

and = of the Wigner functionyield the particle number densities in the momentumand coordinate representations, respectively. The rela-tive velocity of two particles was introduced ad hocinto the resulting formula for the number of interac-tions (e.g., Eq. (4.23)). The following approximationwas also used: final-state particles were regarded asplane waves.

We deal with covariant WPs in the initial and finalstates and do not use the Wigner function formalism,performing calculations for a collision of WPs definedby arbitrary mean 4-coordinates , and 4-momenta

, . The application of this simpler approach ismade possible by the assumption of a weak depen-dence of the matrix element upon a momentumchange within the interval of the order of the WP

+ − + −→ γe e e e

ρ+ − + −→ γe e e e

ρ ≤ a

, ,( )n tr p

, , ( )d n tr r p ,( )n tp

π , ,3(2 ) ( )d n tp r p ,( )n tr

ax bxap bp



momentum dispersion. This assumption, however, isnot universally valid. For example, both the modulusand the phase of the matrix element are of interest incertain problems (e.g., scattering). The matrix ele-ment variation on a scale of the momentum variationin a WP should not then be neglected. Factoring thisvariation in as a correction at the level, one maycalculate the correction to the plane-wave cross sec-tion as a function of phase of the matrix element. Thisis a novel and potentially interesting method for mea-suring this phase by varying the impact parameter ofthe particle collision [122]. The case of a strongmomentum dependence of the matrix element isanother important exception. A strong dependence isexpected if an intermediate state relating the sets ofinitial and final particles with a 4-momentum on themass shell exists. By virtue of the singular propagatorof the intermediate state, the strong dependence of thematrix element on the combination of momenta cor-responding to the 4-momentum of the intermediatestate is then evident. This is the case corresponding toa virtual neutrino in the quantum-field theory of neu-trino oscillations discussed in the next section.

Thus, limiting ourselves to weak variations of thematrix element on a scale of the momentum variationin a WP, we calculate the probability of interaction oftwo WPs. The formula reveals the suppression of theinteraction probability at a nonzero impact parameterof colliding packets. We also demonstrate explicitlyhow the dimensionless interaction probability isexpressed in terms of the product of the plane-wavecross section (with a dimension of area) and themicroscopic luminosity (with a dimension of recipro-cal area). Macroscopic averaging over the impactparameter leads to the well-known expression for theevent number expressed in terms of the product of theplane-wave cross section and the f lux.

6.1. Scattering Amplitude

Let us consider the interaction of two colliding par-ticles and . Final state and the process dynamicsare arbitrary. For definiteness, each initial and finalstate is assumed a fermion WP. The quantum particlestatistics is irrelevant in this case. We are interested inprocess amplitude , wherethe curly bracket denotes a set of WPs with meanmomenta and mean coordinates at time points for the initial (index ) and final (index f) states. Thisprocess amplitude takes the form

(172)

pp

σ p

a b X

= , − ,S 1f f i ix xp pA

ip ix 0ix

i

−

,

φ ,φ ,= π π

× , , , ,

∏ 3 3

*( )( )1(2 ) 2 (2 ) 2

( )

f fi i

i f

ik xiq xf f fi i i

i f

i f i f

d ed eE E

i r sq k

k k pq q p

q k

A1


where

(173)

is the standard plane-wave -matrix element involv-ing singular factor , whichconsiders the conservation law of energy and momen-tum and the regular matrix element

Factor N, which corresponds to the normalizationof the initial and final states, is given by

(174)

Let us rewrite the -function in the form of an inte-gral over four-dimensional space:

This provides an opportunity to rewrite (172) as

which yields

(175)

The approximate equality is valid for WPs that are suf-ficiently narrow in the momentum space and under theassumption that the variation of matrix element

is negligible on a scale of variation ofform factor . The momenta integrals in thisapproximation turn into functions defined in accor-dance with (108). Overlap integral

was introduced in (175). A more general integral

which depends on external momentum , emerges inthe theory of neutrino oscillations with WPs; there-fore, we use the same notation at . Such an inte-

( )

, , ,= , , , − , , ,

= π δ − , , , 1 1 1 1

4 4

( )1

(2 ) ( )f f f f

i f i f

n n n n

i f i f i fi f

i r ss … s r … r

q k i r s

q kk k q q

q k

S

Sπ δ − 4 4(2 ) ( )i fi f

q k

, , ,( ).i f i fi r sq k

, ,

= , , , ,=

.

∏ ∏∏ ∏

w w

2

2 V 2 V

2 V( ) 2 V( )i f

f f f f i i i i

i i f fi i

i fi i

x x x x

m m

E Ep p

p p p p

p p

1

δ

− − π δ − =

4 4 4(2 ) ( ) .

i fi f

i q k

i fi f

q k d xe

+ −

,− −

,

φ ,= π

φ ,× , , ,

π

ψ , − ψ , −

× , , , ,

∏

∏

( )4

3

( )

3

4

( )1(2 ) 2

*( )( )

(2 ) 2

1 ( ) *( )

(

i i

i

f f

f

iq x xi i

i f

ik x xf f f

i f i f

i i f fi f

i f i f

d ed xN E

d ei r s

E

d x x x x x

i r s

q

k

q q p

k k pq k

p p

p p

A

1

, , , . V1 (0) ( )i f i fi r sp pA 1

, , ,( )i f i fr sq k

φ ,( )i iq pψ

,= ψ , − ψ , −∏

4(0) ( ) *( )i i f fi fd x x x x xp pV

,= ψ , − ψ , −∏

4( ) ( ) *( ),iqxi i f fi f

q d x e x x x xp pV

q

= 0q

1 2020


gral is calculated for the SRGP model in this section.In the next section, the following important relationvalid for the SRGP model is proven:

(176)

where and is theGaussian function with the plane-wave limit corre-sponding to the four-dimensional Dirac function.The explicit form of this function is given in the nextsection. Here, we use its plane-wave limit. Four-dimensional overlap volume may be written as

6.2. Number of Interactions for Noncollinear Collisions of Wave Packets

Using (176), we write the amplitude modulussquared in (175):

(177)

The microscopic probability of interaction which is given by in (177), is propor-

tional to the integral over four-dimensional space-time ofthe product of densities of probabilities of initial

and final WPs to intersect at point . Let us multiply (177) bythe Lorentz-invariant number density of final states

and integrate in spatial coordinates of

packets. This corresponds to the setup of many currentexperiments, where the momenta of particles are mea-sured more accurately than their coordinates. Using thefact that, in accordance with (112), =

we find the number of interactions:

(178)

= π δ − ,V2 4(0) (2 ) ( )VG i fP P

= ,i iiP p = ,f ff

P p δ ( )G K

δ

V

,= ψ , − ψ , − .∏

2 24V ( ) ( )i i f fi f

d x x x x xp p

,

π δ −=

× ψ , − ψ , − .

∏ ∏∏

24 42

2 24

(2 ) ( )2 V( ) 2 V( )

( ) ( )

i f

i f

i fi f

i i f fi f

P PE E

d x x x x x

p pp p

p p

A

+ → ,a b X 2A

ψ , − 2( ) V( )i i ix xp p ψ , − 2( ) V( )f f fx xp px

π ∏ 3(2 )

f f

f

d dp x

ψ ,2( )f f fdx p x

,V( )fp

= π

π δ −=

ψ , −×π

ψ , −× .

∏

∏

23

24 4

24

3

2

(2 )

(2 ) ( )2 2

( )V( )(2 ) 2

( )V( )

a b

f

ff

f

i f

f a a

f a

b b

b

ddN d

P PE E

d x xd xE

x x

p p

p

px

p pp

pp

A

PHYSICS O

The calculation of the last factor in (178), which isa four-dimensional overlap integral of two scatteredWPs and , gives

(179)

where are the velocities of particles and , =

, is aunit vector codirectional with the relative velocity(explicit formulas for relative velocities are presentedin Section 6.3), and = , where

As a result, (178) may be rewrit-ten in the following form:

(180)Here, is the differential effective scattering

cross section in the plane-wave approximation:

where identity was used.Factor in (180) has the physical meaning ofluminosity of scattering of two WPs in the definitionused in the scattering theory and accelerator physics.

is defined in accordance with (179) with

The number of interactions of two WPs and isgiven by (180). In the case of noncollinear scattering,the number of such collisions is suppressed by factor

where should be interpreted as theimpact parameter of scattered WPs.

The distribution of impact parameter inexperiments with scattering of particle beams a and bis set by the mode of production of beams. In the caseof a Gaussian distribution, the number of pairs of par-ticles and with impact parameter is

(181)

where is the dispersion of impact parameter of particle beams and under the

assumption of a fixed direction of their relative veloc-ity ; , are the numbers of particles and .

Averaging over the impact parameter, we arrive atthe following expression for the number of interac-tions of particles and :

(182)

a b

,− × σ

,

ψ , − ψ , −

×= = ,πσ

vv

2 2

2 24

( ) 2

2

( ) ( )V( ) V( )

( )2

x ab

a a b b

a b

abx ab ab

x x x xd x

Le b n

p pp p

b n

,a bv v a b vab

− − ×2 2( ) ( )a b a bv v v v = − ,a bb x x = ab abn v v

,σ2x ab , ,σ + σ2 2

x a x b

,σ = σ2 21 2 ,x i i = , .i a b

,− × σ= σ × = σ .2 2( ) 2( ) (0) x abdN d L d L e b nb n

σd

π δ −σ = ,

π−∏

24 4

32 2 2

(2 ) ( )(2 ) 24 ( )

i f f

b fa b a b

P P dd

Ep p m m

p

= −2 2 2 2 2 2( )a b ab a b a bE E p p m mp p v

×( )L b n

×( )L b n

,= πσ2(0) 1 2 .x abLa b

,− × / σ2 2( ) 2 ,x abe b n ×b n

×b n

a b ×b n

, ,

×× = − , πΣ Σ

2

2( ) exp2 2

a bab

x ab x ab

n nn b nb n

,Σx ab

= ×Tb b n a b

n an bn a b

a b

= σ = σ , ( ) ( )T ab T TdN d d n L d Lb b b



where luminosity is defined by the effective spatialdispersions of WPs of particle beams and ,

(183)

In realistic experimental configurations, Luminosity given by (183) is defined

in this approximation by the spatial dispersion of par-

ticle beams and , , in accordance with

the plane-wave approximation.

6.3. Relativistic Invariance of the Impact Parameter Squared

The four-dimensional overlap volume

is Lorentz-invariant due to the relativistic invarianceof function . The Lorentz-invariant number ofevents in (180) requires Lorentz invariance of themodulus of vector . Let us demonstrate byexplicit calculations that, although , , and arenot invariant, modulus is Lorentz-invariant.

Let us denote vector in the rest framesof particles and as and . Their explicit form is

where The relative velocities in the same notation are

written as

Thus,

La b

, ,

= .π σ + Σ2 2

12 ( )x ab x ab

L

, ,σ Σ2 2 .x ab x ab! L

a b,πΣ

21

2 x ab

L

= ψ , − ψ , −2 24V ( ) ( )a a b bd x x x x xp p

ψ( )x

×b nb n ×b n

×b n= −a bb x x

a b ( )ab ( )bb

( )

( )

Γ= +Γ +

Γ= + ,Γ +

2( )

2( )

1

and1

a aa a

a

b bb b

b

b b bv v

b b bv v

− /Γ = − 2 1 2(1 ) ,i iv = ,( ).i a b

( )

( )

= =

Γ − + = − − Γ Γ +

= =

Γ − + = − . − Γ Γ +

( )( )

( )

( )( )

( )

1 11 1 ,1 1

1 11 11 1

aa b

ab b ab

a a bb a

a b a ab

b aba a b

a

b a aa b

a b b b

E

E

pv v

v vv v

v vpv v

v vv v

v v

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

Γ − +× = × −− Γ Γ +Γ × × + ×

Γ + Γ − +× = × −

− Γ Γ +Γ × × + ×

Γ +

( )

( )

1 11 11 1

,1

1 11 11 1

.1

a a a bab b

a b a a

aa a a b

a

b b a bba a

a b b b

bb b b a

b

v vb v b v

v v

b v bv v v

v vb v b vv v

b v bv v v


Useful formula

demonstrates that although Italso shows that the angle between the relative veloci-ties in the rest frame of particles is nonzero. This is apurely relativistic effect. Relative velocities and are collinear in the nonrelativistic approximation.

It can be proved that with is a relativistic invariant. It is instructive to note that

Let us first calculate where and in the laboratory frame with

. In the rest frame of particle , one considersthe 4-vector of impact parameter withcomponents

where . We then calculate

Thus,

(184)

Since (184) is written in an invariant form, we may

calculate vector product in the rest frame ofany particle ( or ). For example, the following isobtained in the rest frame of particle :

where the right-hand part is the square of the vectortransverse to vector :

( )( )

( ) ( )( ) ( )

× × = − Γ Γ−

Γ Γ Γ− + + Γ + Γ + Γ + Γ +

2

2

111

1 1 1 1

a bab ba

a ba b

a b ab a ba b

a b a b

v vv vv v

v vv v

≠ ,ab bav v = .ab bav v

abv bav

×2( )a

abb n =ab ab abn v v

× = − .2 2 2( ) ( ) ( ) 2( )a a a

ab ab abb v b v b v

= −2( ) 2 2( ) ,a

abu bb= Γ ,(1 )i i iu v = ,(0 )b b= −2 2b b a

= ,( ) ( )0( )a ab b b

Γ= , = − ,Γ +

( ) ( )0

( )( )1

a a a a aa

a

bub bu vb b

Γ = −( ) ( )a a abubv

−=

−= .

( )

22

2

( )( ) ( )( )( )

( ) 1and( )

a a a b bab

a b

a bab

a b

bu u u buu u

u uu u

b v

v

[ ]

[ ]

−× = + −

−−

= + − = × .+−

22( ) 2 2

2

222 ( )

2

( )( ) ( )( )

1 ( )( ) ( )( )

( ) 11 ( )

a a a b bab a

a b

ba b a bba

a ba b

bu u u bubu b

u u

b u u bu bu bu uu u

b n

b n

×2( )a

abb na b

a

Γ× = − ,Γ −

2 22( ) 22( )

1a b b

abb

bvb n b

bv

= − = −

Γ= − .Γ −

22 2 2 2

2

2 22

2

( )

( )1

b bT L

b

b b

b

bv vb b b bv

bvb

1 2020


Fig. 6. Macroscopic Feynman diagram of the general formwith the exchange of massive neutrinos.

Is Xs Fs

x

y

νi

XdId Fd

Thus, the relativistic invariance of expression was proven.

7. MACROSCOPIC DIAGRAMS

The modification of Feynman rules, which areneeded to calculate the scattering amplitudes corre-sponding to macroscopic Feynman diagrams withtheir external legs characterized by relativistic WPsrather than by Fock single-particle states, is discussedin this section.

7.1. Macroscopic Diagram of the General Form

We consider single-particle reducible connecteddiagrams. Their overall structure is presented in Fig. 6.The external legs of such diagrams correspond toasymptotically free initial (“in”) and final (“out”)WPs and in the coordinate repre-sentation (i.e., wave functions and

characterized by the most probablemomenta , ; space-time coordinates , ;masses , ; and parameters13 , (“disper-sions”) characterizing the uncertainty in momenta).The following designations are used in what follows: ( ) is the set of in (out) packets in block (“source”), and ( ) is the set of in (out) packets inblock (“detector”). Blocks and denote theregions of interaction of fields and may contain arbi-trary internal lines and loops. The internal line con-necting blocks and denotes the causal Green’sfunction of massive neutrino with mass ( ). It is assumed that vertices and are

13And, in a more general case, by sets of parameters , .

× = ×( ) ( )a bab bab n b n

, ,a a as xp , ,b b bs xp( )ψ ,a a axp

( )∗ψ ,b b bxpap bp ax bx

am bm σa σb

σa σb

sIsF sX

dI dFdX sX dX

sX dXνi im

= , , ,1 2 3i … x y

PHYSICS O

characterized by the SM Lagrangian complementedphenomenologically with the Dirac or Majorana massterm and the corresponding kinetic contribution.Therefore, the initial or the final states (or both ofthem) should feature a charged lepton or an (anti)neutrino14. Blocks and are assumed to be mac-roscopically separated in space-time (hence the term“macroscopic diagram.”)

7.2. Examples of MacrodiagramsThe simplest example of a macrodiagram of the

type presented in Fig. 6 is shown in Fig. 7. The sum ofsuch diagrams (over index ) gives the process amplitude:

(185)where symbol denotes that the diagram verticesare macroscopically separated in space-time. Pro-cesses (185) violate two lepton numbers, but their sumis conserved. Such simple reactions often produce thegreatest contribution to the number of events in exper-iments on neutrino oscillations with atmospheric andaccelerator (anti) neutrinos. In the standard quantum-mechanical approach, process (185) is regarded as asequence of three independent subprocesses:

(i) pion decay in the source (at point );(ii) neutrino propagation with the “transforma-

tion” of into ;

(iii) quasi-elastic interaction in thedetector (at point ).

In the diagrammatic QFT approach, process (185)is regarded as a unified whole, and the intriguingquantum-mechanical phenomenon of f lavor transi-tions (“oscillations”) is reduced to trivialinterference of diagrams with virtual neutrino fieldsof a certain mass. It is not even necessary to use statesor neutrino fields with definite f lavors and .

Let us examine several other examples of poten-tially interesting macrodiagrams. Figure 8 shows fourdiagrams of the lowest order in electroweak interac-tion, which characterize the decay in the sourceaccompanied by quasi-elastic scattering of a virtualneutrino (antineutrino) off a neutron (proton) withthe production of a lepton at the detector vertex.Naturally, all these processes are strictly forbidden bythe lepton-number conservation law in the SM withmassless neutrinos, but they become possible if aDirac or Majorana mass term is present in theLagrangian. The diagrams in Figs. 8a and 8b are validboth for Dirac and Majorana (anti) neutrinos, whilethe diagrams in Figs. 8c and 8d are for Majorana neu-trinos, which are the same as antineutrinos, only. In

14To avoid unnecessary complications, we discount the possibilitythat the external one-packet states contain gauge or Higgsbosons.

sX dX

i+ + −π ⊕ → μ ⊕ τ ,n p

⊕

+ +μπ → μ ν 1x

μν τν+

τν → τn p2x

μ τν → ννi

μν τν

μ 3e

τ


PHYSICS OF PARTICLES AND NUCLEI Vol. 51 No. 1 2020


Fig. 7. Example of a macroscopic Feynman diagram describing the decay at space-time point and the subsequent quasi-elastic neutrino production of a lepton at point . Points and may be separated by a macroscopically large space-timeinterval.

π+

μ+

x1

νi

x2

np

τ–

μπ 2 1xτ 2x 1x 2x

Fig. 8. Diagrams of the lowest order for electroweak interaction that describe the decay in the source and the quasi-elasticproduction of a lepton in the detector.

μ–

μ– μ–

μ–

e–

e–e–

e–

τ+

τ+

τ–

τ–

p pn

p n

n

pn

(a) (b)

(c) (d)

W –

W – W –

W –

W –

W – W +

W +

νj

νj

νj

νiνi

νiνi

νj

μ 3eτ


Fig. 9. Diagrams of the lowest order for electroweak interaction that describe the deuteron synthesis in the -reaction in thesource and the neutrinoproduction of an electron off (a) a gallium nucleus and (b, c) a free electron in the detector.

Z

pp 2D

pp 2D

W + W + W +

W +W +

νi νi νi

νj

νj

e+ e+ e+

e–

e– e– e–

e–

71Ga 71Gexd xd xd

xs xspp 2Dxs

(a) (b) (c)

pp

Fig. 10. Diagrams of the lowest order for electroweak interaction that describe the deuteron synthesis in the -reaction in thesource and the neutrinoproduction of an electron off (a) a gallium nucleus and (b, c) a free electron in the detector.

pp d

W – W – W –

W +W +

νi νiνi

νj

νj

e–

e–

e–

e–

e–e–

e–

71Ga 71Ge

xspp dxs

pp dxs

(a) (b) (c)

pep

the latter case, the diagrams in Figs. 8a, 8d and 8b, 8cinterfere by pairs. Within our formalism, massive neu-trinos at the external legs of these diagrams should becharacterized by WPs of the same type that is used forall the other massive fields.

The diagrams in Fig. 9 describe the primary reac-tion of the -cycle in the Sun

and the detection of neutrinos in a Ga–Ge detectorand a free-electron detector. The diagrams in Figs. 9band 9c interfere. The diagrams in Fig. 10 describe(in the lowest order in electroweak interaction) thedeuterium nuclear synthesis in the so-called -reaction

in the source (the Sun) and the subsequent interactionof virtual neutrino with a gallium target (Fig. 10а)and an electron target (Figs. 10b and 10c). The excep-tionally “slow” -reaction produces approximately

pp+

ν+ → + + ν <1 1 2H H D ( 420 keV)ee E

pep

−ν

∗+ + → + ν = .1 1 2H H D ( 1 44 MeV)ie E

∗νi

pep

PHYSICS O

0.25% of deuterium in the Sun and manifests itself inone of the branches of the proton–proton chain oftransformation of hydrogen into helium. Although itscontribution to the energy balance of the Sun is negli-gible, it produces a tiny, but measurable contributionto the number of events detected in gallium–germa-nium experiments (SAGE, GALLEX, GNO). Unfor-tunately, the detection thresholds for solar neutrinosin modern Cherenkov underground detectors (Super-Kamiokande and SNO) is considerably higher thanthe expected mean energy of neutrinos ( MeV)from the -reaction in the Sun. Therefore, the pro-cesses corresponding to the diagrams in Figs. 10b and10c are still beyond the reach of experimenters. As inthe previous example, the diagrams in Figs. 10b and10c interfere.

The exotic processes demonstrated in Fig. 10 illus-trate expediency of examining macrodiagrams withmore than two in-packets in each of the macroscopi-cally separated blocks. Such diagrams are also of inter-est for the study of reactions involving neutrinos indense hot plasma of the early Universe and in ultra-

≈ .1 44pep



dense astrophysical objects such neutron or quarkstars. In the context of neutrino oscillations, the keyreason for examining diagrams with an arbitrary num-ber of in- and out-packets at the external legs is some-what more pragmatic: it will be shown below that theadditional symmetry of formulas for amplitudes of thegeneral form allows one to simplify their analysis byunifying the notation.

7.3. Feynman RulesThe formal definition of the initial and final states

for the diagram in Fig. 6 may be written as

(186)

Here, just as in (185), symbol denotes that theWPs from and (and from and ) are macro-scopically separated. In addition, it is assumed that allin-packets ( ) are in the distant past, while all out-packets ( ) are in the distant future relative to themoment (or the time interval) of interaction in blocks

and . In order to use the standard diagrammatictechnique based on Wick’s theorems, we should con-sider only the configurations of coordinates and with noninteracting packets in each substate ,and . In other words, the spatial coordinates of thepacket centers should be so distant that states (186)could be approximated well by direct products ofasymptotically free one-packet states and

; i.e.,

(187)

The sequence order of creation operators in (187) isnot significant even if they correspond to identicalfields, because, as it was demonstrated in Section 4.5,quantum correlations between identical WPs with asufficient separation in space-time may be neglected,and the sign of the scattering amplitude is not import-ant. The above requirements (including the macro-scopic separation of interaction regions and ) arephenomenologically justified and intuitively clear butare rather vague in the mathematical sense. Mathe-matically strict conditions will be formulated in Sec-tion 7.9 after the derivation of all the needed interme-diate formulas.

The norm of in- and out-states may be written as

(188)

as

= , , , ∈ ⊕ ,= , , , ∈ ⊕ .

inout

a a a s d

b b b s d

x s a I Ix s b F F

pp

⊕sI dI sF dF

ab

sX dX

ax bx,sI ,dI sF

dF

, ,a a as xp, ,b b bs xp

∈ ⊕

∈ ⊕

φ ,= , π

φ ,= .π

∏

∏

( )†3

( )†3

( )in 2 0(2 ) 2

( )out 2 0(2 ) 2

a a

a a a

s d a

b b

b b b

s d b

ik xaa a a a

sa I I

ik xbb b b b

sb F F

d e E aE

d e E aE

k kk

k kk

k k p

k k p

sX dX

∈ ⊕

∈ ⊕

= ,

= ,

∏∏

in in 2 V

out out 2 Vs d

s d

a aa I I

b bb F F

E

E


where and are the meanenergy and the effective volume of packet ; here and else-where, index is used to denote both initial ( ) and final( ) packets. The external in- and out-legs of diagramsassociated with free fields ( ) and

( ) produce factors

in the scattering amplitude. Therefore, in accordancewith the adopted approximation of narrow packets inthe momentum space, the standard (plane-wave)Feynman factor corresponding to an external legshould be multiplied by

(189)

where each wave function is given by (108) and isdefined by form factor , which depends onmass and dispersion , and 4-vector (integra-tion variable) is the coordinate of the internal point(elementary interaction vertex) of intersection ofexternal leg with a certain internal line. Therefore,the amplitude corresponding to any macrodiagramshould contain factors

(190)

where is the structural factor correspond-ing to the interaction of an internal line with external(in) field at point (for brevity, possible ten-sor or spinor indices and all arguments of function except are omitted). With the definition offunction taken into account, integral (190) may berewritten as

where is the Fourier transform of function. Owing to the expected -like behavior of form

factor , argument of function may bereplaced by . Integral (190) then assumesthe form

(191)

This result holds true for narrow WPs of the generalform, but the SRGP approximation will be used inactual calculations. A relation similar to (191) may also

≡ ≈E E Ep pû û û=V V ( )p

û û û

û ab

Φ ( )a ay ∈ ⊕s da I IΦ ( )b by ∈ ⊕s db F F

Φ , , , , Φ ,†0 ( ) and ( ) 0a a a a a b b b b by s x s x yp p

( )

( )

ψ , − ∈ ⊕ , ∗ψ , − ∈ ⊕ ,

for

for

a a a a s d

b b b b s d

y x a I I

y x b F F

p

p

φ ,( )k pû û

mû

σû

yû

û

( ) ( )ψ , − − , ( )dy y x f z ypû û û û û û û û

( )−( )f y zû û û

( )Φ yû û

yû

( )fû

−z yû û

ψû

( ) ( )

( )

− −

− −

− −

φ ,π

×π

= φ , ,π

3

( )( )4

( )( )3

(2 ) 2

( )(2 )

( )(2 ) 2

ip y x

ik z y

ip z y

ddy eE

dk e f k

d e f pE

p

p

p p p

p p p

û û

û û

û û

û û û

û

û û û

( )( )f pû

( )( )f xû

δ( )φ ,p p

û ûp

( )( )f pû

= ,( )p Ep pû

û û

( ) ( )( )

ψ , − −

≈ ψ , − .

( )

( )( )

dy y x f z y

z x f p

p

p

û û û û û û û û

û û û û û û

1 2020


be obtained for out-legs of the diagram. Since theright-hand side of (191) in the plane-wave limit is thestandard Feynman factor, it follows from (189) thatthe Feynman rules for internal lines remainunchanged.

7.4. Overlap IntegralsIn what follows, we restrict ourselves to the simple

case with interactions in blocks and being char-acterizable by local lepton or hadronic currents(explicit formulas will be derived in the next section fora special class of diagrams). Since the Fourier repre-sentation of the neutrino propagator contains phasefactor (where is the 4-momentum of

virtual neutrino ), all functions and can befactorized off the “dynamic” diagram part in the formof the following two common factors in the integrandfor the scattering amplitude:

(192)

Lorentz-invariant functions (192) characterize thespace-time overlap of in- and out-packets in thesource and the detector averaged over the entire space-time15 and are hereinafter called the overlap integrals.Naturally, and are functions of both4-momentum and 4-momenta and 4-coordinates ofall external packets involved in the reaction; theseparameters are omitted for brevity. Let us examine thekey properties of overlap integrals needed for subse-quent analysis.

7.5. Plane-Wave LimitIt is easy to see that overlap integrals (192) in the

plane-wave limit ( , ) evolve into commonsingular factors

(193)

where and are the transferred 4-momenta in thesource and the detector. They are defined in the fol-lowing way:

(194)

15Phase factors and in (192) may be interpreted as WPsof outgoing and incoming neutrinos.

sX dX

[ ]−exp ( )iq x y q

νi ψû

∗ψû

( )

( )

( )

+

∈

∈−

∈

∈

= ψ , −

∗× ψ , − ,

= ψ , − ∗ × ψ , − .

∏

∏∏

∏

V

V

( ) ( )

( )

s

s

d

d

iqxs a a a

a I

b b bb F

iqyd a a a

a I

b b bb F

q dxe x x

x x

q dye y x

y x

p

p

p

p

+iqxe −iqye

V ( )s q V ( )d qq

σ → 0û

∀û

( )( )

→ π δ − ,→ π δ + ,

V

V

4

4

( ) (2 )

( ) (2 )s s

d d

q q q

q q q

sq dq

∈ ∈ ∈ ∈= − = − . and

s s d d

s a b d a ba I b F a I b F

q p p q p p

PHYSICS O

The functions in (193) ensure exact energy-momentum conservation at vertices and (i.e., insubprocesses and ) and, as aresult, energy-momentum conservation in the

process as a whole:

Note that the information on space-time coordi-nates of external packets is lost completely in theplane-wave limit.

7.6. Overlap Tensors

The overlap integrals are nonsingular at ,and they generalize the Dirac delta functions account-ing for the energy-momentum conservation. In orderto obtain a quantitative description, we pass to theSRGP model and introduce tensors

(195)

where is the 4-velocity of packet. Overlap integrals (192) then take the form

(196)

where

Let us also define tensors

(197)

Since the Lorentz-invariant quadratic form

(where denotes variable in the intrinsic frame ofpacket ) is nonnegative, quadratic forms

and are also nonnegative. In addition, theyare positively defined almost everywhere, because theycan vanish only if 3-velocities of all in- and out-packets are the same (in other words, if there exists areference frame in which 3-momenta of all packetsin the source or the detector turn to zero). The latercondition implies that the 3-momentum of a virtualneutrino is effectively small in magnitude16; specifi-cally, the amplitude contribution for the process withzero momenta of all external particles in the exactenergy-momentum conservation limit may be non-

16Generally speaking, four-momentum in (196) is arbitrary andsimultaneous vanishing of momenta implies only that themost probable value of is small. The order of smallness of isgoverned by the smallness of ratios . We will discuss thisin more detail later.

δsX dX

→ + νs s iI F ν + →i d dI F

⊕ → ⊕s d s dI I F F

∈ ⊕ ∈ ⊕= .

s d s d

a ba I I b F F

p p

σ ≠ 0û

( )μν μ ν μν= σ − ,2T u u gû û û û

= = ,(1 )u p m vû û û û

û

( ), , ,= ± − − ϒ ,V ( ) exp[ ( )]s d s d s dq dx i qx q x x

( ) ( )μν, μ ν

∈ ,ϒ = − − ,

= ⊕ , = ⊕ .

( )s dS D

s s d d

x T x x x x

S I F D I F

û û û

û

μν μν μν μν

∈ ∈= = .ℜ ℜ ands d

S D

T Tû û

û û

[ ]μνμ ν = σ − = σ w

2 2 2 2 2( )T x x u x x xû û û û

wx xû

μνμ νℜs x x

μνμ νℜd x x

vû

pû

qpû

q qσ mû û



zero only if . With the exception of this physi-cally irrelevant case, tensors and are positivelydefined, which is equivalent to the positivity of eigen-values of matrices and Therefore, thereexist positively defined tensors and such that

(198a)

or, in the matrix form,

(198b)

where . Naturally,

and . We call and theoverlap and inverse overlap tensors, respectively. Theexplicit form of these tensors (both in the general caseand for specific processes in the source and the detec-tor) is rather cumbersome, and the corresponding cal-culations are tedious. Therefore, these are given inAppendix A. The properties of certain other quantitiesconstructed from the overlap tensors and 4-momentainvolved in the calculation of macroscopic diagramsare also examined there.

One may calculate integral (196) in the explicitform using the well-known formula for the 4-dimen-sional Gaussian quadrature in the Minkowski space(see (B.6), Appendix B):

(199)

Here,

(200)

(201)

(202)

Tensors take the form

(203)

where indices are not written explicitly but areimplied. Let us clarify the physical meaning of quanti-ties (200)–(202).

7.7. Factors Governing the Energy-Momentum Balance

As one can see from the integral representation offunctions (200)

≡ 0qμνℜs

μνℜd

μνℜsμνℜ .dμνℜ s

μνℜ d

( ) ( )μλ μ μλ μν νλν λν= δ = δℜ ℜ ands s d dR R

μν − μν, , , , ,= = , = ,ℜ ℜ ℜ ℜ ℜ

1s d s d s d s d s dg g

μν= = , − , − , −diag(1 1 1 1)g g

, >ℜ 0s d−

, ,=ℜ ℜ

1s d s d

μν,ℜs d

μν,ℜ s d

( )( )[ ]

, , ,

, , ,

= π δ× − ± .

∓

∓

V4( ) (2 )

exps d s d s d

s d s d s d

q q qi q q XS

( )μν, , μ ν

,

δ = − ,π

ℜℜ

21 1( ) exp

4(4 )s d s d

s d

K K K

[ ]

μ μν λ, , ν λμν, ν ν

=

= σ − ,

ℜℜ

û û

û û û û û

2 ( )s d s d

s d

X T x

u x u x

μν, μ ν= , , ∈ . , ' '

, '' ors d W x x S D

û û û û

û û

û ûS

μν'W

ûû

μν μν μ μ ν ν μν νμμ , ν= δ − , = ,ℜ ' '

' ' ' ' ' ' 's dW T T T W Wûû ûû û û û ûû û û

,s d

( )μν, , μ νδ = − ,

πℜ

4( ) exp(2 )

s d s ddxK iKx x x


factors and in (199) turn intocommon functions and in theplane-wave limit. The probability of the

process at small (but nonzero)values will be suppressed strongly at small deviations

from the exact 4-momentum conservation (i.e., at asmall imbalance between the transfers of 4-momenta

and at the macrodiagram vertices and the4-momentum of a virtual neutrino). The processprobabilities averaged in a certain way over the exter-nal momenta configurations (including, in general,the momenta of detected secondary particles averagedover finite intervals set by the experimental condi-tions) are measured in real experiments. Therefore,factors and account for the sta-tistically approximate energy and momentum conser-vation in subprocesses of neutrino production andabsorption. The allowed “imbalance” is defined bythe inverse overlap tensors (i.e., the spread of4-momenta of in- and out-packets). In what follows,only the configurations of external momenta produc-ing a significant contribution to the total processprobability averaged over all possible configurationsare assumed when we talk about the approximateenergy-momentum conservation at the vertices. Aspecific implementation of such averaging was exam-ined in [58].

7.8. Impact Points, Geometric Suppression Factors, Symmetries, and All That

Let us transform quadratic forms and usingdefinitions (197), (202), and (203):

Inserting a unit matrix into the last terms and tak-ing (201) and (202) into account, we find

whence it follows that

(204)

The obtained expression is the weighted mean(over all external packets at the corresponding vertex)of quadratic deviations of the components of 4-vectors

( ) from the corresponding components ofthe space-time point . Tensor components act

( )δ −

s sq q ( )δ +

d dq qδ ( )δ − sq q ( )δ + dq q

⊕ → ⊕s d s dI I F Fσ

sq dq

( )δ −

s sq q ( )δ +

d dq q

sS dS

( )μμ ν ν, , μ ν μ ν

μμ ν νμνμ ν , μ ν μ ν

= −

= − .

ℜ

ℜ

' '' ' ' '

, '

' '' ' ' '

, '

( )

( )

s d s d

s d

T T x x x

T x x T T x x

û û û û û

û û

û û û û û û û

û û û

S

( )( )

μν μ, μ ν μ μ

μ λ ρν ν, , λρ , ν ν

μ μ νμ μ ν , μν , ,

= −

×

= − ,

ℜ ℜ ℜ

ℜ

'

' '' ' '

'

'

( )

( )

s d

s d s d s d

s d s d s d

T x x T x

T x

T x x X X

û û û û û

û û

û û

û

û û û

û

S

( )μ ν μ ν, μν , ,= − .s d s d s dT x x X X

û û û

û

S

xû

∈ ,S Dû

,s dX μνTû

1 2020


as “weights.” Thus, points Xs and Xd are the centers ofspace-time regions of packet interactions in the sourceand the detector. One needs to study the properties ofsymmetry of and to clarify the physical andgeometrical meaning of this result.

7.8.1. Translation group. It is easy to prove thatquadratic forms and are invariant with respectto translations of all coordinates by one and thesame arbitrary 4-vector ,

(205)

Indeed, under this trans-formation; therefore

Since, according to (201),

it follows that, as stated,

It should be noted that quadratic forms and are also invariant with respect to inversion of space-time coordinates,

Performing transformation (205) with in (204), we obtain the following representation forfunctions :

(206a)

It follows immediately from (206a) that quadraticforms and are nonnegative and, consequently,factors and in (199) suppress theamplitude for certain configurations of momenta andcoordinates of in- and out-packets. Using definition(195), we rewrite (206a) as

(206b)

(206c)

Here and elsewhere, index “ ” means that thecorresponding vector is written in the intrinsic refer-ence frame of packet . It follows from (206c) the sup-pression is weak (i.e., ) for configurations of

coordinates and momenta such that aresmall at any

,s dX ,s dS

sS dS

xû

y

= + . 'x x x yû û û

, , ,= +'

s d s d s dX X X y

( )( )

, , , μν

μ ν μ ν μ ν μ ν, ,

μ ν ν, μ ν ,

= +

× + − −= + − .

'

2

s d s d s d

s d s d

s d s d

T

y x x y y X X y

y T x X

û

û

û û

û û

û

S S S

S

μ ν μ νν , , ν ,

μ νν λ μν, ν , ν λ ν

=

= = ,

ℜ

ℜ ℜ

''

( )

( )

s d s d s d

s d s d

T X X

T x T x

û

û

û û û û

û û

, ,= .'s d s dS S

sS dS

− .x xû û

,= − s dy X

,s dS

( ) ( )μν, , ,μ ν= − − .s d s d s dT x X x X

û û û

û

S

sS dS

( )−exp sS ( )−exp dS

( )[ ] ( ) , , ,= σ − − − 2 22s d s d s du x X x X

û û

û

S

( ),= σ − .

22 ( )s dx Xû û

û û

û

( )û

û

, 1s d !S

,σ −22 ( ) ( )

s dx Xû û

û û

∈ , .S Dû

PHYSICS O

7.8.2. Group of uniform rectilinear motions. It isevident that when all 4-coordinates arethe same (it follows from equality , that ). Still, is this condition necessary for to turn to zero? Fortunately, the answer is negative17. Tosee this, note that 4-vector and itssquare are invariant with respect totransformation

(207)

where is an arbitrary real parameter with a dimen-sion of time. Therefore, 4-vectors and, as is easilyseen from (204), quadratic forms are invariant withrespect to -parametric set of transformations (207),where ( ) is the number of one-packet statescontained in the initial and final states in the source(detector). All transformations of the form (207) com-bine into a group describing uniform rectilinearmotions of packets (i.e., shifts along the classical worldlines of the packet centers). Therefore, both and

are defined uniquely by specifying velocities and arbitrary space-time points on these worldlines. Specifically, if the classical trajectories of all in-and out-packets intersect at the impact point,

at any point of each trajectory. Thus, packetsmay be separated macroscopically in space-timebefore and after the interaction and be consideredasymptotically free, but if velocity vectors of in-packets and vectors contradirectional to the veloc-ities of asymptotically free out-packets are all“pointed” at point (in the source) or (in thedetector), or turn to zero. In view of this, itseems natural to call 4-vectors and impactpoints of in- and out-packets in the source and thedetector, respectively.

If one needs to clarify the pattern of interaction ofpackets, it is instructive to introduce the notion of aWP world tube, which is defined as the cylindricalspace-time volume swept by an oblate ellipsoid (themodel of a WP18) moving along a classical trajectory.Evidently, the suppression of the overlap integral (andamplitude) produced by factor ( )is weak if all world tubes of in- and out-packets in thesource (detector) intersect with each other (see

17“Fortunately”, since otherwise there would be no asymptoti-cally free WPs, and it would be impossible to use the perturba-tion theory.

18It bears remembering that the diameter of this ellipsoid perpen-dicular to the vector of the most probable WP momentum (and, consequently, the diameter of the classical world line) is

, while the diameter parallel to is suppressed by fac-tor emerging as a result of Lorentz contraction.

, = 0s dS xû

,= s dx xû

∀ ∈ ,S Dû

, ,=s d s dX x ,s dS

= −( )X u x u xû û û û û

= −2 2 2( )X x u xû û û û

= + θ , = + θ , 0 0 0x x x x x x vû û û û û û û û û

θû

,s dX

,s dS

,s dNsN dN

,s dX

,s dS vû

xû

, = 0s dS

av− bv

sX dXsS dS

sX dX

p

∝ σw 1d pΓp

−exp( )sS −exp( )dS



Fig. 11. Schematic representation of classical world tubes of colliding WPs. These tubes are space-time cylindrical volumes sweptby classically moving spheroids modeling the WPs. Impact point is defined unambiguously by packet velocities and space-time points , which were chosen arbitrarily at the symmetry axes of tubes (the centers of packets move, on the aver-age, along them). These axes are by no means required to go through point , but if the configuration of coordinates and veloc-ities of in- and out-packets is such that the impact point is located within the region of overlap of all world tubes of packets, thesuppression of this configuration is weak (and does not take place if the axes intersect exactly at point ). In the opposite case,the suppression is strong.

Tim

e

Space

X

xb

xc

xa

X vû

( )= θ x xû û û

X

X

Fig. 12. Schematic illustration of two configurations of a pair ofoverlapping WPs in their center-of-inertia frame. The left config-uration corresponds to factor =

in the integrand in the

numerator of (208); the right one, to factor

= in the inte-

grand in the denominator. Arrows denote momenta

ψ1(p1, y1 – x )* * *

ψ2(–p1, y2 – x )* * *

ψ1(p1, x )* *

ψ2(–p1, x )* *

ψ , − ψ , −1 1 1 2 2 2( ) ( )y x y xp p∗ ∗ ∗∗ψ , − ψ − , −1 1 1 2 1 2( *) ( *)y x y xp p

ψ , ψ ,1 1 2 2( ) ( )x xp p ∗ ∗ψ , ψ − ,1 1 2 1( *) ( *)x xp p

∗± 1 .p

Fig. 11). WPs behave somewhat like interpenetratingclouds. In order to illustrate this, we present functions

and as ratios of overlap integrals of the specialform:

(208a)

In view of the invariance of functions with respect to transformations (207),

one may replace coordinates in (208a) by , where

and rewrite (208a) as

(208b)

It then follows from geometric considerations that if any pair of coordinates and do

not coincide at one and the same (“impact”) point intime, since the integrand in the numerator of (208b) isno greater than the integrand in the denominator. Thisis illustrated schematically in Fig. 12, where a pair of

sS dS

( ) ∈ ,,

∈ ,

ψ , −− = .

ψ ,

∏

∏

( )exp

( )S D

s d

S D

dx x x

dx x

p

p

û û û

û

û û

û

S

ψ , −( )x xpû û û

xû

yû

( ), ,= , = + − ,∈

0 0 0 0

ors d s dy X X x

S Dy x v

û û û û û

û

( ) ∈ ,,

∈ ,

ψ , −− = .

ψ ,

∏

∏

( )exp

( )S D

s d

S D

dx y x

dx x

p

p

û û û

û

û û

û

S

,− <exp( ) 1s dS yû 'y

û


overlapping packets 1 and 2 is shown. These packetsare presented in Fig. 12 in their center-of-inertia frame( ), where they take the form of ellipsoids∗∗ + =1 2 0p p

1 2020


with effective volumes f lattened

along vector . It can be seen that the followingequality holds true for any (and, consequently, forany values of the integration variable in (208b)):

Naturally, the value of (208b) is small if 4-coordi-nates and of any pair of packets and differsubstantially from each other.

Identity (208b) may be rewritten in the followingequivalent form:

(209)

Factor is included in the scat-tering amplitude squared.

7.8.3. Impact vectors. Let us find the spatial dis-tance between the impact point and classical WPworld line . In other words, we aimto determine the classical impact parameter

It is evident that for a motionlesspacket . The impact parameter for a moving packet isdefined by the following condition:

Relying on this condition and assuming temporar-ily that we construct 4-vector with compo-nents

(210)

The last equality implies that

(211)

whence it follows that

Thus, if velocity vector is collinear tovector at any value of (i.e., ifthe classical world line of the center of packet goesthrough the impact point). In addition, if

= . Since 4-vector (determined to within its sign) is a straightforward rel-ativistic generalization of a common nonrelativistic

( ) / ,π Γ σ3 2 31 22 ( )*

∗1p

*x

∗∗ ∗ ∗ψ , − ψ , −

∗ ∗∗ ∗< ψ , ψ , ≠ .

1 1 1 2 2 2

1 1 2 2 1 2

( *) ( *)

( *) ( *) at

y x y x

x x y y

p p

p p

*yû '

*yû

û 'û

( ) ∈ ,,

∈ ,

ψ , −− = .

ψ ,

∏

∏

2

2

( )exp 2

( )S D

s d

S D

dx y x

dx x

p

p

û û û

û

û û

û

S

( )[ ]− +exp 2 s dS S

θ = + θ ( )x x vû û û û û

,−∞<θ <∞= θ − .min ( ) s db x X

û

û û û

,= − s db x Xû û

û

( ),θ = − , = .s dv n X x n v vû û û û û û û

≠ 0,vû

bû

( ) ( )( ) ( )[ ]

−, ,

, ,

= − − − ,= − − − .

10 0 0s d s d

s d s d

b x X v n x Xb x X n x X nû û û û û

û û û û û

( ),= × = × − ,0 and s dn b n b n x Xû û û û û û

( ),= × − .s db n x Xû û û

= 0bû

vû

,− s dx Xû ,− < ∞s dx X

û

û

= =0 0b bû û

,− s dx Xû

( ),−0 0s dx Xv

û û= ,0( )b b b

û û û

PHYSICS O

impact parameter, it natural to call it the impactvector. Taking (211) into account, we find

(212)

Inserting instead of into (206b) (thevalue of remains unchanged) and using (212),we find

(213)

The contribution of each packet is written here inits intrinsic frame, where ; there-fore, as expected, (213) coincides with (206c). Bothequalities in (213) remain true if certain packets are atrest in the laboratory frame, since it follows from (210)that

(214)

Therefore, provisional constraint , whichwas used in the derivation of (213), may be removed.

The physical meaning of this result is clear: theinteraction of in- and out-packets is not suppressed(i.e., the value of is small) if all impact parame-ters are small compared to the effective sizes of packets( ) in their intrinsic reference frames. If all

impact parameters are of the same order of mag-nitude, the greatest contributions to are pro-duced by packets with the largest momentum spreadand, consequently, the smallest effective size19. If wefactor out the effects of the phase space of reaction ordecay and the possible influence of the interactiondynamics on the momentum spread of secondarypackets, the geometric amplitude suppressionbecomes stronger as the number of particles (in actualpractice, secondary particles) involved in the processincreases. The factor of suppression of “incorrect”configurations of world lines of packets is defined inthe laboratory frame by both space and time compo-nents of impact 4-vectors . The contributions ofnonrelativisitc and ultrarelativistic packets to takethe form

(215a)

19The physical inadequacy of the notions of point-like particles(maximally localized states) and plane waves (states with defi-nite momenta) in the QFT perturbation theory then becomesclear: the former are unable to interact (the amplitude turns tozero if a point-like particle is present in any substate or

), and the latter interact at arbitrarily large distances.

is

( ) ( ) ( ) ( )− = Γ − = Γ − + .2 22 2 2 0 2 2 0 21u b b b b b b

û û û û û û û û û

bû ,− s dx X

û

,s dS

( ) ( ),∈ ,

∈ ,

= σ Γ − +

= σ .

22 2 0 2

22 ( )

1s dS D

S D

b b

b

û û û û

û

û

û û

û

S

,= −( ) ( ) ( )s db x Xû û û

û û

( ) ( ) ,→ Γ − + = − .

2 22 0 2

0lim 1 s dbv

b x Xû û û û

≠ 0vû

,s dS

σ∼ 1û

( )b û

û

,s dS

,s dI

,s dF

bû

,s dS

( ) ( )( )[ ] ( ) ( )

, ,

,

σ − − −

+ − +

22 0 0

2 3 1s d s d

s d

x Xx X v

v x X v vû û û û

û û û û!2



and

(215b)

respectively. It follows from (215a) that the suppres-sion for nonrelativistic WPs depends only weakly onthe velocity vector and time interval and isdetermined by the spatial distance between the packetcenter and the impact point. In contrast, the suppres-sion for ultrarelativistic packets depends strongly onthe velocity direction and magnitude and on the dif-ference between times and . The emergence of

large factor in the leading term in (215b) is relatedto the Lorentz contraction of the effective volume of apacket, which results in significant contraction of theregion of its overlap with other packets. The reason forthe strong dependence on the direction of the velocityvector is the same: one needs to “aim well” to hit asmall target.

It follows from the above analysis that impactpoints and characterize the space-time posi-tioning of the effective regions of packet interaction inthe source and the detector. The interaction of packetsgrows stronger as the world lines of their geometriccenters move closer to these points. The configurationof world lines and the coordinates of impact points arein no way related to the dynamics (i.e., to the interac-tion Lagrangian), since they are defined uniquely bythe coordinates and group velocities of asymptoticallyfree in- and out-packets and their effective sizes.However, if the interaction dynamics is neglected, itwill not be possible to calculate the amplitude of pro-cess characterized by the macro-scopic diagram in Fig. 6.

7.9. Asymptotic Conditions

We can finally return to the physical requirementsformulated at the beginning of Section 7.3 and specifythe conditions under which in- and out-packets maybe considered free. If the geometric suppression fac-tors are not too small (only these configurations ofcoordinates and momenta contribute to the measuredquantities), the requirement that the effective regionsof WP interaction in the source and the detector bemacroscopically separated in space-time is equivalentto the required macroscopic separation of impactpoints and . Let us assume that space and timeintervals and are large compared tospace and time intervals and for

and ; i.e., packets from and do

( ) ( )( )[ ] ( )

( ) ( ) ( )( )

, ,

, ,−

, ,

σ Γ − − −

+ × − − − × − − − +

ΓΓ

22 2 0 0

2 0 0

0 0 2

2 1 ,

s d s d

s d s d

s d s d

x X

x X

x X

n x X

n x X

n x X

û û û û û

û û û

û û û û

û@

2

,−0 0s dx X

û

0xû ,

0s dX

Γ2û

sX dX

⊕ → ⊕s d s dI I F F

sX dX−0 0

d sX X −d sX X−0 0

'x xû û

− 'x xû û

∈, ' Sû û , ∈' Dû û S D


not overlap. The sought-for conditions for packetsfrom and should then be independent. We assumethat the effective sizes of packets are large compared tothe characteristic radius of interaction at the diagramvertices. Therefore, only the properties of geometricsuppression factors and inde-pendent of the interaction dynamics are examinedbelow.

First of all, the requirement that time intervals ( ) and ( ) be suffi-

ciently long should be set. However, they cannot be arbi-trarily large, since packets remain stable (i.e., do notspread) within only under the condition that

(216)Since the geometric suppression factors do not

depend on and , one may formulate therequirement that the left-hand side of (216) be largecompared to the effective packet size squared:

(217)This requirement does not contradict stability con-

dition (216), since . If, in addition, is anunstable particle, it is expected that

(218)where is the lifetime of . Conditions (217) and(218) do not contradict each other if , whichis one of the SRGP applicability conditions. The“complete set” of these conditions was obtained inSection 5.8 and has the form

(219)Since for all known long-lived elemen-

tary particles and atomic nuclei, the allowed values of may vary within a fairly wide interval (see Table 1).

Thus, the asymptotic conditions for time parame-ters , which agree completely with SRGP applica-bility conditions (219), are given by (217), and the cor-rect time sequence in the laboratory frame is definedby the following inequalities:

(220)These inequalities are Lorentz-invariant if points and are separated by time-like intervals. If

intervals are space-like for certain ,inequalities (220) are valid only in the laboratoryframe, because a time sequence of two events sepa-rated by a space-like interval is not a Lorentz-invariantnotion.

This should be taken into account, since a packet(for definiteness, packet in the source) may beinvolved in the interaction (i.e., may not produce asignificant suppressing contribution to ) even if

S D

( )−exp sS −exp( )dS

, −0 0s d aX x ,∈ s da I ,−0 0

b s dx X ,∈ s db F

û

, −0 0s dX x

û

, − σ , ∀ ∈ , .20( ) 0( ) 2 4

s dX x m S Dû û

û û û û!

,0( )s dX û 0( )x û

û

, − σ .20( ) 0( ) 21s dX xû û

û û@

σ2 2mû û! û

, − τ ,∼

0( ) 0( )s dX xû û

û û

τû

û

σ τ2 2 1û û

@

τ σ τ .2 2 21 m mû û û û û! !

τ 1mû û

σ

0xû

, , ,< < ∈ , ∈ .0 0 0 ( )a s d b s d s dx X x a I b F

,a bx ,s dX

,− 2( )s dx X û

a

sS

1 2020


Fig. 13. Schematic illustration of an event in which the ini-tial 4-coordinate of the center of in-packet is separatedby a space-like interval from impact point , but a part ofthe effective packet volume is located within the light conecentered at . The other in- and out-packets, the coordi-nates and velocities of which define (together with and

) the position of the impact point, are not shown.

a

Light cone

X

aX

Xax

av

under the condition that points withina part of its effective volume are separated by time-likeintervals from the corresponding impact point: theimpact point may then be located within the classicalworld tube of packet (see Fig. 13)20. The microcau-sality condition is not violated here, since all signalspropagate within the light cone. However, such inter-actions may potentially produce observable effectsimitating causality violations.

The requirement of spatial separation of packetsfrom impact points is, in general, not a necessary one.Certain packets (e.g., a decaying meson or a chargedlepton produced in its decay in the source, a targetnucleus in the detector, etc.) may indeed be at rest inthe laboratory frame before or after the interaction. Inthis case, they should be spatially close to the corre-sponding impact points, otherwise, the scatteringamplitude is, according to (214), small due to thesmallness of factors or . It isclear, however, that all packets in in- and out-statesshould be spatially separated; i.e., the differences inspatial coordinates of each pair of packets shouldbe large compared to the size of these packets. Repre-sentation (208b) and Fig. 12, which illustrates it, sug-gest that this requirement is formulated in the simplest

20It follows from geometric considerations that

( is the effective transverse size of a packet) for such events.

Therefore, and .

− <2( ) 0a sx X

a

< <w w0 2a adbw

ad

σ < .w22 0 605a ab ( )−σ .w

22exp 0 546a ab

( )−exp sS ( )−exp dS

, 'û û

PHYSICS O

way in the center-of-inertia frame of the pair. Sincethe momenta of packets in the center-of-inertia frame

are collinear and the only case ofinterest is when classical impact parameter

does not exceed significantly thetransverse size of each packet, the distance betweenpackets should be large compared to their longitudinalsize. Omitting a constant common factor of the orderof unity, we write this condition as

(221)

where . If packets and are thestates of identical particles with equal momenta( ) and spin projections, condition (217) isthe same as the condition of vanishing of quantumcorrelations at large distances between the packets ofidentical bosons or fermions. The impact parametersof such packets may be arbitrarily small if one of themis in the in-state, while the other is in the out-state(i.e., the packet pair essentially describes the state ofone spectator particle). If both identical packetsbelong to the in- or out-state of one of the diagramvertices, the corresponding amplitude is suppressed,because the distance between them does not changewith time21. It follows from (221) that ultrarelativistic(in the center-of-inertia frame) packets forming a pairdo not interact with each other even at relatively shortdistances.

In the general case, conditions (217) and (221) arenot necessarily independent (a simple example consid-ered below illustrates this); the only important factor isthat they are not mutually exclusive. Inequalities (221)should be matched with the nonspreading conditions,that is, with (216) and similar inequalities for spatialcoordinates

(222)

If (as assumed) for

each pair from or , impact point in thecenter-of-inertia frame of any such pair is locatedbetween packets and close to their world tubes22.Therefore, with (217) factored in, we find

21This suppression is in no way related to the Pauli ExclusionPrinciple: if any two packets (not necessarily those characteriz-ing the states of identical fermions) have equal velocities andare sufficiently separated at a certain point in time, their classi-cal world tubes do not intersect. It should also be added thatparameters and for packets of identical particles shouldnot be regarded as fundamental characteristics of the corre-sponding quantum field and are not necessarily equal.

22More specifically, the distance from to the classical trajec-tories of centers of packets and in the center-of-inertiaframe is much shorter than the distance between these centers.

( )∗ ∗ ∗= − =' *p p n pû û û

∗ ∗× − '( )*n x xû û

( ) ( )∗ ∗− + ,

∗ ∗σ Γ σ Γ

2

' 2 2

' '

1 1x xû û

û û û û

@

, , ,∗ ∗Γ =' ' 'E mû û û û û û

û 'û

∗Γ = Γ ='* 1û û

σû

σ 'û

, − σ .2( ) ( ) 2 4

s d mX xû û

û û û!

∗∗× − σ + σ2

2 2' '( ) 1 1*n x x

û û û û&

, 'û û S D ,∗s dX

û 'û

,∗s dX

û 'û



where and are the veloci-ties of packets and in the center-of-inertia frame.Since it follows from Lorentz transformations that

= (the relation for is similar), the following approximate inequality holdstrue:

where the mutual orientations of spatial vectors andvelocities in different reference frames were consid-ered. Thus, taking conditions (216) and (222) intoaccount, one may state that

(223)

at any .Comparing (223) with (221), we find that SRGP

applicability conditions (216) and (222) do not contra-dict the requirement of spatial separation of packets;i.e., no additional constraints are imposed on param-eters .

Let us illustrate condition (221) by a simple, butimportant (for experiments on neutrino oscillations)example in which states and contain just a singlepacket. For definiteness, we consider two-particledecay in the source (see Fig. 7). The two-packet integral corresponding to this process in thecenter-of-inertia frame of a pion and a muon

( ) is23

Changing the integration variable( ), we obtain the followingintegral:

Its calculation the SRGP model comes down tothe consecutive computation of four standard Gauss-ian quadratures, which yields

(224)

where and are theenergies of a pion and a muon, is a unitvector, and

is the classical impact parameter in the center-of-iner-tia frame of a pion and a muon. Inserting (224) into(208b), we find24

(225)

If vectors and are collinear, it follows from(225) that at any distance between the centers ofwave packets of a pion and a muon. If this distance is suf-

ficiently large (e.g., , so thatcondition (221) is fulfilled), function is small inmagnitude only at small angles between vectors and , viz., , where

as increases, function grows rapidly, while fac-tor decreases, ruling out the possibility of

decay for . The dependence of

( ) ( )( ) ( )( )

∗ ∗ ∗ ∗ ∗ ∗ ∗, , , ,

, , ,

∗ ∗ , , ,

∗ ∗ ∗− = − + − − + −

∗ ∗= Γ − + − − × −

∗ + Γ − + − − × − ,

' ' '

2 2( ) ( ) 0( ) 0( ) ( ) ( )

2 2( ') ( ') 0( ') 0( ') ( ') ( ')' ' ' ' ' '

*

s d s d s d s d

s d s d s d

s d s d s d

x X

x X

x x x X X x x X x X

x X v v x X

x X v v x X

û û û û û û

û û û û û û

û û û û û û

û û û û û û

û û û û û û

23Through to the end of the present section, and denote par-ticle types rather than Lorentz indices.

∗=* * Ev pû û û

∗ ∗∗ =' ' 'Ev pû û û

û 'û

( ),∗ ∗ ∗× − s dv x Xû û

( ),∗ × −( ) ( )s dv x Xû û

û û'û

( )( )

, ,

, ,

∗ ∗∗− Γ − + −∗+ Γ − + − ,

( ) ( ) 0( ) 0( )'

( ') ( ') 0( ') 0( ')' ' ' '

*

*s d s d

s d s d

x X

x X

x x x X v

x X v

û û û û

û û û û û û

û û û û

û û û û

&

( )∗ ∗ ∗∗− σ + σ22

2 2' '4 E Ex x

û û û û û û!

∗, 'v

û û

,σ 'û û

sI sF

π → μν

π μ∗ ∗= − ≡ *p p p

π μ

πμ π π π μ μ μ= ψ , − ψ , −V ( ) ( ) .dx x x x xp p

π μ∗∗+ + ( ) 2x x x x

πμ

μ π π μπ μ

=

∗ ∗∗ ∗− − × ψ , + ψ − , + .

V

* *2 2

dx

x x x xx xp p

24Comparing (225) with (213), we also find the relation betweenimpact parameters

which implies, among other things, that at

in

( )π μ

πμ

π μ π μ π μ

π μπμ

π μ

π=

∗ ∗σ σ σ + σ + σ σ ∗× − , σ + σ

2

2 2

2 2 2

2 2

V( ) *

exp

m m

E E p

b

π π∗ = +∗

2 2E mp μ μ∗ = +∗

2 2E mp=* * *n p p

πμ μ π∗ ∗ ∗= × −( )*b n x x

μπ μππμ π μ

μ π

σσ∗ = + + + , σ σ

222 2 2( ) ( )2 2

1 1b b b

π μπ μ= =( ) ( ) 0b b

πμ∗ = 0.b

−

πμπ μ

∗= + . σ σ

12

2 21 1

s bS

*n μ π∗ ∗−x x

= 0sS

μ π π μ∗ ∗− σ + σ

22 21 1x x @

sSα* *n

μ π∗ ∗−x x −α ρ∗ ∗

2 2!

− /

μ ππ μ

∗∗ρ = + − ; σ σ

1 2

2 21 1

* x x

α* sS

−exp( )sS

μπ 2−α ρ∗ ∗

2 2@ −exp( )sS



Fig. 14. Geometric suppression factor for the two-particle decay as a function of and .

1

10–1

10–2

10–3

10–4

10–5

10–6

10–4

0

–10–4 104

103

102

α∗ ρ∗

−exp( )sS ρ* α*

on and is shown in Fig. 14 for intervals and In experiments, a muon

produced in the decay is always seen emitted fromthe same point (more specifically, from the spatialregion limited by the instrumental resolution) inwhich a pion vanished. In the center-of-inertia frameof a pion and a muon, this implies that impact param-

eter is zero or very small, thus providingan explanation for the obvious experimental fact thatremains unexplained in the theory with plane waves25.

It is evident that conditions similar to (217) and(221) are impossible to formulate for single-particleFock states of the standard QFT perturbation theory,which utilizes a rather meaningless condition ofasymptotic freedom at infinity. This condition is inap-plicable to unstable particles, which serve as the pri-mary source of (anti) neutrinos in many experimentson neutrino oscillations.

The size of the interaction region ( ) at verticeswas neglected in the above analysis. This is justifiableif the effective size of wave packets is much larger than

. This requirement is satisfied when weak and stronginteractions are considered. If electromagnetic inter-actions are considered, this analysis (just as the stan-dard QFT approach) is, apparently, nonexhaustive,and the formulated conditions may not be sufficient.

25We leave open the question of the possibility of experimental

measurement of impact parameter , which could help ver-ify the above theoretical constructs.

ρ* α*ρ2 410 10*& &

−α 410 .*μπ 2

πμ∗ ≈ α ρ* *b

πμ∗b

Ir

Ir

PHYSICS O

7.10. Phase FactorsOverlap integrals (199) are not translationally invari-

ant due to the presence of phase factors Note, however, that the -depen-

dent factor in product is proportional totranslationally invariant phase factor

(226)As a result (see the next section), the square of the

amplitude modulus is translationally invariant. It wasdemonstrated in [58] that factor (226) governs theoscillation behavior of the amplitude squared (as afunction of ) under the condition that impactpoints and are macroscopically separated.

7.11. Overlap Volumes

A representation of quantities and differing from the one obtained by applying formula(199) directly is better suited for the analysis of suchmeasurable characteristics as the count rate of neu-trino events. It is more convenient to go back to defi-nition (196) and write as

Performing a change of integration variables and (with a unit Jaco-

bian), one may rewrite the last integral as

(227)

( )[ ], ,± ∓exp .s d s di q q X qV V( ) ( )s dq q

( )[ ]−exp .s diq X X

−s dX XsX dX

V2( )s q V

2( )d q

,V2( )s d q

( ) ( )[ ],

, , ,

=

× ± − − ϒ − ϒ V

2( )

exp ( ) ( ) .s d

s d s d s d

q dx dy

i q q x y x y

= +' ' 2x x y = −' ' 2y x y

( )

[ ]

μν, , , μ ν

,

= ± −

× − ϒ .

ℜ

2 1 ''( ) ' exp '2

' exp 2 ( ')

s d s d s d

s d

q dy i q q y y y

dx x

V



Fig. 15. Macroscopic diagram describing process (231).

W

W

q

Hadrons Hadrons

Hadrons Hadrons

QCD

QCD

Is Xs

Xd

q q'

q'

qs

qd

,α+

Fs

Id

Fd

F '

F '

νj(qs, d = pin – pout)

–,β

s

d

Introducing notations

(228)

(229)

we present (227) in the following compact form:

(230)

Functions and obviously do not matchfunctions and used earlier, but have thesame plane-wave limit (i.e., inthis limit) and similar properties. The physical meaningand the symmetry properties of functions (229) followclearly from the above analysis, and their integral rep-resentation suggests that quantities and may beinterpreted as 4-dimensional overlap volumes of in-and out-packets in the source and the detector. It fol-lows from the explicit form of these functions that theyassume their maximum values, when the classical world lines of packets intersect atimpact points.

8. AMPLITUDE OF THE PROCESS WITH THE PRODUCTION OF CHARGED LEPTONS

IN THE SOURCE AND THE DETECTOR AND A VIRTUAL NEUTRINO

Let us examine the class of processes26

(231)which are sustained by the weak charged current, as apractically important application of the formalism.Here, and denote the sets of asymptoti-cally free WPs characterizing hadronic states, and and are packets of charged leptons ( ). If

, lepton numbers and are violated in pro-cess (231). This is made possible by the exchange ofmassive (Dirac or Majorana) neutrinos. In the leadingnonvanishing order in electroweak interaction, pro-cess (231) is characterized by the sum of diagramsshown in Fig. 15. Let and be impact points

26Note that the examples presented in Figs. 8 and 10 (Section 7.2)do not belong to this class, although the technique of calcula-tion of the corresponding amplitudes does not differ much fromthe one developed below.

( )( )

μν, , μ ν

μν, μ ν

,

δ = −π−

= ,π

ℜ

ℜ

ℜ

4

2

1( ) exp2(2 )

1exp2

(2 )

s d s d

s d

s d

dxK iKx x x

K K

( )

( )

,∈ ,

,

,

= ψ , −

π −= ,

ℜ

∏2

2

V

exp 24

s dS D

s d

s d

dx x xpû û û

û

S

( ), , , ,= π δ .∓V2 4( ) (2 ) Vs d s d s d s dq q q

δ ( )s K δ ( )d Kδ ( )s K δ ( )d K

, ,δ = δ = δ( ) ( ) ( )s d s dK K K

Vs Vd

, ,= π ,ℜ0 2V (4 )s d s d

+ −α β⊕ → + ⊕ + , ''

s d s dI I F F

,sI ,dI ',sF 'dF

+α

−β α,β = ,μ, τe

α ≠ β αL βL

sX dX


defined by formula (201) (where and). It is assumed that they (and, conse-

quently, the effective interaction regions in the sourceand the detector, which are denoted by dashed curvesin Fig. 15) are macroscopically separated in space-time and that all asymptotic conditions discussed inSection 7.9 are fulfilled. The initial and final statesmay then be regarded as direct products of free one-packet states, and normalization (188) may be used.

In the standard model minimally extended by theinclusion of a neutrino mass matrix, quark-leptonblocks of the diagram in the lowest nonvanishing orderare described by Lagrangian

where is the coupling constant; and arethe lepton and quark charged currents,

( , ) and ( ,) are the elements of the neutrino and quark

mixing matrices ( and , respec-

tively); ; are the charged leptonfields; and standard notations are used for the otherfields and Dirac -matrices. Hadronic blocks (colored

+α= + '

s sF F−β= + '

d dF F

[ ]= − + + . ,( ) ( ) ( ) ( ) ( ) H.с2 2W q

gx j x W x j x W x+

g (2)SU j qj

μ ∗ μα α

α

μ μ

= ν ,

∗= ;

''

( ) ( ) ( )

'( ) ( ) '( )

i ii

q qqqq

j x V x O x

j x V q x O q x

αiV α = ,μ τ,e = , ,1 2 3i 'qqV = , ,q u c t= , ,'q d s b

= PMNSV V = CKM'V Vμ μ= γ − γ5(1 )O α,β ( )x

γ

1 2020


regions in the diagram in Fig. 15), which are impossi-ble to calculate within the perturbation theory, arecharacterized by phenomenological hadronic cur-rents. The normalized dimensionless amplitude ofprocess (231)

is given by the fourth order of the perturbation theoryin constant :

Here,

is the Lagrangian of strong and electromagneticSM interactions governing the nonperturbative pro-cesses of fragmentation and hadronization, and and

are the standard symbols of chronological andnormal ordering of local operators. Normalizationfactor in the SRGP approximation is given by

(232)

Let us consider hadronic matrix element

(233)

Here, the WP definition

(234)

was used, and a contracted notation for many-particleFock states was introduced:

We may write the following for stable relativelystrong and electromagnetic interactions of the initial

( )− / βα=def1 2out in in in out out AS

g

( )βα−= ⊕ :

× :: :× : :: : ⊕ .

4

†

† † †

1 ' '2 2( ) ( ) ( ') ( ')

( ) ( ) ( ') ( ')

s d

q

q h s d

ig F F T dxdx dydy

j x W x j x W x

j y W y j y W y I I

A1

S

= , hexp ( )h T i dz z+S

h( )z+

T: :…

1

∈ ⊕ ⊕ ⊕= = .∏2 in in out out 2 V

s d s dI I F F

Eû û

û

1

μ ν

−

μ ν

⊕ : :: : ⊕

φ ,= π

∗φ ,× π

× : :: : .

∏

∏

†

3

3

†

' ' ( ) ( )

( )(2 ) 2

( )(2 ) 2

( ) ( )

a a

a

b b

b

s d q q h s dik x

a a a a

a

ik xb b b b

b

b q q h a

F F T j x j y I I

d eE

d eE

k T j x j y S k

k

k

k k p

k k p

S

( ) −φ ,, , = ,π

( )

3(2 ) 2

i k p xd es x s

Ek

k k pp k

û û

û

û û û ûû û

, , , ; ∈ ⊕ = ,

, , , ; ∈ ⊕ = .' 'a a s d a

b b s d b

… s … a I I k

… s … b F F k

k

k

PHYSICS O

and final hadrons separated macroscopically in spaceby the source–detector distance:

(235)

where

and and are the corresponding -numberhadronic currents in the source and the detector. Theproof of “factorization formula” (235) is given inAppendix C. The explicit form of hadronic currents isnot needed in this study.

Let us introduce the causal Green’s functions for aneutrino and a boson:

(236)

Here,

(237)

and

(238)

are the bare propagators of neutrino (with mass )and a boson (with mass ), respectively. The lat-ter propagator is written in the unitary calibration.Performing standard transformations, we arrive at thefollowing expression for amplitude:

(239)

( ) ( ) ( )

μ ν

μ ν

⊕ : :: : ⊕ = Π , ; ,

∗× ; , ; , ,

†' ' ( ) ( )'

s d q q h s d

s s a b d d a b

F F T j x j y I Ix y x

X p p X p p

pû û

) )

S

( ) −

∈

∈−

∈

∈

Π , ; , = ψ , − ∗ × ψ , −

× ψ , −

∗× ψ , − ,

∏

∏∏

∏

' ( )

( )

( )

( )

a a

s

b b

s

a a

d

b b

d

ip xa a a

a I

ip xb b b

b Fip x

a a aa I

ip xb b b

b F

x y x e x x

e x x

e x y

e x y

p p

p

p

p

s) d) c

W

[ ]− −

μν μ ν

− −μν

− = ν ν

= Δ ,π

− =

= Δ .π

0

( )4

†

0

( )4

( ) ( ) ( )

( )(2 )

( ) ( ) ( )

( )(2 )

jj j

j iq x y

W

ik x y

G x y T x ydq q e

G x y T W x W y

dk k e

μμ

+Δ = = ,

− + − += γ

2 2

ˆ( )( )

ˆ 0 0

ˆ( )

j j

j j

i q miqq m i q m i

q q

μν μ νμν

−Δ = −

− +

2

2 2( )0W

W

g k k mk i

k m i

j jmW Wm

βα α β∗= , , , ,

4

' ' ( ' ')64

jj j

j

g V V dxdx dydy x x y yA A1



where

spin indices and arguments of functions and areomitted for brevity. The integration of function

over yields

Using the definitions in (236) and our standardapproximation, we can rewrite this integral as follows:

The integration over is performed in much thesame way and yields factor

in the integrand in (239). Collecting all factors, usingthe definition of overlap integrals (192), and introduc-ing tensor function

(240)

we arrive at the following expression for amplitude:

(241)

8.1. Amplitude Asymptotics at Large

The asymptotic behavior of function (240) at alarge spatial distance between impact points and

is relevant to our purpose. One may use the Gri-

( )

ννν β

ν μ μμ μ α

α α α β β β

∗, , , = −

× − −∗ ∗× ψ , − ψ , − Π , ; , ;

'

' ''

( ' ') ( ') ( )

( ' ') ( ' ) ( )

( ' ) ( ' ) '

j Wd

j Ws

x x y y G y y u

O G y x O G x x

x x y x x y x

p

p

p p p

v

A )

)

s) d)

, , ,( ' ')j x x y yA 'y

β

β

νν β β β νν

−−ννβ β

− −− +β β

−νν

∗= ψ , − − −

∗ Δφ , Δ=π π π

∗× = φ ,π

× Δ − Δ .π

' '

''

3 4 4

( )( ) '3

( ' )'4

' ( ' ) ( ') ( ' ')

( )( ) ( )(2 ) 2 (2 ) (2 )

' ( )(2 ) 2

( ) ( )(2 )

j W j

ikyipx iqx j

ip x yi k q p y

iq x yj

I dy y x G y y G y x

dke kd e dqe qE

ddy e eE

dq q p q e

p

p

p

p p p

p p p

νν β β β

−νν β

∗= ψ , −

× Δ − Δ .π

'

( ' )'4

( )

( ) ( )(2 )

j

iq x yj

I y xdq q p q e

p

'x

β β β α α α

−νν β μ μ α

∗ ∗ψ , − ψ , −

× Δ − Δ Δ +π

( )' '4

( ) ( )

( ) ( ) ( )(2 )

j iq x y

y x x xdq q p q q p e

p p

( )νν μ μ νν

β μ μ α

, = Δπ

× − Δ Δ + ,

' ' '4

'

( ) ( )(2 )

( ) ( ) ( )

jd s

j

dqx q q

q p q q p

pG V V

( )

ννβα β β

μ μνν μ μ α α

∗=

∗× , .

4

'

'' '

( )64

( )

j dj

js j

g V u O

x O V

p

p pv

A )1

)G

L

sXdX


mus–Stockinger (GS) theorem [33] to study thisbehavior. The theorem states the following27:

Let be a function that decreasestogether with its first and second derivatives no slowerthan at , , , and

(243)

Then, in the asymptotic limit for

(244)

whereas integral (243) decreases as at . Con-sidering the definition in (240) and the explicit form ofintegrals in (199) and the neutrino propagator in (237),we find that, in the present case,

The function corresponding to in the GS the-orem to within an insignificant (independent of )factor is written as follows:28

(245)

The first requirement of the theorem may be vio-lated formally by the poles of bare boson propagators(238). In order to eliminate this possibility, we use therenormalized (full) propagator, which has no singular-ity in the resonance region, instead of (238). Thispropagator is obtained by performing the standard

substitution, where is the totalwidth of a boson, in the denominator of (238)29.

Since functions and decreasefaster than any power of as , we concludethat function (245) satisfies the GS theorem condi-

27It was proven in [123] that the asymptotics of integral (243)under somewhat more stringent conditions imposed on func-tion takes the form

(242)

If (which is true in almost all practically relevantcases), integral (243) is an asymptotic series in inverse powers

of , and is an asymptotic series in inverse powers of

. The properties of these series were studied in detail in [123–125], and several physical applications of these results were dis-cussed in [126–129].

28Note that the term in (245) does not contribute to ampli-

tude (241) due to matrix factors and .29A more accurate modification was discussed, e.g., in [130, 131]

(see also the references therein).

Φ( )q

( ) ( )

κ

=−κ

, κ =π

κ × − ∇ + × ∇ + Φ .

22

( )4

11 ( )2

i LeJL

iL L

q qq l

L

l l q2

Φ ∈ 3( ) ( )Sq

L , κ 2( )J L2L

Φ ∈ 3 3( ) ( )Cq

−2q → ∞,q κ ∈2 = Ll L =L L

−Φ, κ = .π − κ − 3 2 2

( )( )(2 )

iedJi

qLqqLq e

→ ∞L κ > 0,κ

/−κ , κ = + , π 1 2

( ) 1( ) 14

i LeJL L

lL 2

−2L κ <2 0

= − , = − , κ = − .0 0 2 2 20d s d s jT X X q mL X X

Φ( )qq

∝ jmμ 'O ν 'O

( ) ( ) νν β

μ μ α

δ − δ + Δ −× + Δ + .

'

'

( )ˆ( ) ( )

s s d d

j

q q q q q pq m q p

− Γ2 2W W W Wm m im ΓW

W( )δ −

s sq q δ + ( )d dq q−1q → ∞q

1 2020


tions. Therefore, function (240) behaves in the leadingorder in as

(246)

where

Since factors and under theintegral sign at the right-hand side of (246) have, asfunctions of , sharp and close maxima, the integral issaturated by a small neighborhood of these maximaand can be evaluated by the regular saddle-pointmethod. All calculations below are performed withinthe SRGP model.

The corrections to the GS theorem were discussedin [123].

8.2. Integration over According to (200),

Introducing notations

(247)

we rewrite the expression for in the followingform:

(248)The extremum of this function is defined by

(249)

where

(250)

1 L

( ) ( )

( )

∞− − Θ − −| |νν μ μ

−∞

νν β μ μ α

, = δπ

× − δ +× Δ − + Δ + ,

0' ' 02

' '

8( ) ( )

ˆ( ) ( )

ji

i q T Ljs

j s d j d

j j j j

ex dq eL

q q q qq p q m q p

qpû û

S

G

= , , = − , = ,= + Θ = + .

2 20 0( )

andj j j j

s d s s d d

q q q m LX q X q

q q l l LS S S

δ − ( )s j sq q δ + ( )d j dq q

0q

0q

( ) ( )( ) ( )

μνμ ν

μνμ ν

δ − δ + = − , π

= − −

+ + + .

ℜ ℜℜ

ℜ

04

0

( ) ( )1 1exp ( )

4(4 )( )

s j s d j d

js d

j s j s j s

d j d j d

q q q q

F q

F q q q q q

q q q q

μν μνμ ν μ ν

μ μν μν μν μν μνν ν

= + ,= − , = + ,

ℜ ℜℜ ℜ ℜ ℜ

0 s s s d d d

s s d d s d

F q q q q

Y q q R

0( )jF q

μ μ νμ μν= − + .0 0( ) 2j j j jF q F Y q R q q

( )]

=

× − −− + = ,

0

02

0 0

0 0

( ) 2

( )( ) 0

j

j

j j

j

dF qdq

Rq qY q

q

q Rl qq Yl

( ) ( )

μνμ ν= = − + ,= ,

= , , , = , ,= , .

00

01 02 03 1 2 3

2( )

and (1 )

kn k n

R R l l RR l l

R R R Y Y Yl

Rl

R Yl

5

5

PHYSICS O

Here and elsewhere, summation over repeatedLatin indices ( ) is assumed. The rootof (249), , is the saddle point if the secondderivative

(251a)

(251b)

(where and ) is positive at thispoint. Two intriguing special cases with relatively sim-ple approximate solutions of (249) corresponding tothem are considered below. The general solution isexamined in Appendix D.

8.2.1. Ultrarelativistic case. Let us examine theconfigurations of momenta of external packets satisfy-ing conditions

(252)These relations hold true in all current neutrino

experiments. Therefore, this special case is of princi-pal interest.

In the plane-wave limit ( , ) and under theassumption that (in what follows, this specialcase is referred to as the PW limit), equalities

need to be satisfied to achieve exact energy-momen-tum conservation at each vertex of the diagram.

According to (247) and (250), the root of (249) is

(253)

which is the energy of a real massless neutrino( ).

In the general case with , if conditions (252)are satisfied and a natural additional assumption thatneutrino masses are small compared to the mini-

mum absolute values of energy transfers and atthe diagram vertices30 is fulfilled, the solution of (249)can be presented in the form of a series in powers ofsmall dimensionless parameter

(254)

30It is assumed that the minimum is taken over the entire set(defined by the specific experimental conditions) of the mostprobable momenta of external packets in the source and thedetector.

, = , ,1 2 3k n=0 jq E

( )

( ) ( ) ( )[ ]

−= +

× + − − + |

200

2 30

0 0 0

2( )2

( ) 2 ( )

jj

j

j j j

qd F qR

dqq q q

q

qRl q q Yl q

= + + − −

Γ

23 3

( )22 ( )(1 2 )(1 )j jj j j

Rm

YlRl v v

v

= 0j j qqv Γ = 0j jq m

− .∼ ∼ ∼

0 0s d s dq q q q

σ = 0û

∀û

= 0jm0

= − , = − =0 0 0s d s d sq q qq q l

σ = ,∀

−= = 000 0

( )lim ,sYq q

RYl

û û

= 0 ,jq E l =2 0jq

σ ≠ 0û

jm0sq 0

dq

pû

ν

= .2

22j

jm

rE



The definition in (254) features “representative”virtual neutrino energy31

(255)

which becomes the energy transfer in the PW0 limitand is close to it in magnitude at sufficiently small (this statement is clarified below). According to (255),

is a rotationally invariant function of momenta,masses, and dispersions of momenta of all externalpackets. Owing to the approximate energy-momen-tum conservation, this quantity is nonnegative and istransformed as the zeroth 4-momentum component.

Thus, let us write and as powerseries

(256)

It is convenient to rewrite (249) in the followingform:

(257)

which features dimensionless rotationally invariantfunctions

It can be seen from (257) that coefficients and are expressed in terms of these two functions only

at all . It is easy to determine the coefficientsusing the standard recurrent procedure: one needs toinsert series (256) into (257), expand the obtainedexpression into a series in , and set the factors atpowers of to zero. The first three pairs are as follows:

(258)

(259)

It is evident that the coefficients satisfy the symme-try relation

31Here, is not a Lorentzian index.ν

ν = ,( )YlER

0sq

σû

νE

0q = − 20j jq mq

∞

ν=∞

ν=

≡ = − ,

≡ = − .

01

1

1

1

E nj n j

n

P nj j n j

n

q E E C r

P E C rq

( )[ ]ν

− −− + − = ,

20 0

0( 1) 0j j

j

q qE q

q qq

m

n n

= = .( ) ( )and( )Yl RYl Rl

n m

EnC

PnC

≥ 1n

jrjr

( )( )= , = + − ,

= + + − + ;

1 2

23

322

57 9 (5 2)2

E E

E

C C

C

n n n m

n n n n m

( )( )

= + , = + + − ,

= + + + − + .

1 2

23

11 ( 1) 22

1( 1) 7 5 (5 3)2

P P

P

C C

C

n n n m

n n n n m

− += − .

( 1)( 1)P n E

n nC Cn n


It follows from (257) that this holds true for all .Quantities and are naturally interpreted asthe effective (or the most probable) energy and3-momentum of virtual massive neutrino , respec-tively. They also provide an opportunity to defineeffective neutrino velocity which isgiven by

(260)

As expected, ; i.e., neutrinos areultrarelativistic. Since, in addition32,

(261)

where

(262)

the second derivative in (251) at and ispositive,

(263)

and, consequently, function has an absoluteminimum at this point. We note once again that quan-tities , , and are defined uniquely not only bythe most probable momenta of external packets inthe source and the detector, but also by their massesand dispersions of momenta.

A special configuration of external momenta. Inorder to illustrate the obtained results, we consider thefollowing special configuration of external momenta:

(264)

which corresponds to the exact conservation of energyand momentum “flowing” from to . Quantity

is called the virtuality of a neutrino. Theultrarelativistic case is defined by conditions and but the virtuality is not bound to match

32It bears reminding that quadratic forms and are positive for arbitrary 4-vector .

njE =j jPp l

ν j

= = ,j j j jEv l pv

( )( ) ( )

= − − +

− + + − + .

2

2 3 4

11 22

17 5 22

j j j

j j

r r

r r

v n

n n m 2

< −0 1 1jv !

μνμ νℜ s q q μν

μ νℜ d q q≡/ 0q

μν −μ ν ν= = ,2R R l l EF

ν

μνμ ν =

≡ > , 0q E l

R q qF

=0 jq E 1jr !

[ ][ ] ( )

=

= − − + −− + + − + +

0

20

20

2

3 4

( )

2 1 2 3 2 ( 1)

3 ( 1)(8 5) 2(9 5) ,

j

j

q E

j j

j j

d F q

dq

R r r

r r

n n n m

n n n n m 2

0( )jF q

jE jp jvpû

= − ≡ > , = − ≡ , > ,0 0 0 0s d s dq q q q l% 3 3

S D

= −2 2 24 % 3

2 2!4 %

2 2,jm ! %2jm

1 2020


even in the order of magnitude. It is easy to demon-strate that the following is true for configuration (264):

where

(265)

Expanding and with respect to small parame-ter , we obtain

where, as usual, dots denote higher-order corrections.Thus, and at . Reexpandingthe above expressions for and with respect to two

small (independent) parameters and ,we find

It can be seen that the effective energy (momen-tum) of a neutrino may be both smaller and larger thantransferred energy (transferred momentum ); nat-urally, and at (and in this caseonly). In other words, even if the transferred4-momenta at the diagram vertices are perfectly bal-anced, the effective 4-momentum of a virtual neutrino

is, generally speaking, not the same as .Since the expansion for the effective neutrino velocitytakes the form

the leading correction to the ultrarelativistic limit is independent of the neutrino virtuality.

( )( ) ( ) ( )ν

−

= + − ,

= − − − + − ,

0

12

0 0 0

1 1

1 1 1

E

PR

3% n

%

@ 3n n n n

% %

−= = − .0( )

R RRl @ @

n m

νE n

2 24 %

( ) ( )

ν = + + + + ,

= + − −

× − − + − + + ,

2 2 4

0 2 2 4

22

0 0 0 2

2 42

0 0 02 4

1 12 2 8

( )2

1 2 1 2 2 14 8

E …

…

4 4 4% n

% % %

4n n m n n

%

4 4n n n

% %

ν →E % → 0n n →2 04

jE jP2 2

4 %2 2jm %

( )

( )

−= +

× + + + − + ,

−= +

× + +

+ + − + + .

2 2

222

0 0 02 2

2 2

2

0 2

220 0 2

2

1 4 3 24 4

2

( 1) 14

4 5 2 14

jj

j

jj

j

mE

m…

mP

m…

4%

%

4n n n m

% %

43

%

4n

%

n n m%

% 3

=jE % =jP 3 =2 2jm4

,( )j jE P l ,( )l% 3

( )

= − − + + + ,

2 22

0 02 2 21 1 4 12 4

j jj

m m…v

4n n

% % %

= 1jv

PHYSICS O

8.2.2. Nonrelativistic case. Let us now examine thecontrary case corresponding to the following configu-ration of external momenta:

(266)

This case is of potential interest for future experi-ments on (hypothetical) heavy neutrinos and for cos-mological applications. It is convenient to rewrite (249)in terms of velocity of a virtual neutrino:

(267)

Let us introduce dimensionless 4-vector with components

(268)

It is evident that these components are small inmagnitude if conditions (266) are satisfied. Insertingthe expression for 4-vector , which is written compo-nent-wise as

into definition (268), we indeed find

(269)

Since all the terms in (269) contain small factors( etc.), one may conclude that

. Taking this into account, we seek the solutionof Eq. (267) in the form of a double power series

(270)

where

The first six dimensionless coefficient functions are as follows:

(271)

− .∼ ∼ ∼

0 0s d j s dq q m q q@

= 0j j qqv

−− = − .

−

2

02

(1 ) ( )( )1

j j

j jj

mR Y YlRl

v

v vv

= ,0( )j j j

μ μ μ = − .

01 1

jj

R Ym@

Y

μ μ μ μ μ= − + − ,ℜ ℜ ℜ ℜ

0 0 0 0 k k k ks s d d s s d dY q q q q

μ μ

μ μ μ

= −

+ + − + .

ℜ

ℜ ℜ ℜ

0 0

0 0

1 ( )

( )

j s j sj

k k k kd j d s s d d

m qm

m q q q

5

− 01 ,s jq m ,ks jq m

μ 1j !

∞ ∞

= =

= + ,

( )

01 0

1 ( )n mj j nm j j

n m

C lv

v v

= .+ 0

( )1

jj

j

lv

( )nmC v

= − = = ,

= − + ,

= − + + ,

= − − .

( ) ( ) ( )10 11 12

2( ) 0020 2

2( ) 0021 2

2( ) 0030 2

3( )1 32 2

9( ) 12 22318( ) 32 2103( ) 45( ) 23

8 3

C C C

RC

RC

RC

Rl

Rl

Rl

Rl Rl

v v v

v

v

v

5

55

55

5 55



The following is obtained from (270) and (271):

(272)

Here, function

defines the leading relativistic corrections, and dotsdenote higher-order corrections in and . It canbe seen that the nonrelativisitc relation between theeffective velocity, energy, and momentum remainsvalid up to the second order in and that the rela-tivistic corrections to and are positive.

It can be shown that function is positive. If thisis taken into account, one needs just to insert (270) and(271) into (251b) to find that the second derivative in(251b) is positive at the stationary point:

(273)

The singularity at is hardly a surprise, sinceit only reinforces the intuitive expectation that theamplitude of a process with a neutrino “at rest” in theintermediate state should be equal to zero. However,this case requires a more detailed examination of theconditions of applicability of the saddle-point methodand the GS theorem. These issues will be discussed ina separate paper.

Let us turn to the special case of the accurate bal-ance of the energy and momentum transfer at themacrodiagram vertices. We use designations (264)and, in accordance with (266), assume that

Thus,

and, consequently,

Inserting these relations into (270), taking (271)into account, and reexpanding the obtained expres-sion in powers of two small independent parameters

( )( )

= + + δ + ,

= + δ + .

231

2 4112

j jj j j

j j j j

mE m …

P m …

v

v

δ = + −

20

3( )( ) 1 ( )j j j jRll l@

( )jl 0j

( )jl

jE jP

5

=

= +

× − + + > .

0

20

2 20

0

( ) 22

6( )1 ( ) 0

j

j

jq E

j j

d F qR

dq

…Rl l

v

@

@

= 0jv

≤ − ≤ .0 1 1 and 0 1j jm m! !% 3

μ μ μ = − −

0

1 1j k kj j

R l Rm m3 %

5

= − − ,

= − − .

000

( ) 1

( )( ) 1

jj j

jj j

Rm m

m m

Rl

Rll

3 %

5 5

3 %

5


and , we arrive at the following expres-sion for the effective velocity of a virtual neutrino:

(274)

The effective energy and momentum of a virtualneutrino in the leading order in and are determined from (274):

These simple formulas agree completely with intu-itive expectations only at . It isonly in this special case that the following relationshold true:

8.3. Resulting Formula for the Amplitude

It was proven above that function has anabsolute minimum at both in ultrarelativisticand nonrelativistic cases; in the neighborhood of thisminimum, it may be approximated by a parabola:

(275)

Positively defined function

(276)

was introduced here. It will be demonstrated belowthat this function may be interpreted as the effectiveenergy uncertainty of a virtual neutrino. In the ultra-relativistic case, which will be the only one consideredbelow,

(277)

Within our approximations, function may beconsidered essentially independent of (i.e., universalfor all neutrinos ), since and function isitself small compared to the neutrino energies availablein current experiments. Let us also take into account thefact that all factors of the integrand at the right-handside of (246), with the exception of exponent

jm3 −1jm%

= + + −

× − + , = .

200

2( ) 3( )12 2

1 ( )

j jj

j jj

Rm

…m

Rl Rl

l

v v

v

3

5 5 5

%

jm3 −1jm%

≈ + , ≈ .2 2j j j j j j jE m m P mv v

− 2 21j jm m&% 3

≈ , ≈ +≈ .

2 (2 )and

j j j j j

j

m E m mP

v 3 3

3

0( )jF q=0 jq E

( )−+ .

20

0 2( ) ( )4

jj j j

j

q EF q F E

D

= = 0

2 20 0

1 2 ( ) ,j

j q Ej

d F q d qD

( )

ν

+ += ,

= = .

212

12 2

j jj

j

r rR

ER

v

n 2 DD D

DF

jD

jν j 1jr ! D

( ) − − − − ,

2 20 0 0

1exp ( )4 j jF q i q T q m L

1 2020


are weakly varying functions of integration variable in the neighborhood of stationary point and may,therefore, be taken out of the integral at point .Using (275) and expansion

we arrive at the following simple integral:

Introducing complex-valued phase function

(278)

which features

(279)

where parameter is exactly the same as in thequantum-mechanical case in (53), we obtain

Complex dispersion depends on the effectiveneutrino energy and spatial distance between theimpact points in the source and the detector; its mod-ulus and argument are given by

Collecting all factors, we obtain the followingresulting expression for the function in (246):

(280)

Here, 4-vector was introduced, andthe contribution proportional to (see footnote (28))

0qjE

=0 jq E

− = + −

− − +

2 20 0

22

03

1 ( )

( ) ,2

j j jj

jj

j

q m P q E

mq E …

P

v

( )∞

−∞

= − − + −

× − −

− + − .

0

0

22

02 3

exp

1( ) ( )4

1 ( )4 2

j j jj

j j j

jj

j j

LI dq i E T P L i T

q E F E

im Lq E

P

v

D

( ) Ω , = − + − ,

22( )j j j j

j

LT L i E T P L Tv

D

ν

= ,+ +

= = ,τ

2 22

2 2 2 2

3 3

1 1

2 2

jj

j j

j j jj

Ljj

i i

m L m L LP E

D DD

r r

D Dr

,τ j L

= π − − Ω , .

12 exp ( ) ( )4j j j j jI F E T LD

jD

L

( )( )

− /+ ,

.

1 4211arg( ) arctan2

j j

j j

rD D

D r

( )νν μ μ νν β μ μ−Ω − Θ

α /

, = Δ − Δ

× + .π

' ' ' '

3 2

ˆ( )

( ) ( ) ( )4

j

jj j

ij

j d j s j

x p p p

ep p p p

i L

p

D

û ûG

V V

= ,( )j j jp E P l

jm

PHYSICS O

was omitted. Phase factor in (280) is insignifi-cant, since it vanishes in the amplitude modulussquared.

Owing to the presence of smeared functions and , which are involved in the

expressions for overlap integrals and and establish the approximate energy-momentumconservation ( ), and to the assumedsmallness of neutrino masses compared to representa-tive energy , we may set in the entire preex-ponential factor at the right-hand side of (280). Let usnow use identity

(where , , and is thecommon Dirac bispinor for free left-handed masslessneutrino ) to define matrix elements

(281)

which characterize the production and absorption of areal massless neutrino in reactions and

, respectively33. Taking the above resultsinto account, we then obtain the final expression forthe amplitude in (241):

(282)

It is instructive to isolate the -independent com-mon factor establishing the approximate energy-momentum conservation at vertices. To this end, weuse (200) and definitions (247) to obtain

33Under additional conditions and

, the -boson propagator may be written

approximately as , which corresponds to the four-fermion theory of weak interaction. Using well-known SM

identity , one may then rewrite matrix ele-ments (281) in the following form:

However, this slightly restrictive simplification (inapplicable atultrahigh energies) is not required and will not be used in subse-quent analysis.

− Θ− iie

δ( )δ −

s j sp q ( )δ +

d j dp q( )s jpV ( )d jpV

≈ ≈ −j s dp q q

νE = 0jm

− ν + − − ν − ν +=ˆ ( ) ( )P p P P u u Pp p

± = ± γ51 (1 )2

P ν ν= Ep l − ν( )u p

ν

μμ′− ν μμ ν α α

μ μβ μ μ ν β − ν

= Δ + ,

∗ ∗= Δ − ,

2'

2'

'

( ) ( ) ( )8

( ) ( ) ( )8

s s

d d

gM u p p O u

gM O p p u

p p

p pv

)

)

+α→ ν's sI F

−βν → 'd dI F

ν α+ 2 2( ) Wp p m!

ν β− 2 2( ) Wp p m! W

μν− 2Wig m

=2 28 2F Wg G m

μ− ν μ α

μβ μ − ν

≈ − ,

∗ ∗≈ − .

( ) ( )2

( ) ( )2

Fs s

Fd d

GM i u O

GM i u O u

p p

p p

v)

)

βα βα

−Ω − Θβα α β/

= ,

∗∗= .

π

3 2

( ) ( )4

j

j

j

ij s j d j s dj j j

p p M MV V e

i L

A A

A D1

V V

j

−Θν ν

δ − δ +

= δ − δ + ,

( ) ( )

( ) ( ) j

s j s d j d

s s d d

p q p q

p q p q e



where

Amplitude (282) may then be presented in the fol-lowing form:

(283)

Using (256), one can present function as anexpansion in :

We should recall that function is defined inaccordance with (265) and matches in the case ofthe exact energy-momentum conservation at vertices(see Section 8.2.1). If

(284)

the following approximate relation is valid:

and sign-indefinite difference may beneglected in the neighborhood of the maximum ofproduct (i.e., at ).Then,

(285)

It can be proven that this quantity is positive.It can be seen from the derivation of formula (283)

and its structure that it holds true both for the consid-ered class of processes and, if matrix elements (281) areredefined accordingly, for all other processes sustainedby the exchange of virtual neutrinos between macrodi-agram vertices. It is easy to generalize formula (282) tothe case of reactions with the exchange of antineutri-nos: one needs just to perform the substitu-tion (i.e., ) and modify matrixelements (281) accordingly.

( ) ( )

( ) [ ] ( )

[ ] ( )

( ) ( )

( ) ( )

μ μμ μμ ν μμ ν

μν ν ν

ν ν

ν ν

ν ν

Θ = − + −

× − = − + −

+ − − −

+ − − −

× − + − .

' '' '

00 0

200

2

1 24

1 ( )2

1 ( ) ( )2

1 2( )4

j j

j j

j

j j

j j

Y R p R p p

p p E R Y E E

E R E P

R E E E E

E P R E P

Rl

Rl Yl

Rl

ν νβα /

−Ω −Θ − Θα β

∗=

π∗× .

3 2( ) ( )

4j j

s d s d

ij j j

j

p p M Mi L

V V e

A1

D

V V

Θ j

jr

( )

( ) ( )

Θ = − + − −

+ + − − + .

2 20

2 2 3

1( )2

1 ( )2

j j j

j j

m R r

r r

n n m n n

n m n n 2

0n

n

, ∀ ,1 andj jr r j! !n m n

( ) Θ ≈ − + − − ,

2 20

1( )2j j jm R rn n m n n

−0n n

ν νδ − δ + ( ) ( )s s d dp q p q ν≈ − ≈s dq q p

( ) [ ]ν ν

− − −Θ ≈ = .

4 2 4 20 0 00

2 2

( )4 4

j jj

m R m R

E RE

Rlm n n 5

†V V

α α∗ ,j jV V β β

∗j jV V


8.4. Effective Wave Packet of an Ultrarelativistic Neutrino

Let us return to expression (282). Phase function (278)involved in it may be rewritten in the followingapproximately34 Lorentz-invariant form:

(286)

where . Obviously, at However, it is clear that quantities

cannot simultaneously vanish at all values if the neutrino mass spectrum is nondegener-ate. Therefore, relation , which is set bythe space-time configurations and velocities of exter-nal wave packets and is in no way associated with theeffective velocities of virtual neutrinos, should not beinterpreted as a certain mean neutrino velocity. Thus,with the and values fixed, vector is,in general, nonzero, but is collinear to velocity vector

. It is easy to show that if the direction of vec-tor is arbitrary, the following identity holds true:

(287)

where and are the components of that areparallel and perpendicular to vector . It then followsfrom (287) that the value of at

(288)

(where, naturally, ) decreases bya factor of no more than with respect to its maximumvalue at . The region defined by condition (288)is the interior of an oblate spheroid with radius

that is f lattened by a factor of along the direction.

Let us now examine the transformation underwhich the positions of centers of all external packets inthe source, the value of , and unit vector remainunchanged, while all packets are shifted by ;for simplicity, we set . This transformationshifts impact point and vector by the sameamount:

Since suppression factor in (282) isinvariant with respect to spatial translations

and all external momenta remain

unchanged, quantity may be inter-

34Function is invariant to within corrections .ν 2 2

j ED ( )j2 r

ν

Ω , = + − ,

2

2 2 22

2( ) ( ) ( )j

j j j jT L i p X p X m XE

D

= −d sX X X − =2 2 2( ) 0j jp X m X= .jTL v

−2 2 2( )j jp X m X j

= 0L T XX

j T δ = − jTx L v

=j jv lv

δx

⊥

− = δ ⋅ + δ ×= δ + δ ,

2 2 2 2 2 2

2 2 2 2

( ) ( ) ( )j j j

j j

p X m X E

E m

x l x l

x x

δ x ⊥δx δxl

− Ωexp( )j

( )−⊥δ + Γ δ <

2 2 2 22 1,j jx xD

νΓ = j j j jE m E me

δ = 0x

( )1 2 jD Γ j

l

T l∈ dFû δx⊥δ = δx x

dX L

= + δ , = + δ . ' 'd d dX X X x L L L x

−exp( )dS

+ δx x xû û

βα βα= ''

d d

j jX X

A A

1 2020


Fig. 16.

l

L

L'

Xs Xd

δx

Xd'

preted as the amplitude of the probability of interac-tion of neutrino with 4-momentum at

point (see Fig. 16). This interpretation may seemartificial, but we will provide certain arguments infavor of it below (in the analysis of the expression forthe count rate of neutrino events). If distance is suf-

ficiently large (i.e., ), its relative varia-tion induced by shifts satisfying conditions (288) isinsignificant,

and the amplitude suppression is determinedexclusively by factor , which is close to unity

at . The absolute values of transverse

shift and length difference maybe so large that the ratio exceeds unity. If thisratio is interpreted erroneously as the velocity of apoint-like neutrino, one may conclude incorrectlythat special relativity is violated. It is instructive toclarify this purely quantum effect within a more intui-tive approach by introducing an effective neutrino WP.To this end, we isolate the following factor fromamplitude (282):

(289)

(note that spinors and are contained inmatrix elements (281)). Neglecting the imaginary partof function , we rewrite expression (289) in theform of a product:

(290)

where

j = ,( )j j jp E P l

'dX

L

2 24 1j L @D

δx

− ≈ δ ,2 2' 1 (2 ) 1L L Lx !

βαj

A

− Ωexp( )j

⊥δ Γ

2 2 22 j jx !D

⊥δx − ≈ δ 2' (2 )L L Lx'L T

−Ω , −Ω ,− ν − ν − −≈( ) ( )1 1( ) ( ) ( ) ( )j jT L T L

j ju e u u e uL L

p p p p

− ν( )u p − ν( )u p

2jD

Ψ , − Ψ , − ,−1 ( ) ( )

s d

j jX j d s X j s d

d s

X X X Xp pX X

−ν

Ψ ,

= − − −

2

2 2 22

( )

exp ( ) ( ) ( ),

jy j

jj j j j

x

i p y p x m x uE

p

pD

PHYSICS O

and

Comparing spinor function with thewave function of a fermion packet of the general formin the SRGP approximation

we see that spinor function may beinterpreted as the wave function in the -represen-tation characterizing an incoming WP of a real mas-sive neutrino with acting as parame-ter . The second spinor cofactor in (290),

, is naturally interpreted asa spherical neutrino wave outgoing from the source atdistance from production point .

It is instructive to present the result obtained in asimpler quantum-mechanical formulation by per-forming calculations similar to the transition ampli-tude calculation in the QM approach (see (61)). Let usdetermine the amplitude of transition from the WPstate in the source defined by the most probablemomentum , momentum dispersion , and 4-coor-dinate to the WP state in the detector with thesame momentum, dispersion , and 4-coordinate

(the WP spread in the configuration space isneglected; i.e., the SRGP approximation is applied):

(291)

Here,

(292)

−ν

Ψ , = Ψ , γ

= − − .

†0

22 2 2

2

( ) ( )

( )exp ( ) ( )

j jy j y j

jj j j j

x x

u i p y p x m xE

p p

pD

Ψ ,( )y j xp

[ ] −Ψ , , = ψ , −σ= − − − ,

22 2 2

2

( ) ( ) ( )

( )exp ( ) ( )

ipxy s

s

s x e u x y

u i py px m xm

p p p

p

Ψ , −( )d

jX j s dX Xp

x

ν j σ = Γj j jD

σΨ , − −( )

s

jX j d s d sX Xp X X

−d sX X sX

ν j

jp σsj

sX ν j

σdj

dX

/

, ;σ , ;σ

σ σ= ψ , − ,σ . σ + σ

3 2

2 2

1

2( )

j d dj j s sjsd

sj djj d s j

sj dj

X X

X X

p p

p

1

= +σ σ σ2 2 21 1 1 ,

j sj dj



and

We obtained an expression proportional to function in the SRGP approximation, which should be

compared to The result is . Thus, is the effective neutrino energy uncertainty with

interactions in the source and the detector factored in.It should be compared to defined in accordancewith (63). Since and are the neutrino WP dis-persions in the source and the detector, respectively, itis easy to determined their explicit form by rewritingthe first formula in (250) as where

, . It is clear that quantities and have the physical meaning of spatial dis-

persions of a neutrino WP in the source and the detec-tor, which are related to the momenta dispersions bythe following standard (for a Gaussian packet) expres-sions:

(293)

Inserting and from (293) into (292) to calcu-late the amplitude in (291), we indeed obtain, to within adimensionless factor of the order of unity, withfunction defined in accordance with (276).

Since is a complex-valued function, factoriza-tion (290) is not possible in the general case, but thecorresponding correction, which is clearly of interestfor neutrino astrophysics, can be interpreted as theresult of a peculiar interference of spreading in- andout-neutrino packets. This correction becomes signif-icant at very large distances and induces overall sup-pression of amplitude (241) and modification of oscil-lation factors . To study theseeffects in detail, one needs to analyze observables(such as the count rate for a given type of event) thatare obtained after appropriate averaging of the ampli-tude modulus squared over all unmeasured variableson which amplitude (241) is dependent. This averag-ing depends on the statistical distributions (or, in amore general case, on the kinetics) of ensembles of in-packets and on the detection procedure [57, 58]. Here,we limit ourselves to the case when the imaginary partof is negligible. Let us determine the applicabilityconditions of this approximation. Using (277) and(279), we write

(294)

= , ;σ , ;σ , ;σ , ;σ .sd j d dj j d dj j s sj j s sjX X X Xp p p p1

ψ ,( )xp−Ω ,( ).j T Le σ = Γj j jD

jD

σp

σsj σdj

= σ + σ2 2 ,xs xdRμνμ νσ = ℜ2

xs s l l μνμ νσ = ℜ2

xd d l lσxs σxd

− μνμ ν

− μνμ ν

σ = σ = ,σ = σ = .

ℜℜ

2 2

2 2

4 4

4 4sj xs s

dj xd d

l l

l l

σsj σdj

−Ω ,( )j T LejD

jD

L

[ ]∝ Ω ,exp Im ( )ji T L

jD

ν

.

25

4

1 GeV101 eV 10 km

jj

m LE

rF


It can be seen that for all current experi-

ments with reactor ( MeV, km), accel-erator ( MeV, km), and atmospheric( MeV, km) (anti) neutrinos ifmj 1 eV and . Using typical examples, we

will demonstrate in Appendix А.3.2 that the condition is definitely fulfilled in the domain of appli-cability of the SRGP model.35 In addition,

; therefore,

(295)

With the adopted conditions of narrowness ofexternal packets in the momentum space, ,factored in, it follows automatically that

and (packet stability condition).It was demonstrated in Section 5.7 that the uncer-

tainties of the energy and the momentum compo-nents of an ultrarelativistic WP in the SRGP approx-imation are

( ); thus, the correspondinguncertainties for an ultrarelativistic neutrino packet are

i.e., function , which depends on the masses,momenta, and momentum spreads of external in andout packets, characterizes the neutrino energy uncer-tainty, and defines (to within a numerical factorof the order of unity) the effective size of a neutrinowave packet transverse to momentum . A huge andvery thin disk with ratio of the transverse size tothe longitudinal one may serve as an illustration of anultrarelativistic neutrino WP. The relative neutrinoenergy and momentum uncertainty

is always very small and independent of the neutrinoenergy and mass. This is how one should interpret thestandard quantum-mechanical assumption that thestates of neutrinos with definite masses (and, con-sequently, the states with definite f lavors ) havedefinite 4-momenta.

35In particular, taking into account modern cosmological con-straints on the sum of neutrino masses (see, e.g., [132]).

1j !r

ν 1E *310L &

ν 100E *310L &

ν 100E * . × 41 3 10L &

&710F @

710F @

2 2 2j jD D D

ν ν

σ= .

2 2

2 2 21 1

22j

jm E E RD

F

σ2 2mû û!

−σ2 2 710j jm !

σ σ2 2 2 2j j jL m!

⊥

δ ≈ δ ≈ δ ≈ Γ σ,δ ≈ σ

2 ln 2

2 ln 2

Ep pp p

p

δ × = 0,p p ⊥δ ⋅ = 0p p

⊥

δ ≈ δ ≈ δ ≈ ,δ ≈ Γ δ ,

2 ln 2

2 2 ln 2j j j

j j j

E p p

p p!

D

D

D

1 D

jpΓ 1j @

νδ δ ∼ ∼ 1j j j jE E P P ED F

νi

αν

1 2020


Just as any other SRGP, a neutrino WP moves, inthe average, along classical trajectory Quantum deviations from this trajectory are sup-pressed (in amplitude) by factor

Owing to the smallness of ratio ,transverse deviations may be macroscopically large (orinfinite, if neutrinos are massless). This conclusionagrees with the result formulated at the beginning of thissection without invoking the notion of a neutrino WP.

Thus, we verified that the effective WP of an ultra-relativistic neutrino reproduces all the properties of anSRGP of the general form, with the only caveat thatparameter actually depends on the momenta (aswell as masses and momentum dispersions) of allexternal packets. It should be noted here that thisdependence is not a specific feature of neutrinos or thecovariant formalism, since, as was discussed in Sec-tion 4.3, a WP characterizing the state of any massiveparticle should depend on the momenta of particlesinvolved both in its production and its absorption, andthe assumption used here is just a simpli-fying approximation.

9. MICROSCOPIC PROBABILITY OF MACROSCOPICALLY SEPARATED EVENTS

Using (282) and the formulas for four-dimensionaloverlap volumes derived in Section 7.11, we obtainthe following expression for the microscopic probabil-ity of process (231):

(296)

This expression depends on coordinates andmean momenta of all WPs involved in the reactionand on parameters . Probability (296) is vanishinglysmall if the product of overlap volumes

is small (i.e., if in- and out-packets in the source andthe detector do not overlap in space-time regions sur-rounding impact points and ).

= .j jTL vδ jL

( ) − δ Γ + δ . 22 2 2 2exp 2 ( )j j j LL L LD

Γ ∼

2 2 2j jmD F

σ j

σ = constû

,Vs d

νβα

∈

ν

∈

−Ω −Θα β

π δ −=

π δ +×

∗× .π

∏

∏

242

24

2

3 2

(2 ) ( )V2 V

(2 ) ( )V2 V

12(2 )

j j

s s s s

S

d d d d

D

j j jj

p q ME

p q ME

V V eL

û û

û

û û

û

A

D

xû

pû

σû

( ) ( )( )( )[ ]

− /π=

× − + ,

ℜ ℜ4

1 2V V2

exp 2

s d s d

s dS S

sX dX

PHYSICS O

Note that 4-vector pν is also a function of pû and σ

û,

and in the , limit. Therefore,at sufficiently small ,

What determines the approximate equality of and ? To answer this question, we transform expres-sion (296) in a way proposed by Cardall [42]. Usingthe explicit form of functions and , one readilyderives an approximate relation36

where is an arbitrary slowly varying function of, and . This relation yields

(297)

where the prime on “silent” integration variable isomitted, but it (just as vector ) is no longerrelated in any way to the parameters of external pack-ets. Formulas (296) and (297) are equivalent within theadopted approximations, but it follows from (297) thatthe energy-momentum conservation law is governed bysubintegral factors and , whichmay be replaced by common functions at sufficientlysmall . The obtained result will serve as a basis forcalculation of the quantities measured in currentexperiments on neutrino oscillations.

10. PROBABILITY AND COUNT RATEExpression (297) is the most general result of this

study. At the same time, it is too abstract for directprocessing and analysis of the data provided by currentneutrino experiments. In the majority of such experi-ments, information on the particle coordinates in thesource (and, more often than not, in the detector) isunavailable or is not used in data processing. In addi-tion, it is often the case that only the momenta of sec-ondary particles in the detector are measured. Toobtain an actually observable quantity, one needs toaverage (sum) probability (297) over all variables of in-

36The accuracy of this relation is the same as that of formula(282) for the amplitude: it is the accuracy of the saddle-pointmethod used in the derivation.

ν = = −s dp q q σ = 0û

∀ûσû

( )ν ν

− /−

δ − δ + ≈ δ δ

= π .ℜ ℜ 1 24

( ) ( ) (0) (0)

(2 )s s d d s d

s d

p q p q

sqdq

,δs d D

( ) ( )( ) ( )

ν ν ν

ν ν ν ν

π δ − δ +

= − δ + ,

2 ( )

' ' ' '( )

s s d d

s d d

p q p q F p

dE p q p q F p

D

ν( )F p

νp ν ν ν ν= , =' ' ' '( )p E E lp

νβα ν

∈

ν

∈

−Ω −Θα β

π δ −=

π δ +×

∗× ,+π π

∏

∏

242

24

2

3 2

(2 ) ( )V2 V

(2 ) ( )V2 V

112 2 (2 )

j j

s s s s

S

d d d d

D

j jj j

p q MdEE

p q ME

V V eiL

û û

û

û û

û

A

D

r

νEν ν= Ep l

νδ −( )s sp q νδ +( )d dp qδ

σû



(out-) particle states that are unmeasured unused inthe analysis. We call this procedure the macroscopicaveraging. In each particular experiment, one needs tofactor in the actual physical conditions (by constructingan adequate mathematical model of the experiment) toperform macroscopic averaging. The subsequent analysisbecomes model-dependent in this respect.

10.1. Macroscopic Averaging

In what follows, we use a simple but fairly realisticmodel. It is assumed in this model that the distribu-tions of WPs of in-particles over mean momenta, spinprojections, and coordinates in the source and thedetector (regarded as physical macroscopic objects)may be characterized by classical single-particle distri-bution functions normalized at each pointin time in accordance with the following condition:

where is the total number of particles at time. We may now clarify (or, more exactly, redefine)

the terms “source” and “detector” that were used incalculations of the amplitude to denote the blocks ofmacroscopic Feynman diagrams. In what follows, theterms source and detector are used to denote the cor-responding experimental devices and (in an abstract,but mathematically more rigorous sense) supports and of products of the distribution functions of overspace-time variables (i.e., ;the definition of is similar), which are assumed tobe bounded and nonoverlapping. It is assumed thatthe characteristic spatial sizes of and are smallcompared to the distance between them but are verylarge compared to the effective sizes of all wave packetsmoving within them. For definiteness, we assume thatonly the mean momenta of secondary particles in are measured in our experiment and that the back-ground of secondary particles from in is negligi-ble (due to the fact that is located far from ). Underthese assumptions, the macroscopically averaged proba-bility of process (231) may be written as follows:

(298)

or

, ,( )a a a af s xp

,, , = ∈ ,π

03 ( ) ( ) ( )

(2 )a

a aa a a a a a s d

s

d d f s x N x a Ix p p

0( )a aN x a0ax

6

$

= ∏supp ,ax aa

f6 ∈ sa I$

6 $

$

6 $

6 $

βα∈

∈ ∈

ν ν∈

ν

−Ω −Θα β

, ,=π

, ,×π π

× π δ −π

× π δ +

∗× .π π

∏∏ ∏

∏

23

spins

3 3

43

2 24

2

3 2

( )(2 ) 2

( )(2 ) 2 (2 ) 2

[ ] (2 ) ( )(2 ) 2

(2 ) ( )

2 (2 )

s

s d

d

j j

a a a a a a

a I a a

b b a a a a a as

b F a Ib b a a

b bd s s

b F b b

s d d d

j jj

d d f s xE V

d d d d f s xVE V E V

d d V dE p qE V

M p q M

V V eL

x p p

x p x p p

x p

A

D


Here and elsewhere, symbol denotes averag-ing over spins of all in-particles and summing overspins of all out-particles in and , and symbol indicates that integration over is not performed;i.e., . It is thus implied in (298) that onecan measure the momenta of all secondary particles inthe detector37. It is evident that (298) is the total num-ber, , of events that are detected in and involveparticles with their momenta falling within theinterval from to (i.e., 38).For simplicity, we neglect effects of the order of (see (279)) that are related to the spreading of an effec-tive neutrino WP at very large distances. These effectsare of potential interest for experiments with astro-physical neutrinos, but are usually insignificant in ter-restrial oscillation experiments (including the experi-ments with solar neutrinos). Certain results corre-sponding to very long baselines are presented inAppendix E.

The integration over packet coordinates in (298)can be performed explicitly if one uses integral repre-sentation (229) for overlap volumes and takes into

account the fact that factor and distribu-tion functions in and vary considerably in timeand space only on macroscopic scales, while the sub-integral factors in and differ significantly fromzero only in a small neighborhood of the correspond-ing integration variable.

Let us formulate this more rigorously. In effect, wedemonstrate that a configuration of spatially separatedpoints such that

(i.e., and reach their maxima) may be foundfor any configuration of external momenta and arbi-trary x. For definiteness, . In view of the invari-ance of functions with respect to shiftsalong the world lines of packet centers, one mayreplace coordinates by , where

37Naturally, this assumption is optional; the “placement of squarebrackets” in (298) is dictated by the conditions of a specificexperiment. For example, in the hypothetical experiment with“labeled” neutrinos, the momenta of all secondary particlesboth in the source and in the detector should be measurable.

38For simplicity, we consider an “ideal” experiment with a detec-tion efficiency of 1, although real efficiency and acceptance val-ues, trigger conditions, and event selection criteria are easy tointroduce into the formalism.

spins

6 $ [ ]bdpbp

=[ ]b bd dp p

αβdN $

∈ db F

bp +b bdp p βα αβ=2 dNA

τLjL

,s dV∗−Ω −Ω 2j ie L

af 6 $

sV dV

x

( ),∈ ,

≡ ψ , − = ,∏ 2 1s dS D

x xpû û û

û

3

s3 d3

∈ Sû

ψ , −( )x xpû û û

xû

yû

( )= , = + − .0 0 0 0s sy X X xy x v

û û û û û

1 2020


It is evident that at; i.e., at one and the same point in time, but with

mismatched spatial coordinates ,which are sufficiently separated by construction, sincevariables are external and we have set the neededconditions for them. Naturally, (since remains unchanged if is replaced by ). It isimplied here that a maximum of one packet may havezero velocity. However, it is clear that the contribu-tions from configurations with two (or more) packetswith zero velocities are insignificant. Neglecting themand edge effects, we rewrite (298) in the followingform:

(299)

where differential forms are defined as

(300a)

(300b)

Definition (278) and its Lorentz-invariant form (286),where

and all the other designations remain unchanged, arestill valid for complex phase in (299). It should bestressed that the derivation of (299) relies on the prop-erties of the adopted mathematical model, which doesnot necessarily characterize faithfully the real experi-ments on neutrino oscillations. The experimental con-straints on neutrino masses, which make it natural toassume that varies significantly only onmacroscopically large spatial scales and

varies on scales much larger than ,were also used implicitly. It bears recalling, however,that this interpretation is based on the quantum-mechanical analysis of experimental data. Therefore,simple approximate formula (299) is actually notequivalent to more general expression (298) and cer-tainly not to (297).

( )= ψ , − =∏ 2 1s y xp3û û û

û

=y xû

( )= + −0 0x xx x vû û

0xû

=sX x sXxû

yû

αβ

−Ω −Θα β

ν /

=

∗

× ,π −

spins2

27 216

j j

s d

j jj

dN dx dy d d

V V edE

y x

P P

D

,s ddP

∈

ν∈

, ,=π

× π δ − ,π

∏

∏

3

243

( )(2 ) 2

(2 ) ( )(2 ) 2

s

s

a a a as

a I a

bs s s

b F b

d f s xdE

d p q ME

p p

p

P

∈

ν∈

, ,=π

× π δ + .π

∏

∏

3

243

( )(2 ) 2

[ ] (2 ) ( )(2 ) 2

d

d

a a a ad

a I a

bd d d

b F b

d f s ydE

d p q ME

p p

p

P

= = − , = − ,= , = , ,0 0 0

0( )T X y x

L X XX y x

X X

Ω j

∗Ω + ΩIm( )j i

ijL∗Ω + ΩRe( )j i ijL

PHYSICS O

The integration in and in the source and thedetector in (299) is formally performed over timeintervals in which functions are defined. Thus,(299) is applicable to a wide range of experiments withboth stationary and nonstationary sources and detec-tors. To simplify the analysis further, we integrate withrespect to these variables in (299). This is done for asimple model, which is easy to generalize to a morerealistic case. More precisely, we assume that distribu-tion functions in and depend only weakly ontime in the course of the experiment and may beapproximated with a sufficient accuracy by “rectangu-lar” dependences

(301)

where functions are time-independent. Step func-tions for particle distributions in the detector shouldbe understood as hardware or software trigger condi-tions. The steady periods and may be very long (in experiments with solar or atmo-spheric neutrinos, where ) or very short (inexperiments with pulsed accelerator beams, where the

condition is usually satisfied). In either case, itis assumed in model (301) that the “on” and “off”time intervals of the source (detector) are negligible

compared to ( ). If model (301) is used, isthe only factor in the integrand in (299) depending on

and . Thus, the problem is reduced to calculatinga relatively simple integral

(302)

The following designations were used in (302):

(303)

(304)

0x 0y

af

af 6 $

( ) ( )

( ) ( )

, ; = θ − θ −× , ; ∈ ,, ; = θ − θ −× , ; ∈ ,

0 0 0 01 2

0 0 0 01 2

( )( ) for

( )( ) for

a a a

a a a s

a a a

a a a d

f s x x x x xf s a I

f s y y y y yf s a I

pp x

pp y

af

τ = −0 02 1s x x τ = −0 0

2 1d y y

τ τs d

τ τs d&

τs τd

∗−Ω −Θj ie

0x 0y

( )

=

∗ × −Ω − , − Ω − ,

π≡ τ ϕ − .

ℑ 0 02 2

0 01 1

0 0

0 0 0 0

2

exp ( ) ( )

exp2

y x

y x

j i

d ij ij ij

dy dx

y x L y x L

i S!D

( ) + +

, =

−= −

τ

× − + − ,

2 2

' 1

' 1

0 0'

exp( 1)

4

Ierf 2

l lijij

l ld

l l ijij

S

Lx y iv

@

D

D @

ν

π πϕ = , = − = ,

Δ π= = ,

2 2( )

4 2

ij ij j iij ij

ijij

ij

L LLL E L

EL

v vD

! D

n@

D D



(305)

(306)

where is the error function integral. To obtain amore realistic description of experiments with a pulsedparticle beam in the source (e.g., mesons in the decaychannel), one may generalize model (301) by intro-ducing, e.g., a series of rectangular (or more complex)pulses with pauses between the pulses, in which

. Here, we focus on the simplest case (301),which reproduces the most significant effects. Taking(302) into account, we obtain the following insteadof (299):

(307a)

(307b)

We should recall that the effect of spreading of aneutrino WP is neglected in base formula (297), whichwas used to derive expression (307a). As was demon-strated, this approximation is valid at distances sat-isfying condition (294). It can be seen from (294) thatformula (297) is applicable to all current experimentswith reactor ( MeV, km), accelerator( MeV, km), and atmospheric( MeV, km) (anti)neutrinos ifmj 1 eV and ; the last two conditions aredefinitely satisfied. A more general case with signifi-cant spreading of a neutrino packet is considered inAppendix E. This regime is of potential interest forexperiments with astrophysical neutrinos, but here welimit ourselves to the conditions of “terrestrial” exper-iments.

Differential forms in (307a) are defined inaccordance with (300), where functions should bereplaced by , and expression (307b) is written usingidentity

νπ= , Δ = − ,Δ

Δ = − , = + ,

2 2 22

4

2 1 1

ij ij i jij

ij i jij i j

EL m m mm

E E Ev v v

−= + = + ,π π

2

0

1 1Ierf( ) ' erf( ') erf( )z

zz dz z z z e

erf( )z

= 0af

αβ

αβ νν

= τ

, −× ,

π −

spins

23

( )

4(2 )

d s ddN d d d d

EdE

x y

y x

y x

P P

3

ν αβ νντ= Φ σ , − . ( )d d d d d E

V Vx y y x$

$ 6

3

L

ν 1E *310L &

ν 100E *310L &

ν 100E * . × 41 3 10L &

&710@F

,s ddPaf

af

ν

ν

∈ ν ν

ν ν

∈ ν

π −

=π − Ω

Φ σ× ≡ ,

23spins

23spins

spins

4(2 )

(2 ) 2

2

s d

s

S

d

D

d d dE

d dE d

d ddE V V

y xp

y x

$

$ 6

P P

P

P


where and are the spatial source and detectorvolumes, taken into account. Differential form isdefined in such a way that integral

(308)

is simply the f lux density of neutrinos, which are pro-

duced in S in reactions , in 39. Quantity is defined so that

(309)

is the differential transverse cross section of neutrinoscattering off detector as a whole. In the special(and practically important) case of neutrino scattering

in reaction with a sufficiently narrowmomentum distribution of target particles , differen-tial form becomes the elementary differentialtransverse cross section of this reaction multiplied bythe number of particles of type in the detector.

Let us finally turn to subintegral factor

(310)

where

(311)

(312)

Let us recall that function is the same as in thecase of the exact energy-momentum conservation atthe diagram vertices in Fig. 15. Therefore, sign-vari-able difference in (312) may be neglected in theneighborhood of the maximum of product

(i.e., at ), whichproduces the greatest contribution to the count rate. Inview of the properties of function , contributions ofthe order of or higher can also be neglected in(312). As was demonstrated, in this approxi-mation. At , , and , factor (310)matches the well-known quantum-mechanical

39In more correct terms, integral (308) is the number of neutrinosproduced in unit time within volume , which is centered atpoint , propagating within solid angle in direction

and crossing a unit area that is centered atpoint and perpendicular to .

V6 V$

νΦd

ν ν

∈ν

Φ =π − 23

spins 2(2 )s

SS

d d Ed dV dE

x xy x

P

+α→ ν's sI F $

dx∈x 6 νΩd

= − −( )l y x y x∈y $ l

νσd $

ν∈ ν

σ = spins

12

d

D

d dd dV E

yy $

$

P

$

−βν → 'da F

aνσd $

a

( )αβ ν

α β α β

,∗ ∗= ϕ − − Θ , 2

( )

expi j j i ij ij ij ijij

E L

V V V V S i

3

!

Θ = Θ + Θ ,ij i j

( )Θ = − +

+ +

− −

+ − −

22

02

2 2 3

1( ) ( )22

1 ( ) ) .2

(

jj j

j j

mr

r r

n n m n nD

n m n n 2

0n n

−0n n

ν νδ − δ + ( ) ( )s s d dp q p q ν≈ − ≈s dq q p

n

2( )jr2

Θ > 0j

= 1ijS Θ = 0ij = 0ij!

1 2020


Fig. 17. Simplified (symmetrized) layout of a nonsynchronized measurement. The time “windows” of neutrino emission andabsorption are highlighted. It is assumed that and .

x0

y0

–τ/2 0 τ/2

t' – τ/2 t' t' + τ/2

= +'t L t τt @

expression for the probability of neutrino f lavor tran-sitions:

(313)

It is thus natural to regard (310) as a generalizationof the quantum-mechanical result.

It will be demonstrated in Section 11.3 that .Therefore, all quantum-field corrections (with theapproximations used) reduce the value of (310) com-pared to its quantum-mechanical version (313). Inaddition to this quantitative difference, one shouldbear in mind that the probabilistic interpretation of

becomes technically impossible, since functions, , and involved in it (the last two functions are

contained in factors and phases ) depend on themomenta of external packets and the neutrinoenergy; notably, momenta with and are integration variables. The phase space ofprocesses (231) and the behavior of functions , ,and within these volumes do indeed differ for dif-ferent pairs of leptons . This is attributable bothto the reaction kinematics in and (in particular,the reaction thresholds) and to the uncertainties ofmomenta of lepton packets , , and , which maydiffer even by orders of magnitude. Thus, quantity

does not satisfy the expected “standard” relationslike Eq. (19), which are simply the corollary of the uni-tary property of mixing matrix in the quan-tum-mechanical theory of neutrino oscillations.

In real experiments, this scenario is complicatedfurther by the fact that the count rate of neutrinoevents (in the special case considered here, events withthe production of a lepton) is not determined byreactions of just a single type; instead, it is affected bya large number of processes of different types such as(in experiments with atmospheric and acceleratorneutrinos) decays of mesons ( , , , , etc.)and muons in the source (atmosphere or decay chan-nel of the accelerator) and various neutrino interac-tions in the detector ranging from (quasi) elastic todeep-inelastic ones. The phase space and functions

( )αβ ν α β α β∗ ∗, = ϕ .(QM)( ) expi j j i ij

ij

E L V V V V i3

< 1ijS

αβ3

D n m

ijS Θijû

pû

∈ ⊕ ⊕s d sI I Fû

νED n

m

α β, ( )6 $

σe μσ τσ

αβ3

α= iVV

β

μπ 2 μ2K μ3K 3eK

D

PHYSICS O

and for different combination of such subprocessesin and differ considerably. It helps comprehendthis better if one imagines functions , , and andall quantities depending on these functions as havingindices and attached to them.

Another technical and rather nontrivial obstruc-tion to interpreting formula (310) and analyzing exper-imental data stems from the dependence of factors ,which are called the decoherence factors below, oninstrumental parameters and . Formula (303)was derived with no assumptions regarding the syn-chronization of time intervals and . It isthen hardly surprising that the decoherence factorsmay be arbitrarily small in magnitude if these intervalsare not matched to allow for the obvious fact that thecharacteristic time of arrival of ultrarelativistic neutri-nos from to is approximately equal to mean dis-tance between and . The example of a nonsyn-chronized measurement is presented in Fig. 17, where

and is assumed (for simplicity, it isalso assumed that ). The QFT causality princi-ple and common sense suggest that factor shouldtend to zero under such conditions. The general for-mula for substantiates these expectations andallows one to determine the law of suppression of thecount rate. We will not dwell on this issue, since syn-chronized measurements are our main concern.

Prior to discussing the role of decoherence factors,we introduce another (final) simplifying modificationto the formula for the number of events. This simplifi-cation is again based on the assumption that the char-acteristic sizes of and (at the very least, theircharacteristic linear sizes along the neutrino beam) aresufficiently small compared to . Note that this con-dition is not satisfied for several short-baseline accel-erator and reactor experiments; the analysis for suchexperiments is much more complicated40. If the sim-plification is applicable, one may substitute by

40Naturally, this complication should not be regarded as a draw-back of our formalism, since quantum-mechanical formulasare simply inapplicable in this case.

n

6 $

D n m

α β

ijS

,01 2x ,

01 2y

,0 01 2( )x x ,0 0

1 2( )y y

6 $

L 6 $

= +'t L t τt @

τ = τs d

ijS

ijS

6 $

L

−y x



and differential forms and by their aver-aged (over and , respectively) values and

in (307b). The result is

(314)

Evidently, the domain of applicability of approxi-mation (314) is much narrower than that of (307). Thisis attributable to the additional constraints on distri-bution functions and on the absolute sizes and geom-etry of and , which were used implicitly in the tran-sition from (307) to (314). The following is obtained inthe “nonspreading” regime of a neutrino WP:

(315)

Let us discuss the function involved in (315). Func-tion governs the suppression of interference at dis-tances exceeding the coherence length in the regimewith a negligible contribution of the effect of spatialdispersion of a WP:

(316)

where

(317)

is the quantum-field generalization of the quantum-mechanical coherence length in (73b). This quantitymatches (73b) if one performs the substitu-tion. Dimensionless factor , which is new comparedto the quantum-mechanical approach, covers thefinite intervals of integration over the operating timesof the source and the detector and the probable lack ofsynchronization between the times of emission anddetection of neutrinos. Let us examine this factor,which will be called the decoherence function, inmore detail.

11. DECOHERENCE FUNCTIONThe decoherence factors and the method for their

measurement in neutrino experiments are discussed inthis section. Some complementary results are pre-sented in Appendix F. The above analysis of interac-tions in the source and the detector was rather generic.In particular, it was not assumed that the time intervalsof emission and detection of neutrinos are synchro-nized in some way. It is easy to demonstrate that fac-tors may be arbitrarily small if the effective times of

emission and detection of neutrinos and arenot synchronized (an exact condition will be formu-lated below). This important factor alone explains whythe “oscillation probability” is written in quotes. Evi-

L νΦd νσd $

6 $ νΦdνσd $

αβ ν αβ νν= τ Φ σ , . ( )ddN d d E L$3

af6 $

( )αβ ν α β α β∗ ∗, = ϕ − . 2( ) expi j j i ij ij ij

ij

E L V V V V S i3 !

ij!

ν

− π= = = coh

( ) 2 ,2

j iij

ij ij

L L LE L L

v v D D!

ν=π

osccoh

2ij

ijE L

LD

σ →p D

ijS

ijS0sX 0

dX


dently, quantities are not the probabilities of f la-vor transitions, since they do not satisfy unitarity rela-tions (19). The reason for this is that factors and depend on the momenta, masses, and dispersions ofexternal WPs; i.e., they depend (formally) on indices

and . The unitarity relations are satisfied approxi-mately only if these dependences are rather weak or ifthe decoherence factors themselves are insignificant( , ). This effect is a direct and rathernontrivial corollary of the quantum-field approach. Itwill be demonstrated below that diagonal functions are independent of neutrino masses and, conse-quently, of index . Universal function describes the suppression of the event number in thecase of nonsynchronized processes of emission anddetection of neutrinos and when the detector exposuretime is either much longer than the source operationtime or much shorter than the space-time “width” ofthe effective neutrino WP. Since these effects are notrelated to the interference of neutrino states, it seemsreasonable to redefine the oscillation probability ofneutrinos by isolating factor :

where

(318)

However, this redefinition is immaterial and a mat-ter of taste, since relations (19) are also not fulfilled forquantities in the general case.

In the next subsection, we assume that the times ofemission and detection of neutrinos are synchronizedand examine novel effects related to the finiteness ofintervals of neutrino emission and detection.

11.1. Synchronized MeasurementsNote that general suppression factor

in the integrand in (302) is minimized at .Therefore, time may be interpreted as the effec-tive (or the most probable) delay time between theproduction and absorption of virtual neutrinos and

. Since

this delay time is almost equal to . The count rate isthus expected to be maximized if the “on” periods ofthe source and the detector are shifted so that timewindows and overlap. The

αβ3

ijS ij!

α β

≈ δij ijS 1ij !!

iiS

i =0 iiS S

0S

αβ ν αβ ν, = , ,0( ) ( )E L S P E L3

( )αβ ν

α β α β

,∗ ∗= ϕ − . 2

0

( )1 expi j j i ij ij ij

ij

P E L

V V V V S iS

!

αβ ν,( )P E L

( ) ( ) − − + −

2 222

2 2exp 2 j ji j

i j

L T L Tv v

v v

D D

= ijT L v

ijL v

νi

ν j

ν

+ +≈ ≈ − ,

2 2

212 4

i j i jij

m m

E

v v

v

L

,0 01 2[ ]x x − , −0 0

1 2[ ]y L y L

1 2020


Fig. 18. Simplified (symmetrized) layout of a synchronized measurement. The time “windows” of neutrino emission and absorp-tion are highlighted.

x0

y0

–τs/2 0 τs/2

L – τd/2 L L + τd/2

dependence of the count rate on time intervals involved in function in (303) may be studiedfor such synchronized measurements. In order toreduce the number of independent variables in thedecoherence factors, we set (without any significantloss of generality) a certain “timing” symmetry:

where, naturally, . This timing diagram isshown schematically in Fig. 18.

Since is an even function, decoherence fac-tors (303) may be written as

(319)

where

In the majority of current neutrino experiments(astrophysical ones included), terms in the argu-ments of functions involved in (319) can beneglected. Indeed, using the current (model-depen-dent41) cosmological constraint [132] ,we obtain the following estimate:

On terrestrial scales, this value is exceptionally smallcompared to the duration of even the shortest neutrinopulses in accelerator experiments. In addition, termsfor sufficiently high neutrino energies remain negligibleeven at distances exceeding the size of the Galaxy. For

41The available estimates of the upper bound of , whichwere obtained by analyzing cosmological data within the

model and its extensions, span (roughly) theinterval from 0.07 to 0.30 eV.

−0 0'l lx y

Ierf( )…

τ τ τ τ= − , = , = − , = + ,0 0 0 01 2 1 22 2 2 2

s s d dx x y L y L

,τ > 0s d

Ierf( )z

( ) ( )[ ]

( )[ ]( )[ ]( )[ ]

−= τ + τ + −

τ+ τ + τ − +− τ − τ + −− τ − τ − + ,

2expIerf

4IerfIerf

Ierf

ijij s d ij ij

d

s d ij ij

s d ij ij

s d ij ij

S L i

L iL i

L i

@D @

D

D @

D @

D @

ν

+= + = .

2 2

22( ) i jij i j

m mr r

E

ijL

...Ierf[ ]

. 0 12 eViim &

ii m

νΛ + CDM m

−

ν

× .

222

4

1 GeV5 10 s

10 kmij

LLE

&

ijL

PHYSICS O

example, the value of for the (anti)neutrino signalfrom the SN1987A supernova explosion in the LargeMagellanic Cloud ( MeV, kpc) is nolarger than ~ s. This estimate should be com-pared to the typical duration of a neutrino burst in asupernova explosion, which is s. Let us bring for-ward one more argument as to why the termsshould be neglected. The formal sufficient conditionfor this is written as

This inequality should be fulfilled within our ideal-ized model if for no other reason than because timeintervals and cannot be set with a resolution bet-ter than the “on/off” times of the source and thedetector, which are neglected. Thus, with these reser-vations42, decoherence factors (319) are expressed interms of universal (independent of indices ) real-valued function of three dimensionless realvariables:

(320)

Let us examine the properties of function relevant to its experimental measurement.

11.2. Diagonal Decoherence FunctionThe mean event count rate of the detector,

, is proportional to function oftwo dimensionless variables, which is defined as

(321)

42Note that all these reservations bear no relation to the applica-bility domain of our formalism; they just outline the region ofapplicability of the simplified model used for qualitative analy-sis of the simplest corollaries of the theory. Models that aremore realistic simply require calculations that are more tedious.

ijL

ν 20E . 51 4L−× 42 10

≈10ijL

τ − τ .s d ijL@

τs τd

,i j, ,( ' )S t t b

( ) ( )[ ]

−

= τ , τ , ,

, , =

× + + − − + .

2

( )

( ' )2 '

Re Ierf ' Ierf '

ij s d ij

b

S S

eS t t bt

t t ib t t ib

D D @

, ,( ' )S t t b

αβ τddN τ , τ0( )s dS D D

( ) ( )[ ], = , ,

= + − − .

0( ') ( ' 0)1 Ierf ' Ierf '

2 '

S t t S t t

t t t tt



Fig. 19. Decoherence function (left) and its distribution density in plane (right). Dark and light regionsin the right panel correspond to smaller and larger values of , respectively.

10

10

8

8

6

6

4

4

2

20

t '

t

S(t, t ', 0) < 0.5

S(t, t ', 0) > 0.995

1.00

0.75

0.50

0.25

012.5

12.5

10.010.0

7.57.5

5.0 5.02.5 2.5

0 0

S(t,

t ', 0

)

tt '

, = , ,0( ') ( ' 0)S t t S t t ,( ')t t,0( ')S t t

It is easy to demonstrate that this function, whichcorresponds to noninterference (independent of theneutrino mass) decoherence factors , varies in theinterval from 0 to 1. Since

is a steadily increasing function of . It followsfrom (388a) (see Appendix G) that at

for any ; thus, at . It canbe seen from asymptotic formula (389) that

at and . This proves that

for all positive values of and .

Formulas (388) and (389) define the asymptoticbehavior of in different regimes. In particular,at small and ,

If is large but finite (i.e., ), we obtain

iiS

( ) ( )∂ ,= + − − >

∂, >

0( ')2 ' erf ' erf ' 0

( ' 0),

S t tt t t t t

tt t

,0( ')S t t t, →0( ') 0S t t

→ 0t >' 0t , >0( ') 0S t t , >' 0t t

, →0( ') 1S t t → ∞t < < ∞0 't

< , <00 ( ') 1,S t t

t 't

,0( ')S t tt 't

( ) ( ) ( ), ≈

π × − + + + + .

0

2 2 2 2 2 2

2( ')

1 1' ' '1 3 33 30

S t t

t t t t t t t

t − ' 1t t @

− − − + , ≈ − − ≈ π − +

2 2( ') ( ')

0 2 21( ') 1 1

4 ' ( ') ( ')

t t t te eS t tt t t t t


and in particular

if In the contrary limiting case, we have

Figure 19 presents the behavior of function and its distribution density in the plane. Thevalidity of the following important inequalities is easyto prove:

(322)

They help isolate the regions of weak and strongsuppression of the total number of events due to thedecoherence effects43. The right panel of Fig. 19,which presents the two-dimensional distribu-tion density (the darker the shade, the smaller thevalue of function ), illustrates these inequali-ties. The regions of transition between the abovemen-tioned asymptotic regimes are seen clearly.

It follows from Fig. 19 that mean event count rate of the detector decreases as the

ratio increases (note that this is not true for the totalnumber of events). The reason for this is clear: the

43For example, the suppression does not exceed ( ) if (i.e., ).

−, ≈ − π → ,2

0( ') 1 ( ) 1tS t t e t

' 1.t ! −' 1,t t @

− − − + , ≈ − − . π − +

2 2( ' ) ( ' )

0 2 21( ') 1 1

' 4 ( ' ) ( ' )

t t t tt e eS t tt t t t t t

!

,0( ')S t t,( ')t t

, < ≥+ δ , > δ δ > .

0

0

( ') ' for 'and ( ) erf( ) for 0

S t t t t t tS t t t t t

−410, > .0( ') 0 999S t t > + .' 2 3273t t τ > τ + .2 3273s d D

,0( ')S t t

,0( ')S t t

αβ τddN τ τ > 1d s

1 2020


Fig. 20. Function (solid curve) and its asymptotics at large and small values of (dashed curves) plotted in accordancewith (323).

1.0

0.8

0.6

0.4

0.2

0

S(t,

t, 0)

Small t asymptotics

Large t asymptotics

10–3 10–2 10–1 100 101 102 t

,0( )S t t t

number of detected neutrinos cannot exceed the num-ber of particles emitted by the source. The secondinequality in (322) has a less evident corollary:

is not suppressed at sufficiently large ratios44. Two dashed lines in the right panel of Fig. 19separate the regions in which ( ) and

( ). Thus, the condition nec-essary (although still not sufficient) to set isthat detector exposure time is either short comparedto or (at ) much longer than effective timescale determined in those regions ofthe phase space of reactions in and that producea significant contribution to the count rate. In theintermediate region, the decoherence factor inducesmoderate suppression of noninterference terms in theintegrand for the count rate.

The following is obtained at , which is a spe-cial case of particular interest for experiments withaccelerator neutrinos:

(323)

Figure 20 shows function . Asymptotics(323) at small and large values of are also shown forcomparison. It can be seen that these asymptotics arewell defined everywhere except a relatively narrow

44This condition is definitely satisfied in experiments with solar,atmospheric, and geophysical (anti) neutrinos, but may be vio-lated in accelerator experiments.

αβ τddN τ τs d

< .0 0 5S τ < τ2s d

> .0 0 995S τ > τ + 2s d D

, =0( ') 1S t tτd

τs τ ≈ τd s

ντ = 1 min( )D

6 $

='t t

−−, = −π

− + , π ≈ − . π

24

0

2 4

1( ) erf(2 )2

2 2 81 for 13 15

11 for 12

teS t t tt

t t t t

tt

!

@

,0( )S t tt

PHYSICS O

region of . It is also evident that approaches unity only at (in practice, at

).

11.3. Nondiagonal Decoherence Function

The behavior of function at is muchmore complex than that at . It is hard and point-less to perform a full analytical examination of thisfunction of three independent variables, since it isimpossible to avoid Monte Caro modeling in process-ing the data from a real experiment with its specific fea-tures. We have performed a thorough numerical analy-sis and analytical examination of the most importantspecial cases and found that at .Let us consider several typical examples clarifying thisresult.

Figures 21 and 22 present the profiles of function calculated for 18 values of parameter .

Numerical calculations were carried out with the useof formulas from Appendix G. It can be seen that thebehavior of as a function of and becomesmore and more involved as increases. For example,at , function oscillates rapidly nearzero even under relatively small variations of variables and . This results in strong suppression of nondiag-

onal (with ) contributions to

Figure 23 presents as a function of atfixed values of . At small , this dependence has aquasi-periodic nature, which is manifested against thebackground of rapid decay of with increasing

. Function depends strongly on the ratio ofparameters and . At very large values of ,

ceases to depend on and approaches very

. .0 5 0 8t& & ,0( )S t t1t @

100t *

, ,( ' )S t t b ≠ 0b= 0b

, , < ,0( ' ) ( ')S t t b S t t ≠ 0b

, ,( ' )S t t b b

, ,( ' )S t t b t 'tb

> −3 4b , ,( ' )S t t b

t 't≠i j αβ ν,( ).E L3

, ,( )S t t b bt t

, ,( )S t t bb , ,( ' )S t t b

t 't = 't t, ,( ' )S t t b t



Fig. 21. Profiles of decoherence function calculated for nine different values of parameter varying from 0.1 to 0.9.

1.00

0.75

0.50

0.25

00

01 12

2

3

3

tt '

S(t, t ', 0.7)

1.00

0.75

0.50

0.25

00

01 12

2

3

3

tt '

S(t, t ', 0.8)

1.00

0.75

0.50

0.25

00

01 12

2

3

3

tt '

S(t, t ', 0.9)

1.00

0.75

0.50

0.25

00

01 12

2

3

3

tt '

S(t, t ', 0.4)

1.00

0.75

0.50

0.25

00

01 12

2

3

3

tt '

S(t, t ', 0.5)

1.00

0.75

0.50

0.25

00

01 12

2

3

3

tt '

S(t, t ', 0.6)

1.00

0.75

0.50

0.25

00

01 12

2

3

3

tt '

S(t, t ', 0.1)

1.00

0.75

0.50

0.25

00

01 12

2

3

3

tt '

S(t, t ', 0.2)

1.00

0.75

0.50

0.25

00

01 12

2

3

3

tt '

S(t, t ', 0.3)

, ,( ' )S t t b b

slowly its asymptotic form . Probability (310)in this asymptotic regime takes the following formknown from literature (see, e.g., [45, 46, 66] and refer-ences therein):

(324)

the only difference being that functions , , and involved in , , and depend on the momentaof external packets and the neutrino energy. Thisdependence alters qualitatively the behavior of oscilla-tion suppression factors if at least some of the external

− 2exp( )b

( )αβ ν α β α β

∗∗, =

× ϕ − − − Θ ,

2 2

( )

exp

i j j iij

ij ij ij ij

E L V V V V

i

3

! @

D n m

ij! ij@ Θij


WPs have relativistic mean momenta. Since

Let us clarify the physicalmeaning of the other suppression factors.

Factor in (324) suppresses the interfer-ence of contributions with at distances exceedingthe coherence length

at which neutrino WPs and are already sepa-rated sufficiently in space (due to the differencebetween their phase velocities) and cease to interfere.

Θ ≥ 0,ij

αβ ν αβ ν, ≤ ,(QM)( ) ( ).E L E L3 3

( )− 2exp ij!

≠i j

= , Δ = −Δ

coh 1 ( )ij ij ij j iij

L L v v v

v

@

D

∗ψi∗ψ j

1 2020


Fig. 22. Profiles of decoherence function calculated for nine different values of parameter varying from 1.0 to 9.0.

0.10

0.05

0

–0.05

–0.100

01 12

2

3

3

tt '

S(t, t ', 7.0)

0.2

0.1

0

–0.10

01 12

2

3

3

tt '

S(t, t ', 4.0)

0.5

0.3

0.1

–0.10

01 12

2

3

3

tt '

S(t, t ', 1.0)

0.5

0.3

0.1

–0.10

01 12

2

3

3

tt '

S(t, t ', 2.0)

0.5

0.3

0.1

–0.10

01 12

2

3

3

tt '

S(t, t ', 3.0)

0.2

0.1

0

–0.10

01 12

2

3

3

tt '

S(t, t ', 5.0)

0.2

0.1

0

–0.10

01 12

2

3

3

tt '

S(t, t ', 6.0)

0.10

0.05

0

–0.05

–0.100

01 12

2

3

3

tt '

S(t, t ', 8.0)

0.10

0.05

0

–0.05

–0.100

01 12

2

3

3

tt '

S(t, t ', 9.0)

, ,( ' )S t t b b

Naturally, in the plane-wave limit. Factor

suppresses interference contributions inthe contrary case, when external WPs in or (or in

and ) are strongly delocalized (or, in the plane-wave limit, smeared over the entire space). The totalsize of the neutrino production and absorption regionsin and is of the order of . Interference contri-butions vanish if this size is large compared to theinterference length

In other words, the quantum-field theory predictsthat neutrino oscillations should vanish in the plane-

→ ∞cohijL

( )− 2exp ij@

6 $

6 $

6 $ 1 D

= = .Δ π

int 214

ijij

ij

LL

E n

PHYSICS O

wave limit. The probability f lavor transitions isthen independent of and becomes equal to

Thus, nontrivial ( -dependent) interference ofamplitudes with intermediate neutrinos of differentmasses is possible only at It follows from thedetailed analysis of processes , , and

(see Appendix А) that function is nonzero ifat least two (in- or out-) WPs with interact atboth vertices of the diagram in Fig. 15 characterizingprocess (231). The same conditions necessarily implythat nondiagonal contributions vanish at sufficientlylarge distances between and ( ).

ofL

α β . 2 2i i

i

V V

L

≠ 0.D

→1 2 →1 3→2 2 D

û σ ≠ 0û

6 $cohmax ijL L@



Fig. 23. Dependences of function on at various fixed values of (see legend). The asymptotics of function at is shown by the dashed curve for comparison.

100

10–1

10–2

10–3

10–4

0 10 20 30 40 50

1

2

3

4

5

6

7

8

9exp(–b2)

b

S(t,

t, b)

1: t = 0.052: t = 0.103: t = 0.25

4: t = 0.505: t = 0.756: t = 1.00

7: t = 3.008: t = 5.009: t = 10.0

, ,( )S t t b b t , ,( )S t t b→ ∞t

Thus, the domain of applicability of the standardquantum-mechanical formula for the probability ofneutrino f lavor transitions (adopted by default in alldata processing programs of current oscillation exper-iments) is constrained by rather stringent conditions:

(325)

Here, angle brackets denote the fact that the valuesof functions , , and are defined by the regionsof the phase space of process (231) yielding the pri-mary contribution to the measured number of neu-trino events.

11.4. Additional Methodological RemarksIt follows from the above analysis that the rather

strong dependence of factor on its arguments at provides an opportunity to estimate function

(or, more precisely, its mean values within the above-mentioned regions of the phase space) experimentallyby measuring the count rate as a function of variables

and (at fixed distance ) and comparing the dataobtained with the results of Monte Carlo modeling. Itis evident that the optimum strategy of such an exper-iment should be the subject of a separate study and tai-

ν

π π, , Θ .

2 22 1 1 1

2 ijij ij

LE L L

! ! !D n

D

D n m

,0( ')S t t't t& D

τd τs L


lored to each particular experiment. We consider hereonly the most general technique without going deepinto the features of experiments.

The following ratio is a convenient measured quan-tity for experiments with accelerator neutrino beamsemitted in time window and detected in time :

Here, is the number of events detectedat time within time window , and is the timewindow expanded to the maximum45. Evidently,

grows linearly with as it increases from zero(at ) almost to unity (at ). The mostintriguing region is the narrow interval near ,where linear growth ends with the plateau at

. This transition is smooth rather than step-

45This should be a compromise value in the sense that is needs toexceed considerably to ensure reliable signal detection butneeds not be too large so as to prevent the signal from being lostin the background.

τs τd

( )( )

∈ ττ = .

∈ τ

0

0 max( )i d

id

i di

N yW

N y

( )∈ τ0i dN y

0iy τd τmax

d

τs

τ( )dW τd

τ τd s& τ ≈ τd s

τ = τd s

≈ 1WΔτ ≥ τd d

1 2020


wise, and the measurement of derivative in this region should allow one to measure , since

Such a measurement is very important, since func-tion is included in the probability of f lavor transi-tions (see below) and needs to be known to extract theparameters of neutrino oscillations (i.e., differencesbetween the squares of masses and mixing angles)accurately. Since function depends on both theneutrino energy and the external momenta, quantity

should be measured at least within sev-eral experimentally accessible kinematic intervals.

We should recall that conditions (325) and for-mula (324) were obtained with the use of a number ofsimplifying assumptions and approximations. Theirvalidity in real-life experiments is by no means alwaysapparent. It follows from the above analysis (supple-mented by the results detailed in Appendix F) thatconditions (325) should be used in combination withdata on the periods of operation of the source and thedetector, their size and geometry, the explicit form ofdistribution functions of in- particles in the source andthe detector, and other technical features in the pro-cessing and interpretation of measurements performedin real neutrino experiments.

12. DISCUSSION12.1. How a Virtual Neutrino Becomes Real

A neutrino is characterized as a virtual particle inthe macroscopic diagram corresponding to the pro-duction of charged leptons and in the source andthe detector. On the other hand, it is expected intui-tively that any virtual particle traveling over a suffi-ciently large distance should be “similar” to a real par-ticle with a 4-momentum on the mass shell. For exam-ple, two approaches to the study of light emitted by alamp and reaching sensitive cones in our eye are pos-sible. In the first approach, the propagation of a realphoton with a 4-momentum on the mass shell is con-sidered. In the second approach, both the emission oflight and its detection by the eye are characterizedwithin the QFT formalism by a single macroscopicdiagram with two vertices: the source (lamp) and thedetector (eye). Naturally, these two descriptions donot contradict each other.

A virtual particle46 with mass is described in theconfiguration space by the causal Green’s function[133]

46For simplicity, we limit ourselves to scalar particles.

τ τ( )d ddW dD

τ =τ

τ = τ .τ τ( ) 1 erf(2 )

2d s

ds

d s

dWd

D

D

D

τ τ( )d ddW d

+α

−β

m

−λ += ,−λ +π

12

( )( )4

c K m iimD xi

e

e

PHYSICS O

where and at . is the Hankel function (Bessel function of the

third kind). Function decreases at spatial infin-

ity (in the causally unconnected region) as and at temporal infinity (in the causally connectedregion) as . is concentrated near light cone

and has singularities there.The diagrammatic approach with the source and

the detector characterized in the coordinate space bysome functions and , respec-tively, yields amplitude

where , are points at which and are maximized.

Thus, since differs significantly from zeroonly in the vicinity of the light cone and functions

and are concentrated nearpoints , , amplitude at sufficientlylarge47 corresponds rather accurately to a WPcharacterizing a particle with mass moving alongclassical trajectory from point topoint with an exponentially small deviation from it.Velocity , where ,and the modulus of momentum is the effectivemomentum at which integral (corresponding to the amplitude in the momentumrepresentation) is saturated.

Therefore, a virtual particle becomes almost real atmacroscopic distances, thus resolving the apparentcontradiction.

12.2. Neutrino Wave Function and Process Amplitude

Amplitude (282) allows for a natural interpre-tation in terms of the wave function of a virtual neu-trino (see Section 8.4). The wave function of state ,which was produced at point in the source, at arbi-trary point is a WP in the SRGP approximation:

, where and is given by (286). The momentum dispersion

of neutrino produced in the source is defined by the4-momenta and momenta dispersions of all in- andout-packets in the source (see in (293)). The wavefunction of state interacting at point in the

47Compared to the spatial dispersions of functions .

λ = = −2 2 20x x x −λ + = λi ie λ > 0

ν( )K x( )cD x

− / − |λ|λ 3 4 me

− /λ 3 4 ( )cD xλ = 0

ψ −( )s sx x ψ −( )d dx x

− ∝ ψ −

× ψ − − ,( ) ( )

( ) ( )

d s s s

cd d

x x dxdy x x

y x D y x

A

sx dx ψ −( )s sx xψ −( )d dx x

( )cD x

ψ −( )s sx x ψ −( )d dx xsx dx −( )d sx xA

−d sx x

,ψ ( )s d x

m= + −( )s st tx x v sx

dx= p Ev n = − −( ) ( )d s d sn x x x x

p−∝ ( ) ( ) ikxp dk x eA A

αβA

ν j

sXx

ψ , − −*( )s sx Xp x X −Ω ,ψ , = ( )( ) j T Lx epΩ ,( )j T L

ν j

σsj

ν j dX



detector is also a WP with momentumdispersion defined by the 4-momenta and disper-sions of all in- and out-packets in the detector (293).The projection of state in the source ontostate in the detector defines amplitude

12.3. Ensemble Averaging and Role of Relativistic Covariance

In accordance with intuitive expectations, theinterference of diagrams corresponding to externalstates with well-defined momenta is suppressed.Intermediate neutrinos with different masses corre-spond in this case to external states with differentmomenta. In the limit of exactly determinedmomenta, such states are orthogonal, and, in accor-dance with the QFT rules, diagrams with differentexternal states need to be summed at the level ofsquares of moduli of matrix elements. This eliminatestheir interference. The calculation with WPs corre-sponding to external states reproduces this rule. Toachieve that, one should always sum amplitudes.Interference contributions in the square of the ampli-tude modulus vanish automatically in the plane-wavelimit.

It is possible to determine the probability of neutrinooscillations in the QFT formalism correctly by factorizingexpected number of events (with the production of

charged leptons and in the source and the detector)into three factors .The calculation of the expected number of events is, inturn, possible by the procedure of averaging over theensemble of initial particles and integrating in coordi-nates and momenta of final particles. The Lorentzinvariance of requires covariance of WPs corre-sponding to external particles. By construction, thisrequirement is fulfilled in our formalism. In addition,the WP covariance is of fundamental importance inthe determination of impact 4-vectors. Nonzeroimpact 4-vectors of scattering WPs suppress the inter-action probability. It would be very hard, if not impos-sible, to determine the correct form of suppressionassociated with nonzero impact 4-vectors in a model

with a noncovariant WP , since the object is transformed in a very complex way under

Lorentz transformations.The Lorentz invariance of the neutrino oscillation

probability follows directly from the covariance of theformalism. It is instructive in this context to discuss anapparent paradox related to the interference of neu-trino states. Let there be two neutrino states withmasses and . Let us perform aboost to the rest frame of the heavy neutrino. Themassless neutrino moves (with the speed of light) in

ψ , −( )dX xpσdj

, ;σj s sjXp, ;σj d djXp

αβ ∝ , ;σ , ;σ .j d dj j s sjX Xp pA

im

αβN+α

−β

ν ν αβ ν νΦ , × , ; × σ( ) ( ) ( )E L E L E3 D

αβN

− − / σ∝2 2( ) 4 pe k p

σp

= 0im = > 0jm m


this frame away from the massive neutrino resttowards the target. Is it possible for these two massstates to be in a coherent superposition and induceneutrino oscillations? The massless neutrino reachesthe detector well in advance of the heavy one and, itshould seem, may cease to overlap with it, thus dampingoscillations. The solution to this paradox is simple:under Lorentz boosts, the WP form is transformed insuch a way that two WPs overlapping in a certain refer-ence frame remain overlapping in any other. Thus, oscil-lations do not vanish in any reference frame, althoughtheir “pattern” may change dramatically. The formalismcovariance ensures the reconstructibility of this patternafter the transition from one frame to another.

An important detail should be pointed out: sincethe ultrarelativistic approximation was introduced incalculations, the Lorentz invariance of the resultingformulas is, strictly speaking, an “approximate” one.This is still a fine approximation, since the velocity ofeven a moderately energetic (anti) neutrino is close tothe speed of light:

However, the approximate Lorentz invariance,which holds under Lorentz boosts with gamma factors

, is certainly violated in the formaltransition to the rest frame of a neutrino48. At the sametime, the formalism itself does not require the ultrarel-ativistic approximation and may be applied to any pro-cess with an arbitrary (even the most exotic) configu-ration of momenta of external in- and out-packets49.The ultrarelativistic approximation (together withother simplifying assumptions) was used just to sim-plify calculations and for convenience of comparisonof the obtained results with other approaches andexperimental data.

12.4. Probability of Flavor Transitions

Function is not the probability ofneutrino oscillations in the quantum-mechanicalsense. It depends on the neutrino energy, the source–detector distance, effective dispersion of a neutrinoWP, and the widths of time intervals of operation ofthe source ( ) and the detector ( ). In the generalcase considered here, the neutrino emission anddetection times are not necessarily synchronized.

12.4.1. Synchronized and nonsynchronized mea-surements. In the case of nonsynchronized production

48This is why the “proofs” of Lorentz invariance of the standardformula for the oscillation probability, which one may find inliterature, are erroneous.

49We have prepared some “work pieces” for such potentiallyinteresting problems (and with educational purposes); see Sec-tions 8.2.2 and Appendix D.

at

− − ≈ . × . .

2 214 1 MeV

1 0 5 100 1 eV

mEp

p

v

νΓ max( )jE m!

αβ ν, ;( )E L3 D

D

τs τd

1 2020


and detection processes, is, in accor-dance with intuitive expectations, suppressed expo-nentially. This fact alone explains why one should notexpect unitarity conditions (19) to be satisfied. In thecase of synchronized measurements, formula (315) (or(318)) predicts a number of intriguing phenomenastemming from the hierarchy of time scales , , and

.

Stationary source. The case of a stationary source,which was discussed in Section 3.8, is obtained in the

limit (or the approximation). It differssignificantly from the quantum-mechanical approachin that functions and depend on the event kine-

matics and, consequently, on the 4-momenta of and (i.e., on indices , ). Therefore, the unitarityconditions for can be satisfied only if

the dependence of functions and on their vari-ables and parameters (i.e., on indices , ) is weak.With the exception of these important factors, thequantum-field result is similar to the quantum-mechanical one (see (71)). In particular, the vanishingof interference terms at distances exceeding the coher-ence length is predicted. The exponential loss ofcoherence is compensated in part by the longitudinalneutrino WP spreading. The spreading effect is alsomanifested in the power suppression of interferenceterms.

Function gives rise to the exponential and dis-tance-independent suppression of interference termsin . The lower the neutrino energy-momentum dispersion , the stronger the suppres-sion. Thus, the probability of f lavor transitionsbecomes distance-independent in the plane-wavelimit, and neutrino oscillations vanish. Theformal reason for oscillation suppression is the above-mentioned rule of the squared amplitudes summationcorresponding to diagrams with orthogonal externalstates. The quantum-mechanical experiment with twoslits and an electron, which passes through them andreaches a screen inspected by the experimenter, pro-vides a fine physical illustration of this effect. If theexperimenter is able to determine the slit throughwhich the electron passed, the interference pattern onthe screen vanishes. The case is thus equiva-lent to identifying unambiguously the mass of an inter-mediate neutrino; the interference of diagrams withthe exchange of and ( ) is then destroyed.

Function illustrates why charged leptons do notoscillate. The answer to this question was first pro-vided in [134]; in the present study, we reproduced itby using a different technique.

αβ ν, ;( )E L3 D

τs τd

ντ = 1 minD

τ → ∞s τ τs d@

D2ij@

+α

−β α β

αβ ν, ;( )E L3 D

D2ij@

α β

2ij@

αβ ν, ;( )E L3 D

D

→ 0,D

→ 0D

νi ν j ≠i jm m2ij@

PHYSICS O

The loss of neutrino coherence depends on oron relative dispersion . The available esti-mates of its value for, e.g., reactor antineutrinos varywithin several orders of magnitude. Thus, assuming

cm, which is equal to the size of the uraniumnucleus, one obtains an upper bound: MeV.Atomic or interatomic distances provide the followingestimate: cm and eV.The broadening of lines at a nonzero ambient tem-perature results in cm and eV. Thedecoherence in neutrino oscillations may manifestitself in future experiments with a large number ofoscillation cycles (e.g., the JUNO reactor experiment[135]). Figure 24 shows the survival probability fordistance L = 52 km (the planned experimental base-line) as a function of the antineutrino energy. Calcula-tions were performed in the standard (plane-wave)model of neutrino oscillations and in the model with aneutrino WP for two values of parameter ,which are not exactly realistic, but are also notexcluded by the current experimental constraints. Itcan be seen that the decoherence effect may play a sig-nificant part in the determination of the neutrino masshierarchy at sufficiently large values of . Note thatthe region of allowed values, identified in [135] corresponds to the case when the neu-trino WP effects for distances L 52 km are negligibleand the formula for becomes numericallyequivalent to the formula for the flavor transition proba-bility in the plane-wave approximation.

Nonstationary source. If condition , whichis typical for certain accelerator experiments, is ful-filled, it becomes possible to measure dispersion bycomparing the count rate to the one obtained from theformula for . A new important factorshould also be mentioned here. If and are finite,

the suppression of interference terms (by factor ) issomewhat weaker than that for a stationary source.The functional dependence of the corresponding sup-pression factor, which is independent of distance ,but depends on operating time intervals and ofthe source and the detector, was examined in detail inSection 11. Let us discuss the issue of why the suppression at finite and is below the asymptotic

limit, where depends on the neu-trino energy dispersion, on a qualitative level. Thiseffect may be interpreted by appealing to a quantum-mechanical analogy: the measurement of the energy-momentum at finite times is equivalent to introducingan additional uncertainty, which makes the neutrinostates more coherent.

The proposed theory of f lavor transitions conclu-sively resolves all the paradoxes of naïve plane-wave

D

νσ =rel ED

−σ ∼

1210x

∼ 10D

− −σ −

8 7(10 10 )x −∼

3 2(10 10 )D

−σ ∼

410x .∼ 0 1D

νe

νσ = σrel p

σrel

σrel− −σ16 1

rel10 10 ,! !

&

αβ ν, ;( )E L3 D

τ ≤ τs d

D

αβ ν, ;( )E L3 D

τs τd− 2

ije @

Lτs τd

, ,( )S t t bτs τd

− 2exp( )b = ijb @



Fig. 24. survival probability (at L = 52 km) as a function of the antineutrino energy in the plane-wave model and the modelwith a neutrino WP for two values of .

L = 52 km

1.0

0.8

0.6

0.4

0.2

02 3 4 5 6 7 8 9 10

Surv

ival

pro

babi

lity

Eν, MeV

Without decoherenceσrel = 0.05σrel = 0.2

νeσrel

approximation the quantum-mechanical theory.In addition, it predicts a number of novel effects,which can be studied in experiments with accelerator,atmospheric, and reactor neutrinos and neutrinosemitted by artificial radioactive sources. The results ofthe first study of this kind were presented in [88]. Thepotential influence of decoherence effects on theaccuracy of determination of mixing angle in accel-erator experiments was examined in [91]. However,the possible loss of coherence is not associated herewith the WP nature; instead, it is related to the poten-tial effects of interaction with the environment.

12.5. Wave Packet Observability

Although the model with a WP is undeniably effi-cient in producing a theoretical description of neu-trino oscillations, a reader familiar with classical stud-ies (see, e.g., [98, 117]) on scattering theory can ask alegitimate question: “It is indeed demonstrated intextbooks on scattering theory that a consistent theo-retical description of scattering processes cannot beconstructed without a WP. However, no informationon the WP properties is left in the cross sectionobtained by averaging the interaction cross sectionover an ensemble of scattering particles. At the sametime, event number Nαβ obtained after averaging overa particle ensemble in the theory of neutrino oscilla-tions with a WP depends explicitly on the dispersions

of

θ23


of all WPs via functions , , etc. Is there a contra-diction?” Let us try to answer this interesting question.Let us retrace the key steps of Taylor’s calculations[98] using our notation. WP scattering amplitude

can be presented as (see Section 6)

where is thestate norm, and is the matrix element in the plane-wave approximation. The Taylor’s calculation [98](reproduced later by Peskin and Schroeder [117])assumes that the dependence of matrix element

of the process on momenta isweaker than the momentum dependence of a WP

. The following factorization is possible in thisapproximation:

D n

= , − ,1f f i ix xp pA S

−

,− −

φ ,= π

φ ,× , ,

π

∏ ( )

43

( )

3

( )(2 ) 2

*( )( )

(2 ) 2

i i

i

f f

f

iq x xi i i

i f

ik x xf f f

i f

d ei d xE

d e

E

q

k

q q p

k k pq k

A1

= , , , ,f f f f i i i ix x x xp p p p1

,( )i fi q k ,( )i fq k

φ ,( )i iq p

,

ψ , − ψ , −

× , ,

∏

4 ( ) *( )

( )

i i f fi f

i f

i d x x x x xp p

p p

A1

1 2020


and macroscopically averaged is indeed inde-pendent of the WP type [98].

However, the Taylor’s approximation is not alwaysvalid. Let us list several counterexamples.

(1) The dependence of on momenta may be strong (e.g., hadronic resonances).

(2) The possibility of measurement of the phase ofa matrix element in the linear approximation to theplane-wave cross section is proposed in electron vor-tex beams (a variation of a WP) [136, 137]. The intro-duction of quadratic corrections to the Taylor’sapproximation also allows one to measure the phase ofa matrix element [122]. Such measurements are notpossible in the plane-wave approximation.

(3) The cross sections of certain processes (e.g.,) calculated in the plane-wave approxi-

mation contradict experimental data [118]. As wasnoted in Section 6, the measured cross section turnedout to be ~30% lower than the calculated one in theregion of low photon energies. This is attributable tothe fact that impact parameters up to 5 cm producea significant contribution to this cross section,whereas the transverse size of colliding beams was nolarger than cm. If impact parameters are lim-ited to , the observed number of photonsdecreases. The theory with a finite size of collidingbeams taken into account was developed in [120].

(4) If the matrix element is a sum of singular matrixelements, e.g.,

it is not possible to “defactorize” the WP form at a sin-gle point in the transfer of 4-momentum if masses

are different. Matrix elements of this type corre-spond to the interference of diagrams with virtual neu-trinos in the quantum-field theory of neutrino oscilla-tions (or, more correctly, f lavor transitions). Thus, theanalysis of neutrino oscillations in the quantum fieldtheory with a WP is indeed an important exceptionthat is not covered by the Taylor’s approximation.

13. CONCLUSIONSLet us summarize the results.It has already been noted multiple times (starting

from the 1970s) in the literature that the plane-wavequantum-mechanical theory of neutrino oscillationsis incomplete and self-contradictory. We have dis-cussed critically the hypotheses and approximationsunderlying this theory and tried to outline the domainof their applicability using one of the simplest exten-sions of the plane-wave theory based on the model ofa neutrino WP. The general properties of a WP werestudied within the noncovariant quantum-mechanicalformalism. Several well-known features of a WP such

2A

,( )i fi p p

,( )i fq k

+ − + −→ γe e e e

ρ

−

310aρ ≤ a

α β∗, ,

−∼ 2 21' '( )i f k k d s

k k

i V Vq m

q k

qkm

PHYSICS O

as the quasi-classical nature its trajectory and thegeneral law of spreading in the configuration spacewere reproduced. One novel result is the proof that theWP spreading leads to inverse-square law forthe probability of finding a state at distance from thesource in a sufficiently long observation time. The the-ory of neutrino oscillations was constructed within themodel of a noncovariant Gaussian WP. A correct der-ivation of the formula for the probabilities of f lavortransitions via macroscopic averaging of the squaredmodulus of the transition amplitude was proposed. Ageneral formula for the probabilities depending onoscillation length , coherence length , disper-

sion length , and effective spatial size of a neu-trino WP (see the definitions in (73)) was obtained asa result. This formula predicts the following:

(1) Loss of coherence for a pair of states and

at distances . The probability of oscillations atvery large (astronomical) distances is then reduced tononcoherent sum that is inde-

pendent of energy and distance50.(2) Noncoherent production (or detection) of neu-

trinos, which is governed by factor (see (78)). This

factor explains why charged leptons do not oscillate.(3) Corrections to the oscillation phase associated

with the neutrino WP dispersion effect.(4) Partial compensation of the loss of coherence

for a pair of states and due to packet spreading.(5) Additional suppression of interference terms by

factors of the form (see (79)).A covariant theory of relativistic wave packets,

which are constructed as linear superpositions of sin-gle-particle Fock states with definite momenta andevolve into these states in the plane-wave limit, wasproposed. The properties of such packets were studied.It was demonstrated, e.g., that mean 4-momentum

of a relativistic packet and its effective volume are exact integrals of motion. At the same time,

, where is the effective mass of a WP,which always exceeds the mass of quanta of the corre-sponding free field. A relativistic packet moves, on theaverage, along a classical trajectory with velocity

(where and are its most probable3-momentum and energy, respectively), and quantumfluctuations about this trajectory decay at distances

50This result is also reproduced in the quantum-field theory andis the real reason (as opposed to the meaningless “averagingover sources” mentioned in many papers) why astrophysicalneutrinos do not oscillate, with an important reservation thatthe interactions of neutrinos with matter in the astrophysicalsource are neglected.

of

∼

21 xx

oscijL coh

ijLdijL σx

νi ν j

@cohijL L

αβ α β= 2 2k kk

P V V

( ) ( ) − = −Δ σ @2 2exp exp 8 2ij kj pm p

νi ν j

1 kjq

P V

=2 2P m m

= Ep pv p p Ep



that are large compared to the effective size of thepacket. The suppression of f luctuations transverse tovector is weaker than that of longitudinal f luctua-tions, and ultrarelativistic packets may have onlytransverse f luctuations (i.e., packets move within clas-sical light cones). Multipacket states characterizingsystems of free identical bosons or fermions werestudied. It was demonstrated that the effects of Bose–Einstein attraction and Fermi repulsion are significantonly at distances comparable to the effective size ofpackets and become negligible at larger distances.

A relativistic Gaussian packet characterized by asingle phenomenological parameter , which definesthe scale of quantum 4-momentum fluctuations andthe effective packet size, was examined thoroughly asthe simplest working model of a “memoryless” relativ-istic wave packet satisfying all the requirements of theformalism. The regime in which the RGP spreading isnegligible (SRGP model) was studied in detail. Thisregime provides an opportunity to use an RGP in the

-matrix QFT formalism to characterize asymptoti-cally free states of stable and unstable (but relativelylong-lived) particles. It was demonstrated that allunstable long-lived particles and atomic nucleiregarded as important decay sources of neutrino andantineutrino beams have fairly wide regions of allowedvalues of satisfying all the constraints of the SRGPmodel under which a wave packet does not spreadwithin the particle lifetime.

The developed theory was used to describe thescattering of relativistic WPs. The amplitude of anarbitrary process of the form in the

-matrix QFT formalism with WPs was calculated.The general formula for the rate of this process wasobtained (see (180)):

where is the standard plane-wave cross section,and is the luminosity of scattering oftwo WPs as defined in the scattering theory and accel-

erator physics. Factor introduces geomet-ric suppression of the interaction probability of twoWPs scattered with a nonzero impact parameter

, where is the impact vector and isa unit vector codirectional with relative velocity .

is the effective spatial dispersion ofthe region of overlap of two Gaussian WPs with spatialdispersions and and impact parameter .This result has been obtained for the first time in [120]with the use of the Wigner function. In the presentstudy, it was reproduced within the covariant WP for-malism. Macroscopic averaging of the probability ofinteraction of colliding packets over the impactparameter yields standard formula , whereluminosity is defined by the spatial dispersions of

pv

n

σ

S

σ

+ →a b XS

,− × / σ= σ × = σ2 2( ) 2( ) (0) ,x abdN d L d L e b nb n

σd

,= πσ2(0) 1 2 x abL

,− × / σ2 2( ) 2 x abe b n

×b n b = ab abn v v

abv

, , ,σ = σ + σ2 2 2x ab x a x b

,σ2x a ,σ

2x b ×b n

= σdN d LL


WPs and beams of particles and . The Lorentzinvariance of impact parameter was proven.

The technique of calculation of macroscopic Feyn-man diagrams with the exchange of massive neutrinoswas developed. It is based on the covariant theory ofwave packets, where the states of free fields (externallegs of macrodiagrams) are constructed as linearsuperpositions of single-particle Fock states that passinto these states in the plane-wave limit. The overlapintegrals of packets, which characterize the spatiotem-poral overlap of in- and out-packets in the source anddetector, were studied in detail. These integralsaccount for the key differences between our approachand the standard QFT formalism, which utilizes stateswith definite momenta smeared over the entire space-time. It was demonstrated that the overlap integrals aresmall in absolute value if the classical world tubes ofpackets are located far away from impact points (4-vectors and , which define the position of effec-tive regions of interaction of packets near the macrodi-agram vertices). The conditions under which in- andout-packets may be considered asymptotically freewere formulated.

The amplitude of process (231) of the general formwith the production of two oppositely charged leptons

and at macroscopically separated diagram verti-ces with an arbitrary number of other particles(all external legs are wave packets) was calculated. Itwas demonstrated that diagram blocks characterizingthe interactions of hadrons may be presented as aproduct of two hadronic currents that are associatedwith the source and detector vertices and depend onlyon the corresponding sets of variables (momenta,spins, and coordinates of external packets). As a result,the amplitude was presented as a sum of products ofmatrix elements of the subprocesses of production andabsorption of a real massless neutrino and factor

, which may be interpreted as a spherical wave ofa neutrino with mass with amplitude having theform of a relativistic Gaussian packet with its “disper-sion” being a relativistically invariant function of theeffective neutrino energy as well as momenta, masses,and momenta dispersions of all external packets. Theamplitude includes geometric suppression factors,which are associated with the incomplete overlap ofexternal packets at the diagram vertices, and nonsin-gular factors, which ensure approximate energy-momentum conservation at these vertices, as commonfactors.

Since process (231) is, in the general case, leptonnumber violating, experimental studies of this processare an important source of information on the mixingparameters and the differences between squared neu-trino masses. A Lorentz-invariant formula for theprobability of process (231) was derived and subjectedto statistical averaging, which led to an experimentallymeasurable quantity, the differential number of events

a b×b n

sX dX

α β

∗ψ j L

jm ∗ψ j

1 2020


in the detector (dNαβ). The terms “source” and“detector” are used here to denote mathematicalmodels characterizing the conditions of an actualexperiment. With a number of simplifying assump-tions introduced, the formula for can be pre-sented as a multidimensional integral of the product ofdifferential forms and , which characterizethe f lux (energy spectrum) of massless neutrinos andthe differential cross section of their interaction withthe detector, and factor , which is the generaliza-tion of the standard quantum-mechanical expressioncharacterizing f lavor transitions . It wasdemonstrated that quantum-field function didnot have probability features, since it includes, inaddition to the standard oscillation factor, decoher-ence factors that depend on neutrino energy andmomenta of external packets (variables of integra-tion over the phase space of process (231)) and onmasses and parameters (uncertainties ofmomenta of external packets, including the packets ofleptons ).

The value of general decoherence factor , whichsuppresses the mean event count rate in synchronizedmeasurements, is defined within a simple model forthe distribution functions of in-packets in the sourceand detector by the ratio of source operation time and detector exposure time and by space-timewidth of the effective neutrino WP, where

is a function of and . The suppression is weakif or . The strong dependence of

on parameters and provides an opportunity tomeasure the averaged value of function by varyingthese parameters (or one of them) in a specializedaccelerator or reactor experiment51. Such measure-ments (even rough ones) would provide useful refer-ences for planning and processing the data of futureprecision experiments at neutrino factories and exper-iments with -beams, “ultrabeams,” etc. The behaviorof nondiagonal decoherence factors ( ) is morecomplex, which is illustrated by analytical and numer-ical estimates given above. It was demonstrated thatfactor ceases to depend on and in the

asymptotic regime and may inducestrong suppression of oscillatory contributions to .The conditions under which the decoherence effectsare weak (and, consequently, the standard quantum-mechanical formula for the probability of f lavor tran-sitions is applicable) were formulated. It was shownthat the decoherence effects need to be taken intoaccount in current accelerator experiments utilizing

51Only one experiment of this kind, where our model (in its max-imally simplified form) was used in the data analysis on vanish-ing of reactor , has been performed to date [88].

αβdN

νΦd νσ $d

αβ3

α βν → ναβ3

νEû

p

ûm σ

û

α,β

0S

τs

τd

ντ ∼ D1D νE

ûp

τ τ@s d ντ τ τ∼ @s d

0S τs τd

D

νe

βijS ≠i j

ijS τs τd

ντ τ τ* @s d

αβ3

PHYSICS O

short pulsed neutrino beams. These effects are alsosignificant in astrophysical neutrino experiments (dueto the spreading of a neutrino wave packet at large dis-tances). The procedure for measurement of decoher-ence factors in future experiments was discussed.

To conclude, let us formulate the key results of thisstudy.

(1) The covariant WP formalism, which eliminatesartificial singular state normalizations typical of thestandard -matrix QFT approach, was developed.

(2) The new model of a relativistic Gaussian WPwas proposed. The difficulties in generalizing the for-malism with a WP of a more general form are technicalrather than fundamental.

(3) The interaction points of in- and out-packetsare arbitrary. This leads to a four-dimensional gener-alization of the impact parameter. A correct definitionof points and , which characterize the neutrinoproduction and absorption, follows automaticallyfrom the formalism. These points are not set ad hoc,as is done in all the other approaches that we know of.In contrast, they emerge automatically as effectivepoints contained in the phase of the complex-valuedfour-dimensional overlap volume of packets with arbi-trary initial or final coordinates and velocities ,which define uniquely the classical world tubes ofpackets. The factors of geometric suppression of theWP interaction probability for noncollinear collisionsand the overlap volumes (with nonzero impact param-eters factored in) were calculated.

(4) The longitudinal dispersion of the effectiveneutrino WP is automatically taken into account in theformalism.

(5) A correct method for macroscopic averaging,where arbitrary (finite or infinite, synchronized ornonsynchronized) time intervals of operation of thesource and detector are taken into account, was devel-oped and applied. A formula for the number of eventscorresponding to a macroscopic Feynman diagram ofa rather general form was derived. Its generalization toother types of macrodiagrams is fairly evident. Anexpression generalizing the probability of a f lavortransition, , with consideration of all the aboveeffects was obtained.

(6) In all physically reasonable limiting cases, theformulas for either agree with the ones verifiedexperimentally or predict novel nontrivial effects.

The following are examples of novel effects:(1) Dependence of the probabilities of f lavor tran-

sitions (more exactly, the count rate for events withviolation of lepton numbers) on finite time intervals ofoperation of the source and detector. This is import-ant, e.g., for experiments with accelerator neutrinos.

(2) Exponential suppression of the number of non-synchronized events.

S

Sx Dx

κx κv

αβ3

αβ3



(3) Spreading of the effective neutrino WP at largedistances.

(4) Dependence of the neutrino momentum dis-persion on the event kinematics in the source anddetector. Explicit formulas describing this dependencewere derived in the most general case (i.e., for arbi-trary reactions in the source and detector) and ana-lyzed in detail for the simplest and the most importantprocesses (two- and three-particle decay, quasi-elasticscattering) in the plane-wave limit.

APPENDIX APROPERTIES OF OVERLAP TENSORS

A.1. General Formulas for and Let us examine the general properties of overlap

tensors

It is useful in this regard to write matrices in the explicit form:

or, equivalently,

Here and elsewhere, and ( ) are thecomponents of vectors and , respec-tively ( ). As above, index designates packetsfrom all four sets of initial ( ) and final ( ) states.It is evident that , but . This

follows from the positivity of quadratic forms under the assumption that for at least twopackets . Note also that and,

consequently, The positivity of allprincipal minors is another important property of thedeterminant . It then follows that the spatial parts

of matrices (i.e., matrices ( )) arealso positively defined; therefore, quadratic forms

are positive.

μν,ℜs d

μν,ℜ s d

( )μν μν μ ν μν, = = σ − .ℜ û û û û

û û

2s d T u u g

μν, ,=ℜ ℜs d s d

,

Γ − −Γ −Γ −Γ −Γ + = σ −Γ +

−Γ +

ℜ

û û û û û2 û û

û û û û û û

û


û û û û û û

21 3

21 1 1 2 1 32

22 2 1 2 2 3

23 3 1 3 2 3

1

1

1

1

xs d

x

x

u u u

u u u u u u

u u u u u u

u u u u u u

, = σ Γ

− − − − − − × . − − −

− − −

ℜ 2

21 2 3

2 21 2 3 1 2 1 3

2 22 2 1 3 1 2 3

2 23 3 1 3 2 1 2

( )

1

1

1

s d

v v v v

v v v v v v v

v v v v v v v

v v v v v v v

û û

û

û û û û




ûiu vûi = , ,1 2 3i

=û û û

mu pû

v= v

û ûi iu û

,s dI ,s dF

== =û

û û 0 0T T v , >ℜ 0s dμν, μ νℜs d x x

σ >û

0û

μμ= = − σ

û û

2Tr 3T T

, = − σℜ ûû

2Tr 3 .s d

,ℜs d

,ℜs d ,ℜijs d , = , ,1 2 3i j

,ℜijs d i jx x


In what follows, the following definitions are used:

(A.1)

Here and elsewhere, indices and are omitted forsimplicity. Spatial indices assume the values of 1,2, 3; it is assumed, unless stated otherwise, that

, and the triad in each formula con-taining all three indices is a cyclic permutation of (1, 2,3). The determinants of and may be written inthis notational convention as

(A.2)

A.2. Inverse Overlap Tensors

The matrices inverse to are defined in terms of

algebraic cofactors of the matrix elements of :

(A.3)

(A.4)

The matrix elements of are then given by

where

Explicit formulas for the components of inverseoverlap tensors in terms of the most probable 4-veloc-ities were obtained in [123]:

(A.5)

( )ω = σ + , ω = σ ,

υ = σ Γ , = σ .

û û û û

û û

û û û û û û

û û

2 2 2 2

2 2

1i i

i i i j k

u

u w u u

u

s d, ,i j k

≠ ≠i j k , ,( )i j k

ℜs ℜd

( )

( )

, = ω ω + ω

+ υ υ − υ − υ

+ ω υ υ − ω − υ ω ω .

ℜ ∏ ∏

2

2

2

s d i ii i

i i i i j j k ki

i i j k i i j ki

w

w w w w

w w

μν,ℜ s d

,ℜs dμν,A s d ,ℜs d

−− μν, , ,= ,ℜ ℜ A

11s d s d s d

( )( )

( )( )

,

,

,

,

= ω − ω + ,

= υ ω ω − υ ω − υ ω+ υ + υ − υ ,

= ω ω ω −+ υ υ − υ ω − υ ω ,

= υ υ − ω ω+ υ υ − υ − υ + ω .

∏ ∏A

A

A

A

00 2

0

2

2 2

2

2

s d i i i iii i

is d i j k j k k k j j

i j j k k i iiis d j k i

i j k j k k jjks d j k i i

i i i j j k k j k

w w

w ww w w w

w

w

ww w w w w

,ℜ s d

( ) −μν − μν, , , ,μν= = ,ℜ ℜ ℜ

A11

s d s d s d s d

, , , , ,

, , ,

= , = = − ,

= = .

A A A A A

A A A

00 00 0 0 0i i is d s d s d s d s d

ij ji ijs d s d s d

μν μν, ,

, , ∈ ,,= σ σ σ .ℜ ℑℜ

6 $

2 2 21 abcs d a b c s d

a b cs d

1 2020


The multiindex expressions (symmetric in Lorentzindices) appearing in (A.5) take the form

This result was obtained without assuming energy-momentum conservation.

The positive definiteness of symmetric matrices (and, consequently, ) leads to a number of

strict inequalities; in particular52,

(A.6a)

Similar inequalities are valid for the cofactors

and (since ) and for the elements of

matrix (without summation overrepeated indices):

(A.6b)

Inequalities (A.6b) have important corollaries suchas the positivity of functions and .Indeed, the following is true in the reference framewith axis directed along unit vector :

52The left-hand sides of inequalities (A.6a) are the principalminors of the first and the second orders. It is implied thatsummation over repeated indices is not performed in (A.6a)and (A.6b).

[ ],

= Γ Γ −

+ Γ − + ,

ℑ 00

2 2

1( ) ( )( )3

1 11 ( )2 3

abcs d a b a b b c c a

a b c

u u u u u u

u u

[]

, = Γ − −

+ − Γ ,

ℑ 0 21 ( ) 2( )( )2

2( ) ( )

abc is d c a b ci ci a b b c ai

a c ai b a b ai

u u u u u u u u u

u u u u u u

[ ] ( )[ ]

[ ]

, = Γ Γ − δ + Γ −

× − δ + Γ Γ + Γ − Γ

− Γ Γ + + Γ Γ − Γ ;

ℑ 2

2

1( )( ) ( ) 12

1 ( )

( ) ( ) ( )

( ) ( )

abcijs d a b c a c b a b ij c

a b ij

c a b c b c a c a b

a b a b ai bj b b a a b ci cj

u u u u u u

u u

u u u u u u

u u u u u u u u

[ ] [ ]

,, , , ∈ ,

= σ σ σ σ Γ Γ

× − + −

− Γ Γ − + Γ

× − − .

ℜ 6 $

2 2 2 2

2 2

2

( )

1( )( ) ( ) ( )( )( ) 13

1 1( ) ( ) 12 6

3( ) 2( )( )( ) 1

s d a b c d a b c aa b c d

c d d b b c a b b c c a

a b a b c d d

b c a b b c c a

u u

u u u u u u u u u u u u

u u u u

u u u u u u u u

,ℜs d ,ℜ s d

( )( )

μμ, , , ,

, , ,

> , − > ,

− > .

ℜ ℜ ℜ ℜ

ℜ ℜ ℜ

200 0

2

0 0

0

ii is d s d s d s d

jj kk jks d s d s d

μν,A s d

μν,

A s d , >ℜ 0s dμν μν μν= +ℜ ℜ

s dR

μμ > , − > , − > .2 200 00 0 0ii i jj kk jkR R R R R R R

5 − −m n n2

0 0

z l

( )

−= , − − =

−= .− +

55 m n n

22 00

33 0 0 2

200 33 03

200 03 33

( )

2

RRR

R R RR R R

Rl

PHYSICS O

Since these quantities are rotationally invariant, itfollows from (A.6b) that and .It is evident from the latter inequality that quantity (285)is positive. This results in the suppression of probabil-ity (297).

Functions and are constructed from the com-ponents of 4-vector , where

and tensor components . Note thatscalar product and zeroth com-ponent are the only quantities actuallyneeded to calculate function . It is also sufficient todetermine them in the PW limit, where calculationsare simplified considerably. In what follows, we usesymbol to denote that function is calculated inthe PW limit. In these terms,

Let us examine several simple types of processes inthe source and detector, which are relevant to real-lifeneutrino experiments, to clarify these general formulas.

A.3. Two-Particle Decay in the Source

Let us study the simplest process: lepton decay in the source ( ). Here, is a charged

meson ( , , ), is a charged lepton( ), and is a virtual neutrino or antineu-trino. Since such decays are the primary sources ofhigh-energy accelerator, atmospheric, and astrophys-ical neutrinos and antineutrinos, we examine thisexample in sufficient detail53.

A.3.1. Formulas for arbitrary momenta. In the caseunder consideration, the determinant of matrix iseasy to calculate using formula (A.2) written in theintrinsic frame of the meson packet54:

(A.7)

53Naturally, the formulas given in this section hold true for anytwo-particle decay (e.g., decays of relativistic ions via the cap-

ture of an orbit electron (such as ) in thethought experiment on detection of low-energy neutrinos fromthe EC decay. With certain reservations, the formulas are alsoapplicable to the consecutive processes of emission and reso-nance absorption (induced by the electron capture) of a Möss-

bauer antineutrino (e.g., , ).54Subscript denotes the intrinsic reference frame of a meson.

Throughout the present section, indices and denote thecorresponding particles and should not be confused withLorentz indices.

>5 0 − − >m n n2

0 0 0

n m= +s dY Y Y

μ μν μ μνν ν= = − ,ℜ ℜ ands s s d d dY q Y q

= +ℜ ℜ

0 0 0i i is dRν= + =( )s dYl Y Y l E R

= +0 0 0s dY Y Y

n0

f f0

ν

μν − μν, , μ ν ν , μ ν=

μ, , μ

→ =

→

ℜ ℜ

ℜ

1

0 0and .

s d s d s dq E l

s d s d

Y l q l E q q

Y q

→ ν *a 2a a±π ±K ±,sD …

± ± ±,μ , τe ν*

+ +→ ν140 57 140 57Pr Ce *

→ + ν3 3H He * ν + →3 3He H*

ℜs

w

a

= σ σ σ .ℜ

w22 2 4

2s a u



Here, and . Sincedeterminant is Lorentz-invariant, it is trans-formed to the laboratory frame by substitution

where

is the invariant relative velocity of a meson and a lep-ton in the laboratory frame. Note that the kinematicvariables corresponding to different particles are inde-pendent in this formula because they are not con-strained by the energy-momentum conservation law.In the particular case with a virtual neutrino on themass shell and the neutrino mass and the spreads oflepton and meson momenta being negligible, one mayuse the common relations of the two-particle decaykinematics, which state that

A somewhat more complex calculation involvinggeneral formulas (A.4) allows one to determine theexplicit form of cofactors :

(A.8)

An explicitly Lorentz-invariant formula for thecomponents of the inverse tensor may be derivedfrom (A.8) or (A.5):

It follows that

(A.9)

Let us now find the explicit form of 4-vector withits components defined as

With elementary transformations, the following isderived from (A.8):

(A.10)

Useful identities:

A.3.2. PW0 limit. Since kinematic constraints werenot applied in the derivation of formulas in the previ-ous section, they hold true for an arbitrary 4-vector .

Let us now introduce the exact conservation laws (i.e.,pass to the PW limit, which implies in this specialcase that and ):

where

are the energies of a lepton and a neutrino in theintrinsic reference frame of a meson. Inserting theserelations into (A.9), we find the PW limit for qua-dratic form :

(A.11)

Estimation of the parameters of an effective neutrinopacket. In order to illustrate the result, we consider aspecial (although fairly realistic) case when the contri-bution from the reaction in the detector to completefunction may be neglected (i.e., it is assumed that

σ = σ + σ2 2 22 a = = Γ mu p vℜs

( )= − − × = ,

w 2 21 ( )Va a a a aa

E E u um m

u p p p p

( )− − ×=−

2 2

V1

a aa

a

v v v vv v

− −= = = .+

w w

2 2 2 2

2 2V and2

a aa

aa

m m m mm mm m

v u

μνA s

( )

[ ]

[ ]( )

= σ σ σ Γ + σ Γ + σ σ × ,

= σ σ σ Γ + σ Γ

+ σ σ Γ + σ Γ ,

= σ σ σ + σ+ σ σ + σ

+ σ σ Γ − Γ − × ,

= σ

A

A

A

A

200 2 2 2 2 2 2 2 22 2

0 2 2 2 22

2 2 2

2 2 2 222 2 2

2 22 2

( )

( )

( )

( )

s a a a ai

s a a a a ai

a a a i

iis a a ai a i ai

i a a ai i

a a a ajks

u u u

u u u

u u u u u

u u u u u

u u

u u u u

( )]σ + σ + σ σ× + , ≠ .

2 4 4 2 22 ( )a aj ak j k a a

aj k j ak

u u u u u uu u u u j k

( )

μ ν μ ν μν μ ν μ νμν

σ + σ − σ σ − − + = . σ σ σ + σ −

ℜ

4 4 2 2 2

2 2 2 2 2

( ) 1 ( )( )

( ) 1a a a a a a a a

sa a a

u u u u g u u u u u u u u

u u

( ) ( ) ( ) ( ) ( )μνμ ν

σ + σ + σ σ= − .

σσ σ σℜ

w

2 24 4 2 2 2

2 22 2 222

2a a a a as

a

u q u q u u u q u q qq qu

sY

( ) ( )

( ) ( )

νν

νν

= = − − − ,

= = − − − .

ℜℜ

ℜℜ

A A

A A

0 0 00 0

0

1

1

is s s a a i

s

i i ij is s s a aj

s

Y q E E p p

Y q p p E E

μ μμ = − − − . σ σ

w2 2 2 2 2

( ) ( )1 1 1a a as

a a

p p p p p pYm mu

( ) ( )( ) ( )[ ] ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( ) ( )[ ]

= Γ Γ− Γ + Γ + ,

= + +

20

0

.

a a

a a a

a

ai i i i aj j k k ak k j ji

u q u q qq

u q u q u q u q u q u q

u q u q u q u qu q u q

q

0

ν ν= − = =aq p p p E l ν =2 0p

νν= , = , = ,

www a

a am EEu u u q E u q

m m

ν+ −= =

w

2 2 2 2

and2 2

a a

a a

m m m mE Em m

0μν

μ νℜ s q q

μνμ νℜ = + .

σ σ

@

22

2 2 1as

a

mmq q

D



parameters for all are sufficiently large com-pared to and ). This assumption is valid in exper-iments with accelerator and atmospheric muon neu-trinos and antineutrinos. The interactions of mesonsand muons in the source (decay channel, atmosphere)with each other and the medium are normally negligi-ble in such experiments; therefore, the values of and

are very small (close to the limiting ones in theSRGP model). The values of parameters for long-lived in- and out-particles (nuclei, hadrons, leptons)in a typical detector are defined by their mean freepaths in the detector medium rather than by theirdecay widths55; thus, . At the very least, the

condition is satisfied for short-lived particlesand resonances that are produced in the detector anddecay prior to interacting with the medium.

Thus, using (A.11), we find

(A.12)

and, consequently,

(A.13)

It is evident that the effective wave packet of a vir-tual neutrino with a given mass is defined completelyin this simplest case by the masses and momentumspreads of packets and , and the values of for allthree known neutrinos are exceptionally small at any

and allowed by the SRGP approximation. Inaddition, since all the other (known) elementary par-ticles with a nonzero mass are many orders of magni-tude lower than the three known neutrinos, we con-clude that . Although these estimates wereobtained without including the contributions of thereaction in the detector, they explain qualitatively therelevance of the standard quantum-mechanicalassumption that light neutrinos have definitemomenta in spite of the fact that they are produced inprocesses involving particles with relatively largemomentum spreads. It follows from (A.12) that at ; i.e., massless neutrinos may be regarded asplane waves. With obvious reservations, this remark-able fact may be used in the analysis of processes sim-ilar to those depicted in Fig. 8, where light massiveneutrinos act as external wave packets.

The conditions of applicability of the SRGPapproximation for unstable particles

( is the total decay width of particle ) yieldthe following important constraint:

Thus, two-particle decays of any mesons with amuon in the final state ( , , etc.) are constrainedfrom the above:

therefore, . This yields thefollowing lower bounds for the effective sizes of a neu-trino packet transverse and longitudinal with respectto the neutrino momentum56:

The constraints on the characteristics of neutrinopackets emerging in decays depend on the type of thedecaying particle. For example, inthe decay of a meson. At eV, this corre-sponds to a limiting effective transverse size of a neu-trino packet of “just” 6.6 m.

Effective sizes and define (in the order ofmagnitude) the allowed transverse and longitudinalquantum deviations of the center of the neutrino wavepacket from “classical trajectory” . Evi-

dently, transverse deviations may be hugeand exceed in magnitude the size of current neutrinodetectors and the natural accelerator neutrino beamdivergence (all over distances of several hundred orthousand kilometers from the source). This shouldcome as no surprise if one remembers that the stan-dard quantum-mechanical description of a massiveneutrino as a state with definite momentum implies(as a direct consequence of the uncertainty principle)that its transverse and longitudinal “sizes” are infinite.When data from common experiments on neutrinooscillations are interpreted, this description does notyield nonphysical results because neither the trans-verse size of a neutrino packet nor transverse quantum

55Wave packets form anew in each (inelastic or elastic) interactionof a particle with the medium and external fields.

σû

∈ Dσa σ

σa

μσσû

û

,μσ σû@ a

μσ σû@

−

ν ν

≈ + σ σ

D !

1222 2 2

2 22 ,a

a

mmE E

−

σ ≈ + . σ σ

!

12 222 2

2 22j a

j ja

m mm m

a σ j

σa σ

,σ σ

2 2j a

σ = 0j

= 0jm

( )σ σ , σ = Γû û û û û

!4max max1 m

56It should be noted that is by no means the size of a neutrinopacket at rest, since all estimates were obtained in the ultrarela-tivistic approximation and remain true only in those referenceframes where a neutrino is ultrarelativistic. The same importantreservation applies to the Lorentz invariance of neutrino wave

function . The explicit form of the effective wavepacket of a nonrelativistic neutrino requires separate consider-ation, which is beyond the bounds of the present study.

Γ = τû û

1 û

− σ + . Γ Γ

!

122

2j a

ja

m mm

μπ 2 μ2K

−μ

μ

Γσ≈ . ×!

218

2 1 4 10 ;2

j

j mm

μ μσ σ ≈max max 1j jm m

⊥jd

Ψ ,( )jy j xp

⊥

⊥−

ν

..

. = . × . Γ

@

@5

0 1 eV2 6 km

1 GeV 0 1 eVand 2 6 10 cm

jj

jj

j j

dm

dd

E m

τ2a−σ . ×!

2 2 132 2 10j jmsD = .0 1jm

⊥jd

jd

=j jTL v⊥ ⊥δ ∼ 2j jdL



f luctuations are found in the expression for the countrate of neutrino events averaged properly over spatialcoordinates of external packets. This averaging alsoeliminates automatically the dependence on timecomponents ; only the dependence of the count rateon external (“instrumental”) space-time variables setby the experimental conditions57 remains. Large trans-verse sizes of a neutrino packet may manifest them-selves in specialized neutrino experiments, where boththe source–detector distance and the time intervalbetween the neutrino production and its interactionare monitored. Such experiments will be discussed ina separate study.

The effects of noncollinearity of momentum trans-fers in the source and detector are contained in func-tions and . The properties of these functions are dis-cussed below.

Contributions to functions and . Using generalformula (A.10), we find 4-vector Ys in the PW0 approx-imation:

Scalar products needed to calculate the a,2 contri-butions to function take the form

This yields the following:

Function may serve as an estimate for completefunction if, just as with function , we neglect thecontribution to from the detector part of the dia-gram (in the present case, from and ) assumingthat parameters ( ) are sufficiently large com-pared to and . Since function depends linearly

57However, it should be kept in mind that the applicability condi-tions of the mathematical averaging procedure are ratherrestrictive and are by no means guaranteed to be fulfilled in allneutrino experiments. For example, they are definitely violatedin accelerator experiments with a baseline shorter than theeffective transverse sizes of neutrino packets.

ûx

û

0x

n

n m

νν

νν

= + − σ σ σ

= + − . σ σ σ

w

w

2 22

2 2 2

22 2

2 2 2

1

1

a as a

a a a

a

a a

m mmY p pm E

mm mp pm E

n

μν

μ ννν

= = + σ σ ℜ

22

2 21 1a

s sa

mmY l q qEE

ν

ν ν

ν

Γ= − =

× + − − . σ σ σ

w

w

0

2 22

2 2 21

as s s

a a a

a aa a

Y Y Y lE

m m E m EmE E E

l

( )

ν

νν

ν

σ= Γ − σ + σ

× − ≡ , .

nw

2 2

2 2 2 2

1

aa

aa a

s a

m EsY l mm ms

E E EE

Y l

n s

n Dn

0dY dY l

σû

∈û Dσa σ n s


on Ea at a fixed value of Eν, the following inequalityholds true:

where

is the minimum energy of particle needed to producea massless neutrino with energy in an decay.Thus, the absolute minimum of function is negative:

Function increases with neutrino energy andmay be arbitrarily large at :

As is well known, the neutrino energy distributionin an decay is uniform (i.e., independent of )within the following kinematic bounds:

It follows that the mean energy of a decay neutrinois . Therefore, the following is true towithin at high energies of decaying mesons,

:

Consequently,

Under the same assumptions, retaining only theleading terms in and , one may estimate the

-decay contribution to function :

( )ν≥ , ,n nmin

s s aE E

ν ν

νν

= +

w

w

min

2a

am E EE

EE

aνE 2a

n s

( )( )ν

− σ= , = − ,σ + σ

< − .

n n

n

w

2 2 2min

2 2 2 2

2min

2

( )2

1 12

as s a

a a

sa

m mm Em m

mm

sn

ν νw

@E E

( ) ( )ν

ν ν

νν

≥ , = −

× + .

n n n

2

w

w

min min

2

1 1 22

1

s s a sE E

E EEE

2a νE

ν ν νΓ − ≤ ≤ Γ +w w(1 ) (1 ).a a a aE E Ev v

ν ν= Γ w

aE E−Γ2

2( )a

Γ @ 1a

( ) ( )( )

ν

ν

ν

, ≈ Γ −

= −

n n

nw

2 min

2min

1

1 .

s a a s

s

E E

EE

( )ν ν=

ν

≈ − .

n n

w!

2min1 1 1

2j

s j sE E

mr

E

Γa ν νwE E

2a m

( ) ( )ν

ν ν

ν

σ σ≈ Γ + + σ + σ σ + σ

× .

m

w

w

4 4 22

2 2 22 2 2 2

22

21 1a as a

a aa a

a

m Emm m

E EE E

1 2020


It can be seen that ; however, inequali-ties (284) assumed in Section 8.2.1 remain valid at

.

A.4. Quasi-elastic Scattering in the DetectorLet us consider the quasi-elastic scattering of a vir-

tual neutrino , where target particle maybe an electron, a nucleon, or a nucleus and is acharged lepton, as the simplest (and the most import-ant) example of a reaction at the detector vertex. Sincethe velocities of target particles in the laboratory frameare very low (thermal) in a typical neutrino experi-ment, we assume that the laboratory frame coincideswith the intrinsic frame of a wave packet describing thestate of particle . If necessary, all formulas may berewritten in any other reference frame, since we areconcerned just with vectors and tensors, and the lawsof their transformations are well known.

A.4.1. Formulas for arbitrary momenta. The deter-minant of matrix in the intrinsic reference frame ofpacket takes the form

(A.14)

Here, is the relative velocity of particles and, and . One important implication

of this formula is that determinant remains non-negative even if one (but only one) of the particles( , , or ) is described by a plane wave. If, e.g., oneneglects the terms proportional to , formula (A.14)takes the form coinciding with that of (A.7):

(A.15)This important feature provides an opportunity to

simplify the analysis of multipacket in- and out-statesby neglecting the contributions of packets with verylarge spatial sizes (characterized by very small values ofparameters ). However, it should be kept in mindthat approximate formula (A.15) is applicable only if

58. The same is true in the general case; i.e.,when omitting the contributions of packets with verysmall values of , the phase-space regions withinwhich determinants and calculated in thisapproximation vanish, should be cut-off. Such regionsare typically located near the kinematic boundaries ofthe phase space and do not contribute to experimen-tally measureable characteristics.

According to (A.4), cofactors are written asfollows:

58We should recall that in (A.7) is always nonzero.

m n@s s

ν ν∼E E

ν → *a b a

a

ℜda

( ) ( )

[ ]

= σ σ + σ σ σ+ σ σ σ

+ σ σ σ + σ + σ σ + σ .

ℜ

22 2 2 2 2 23 3

2 2 2

2 2 2 2 2 2 2 2 2

V

2 ( )( )

( ) ( )

d a b b b

a b b b

a b a b b a

u u

u u u u

u u

Vb b σ ≡ σ + σ + σ

2 2 2 23 a b

ℜd

a b

σ2

( )≈ σ σ σ + σ .ℜ2 22 2 2 2

d a b a b bu

σû

≠ 0bu

wu

σû

ℜs ℜd

μνAd

PHYSICS O

(A.16)

The following is then obtained for an arbitrary4-vector :

Using (A.16), we determine the components of4-momentum :

The coefficient functions found here are given by

( )[

( )( )

( )( )( )

= σ σ σ + σ Γ + σ Γ+ σ σ × ,

= σ σ σ + σ Γ + σ

+ σ σ + σ Γ + σ ,

= σ σ σ − σ σ+ σ σ σ − σ σ + σ σ

× σ Γ Γ + σ + σ

A

A

A

00 2 2 2 2 2 2 23 3

22 2

0 2 2 2 2 23

2 2 2 2

2 2 2 2 2 2 23

2 2 2 2 2 2 2 2 23

2 2 2

( )

( )

2

( )

d a b b

b bi

d b a b b b bi

a b b i

iid b b a bi

a b b i b

a b b b b

u u u

u u u

u

u

u u u

u u

u

u

( )( )( ) ( )

−+ σ ,

= σ σ σ − σ σ+ σ σ σ − σ σ + σ σ

× σ Γ Γ + σ + σ + , ≠ .

ℜ

A

23

2 2 2 2 2 23

2 2 2 2 2 2 2 23

2 2 2 ( )

i i d

jkd b b a bj bk

a b b j k b

a b b b bj k bk j

u

u u

u u

u u u u u u

j k

u

u

q

( )

( )( ) ( )( ) ( )

( ) ( )

μνμ ν

<

= + − +

+

= σ σ σ + σ Γ + σ Γ + σ σ ×

− σ σ σ + σ Γ + σ Γ

− σ σ σ + σ Γ + σ Γ

+ σ σ + σ + σ σ

ℜ ℜ

A A A

A

00 2 00 0

22 2 2 2 2 2 2 2 2 23 3 0

2 2 2 2 23 0

2 2 2 2 23 0

2 22 4 4 2 23

2

2

2 ( )

2 ( )

2

i iid d d d d i i

ijkd j k

j k

a b b b b

b a b b b b

a b b b

b b b

q q q q q q

q q

q

u u q

u u q

u u

u q

u q

u q u q ( ) ( )[ ] −

+ σ σ σ × − × + σ .ℜ

22 2 2 2 2 2 23

( )

( ) ( )

b b

a b b b d

u u u q u q

u u q u u q q

dY

( )

( )

σ= − − ,

σ= + .

ℜ

ℜ

20 0 0 03

23

d a a b bd

d b bd

Y c m c E c E

c cY u u

( )( ) ( ) ( )

( )[ ]( )[ ]

( )[ ]( )( )[ ]

= σ σ + σ Γ + σ Γ + σ σ

× Γ Γ − − + σ Γ + σ Γ ,

= σ σ + σ + σ Γ − ,= σ σ + σ + σ Γ − ,

= σ σ Γ − σ + σ + σ Γ− σ + σ

+ σ σ + σ Γ − Γ − ,= σ

0 2 2 2 2 2 2 2 23

22 2 2 2

0 2 2 2 2 23

0 2 2 2 2 23

2 2 2 2 2

2 2

2 2 2 2

1

1

1

( )

a a b b b

b b b b

b a b b

a b b b

b b a b b a b b

b b

a b b b a b

c

c

c

c m m u u m

m

m m m m

c

u u

v v

v v

u

( )[ ]( )

( )[ ]

σ Γ − σ + σ× + σ Γ − σ + σ

+ σ σ + σ Γ − Γ − .

2 2 2 2

2 2 2

2 2 2 2

( )a b b b b

b a b

a b b a b b

m m

u u m m

m m m mu



As it should be, the above expressions are symmet-ric with respect to index interchange , and theirexplicitly noncovariant form is attributable to the useof a special reference frame.

A.4.2. PW0 limit. The kinematics of reaction in the PW limit allows one to write quantities ,

, and in terms of two arbitrary indepen-dent invariant variables; we used the standard pair ofvariables

In order to rewrite the expressions from the previ-ous section in terms of these variables, we use the fol-lowing exact kinematic relations:

where and are theenergy and the momentum of a massless neutrino inthe laboratory frame;

are the energies of particles , , , and in the cen-ter-of-mass system of colliding particles and ,which is determined by

and . Lepton scattering angle in the cen-ter-of-mass system is related to :

↔ b

→2 20 ℜd

μνμ νℜ d q q dY l

( )ν ν

ν

= + = += − − .

2

2 2

( ) 2

and ( )a a as p p m E m

Q p p

( )( )( )( )

( )

ν

νν ν

ν

νν ν

νν

∗ ∗ ∗∗= − θ ,

∗ ∗∗ ∗= − θ ,

∗ ∗∗ ∗= + θ ,

∗ ∗ ∗ ∗= + θ ,

∗ θ× = , × = ,

∗− −= , = ,− −

2 2

2 2

1 cos *

cos *

1 cos *

cos *

sin * 0

2V2

b b aa

b b aa

aa

aa

b bb

bb b

b b

E E E E PmE E E E Pm

E E E E PmE E E E Pm

E Pm m

s m m sPu um m s m m

p p

p p

u u u u p

( )ν ν= = − 2 2a aE s m mp ν ν= Ep l

ν− + −∗∗ = , = ,

+ − +∗∗ = =

2 2 2

2 2 22 2

and ,2 2

a b

a ba b

s m s m mE Es s

s m s m mE Es s

ν* a bν* a

ν ν∗ ∗ ∗∗ ∗ ∗ ∗ ∗+ = + , + = + = ; 0a b a bE E E E p p p p

∗ ∗= P p θ*2Q

( )ν∗ ∗∗= − θ − .

2 22 cos *Q E E P m


The kinematically allowed region of the phasespace is defined by the following inequalities:

(A.17)

(A.18)

Performing elementary (although rather cumber-some) algebraic transformations, we find

Nonzero coefficients , ( ), and ( ) have the form

( ) ≥ = , +

22th max ,a bs s m m m

( )− + ± ν∗ ∗ ∗≤ ≤ , = ± − .

2 2 2 2 22Q Q Q Q E E P m

, =

μνμ ν

, =

, =

σ= ,

σ= ,

σ= .

ℜ

ℜ ℜ

ℜ

2223

2 2 20

2223

2 2 20

3223

2 2 20

4

4

8

k ld kl

k la b

k ld d kl

k la b

k ld d kl

k la b

A s Qm m m

q q B s Qm m m

C s Qm m m

Y q

klA klB ≤ , ≤0 2k lklC ≤ , ≤0 3k l

( ) ( )[( ) ( )

( )[( )

( )[( ) ( )( ) ( )[ ]

= σ σ σ + σ −+ σ σ + σ − − σ σ

+ σ σ σ + σ+ σ σ − ,

= σ σ σ ++ σ + +

+ σ + + σ σ + σ= σ σ σ

2 2 2 2 2 2 2 200

2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 23

2 2 4 2 2

2 2 2 2 2 201

2 2 2 2 2

4 2 2 2 2 4 2 2

2 2 202 3

2

2 3

4

2

2

2 ,

b a a a b a b

b b a b

b b b a a

a b a

a b a a b

b a b

b a b b a

a b

A m m m m

m m m

m m m m

m m m

A m m m

m m m m

m m m m

A m( )+ σ ,

2 2 2bm

( ) ( )[( )

( ) ( )[ ]( )[ ]

( )

= −σ σ σ + ++ σ +

+ σ + + σ σ + σ= −σ σ σ + + + σ ,

= −σ σ σ ,= σ σ σ + σ ,

= σ σ σ

2 2 2 2 2 2 210

2 2 2 2

4 2 2 2 2 4 2 2

2 2 2 2 2 2 2 211 3

2 2 2122 2 2 2 2 2

20 32 2 2

21

2

2

2 ,

2

;

b a a b b

a b

b a b a b a

a b a b b

a b

a b b a

a b

A m m m m

m m m

m m m m

A m m m m

A

A m m

A

( ) ( )[( ) ( )]( )

( )( )[ ]

( ) ( )[ ]( )

= σ + σ ++ σ + + σ ++ σ + σ + σ σ ,= σ + σ +

× σ + + σ + σ× σ + + + σ +

= σ σ + σ ,

2 2 2 2 2 2 2 200

2 2 2 2 2 2

2 4 2 4 2 2 2 2

2 4 2 2 2 2 201

2 2 2 2 2 2 2

2 2 2 2 2 2 2

2 2 2 2 2 202 3

2

2

2 2 ,

a b a a a b

b a b

b a a a b

a b b

b a a b b

a a b b a

a b b

B m m m m m m

m m m m

m m m m

B m m m m m

m m m m

m m m m m

B m m m

1 2020


( )[( ) ( ) ( )

( )[ ]( )[

( )]( )

= − σ σ + ++ σ + + σ σ + +

+ σ + σ += − σ σ + + +× σ σ + σ σ + σ σ ,= − σ σ + σ σ + σ σ ,

= σ σ

2 2 2 2 2 2 210

2 2 2 2 2 2 2 2 2 2

2 4 2 4 2 2

2 2 2 2 2 2 211 3

2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 212

2 2 220 3

2

2

2 ,

2

b a a b

a b a b a b a b a

a a b b a

b a a b

a b b a a b

a b b a a b

a

B m m m m m

m m m m m m m

m m m m

B m m m m m

m m m

B m m m

B m m( )+ σ ,= σ σ + σ σ + σ σ ;

2 2 2

2 2 2 2 2 2 2 2 221

b b a

a b b a a b

m

B m m m

( )[( ) ( )

( ) ( )[ ][ ( )

( ) ( )( )

( )

= σ σ + − ++ σ − + σ + σ + σ

× σ + − σ + σ= σ σ + +

+ σ + + σ −

+ σ − ++ σ σ +

2 2 2 2 4 2 2 2 2 2 200

2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2

2 2 2 2 2 2 2 401

2 2 2 2 4 2 2 2

4 2 2 2 2

2 2 2 2

2

2 ,

2

3

2 3

a a b a a a b b

b b a b a b b

b a b b

a a b a b

b b b b

b a b

b b a

C m m m m m m m m m

m m m m m m

m m m

C m m m m m m

m m m m m m

m m m m

m m m( )( )

( )[( ) ( )[

( ) ( )( ) ( )[ ]

− += σ σ + σ ,

= − σ σ − + ++ σ − + σ −

+ − − σ + + σ σ

× + − − + ,

2 2 2

2 2 2 2 2 202 3

2 2 2 4 2 2 2 2 2 210

2 2 2 2 4 2 2 2

2 2 2 4 2 2 2 2 2

2 2 2 2 2 2 2

3 ,

2 2

3 2

3

2 4 3

b

a b b

a b a a b a b

b b a b a a b

a b b a b b

a a b b b

m m

C m m m

C m m m m m m m m

m m m m m m

m m m m m m

m m m m m m m

( )[( ) ( )

( )( ) ( )

( )( )

= −σ σ + ++ σ + + − σ − −

+ σ + + + σ σ + − − + ,

= −σ σ + σ− σ σ − − + σ + σ ,

2 2 2 2 2 211

2 2 2 2 2 4 2 2 2 2

4 2 2 2 2 2 2

22 2 2 2 2 2

2 2 2 2 212

2 2 2 2 2 4 2 4 2

3

3

a b a b

b a b b a b

b a b b

b a a b

a b b

b a b b b

C m m m m

m m m m m m m m

m m m m

m m m m m m

C m m

m m m m m

( )[ ]( ) ( )

( )( ) ( )

= σ σ − − σ+ σ − − σ σ + σ

+ σ σ − σ= σ σ + σ − σ σ +

− σ − σ ,= −σ σ ,= σ σ .

2 2 2 2 2 2 220

4 2 2 2 2 2 2

2 2 2 2 2

2 2 2 2 2 2 2 2 221

4 2 4 2

2 222

2 231

2 ,2

a b a b b

b a b b b

b a b

a b b b b

b b

b

b

C m m m mm m m

m mC m m m m

m mCC

PHYSICS O

Thus, quadratic form and scalar product

are rational functions of variables and :

The following function was introduced here59:

It is also expedient to introduce function defined in the following way:

Although neither nor have a clear physicalmeaning, they help illustrate the behavior of functions

and , which are of interest to us, in the special casewhen the contributions to and from the reactionin the source can be neglected60. This case is definedby the following conditions:

In the simplest particular case with = (these relations characterize the

scaling of effective sizes of packets), one can demonstratethat functions (and, consequently, ) and are independent of parameter and are defined by thekinematics only. The domains of functions and are bounded by kinematic conditions (A.17) and (A.18),and the shape variations of surfaces shown in differentpanels are attributable primarily to the differences inreaction thresholds (A.17) (essentially, the differences inmasses of final leptons). Therefore, these variations aresmoothed out at sufficiently high energies (i.e.,

). The fact that function turns to zero at for thresholdless reaction is irrele-

vant to our analysis limited to ultrarelativistic neutrinos61.

59It is worth noting that, in contrast to , function is not arelativistic invariant, although it is expressed (in the laboratoryframe) in terms of two invariants.

60This case is exactly the opposite of the one discussed as anexample in Section A.3.2.

61We should recall that the formulas for dispersion are modifiedgreatly at ; see Section 8.2.2.

μνμ νℜ d q q

dY q s 2Q

μν ,μ ν

,

= ≡ , ,ℜ

F

2

22 ( )

k lkl

k ld dk l

klk l

B s Qq q s Q

A s Q

,

,

= ≡ , , .

n F

2

2 22

1 ( ) ( )2

k lkl

k ld d dk l

klk l

C s Qs Q s Q

A s QY q

Fd nd

,

,

= =

n

2

21 .2

k lkl

k ldd k l

d klk l

C s Q

Y l B s QY l

Dd

,

ν,

= = .

D

F

22

2 21 1

2 2

k lkl

k ldk l

d klk l

A s Q

E B s Q

Dd nd

D n

D n

μν μν

μ ν μ νℜ ℜ @ @and .d s d sq q q q Y l Y l

σ = σa a b bm mσ = λ = constm

λ F2 d λDd nd

λFd nd

@ thmax( )s s Fd

ν → 0E −ν →n pe

ν ∼ jE m



In the general case, the behavior of functions and is much more complex. Naturally, this assumption,

which was adopted for purely illustrative purposes, is com-pletely arbitrary and unrealistic. In a more realistic case,

, even function varies strongly withinits domain, and the features of its behavior are hard toreproduce in a two-dimensional plot. Let us consider themost important limiting cases, asymptotics, and inequali-ties to gain a better understanding of the properties of func-tions and .

A.4.3. Low-energy limits of functions and .The limits of functions and at the kinematicthreshold of quasi-elastic reaction inthe detector are as follows62:

It is assumed here that . The thresh-old values of and are

The following is then obtained for the thresholdless

reaction ( , ):

Thus, function may assume a value of exactly zeroonly in a thresholdless reaction (e.g., ) at .Naturally, this formal limit goes well beyond the bounds of

ultrarelativistic approximation , whichwas used in derivation of the formulas for functions and

, and is of no practical importance, because ultrarelativ-istric neutrino and antineutrino beams63 only are used inall current neutrino experiments. It is of interest to note

that the limit of at and (for a thresholdless reaction), which is given by

may still be large in magnitude if at least two out of three parameters , , and are small compared to .

A.4.4. High-energy asymptotics of functions and . Under the assumption that and ,

the asymptotic behavior of functions and at high energies is independent of :

These asymptotics satisfy the following inequalities:

( , ), and their limiting values at kinematic boundaries are

62All formulas are written in the PW limit, which implies the exact energy-momentum conservation in reaction plus .It is also assumed (unless stated otherwise) that all parameters are nonzero.

Fd

nd

σû û

1m Flog( )d

Fd nd

Fd n d

Fd nd

ν + → + b b

0 →2 2 = ,∀0jm jσû

( )

( ) ( )( )

+, = + ,σ + σ σ σ + σ − + , = .

σ + + σ + σ

F

n

2 22

th th 2 2 2

22 2 22

th th 22 2 2 2 2

( )

( )2

b ad

b a

b a bd

a b b a a

m m ms Q

m m ms Qm m m m

< + a bm m ms 2Q

63It appropriate at this point to note that a more general analysis covering the nonrelativistic case is of potential interest in the context ofstudies into the possibility of detecting relic neutrinos and for accelerator and astrophysical experiments in the search for hypotheticalsuperheavy neutrinos and sterile neutrinos of keV-scale masses.

= + = − . +

22 2

th th( ) and ab b

b

ms m m Q m mm m

> + a bm m m = 2th ,as m = −

2 2thQ m

( )[ ]( )[ ]

, = , ,σ σ + σ + −= − .σ σ + − + σ + σ

F n2 2th th th th

2 2 2 2 2 2 23

2 2 2 2 2 4 2 4 2

( ) 0 ( )

212

d d

a b b a b

a b a b a b b a

s Q s Q

m m m mm m m m m

Fd

ν →n pe ν = 0E

ν @2 2max( )jE m

Fd

nd

Fd ν ν= ≡max( )jE m m = −

2 2Q m

( )( ) ( )

−ν σ σ + − + σ + σ + + , σ σ σ σ σ σ σ − − + +

12 2 2 2 2 2 2 2 4 2 4 2 2 2 2

2 2 2 2 2 2 2 2 22 2 2 2 2 43

4

2a a b a b b a a b a b

a b a ba b a b

m m m m m m m m m m m

m m m m m m

σa σb σ νm

Fd n d , ,σ ≠ 0a b < ∞2Q

,F 2( )d s Q ,n2( )d s Q s s

−

→∞−

→∞

, + + − + + + , σ σ σ σ σ σ σ σ

σ, + + + + . σ σ σ σ σ σ σ

∼

∼

2 12 2 2 2 2 22 22

2 2 2 2 2 2 2 23

12 2 2 2 2 222 3

2 2 2 2 2 2 2 2

( )

( )2

a b a b a bd s

a b a b a b

a b a bd s

a a b a b

m m m m m mm Qs Q

m m m m mmsn s QQ

F

−

σ σ − σ< , < + + , < , < + + , σ σ σ σ σ σ + σ σ σ

2 2 2 2 12 2 2 22 2 22 2

2 2 2 2 2 2 2 2 2 2 2 2 23

( ) ( ) 12 2

b l b aa b a bd d

a b a b b a a a b

m mm m mm m mss Q s Qm m m m

F n

→ ∞s < ∞2Q

( )( )( )( )

− −→∞ →∞

+ +→∞ →∞

σ − − σ, = , , = ,σ σ + σ

σ − − σ, = , , = .σ σ + σ

2 2 2 2 222 2

2 2 2 2 2

2 2 2 2 2 22 2

2 2 2 2 2

lim ( ) lim ( )2

lim ( ) lim ( )2

b a b bd ds s

a b b a

b a bd ds s

b a a

m m mms Q s Qm m

m m m ms Q s Qm m

F n

F n



It is evident that these quantities are symmetric withrespect to index interchange . The threshold values vanish at specific relations betweenparameters and masses. The next ( ) correctionsneed to be taken into account in these exotic cases.

In the particular case when target particle is anucleon, it follows from dynamic considerations that,at high neutrino energies, the mean scattering angle in

the center-of-mass system, , is equal in order ofmagnitude to the inverse Lorentz factor of a lepton

. Therefore, and ,where is the scattering angle of a lepton in the labo-ratory frame (coinciding with the intrinsic frame of anucleon). It can be demonstrated that the correspond-ing asymptotics of functions and take the form

Since only a narrow region of angles close to produces a significant contribution to the

count rate for quasi-elastic events at high energies, onemay conclude that effective asymptotic value isalmost constant and is defined primarily by momen-tum dispersion of the lepton WP. Arbitrary64 varia-

tions of parameters and may change the asymp-totics only within a factor of two.

The asymptotic behavior of changes dras-tically if one (and only one) of the parameters turnsto zero. If or , the asymptotics is inde-pendent of :

and if , increases quadratically with :

Other variables. Certain properties of function become more evident if it is rewritten in terms of variables and . Let us consider the asymptotic expansion of at and a fixed value of . If the values of

are not too small, it can be written as

At , one finds

As was already noted, at highenergies. It can be demonstrated that the correspond-ing asymptotic expansion takes the form

↔ b ±,n 2( )d s Q

σû

∼ 1 s

a

θ*

∗ ∗Γ = E m ∼

2 2Q m θ ∼ m sθ

Fd nd

( )

( )( ) ( )[ ]

−

→∞

ν→∞

σ, + + + + < , σ σ σ σ σ σ σ σ

σ σ, < .σ + σ σ + σ + σ + σ σ

∼

∼

12 2 2 22 2 2 22

2 2 2 2 2 2 2 23 3

2 22

2 2 2 2 2 2 2 2 2 2

1 2

22 2

a b a bd s

a b a b

bd s

ab a a b a b a b

m m m mm m ms m

s Es mmm m m

F

n

θ = θ

Fd

σ

σa σb

,F 2( )d s Qσû

σ = 0a σ = 0bs

64Naturally, these variations should not violate the conditions of applicability of the SRGP model.

→∞

+ , σ = ,σ + σ σ, ⎯⎯⎯⎯→ + , σ = ,σ + σ σ

F

22

2 2 22

22

2 2 2

at 0( )

at 0

ab

d s

ba

mQ

s QmQ

σ = 0 s

( )→∞

+ + σ + σ σ σ , , σ = .

+ + −∼

2 222

2 2 2 22

22 2 2 2 2( ) at 0

4

a b

a b a bd s

a b a b

m mQ ss Q

Q m m m mF

Fd

νE θ* Fd ν → ∞E θ*θsin *

νν

−

σ σ − σ,θ = + + − + θ + σ σ σ θ σ σ σ + σ

× + + + + + + σ σ σσ σ σ σ σ σ σ

2 2 2 2 2 2 2 22 21 3

2 2 2 2 2 2 2 2 2 2

2 222 2 2 2 22 2

2 2 2 2 2 2 2

( ) cos* *2 sin *2 4

a b b b bd

a b a b b b

a b a b b a b

a ba b b a b

m m a m m mm mEm E m m

m m m m m m m mm m m

F

ν

+ . σ θ

22

2 4 21

sin *

amE

2

θ =sin 0*

ν νν νν ν

− −, = − + , , π = − + . σ σ

2 2 2 2 2 2 22

2 2 2 2( 0) 1 ( ) 1a b a b a ad d

a ab

m m m m m m mmE Em E m EE E

F 2 F 2

∗∗θ Γ = ∼* E m ν

ν ν

∗,θ = Γ σ= − + , σ −

F

22 22

1 302 2 2

0

( 1 )*

( 1)

d

a

a

E

b mm bm b E E



where

It is evident that the effective asymptotic value is almost constant and, since , isdefined primarily by the momentum dispersion of thelepton packet.

The case of strong hierarchy. Function issimplified considerably in the case of a strong hierar-chy of parameters , , and Calculating the cor-responding sequential limits, we obtain the following:

It can be seen that65 the shape and the value offunction are not affected by the smallest andthe largest of parameters σ. This nontrivial propertymay be generalized to the case of processes with an arbi-trary number of particles in the final state. In the stronghierarchy scenario, the only significant parameter is thesecond-largest dispersion. This is useful in the analysis ofmultiparticle processes (with more than two externalpackets), since one may then treat packets with (compar-atively) very small as plane waves. In particular, thecalculation of radiative corrections with plane-wave pho-tons in the external legs of Feynman diagrams is simpli-fied significantly, since the standard QFT calculationmethods become available. We should recall that loopelectroweak corrections do not introduce additionalcomputational complications associated with the WPformalism, since all of them are formally included in the

corresponding matrix elements and are in no way relatedto the characteristics of external in- and out-states.Under the convention adopted in the main text, theexternal legs of diagrams do not feature gauge bosons.

A.5. Three-Particle Decay in the SourceThe general formulas characterizing three-particle

decay agree formally with the ones for the scattering (if they are considered in the intrinsic ref-

erence frame of particle ). The primary difference is ofkinematic origin. Therefore, we discuss this case in brief.Just as in the case of the scattering, functions and may be written in terms of two independentinvariant variables. One may use, e.g., any pair of invariants

( )( )

( )( )

( )

σ σ + σ= + ,σ σ + σ + σ σ

= σ σ + σ − + σ + − + σ

+ σ σ + σ σ + + + σ σ + σ × − − σ

2 2 2 2 2

0 2 2 2 2 2 2 2 23

2 232 2 2 2 2 2 2 2 2 2 2 2 21

2 2 2 2 2 2 4 2 2 2 2 2 2 4

2 2 2 2

1

483 3

2

b a a b

b a a b a b

b a a b b a b a a a b b

b a a b b a a b a b a b

b a b

m mbm m m

m mb m m m m m m m m

m m m m m m m

m m m m ( )[ ]+ − + σ .

2 2 2 2 2 2a a a b bm m m m

ν, θF ( )*d E< <01 2b

,F 2( )d s Q

σa σb σ.

( )( )

( )( )

( )( )

( )( )

+ , σ σ σ , σ − − −

+ , σ σ σ , σ − − −

− , σ σ σ , σ + + −, ≈− , σ σ σ σ − − −

2 22 2

22 2 2 2

2 22 2

22 2 2 2

2 22

22 2 2 2 22

2 22

22 2 2 2

for4

for4

for4( )

for4

aa b

ab a

ab a

ab b

a ba b

ba b a bd

aa

b a

mQ m

s Q m m m

mQ m

s m m m m

s m m

Q m m m ms Q

s m m

s Q m m m

F

@ @

@ @

@ @

@ @

( )( )( )

( )

, − − − , σ σ σ , σ − − − − − − , σ σ σ . σ + + −

2 22 2 2

22 2 2 2

2 22 2 2

22 2 2 2 2

for4

for4

b

ab a

b b

a ab a

aa b a b

s Q m m m

s m m m m

s Q m m m

Q m m m m

@ @

@ @

65In the case of strong hierarchy of scales of momenta dispersionsof external packets, but not in the general case.

,F 2( )d s Q

σû

→ + + ν *a b→2 2

a

→2 2 ℜsμν

μ νℜ s q q

( ) ( )( ) ( )( ) ( )

ν

ν

ν

= + = − ,

= + = − ,

= + = − ,

2 21

2 22

2 23

b a

a b

b a

s p p p p

s p p p p

s p p p p



which are related by identity =, as these variables. The physical domain

for them is defined by conditions

For definiteness, we use the pair. Thedomain of definition for this pair is the Dalitz diagram

where

Using the results obtained in the previous section,we find Therefore, quadratic form is a rational

function of variables and ,

Nonzero coefficients and ( ) take the form

We will not study the limiting cases and asymptoticsin detail, because they may obtained from the formu-las for quasi-elastic scattering in the detector. As an

example, we consider the case of a strong hierarchy ofparameters , , and , where, as in the case of the

scattering, function is especially simple:

+ +1 2 3s s s+ +

2 2 2a bm m m

( ) ( )( )

+ ≤ ≤ , ≤ ≤ − ,

≤ ≤ − .

2 22 21 2

223

b a a

b a b

m m s m m s m m

m s m m

,1 2( )s s

− +≤ ≤ , ≤ ≤ − ,

2 21 1 1 2 ( )as s s m s m m

( ) ( ) ( ) ( )± + − + − − − −= + − .

∓22 2 2 2 2 2 2 2

2 2 2 2 2 21

2

42

b a b a ab

s m s m m s m s m m m ms m ms

, =

μνμ ν

, =

σ= ,

σ= .

ℜ

ℜ ℜ

223

1 22 2 20

223

1 22 2 20

'4

'4

k ls kl

k la b

k ls d kl

k la b

A s sm m m

q q B s sm m m

μνμ νℜ s q q

1s 2s

μν ,μ ν

,

= ≡ , .ℜ

F

1 2

1 2

1 2

'

( )'

k lkl

k ls s

k lkl

k l

B s sq q s s

A s s

'klA '

klB ≤ , ≤0 2k l

( ) ( ) ( ) ( )[ ( ) ] ( ) ( )( )[ ( ) ( )] ( ) ( )

= σ σ σ + σ − + σ σ + σ − + σ σ

× σ + + − + σ + σ − ,

= −σ σ σ + + + σ + σ + σ + σ + σ

= σ σ σ + σ

2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 200

2 2 2 2 2 2 2 2 2 4 2 2

2 2 2 2 2 2 2 4 2 2 2 2 2 2 2 201

2 2 2 2 202 3

'

7 4

' 2 2 2 ,

'

a b a a b a b b a b

a b a b a b a b

a b b a a b b a a

a b b

A m m m m m m m

m m m m m m m m m

A m m m m m m m m

A m m( ) ,2

( ) ( )[ ( )] ( ) ( )( )[ ]( )

= −σ σ σ + + + σ + σ + σ σ + σ +

= σ σ σ + + + σ , = −σ σ σ ,

= σ σ σ + σ , = −σ σ σ ;

2 2 2 2 2 2 2 4 2 2 2 2 2 2 2 210

2 2 2 2 2 2 2 2 2 2 211 3 12

2 2 2 2 2 2 2 2 220 3 21

' 2 2 2 ,

' '2

' '

b a a b a b a b b

a b a b a b

b a a a b

A m m m m m m m m

A m m m m A

A m m A

( ) ( )[ ( ) ( )( )

( )[ ( )] ( )[ ]( ) ( ) ( )

= σ + σ + + σ + + σ + + σ + σ + σ σ

= − σ σ + + σ + + + σ + σ +

+ σ σ + + = σ σ + σ ,

2 2 2 2 2 2 2 2 2 2 2 2 2 200

2 4 2 4 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 4 2 4 2 201

2 2 2 2 2 2 2 2 2 2 2 202 3

10

'

,' 2 2

'2 ,

a b a b a a b b a b

a a b b a b b

a a b b b a b b b a b

b b a b a b b

B m m m m m m m m m mm m m m

B m m m m m m m m m m m m

m m m m B m m m

B ( ) ( )[ ]

( ) ( )

= − σ + σ + σ + σ + + σ × σ + + σ + + ,

= σ σ + + + σ σ + σ σ + σ σ ,

2 4 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 211 3

' 2 22 2

' 2

b a a a a b a b a

a b b a b

a b a b a b b a a b

m m m m m m m m mm m m m m

B m m m m m m m m

( )( )

= − σ σ + σ σ + σ σ ,

= σ σ + σ ,

= σ σ + σ σ + σ σ .

2 2 2 2 2 2 2 2 212

2 2 2 2 2 220 3

2 2 2 2 2 2 2 2 221

'

'

'

a b b a a b

b a a

a b b a a b

B m m m

B m m m

B m m m

be σa σb σ→2 2 ,F 1 2( )s s s



It can be seen that, under strong hierarchy, theshape and the value of function are notaffected by the smallest and the largest of parameters.

APPENDIX B

MULTIDIMENSIONAL GAUSSIAN QUADRATURES

Integrals

(B.1)

where is a symmetric positively definedmatrix and are arbitrary complex constants66, areused often in the main text. Although such integralsare well known (see, e.g., [138]), we detail here a sim-ple calculation of (B.1), since the published paperssuffer from a certain confusion (or, to put it better, lackof agreement) regarding the definition of the matrixinverse to in the Minkowski space. Symmetricmatrix can always be diagonalized with orthogonaltransformation (see, e.g., [139]):

(B.2)

where are (positive) eigenvalues of matrix . There-fore, the quadratic form in the exponent of the inte-grand in (B.1) may be rewritten as

(B.3)

where and, consequently, .The Jacobian of this transform is , so .Inserting (B.3) into (B.1), we reduce the integral to aproduct of standard Gaussian quadratures:

(B.4)

According to (B.2),

The following is then obtained for the 3-dimen-sional Euclidean space:

(B.5)

In the 4-dimensional Minkowski space,

(B.6)

Note that in the latter case, because, It is evident that the

eigenvalues of matrix are ; therefore, it ispositively defined. Naturally, In the case of

most importance to us (with quantities and constituting a tensor and a 4-vector, respectively),

is a Lorentz scalar, since integral (B.1) is alsoa Lorentz scalar.

( )( )( )

( )( )

( )( )

( )

+ − − , σ σ σ , σ − − −

+ − − , σ σ σ , σ − − −

− , σ σ σ , σ + − −, ≈−

− − −

F

@ @

@ @

2 22 21 2

22 2 2 22

2 22 21 2

22 2 2 21

2 221

22 2 21 2

1 2 221

22 2 2 22

for4

for4

for4( )

4

a b aa b

aa a

a b bb a

bb b

a ba b

ba bs

a

a a

s s m m m

s m m m m

s s m m m

s m m m m

s m m

s s m m ms s

s m m

s m m m m

( )( )

( )( )

, σ σ σ , σ − , σ σ σ , σ − − − − , σ σ σ . σ + − −

@ @

@ @

@ @

2

2 222

22 2 2 21

2 222

22 2 21 2

for

for4

for4

a b

bb a

b b

b ab a

aa b

s m m

s m m m m

s m m

s s m m m

66In this study, quantities and are the components of atensor and a 4-vector, respectively, but this is not significant forsubsequent analysis. In addition, (B.4) is independent of thedimensionality and signature of spacetime.

,F 1 2( )s s s

( )μ ν μμν μ, = − + ,&( ) expA B dx A x x B x

μν=A A

μB

μνA μB

AA

μν=O O

μν α μα να μα να μνα α

= , = δ , A a O O O O

αa A

( ) ( )

( )μ ν μ

α μα να μα

α α μ μα αα μ

− +

= − + ,

2

a O x O x B x

a y B O y

μα μα=y O x μ

μα αα=x O y

= 1O =dx dy

( )μ μαμα α α

π , = .

∏&21( ) exp

4A B B O

a a

( )− − μνμνα μα να α

α α= = = . ∏

def1 1 anda O O A A a A

( ) −π, = , = &3

11( ) exp .4 kn k nA B A B B A A

A

( )μν −μ ν

π, = , = . &2

11( ) exp4

A B A B B A gA gA

−≠ 1A A−= = −

10 0 0( )k k kA A A = , ,1 2 3.k

A α >1 0a= 1 .A A

μνA μB

μνμ ν

A B B



APPENDIX CFACTORIZATION OF HADRON BLOCKSLet us demonstrate that, under certain reasonable

assumptions formulated below, hadronic matrix ele-ment (233) may be reduced to form (235).

To this end, we express the matrix element at theright-hand side of (234), which includes elementaryquark currents and , in terms of hadronic cur-rents. Using the definition of a chronological productof local operators, we write67

(C.1)

where

(C.2a)

(C.2b)

and

(C.3)

is the evolution operator for the hadronic part of theLagrangian. Utilizing the well-known properties ofthis operator

we rewrite expression (C.2a) in the following way:

(C.4)

Here, is an arbitrary parameter. Let us nowdefine the hadronic current operator (in the Heisen-berg representation) as

(C.5)

Equation (C.4) may then be rewritten as

(C.6a)In a similar fashion, the following is derived

from (C.2b):

(C.6b)Inserting (C.6) into (C.1), we find

(C.7)

As is known, single-particle hadronic states do notchange under the effect of a hadronic -matrix.Since our states are direct products of single-particle hadronic states, it may be assumed that

to within an insignificant phase fac-tor. Therefore, the following expression derived from(C.7) is true to within this factor:

(C.8)

Let us now discuss matrix element

Owing to the translational invariance, current satisfies equation

(C.9)

where is the complete 4-momentum operator cor-responding to the hadronic part of the Lagrangian(see, e.g., [140]). Using (C.9), one obtains

(C.10)

where

The formal solution of differential equations (C.10)takes the form

(C.11)

where xs and xd are arbitrary space-time 4-vectors, and are the components of a certain tensor68 such

that

A similar result may also be obtained for matrix ele-ment (C.8). Indeed, rewriting (C.8) in the explicit form

67Lorentz indices and normal ordering symbols are omitted forbrevity in this section (when it can be done without ambiguity).

qj†qj

= θ − , + θ − , ,

S†

0 0 0 0

( ) ( )( ) ( ) ( ) ( )

q q hT j x j yx y A x y y x B x y

, = ∞, , , −∞ ,U U U†

0 0 0 0( ) ( ) ( ) ( ) ( ) ( )q qA x y x j x x y j y y

, = ∞, , , −∞ ,U U U†

0 0 0 0( ) ( ) ( ) ( ) ( ) ( )q qB y x y j y y x j x x

τ

τ

τ , τ =

+U

2

1

2 1 0 h( ) exp ( )i dx d xx

τ , τ = τ , τ τ, τ , τ , τ τ , τ = ,∞, −∞ = , ±∞,±∞ = ,

U U U U U

U U

2 1 2 1 2 1 1 2( ) ( ) ( ) ( ) ( ) 1( ) ( ) 1hS

, = ∞, τ τ, , τ τ,× , , τ τ, , −∞

= ∞, τ τ, , τ× τ, , τ τ, −∞ .

U U U U

U U U U

U U U

U U U

0 0 0†

0 0 0 0 0

0 0†

0 0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

q

q

q

q

A x y x j x x x

x y j y y y yx j x x

y j y y

τ

τ→−∞= τ, , τ .U U0 0( ) lim ( ) ( ) ( )qJ x x j x x

, = .S†( ) ( ) ( )hA x y J x J y

, = .S†( ) ( ) ( )hB y x J y J x

= . S S† †( ) ( ) ( ) ( )q q h hT j x j y T J x J y

68In the general case, this tensor may depend parametrically onthe momenta and spins of the initial and final single-particleFock states of hadrons.

S bk

=S†h b bk k

= ≡ .

S†

† †

( ) ( )

( ) ( ) ( ) ( )b q q h a

b a T

k T j x j y k

k T J x J y k J x J y

= .† †( ) ( ) ( ) ( )b aJ x J y k J x J y k

( )J x

μμ∂ = , ,( ) [ ( ) ]i J x J x P

μP

( )

μ

μ μμ

∂ ∂+ = , ∂ ∂

= − ,

† †

†

( ) ( ) [ ( ) ( ) ]

' ( ) ( )

i J x J y J x J y Px y

K K J x J y

∈ ∈

∈ ∈

= + = + ,

= + = + .

' '

's d

s d

s d a aa I a I

s d b b

b F b F

K K K k k

K K K k k

( )( ) ( )( ) − − + − − μ ν

μ ν μν

=× + − ,

' '†

†

( ) ( )

( ) ( ) ( )

s s s d d di K K x x K K y x

s d

J x J y e

J x J x C x y

μν( )C x

μν − =( ) 0.s dC x x

= θ −+ θ − ,

† †0 0

†0 0

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )TJ x J y x y J x J y

y x J y J x



and taking relation and (C.10) into account, we obtain the followingequations:

(C.12)

Their solution coincides with (C.11), where issubstituted by . It is convenient for our purpose tochoose the impact points for in- and out-packets inthe source and detector as 4-vectors and , respec-tively. Since points and are macroscopicallyseparated, currents and are mutuallycommuting: As a result, matrix

element is factorized into two cofac-tors that are associated with the source and detectorvertices and depend on the corresponding variablesonly. Thus, having introduced 4-vectors ( -numberhadronic currents)

one obtains

(C.13)

Tensor components , which satisfy con-ditions

(C.14)

may feature a singularity (no stronger than a func-tion or its derivatives) only at point . At the sametime, by virtue of condition (C.14), they should vanishafter integration over and performed over suffi-ciently small space-time volumes surrounding theimpact points. Functions enter the ampli-tude only via integral

(C.15)

where and are functions describing the WPs

of final leptons and with momenta and ,

respectively. The contribution of product to theintegrand in (C.15) is significant only within the clas-

( )μ μ∂ ∂ + ∂ ∂ θ − =0 0( ) 0x y x y

( )μ μ

μ

∂ ∂+ ∂ ∂

= − .

†

†

( ) ( )

' ( ) ( )

T

T

i J x J yx y

K K J x J y

T

sx dxsX dX

( )sJ X †( )dJ X

μ ν, =†[ ( ) ( )] 0.s dJ X J X

μ ν†( ) ( )s d T

J X J X

c

( ) ( )

( ) ( )

−

−

; , = ,

∈ , ∈ ,∗ ; , = ,

∈ , ∈ ,

)

)†

( )

'

( )

s s s

d d d

i K K Xs s a b b s a

s s

i K K Xd d a b b d a

d d

X k k e k J X k

a I b F

X k k e k J X ka I b F

( ) ( )[ ] ( ) ( )

μ ν

μ νμν

= − + −∗× ; , ; , + − .

†( ) ( )exp

( )

T

s s d d

s s a b d d a b

J x J yi K K x K K y

J X k k J X k k C x y

μν −( )C x y

μν − = ,( ) 0s dC X X

δ=x y

x y

μν −( )C x y

−μν

α α α β β β

∝ −

∗ ∗× ψ , − ψ , − ,

( ) ( )

( ) ( )

iq x ydxdye C x y

x x x yp p

α∗ψ β

∗ψ ψ+α

−β αp βp

α β∗ ∗ψ ψ


sical world tubes located near the correspondingimpact points. Therefore, the integrand in (C.15) isnot negligible only if they have a considerable overlapregion (in the limiting case, if the axes of lepton tubescoincide). However, this configuration is strongly sup-pressed by virtue of the approximate energy-momen-tum conservation. This is evident from the analysis ofthe simplest case with Integral(C.15) is then proportional to the smeared function

and is negligible, since These qualitative arguments enable one to ignore non-physical long-range correlations induced by term

in (C.13). By virtue of (C.8), one canrewrite the right-hand side of (233) as

Taking the properties of the form factor in and into account, one can substi-

tute variables and in functions and by thecorresponding external momenta and . Havingintroduced this (last) approximation, we arriveat (235), thus completing the proof.

APPENDIX D

STATIONARY POINT FOR AN ARBITRARY CONFIGURATION

OF MOMENTA OF EXTERNAL WAVE PACKETS

Let us detail the algorithm for solving Eq. (249) inthe general case (i.e., for an arbitrary configuration ofexternal momenta). The general solution is of interestboth in the methodological context and for purposesof data processing in neutrino experiments at interme-diate (subrelativistic) energies. Although the proposedalgorithm is rather cumbersome, it is easy to imple-ment in the form of a computer program and is thusconvenient primarily for numerical analysis.

Form (267) of Eq. (249), where the velocity of avirtual neutrino is the unknown quantity, is more con-venient to work with. Raising both parts of (267) to thesecond power, we arrive at algebraic equation of thefourth order

(D.1)

μν − ∝ δ −( ) ( ).C x y x yδ

α βδ +( )p p α β α β+ ≥ +0 0 .p p m m

μν −( )C x y

( )[ ]

( )[ ]

( ) ( )

− −

− − −

μ ν

φ , π

∗φ ,× π

∗× ; , ; , .

∏

∏) )

3

3

( )(2 ) 2

( )(2 ) 2

a a a a

a

b b b b

b

i k p x k xa a a a

a

i k p x k yb b b b

b

s s a b d d a b

d eE

d eE

X k k X k k

k

k

k k p

k k p

φ ,( )a a ak p φ ,( )b b bk pak bk ) s )d

ap bp

+ + + + = ,v v v v4 3 2

3 2 1 0 0c c c c

1 2020


with coefficients

Here, ; it is assumed from now on that (the case of a massless neutrino is trivial), and

index “ ” numbering neutrinos is omitted to simplifythe formulas, which are already cumbersome enough.The other designations are the same as in the maintext. Equation (D.1) can be solved using the Des-cartes–Euler method (see, e.g., [141]). Following thecorresponding algorithm, we write Eq. (D.1) in the"incomplete” form

(D.2)

The solutions of this equation are constructed fromthe roots of cubic equation

(D.3)where

Equation (D.3) may also be reduced to the incom-plete (Cardano) form:

Here,

−= ,+ η

+ − η= − ,+ η

+ + + − η= ,+ η

+ + η= − .+ η

η

η

η

η

2 2

0 2 20

20

1 2 20

2 2 2 20

2 2 20

20

3 2 20

( ) ( )( )

( ) 2( ) ( )2( )

6( ) 4 ( ) ( )( )

( ) 2( ) ( )2( )

c

Rc

R Rc

Rc

Rl lRl

Rl Rl lRl

Rl Rl lRl

Rl Rl lRl

μ μη = jY m> 0jm

j

+ + + + + + = .

v v v

4 23 3 3

2 1 0 04 4 4c c cc c c

+ + + = ,3 22 1 0 0z a z a z a

−= − , = , = .

222 01 2

0 1 24

64 16 2c cc ca a a

+ + + + = .

p q3

2 2 03 3a az z

[ ][ ]

[ ]

+ − η += − = − ,+ η

= − + = − , + η

p

q

η22 2 22

021 22 2

03

1 2 20 32 2

0

4 ( ) ( )3 48 ( )

23 3 864 ( )

R Raa

a a a Aa

Rl l

Rl

Rl

= + + + ,2 30 1 2 3( ) ( ) ( )A A A A ARl Rl Rl

[ ][ ] [ ]

[ ][ ]

[ ]

= − η −

+ η + η + − η − ,= − η −

+ η − η + η ,= − η + + η ,

= .

η

η η η

η

η η

η

6 2 2 40 0

34 2 2 4 2 2 20 0 0

4 2 2 21 0

2 2 20 0

2 2 2 2 42 0

33

3 ( )

3 16 ( ) ( ) ( )

12 2 ( )

7 ( ) ( ) ( )

48 ( ) 54( )

64

A R R

R

A R R R

l

A R R l

A R

l

l l l

l

l l

l

PHYSICS O

The number of real roots is defined by the sign offunction

which coincides with the sign of polynomial + with coefficients

Three different real roots are found at ; at, one real root and two mutually conjugate com-

plex roots are present; at , two (or all three) realroots may coincide. The validity of the following use-ful identity may be proven:

(D.4)Del Ferro–Tartaglia–Cardano solution in radicals.

The roots of incomplete cubic equation (D.3) are

where

at and at . The expression for is simplified if identity (D.4) is taken into account:

Solution in the Vieta’s trigonometric representation.For completeness, we also present the more compacttrigonometric solution (Vieta’s representation), whichmay turn out to be more convenient for numerical cal-

[ ] η η − η = + = ,

+ η

q pV

222 30

62 20

( ) ( ) ( )

4 27 27 648 ( )

R l Bl Rl

Rl

= +0 1( )B B B Rl +2 32 3( ) ( )B BRl Rl

[ ][ ] [ ]

[ ][ ][ ]

[ ]

= − η −

+ η + η + − η − ,= − η −

+ η − η − η ,= − η + + η ,

= .

η

η η η

η

η η

η

6 2 2 40 0

34 2 2 4 2 2 20 0 0

4 2 2 21 0

20 0

2 2 2 2 42 0

33

3 ( )

3 7 ( ) ( ) ( )

6 2 4 ( )

2 ( ) 2( ) ( )

48 ( ) 27( )

64

B R R

R

B R R R

l

B R R l

B R

l

l l l

l

l l

l

< 0B> 0B

= 0B

[ ]= + η η − η .22

027 ( ) ( ) ( )A B R ll Rl

+ −

± + − + −

= + + ,

= − + ± − ,

0 ( )

1 3( ) ( )2 2

z a A A

z a A A i A A

[ ]

[ ]

δ

±

+ + += = − ,+ η

± η η − η= − ± = ;

+ ηV

2 30 1 2 32

22 20

203

32 20

( ) ( ) ( )3 12 ( )

( ) ( ) ( )18 3

2 96 ( )

C C C Caa

A Bi R lqA

Rl Rl Rl

Rl

l Rl

Rl

[ ][ ]

= −η − η − ,= − η η − ,= − η η − ,

= ;

η

η

η

2 2 2 20 0 0

1 0 02

2 0

3

2 2 ( )

2 4 3( )

2( ) 5 ( )4

C R

C R

C R lC R

l

l

l

δ = 0 ≥ 0B δ = 1 < 0B±A

[ ]

/δ

±

η − η ±= .

+ η

η2 32

0

2 20

3 3 ( ) ( ) ( )

12 ( )

R l i BA

l Rl

Rl



culations. If nothing else, it is useful for monitoring theaccuracy of calculations by means of comparison withthe canonical solution. The explicit form of the trigono-metric solution depends on the sign of function .

B < 0. As was already noted, Eq. (D.3) has threereal roots in this case (sometimes called the irreduciblecase):

where

B ≥ 0. Equation (D.3) has one real and two com-plex roots. Let us introduce the following notation69:

Then, the roots are

Roots of Eq. (D.1). The roots of incomplete equa-tion of the fourth order (D.2) are given by combina-tions

where four out of the possible eight combinations ofsigns are chosen so that condition

is satisfied. Here,

69Note that and ; the real value of the cubicroot is taken in all cases.

B

( )±

α= + ζ ,

α ± π= − ζ ,

0 0

0

cos3

cos3

z a

z a

[ ]+ − η + ηζ = ,

+ η

α = − .+ − η + η

2 2 20

0 2 20

2 2 20

4 ( ) ( )6 ( )

cos4 ( ) ( )

R R

AR R

Rl lRl

Rl l

α ≤ π' 4 β ≤ π 2

βα = , β = −α

= + − η + η .

3

32 2 20

4tan ' tan sin2 cos

4 4 ( ) ( )R RA

Rl l

( )±

= − ζ α ,

= + ζ α ± α .

0 0

0

cosec2 '1 cosec2 ' 3 cot 2 '2

z a

z a i

− +Ξ = ± ± ± ,0n z z z

[ ]

− +− =

+ + + +=+ η

10

2 3 40 1 2 3 4

32 20

8( ) ( ) ( ) ( )

8 ( )

cz z z

D D D D DRl Rl Rl Rl

Rl

( )[ ]

[ ][ ][ ]

= η η + η ,= η − η + η η − η ,

= η η − η − η× η − η η + η ,= η − η η + η ,

= η .

3 2 20 0 0

2 2 2 21 0 0 0

22 0 0

2 20 0

2 23 0 0

24

( )

3 4 ( ) 2( )

2 3 ( ) ( )

6 3 ( ) ( )

9 8 ( ) ( )

2( )

D R

D R R

D R l

D R

D l

l

l l

l

l l

l l


All four roots of Eq. (D.1) can then be derivedusing formula

The real nonnegative root of interest to us, whichcorresponds to the stationary point, should satisfy thecondition of positivity of the second derivative (251).The solutions found in the main text for two oppositelimiting cases ( and ) may serve as addi-tional criteria of uniqueness of the solution of the gen-eral form based on the algorithm described here, sincethey should “join” the correct numerical solutionsmoothly under the corresponding variations ofmomenta of the external WPs and discrete parametersdefining the effective velocity of a virtual neutrino.

APPENDIX ESPREADING OF A NEUTRINO WAVE PACKET

AT EXTREMELY LONG DISTANCESLet us consider the generalization of some results

from the main text to the case with spreading of aneffective neutrino packet at astronomical distances.This generalization can apply in the processing of datafrom neutrino telescopes (Baikal GVD, IceCube,KM3Net ARCA, etc.), various experiments on radiodetection of ultrahigh-energy neutrinos, and futureorbital experiments that are aimed, among otherthings, at measuring the f lavor composition of neu-trino and antineutrino fluxes from remote astrophysi-cal sources.

The integration over in (302) yields the follow-ing expression for the decoherence factor:

(E.1)

(E.2)

Here,

= Ξ − = , , , .v 3 4 ( 1 2 3 4)n n c n

− v !1 1 ∼v 1

0x

( )π= Φ

Δ× − − +

Δ − − − +

v

v

D

DD

DD

02

01

0

0 01

0 02

exp4

erf 24

erf 2 ,4

y

ij ijij y

ijij

ij ij

ijij

ij ij

S dy

i ELy x

i ELy x

Φ = − −

Δ Δ − + − Δ

v v

v

D D

D

D

22

2

2

*1 1

.4

i jij

i j ij

ij ijij

ijij

L

E Ei P L

( )

( )

Δ = − , Δ = − , = + ,

= + ,

v v v

DD

D

D D D

22

2

22 2

*1 12

1 *2

ij i j ij i j

ji

ij i jij

ij i j

P P P E E E

1 2020


and the other notations are the same as in the maintext. Using the primitive of error function (306) intro-duced in the main text and integrating over in (E.1),we find the generalization of formula (303):

(E.3)

Although this expression is much more complexthan (303), there are no difficulties in analyzing thedecoherence effects numerically. In the present sec-tion, we examine only the key properties of complex-valued phase . To this end, we need to isolate realand imaginary parts of the phase and determine thelength scales governing its behavior.

Using the definitions introduced above and drop-ping evidently small corrections, we find70

Inserting these expressions into (E.2), one obtains

(E.4)

where

(E.5)

It follows from the analysis of (E.4) that the realpart of phase becomes independent of withintwo (strongly) spatially separated regions defined bythe following conditions:

and

The corresponding asymptotic values of cal-culated to an accuracy of take the form

(E.6a)

(E.6b)

Here and elsewhere, the common definitions foroscillation lengths and differences between the neu-trino masses squared are used:

It can be seen that the asymptotics at large ,, is almost independent of neutrino masses and

factor ; both and become arbitrarilylarge if tends to zero. In addition, obvious relations

0y

( ) + +

, =

π= Φ −

Δ× − + − .

v

D

DD

2' 1

2' 1

0 0'

exp ( 1)8

Ierf 24

l lij ij

l lij

ijij l l

ij ij

S

ELx y i

Φij

( ) ( )

( )( ) ( )

( ) ( )

ττ τ − τ − + + − τ + τ − = ,

ττ + + −

+ τ + τ + − τ τ τ − τ = , = . − τ+ τ + τ

22

22 2 2 2

22 2

1 1 1 12

1 1

*2 112 1 1

jii j i j

j i i j i jij

ji

i j j i

i j i j i j i jij

iji j ij

i

D ii

v v v v v v

v

v v v v

D D DD

D

( )( )( )

Δ +Φ = − + − +

− − + +

+− − Δ − Δ +

Δ + − − − , +

2

2

22 2

2

2 22 2

1 1 14 21

1 1 1 1 12

1 14 1

1 111 4

ij i jij ij

ij i jij

iji j i j

ij i jij ij

i jij

ij iji j

ij i j

EL

L i

E P L

i EL

r

v v

v v v v

v v

v v

r rD

Dr

D

r r r

r

rr r D

D

− /

ν

+= + ,

− Δ= ≈ .

r rD D

r r Dr

1 22 2

2 2

3

12

22

i jij

i j ijij

m L

E

Φij L

ν= nLD

! 2 ,4

EL

ν ν

ν ν

+

,= −

− + .

D

L

D

@ @@

!

2

2 2

2

2 2

2( )if or

2if

i j

i j i jij

i j

i j i j

E Em m

m m m mL

E Em m

m m m m

70Note that , and

.

ΦRe ij

=v v ,jj j ( )= +D D2 2 21jj jr

= D D D2

j jj

,2( )i jr

( )

Φ ≈ Φ

π = − + + ,

L

n m

D n

!

0

2

Re Re

12

ij ijL

i jij

r rL

( )[ ]

∞

ν

Φ ≈ Φ

= − − + + .

L

nD

@

2

Re Re

1 4( 1)4

ijij ijL

i jE r r

νπ= , Δ = − ≠ .Δ

2 2 22

4 ( )ij ij i jij

EL m m m i jm

L∞ΦRe ij

n Φ0Re ij∞ΦRe ij

D

( )[ ]∞,

= Φ±

= − Φ +

LL

n

n 2

0and Re

( )Re 1

ij iji j

i j ij i j

r r

r r r



Fig. 25. Variation of function with (in kiloparsecs) calculated at eV, eV, and GeV for

seven values of ratio (indicated next to the curves). Function was estimated under the assumption that the dominantcontribution to it is produced by the decay in the source.

15

10

5

0

–5

–10

–15

log(

−R

e Φ

ĳ)

Eν = 10 GeV

10–21

10–3 10–6 10–9 10–12 10–15 10–18

10–12 10–8 10–4 100 104 L, kps

Φlog Re ij L = .0 1im = .0 01jm ν = 10E

νD E n

μπ 2

lead to model-independent inequalities

Therefore, as is seen from (E.4), a wide interval ofdistances , where increases qua-dratically with , exists for any pair of neutrinos :

All these features of behavior of the real part ofphase are illustrated by Fig. 25, which presents func-tion within a wide range of values, whichvary from terrestrial ( km) to cosmological( Mpc) distances, at several arbitrary values ofdimensionless ratio . The curves in this figurewere calculated for eV, eV, andEν = 10 GeV. In order to estimate the value of import-ant factor , we assumed, for illustrative purposes,that it is saturated by the contribution from piondecays (the typical source of accelerator, atmospheric,and astrophysical neutrinos) and that the contributionof reactions in the detector is negligible. It was alsoassumed that the pion WP in the momentum space ismuch wider than the muon packet; i.e., . It isthen easy to demonstrate that factor has the follow-ing approximate form at ultrarelativistic energies,

:

∞Φ Φ .L L! !0and Re Reij ij ij

L L! ! ijL ΦRe ij

L ν ,νi j

Φ ≈ − − ≈ − − .

v v

D D

22 2 2 2 21 1Re ( )ij i j

i j

L r r L

Φlog Re ij L* 100

∼

410νD E= .0 1im = .0 01jm

n

μ πσ σ!

n

πΓ @ 1

ν ν ν≈ ≈ . ≈ . × .nw 2 2 5( ) ( 29 8 MeV) 1 1 10E E E


The results of a similar analysis for the imaginarypart of phase (E.4) reveal that this part increases lin-early (although with different coefficients) with inregions and . In particular,

(E.7a)

(E.7b)

These two regimes are separated by a relatively nar-row transition region near . Nothing unex-pected occurs in the region. This behavior of

is illustrated by Fig. 26, which shows the

ratio calculated with the same approxi-mations that were used for the real part of the phase.The unconventional behavior of function at

is of a purely academic interest, since it isrendered unmeasurable by an enormous suppressionfactor

which eliminates interference terms ( ) found inthe expression for the event count rate. Thus, theoscillatory behavior of the count rate is encounteredonly at . This is exactly the region thatwas examined thoroughly in the main text.

LL! ijL L@ ijL

[ ]πΦ ≈ Φ = + + ,L

n!

st 2Im Im 1 ( )ij

ij ij i jLij

L n r rL

∞

Φ

π ≈ Φ = + + + .

Im

3 1Im 1 ( 1)( )2 3

ijij L

ij i jij

L r rL

L

n

@

= L ijLL&L

ΦIm ij

Φ ΦstIm Imij ij

ΦIm ij

L* ijL

( )− / ν ∝ + − , r D

1 42 2 21 exp (16 )ij E

≠i j

L! min( )ijL

1 2020


Fig. 26. Ratio as a function of calculated with the same parameters as those used for the real part of the phase (see

the caption of Fig. 25) for three values of ratio (indicated next to the curves).

Im Φ

ĳ/ Im

Φĳst

1.00

0.95

0.90

0.85

0.80

0.75Eν = 10 GeV

10–3 10–610–9

10–12 10–8 10–4 100 104 L, kps

ΦstIm Imij ij L

νD E

APPENDIX F

SPATIAL AVERAGINGIn realistic scenarios, the sizes of the source and

detector are not always negligible compared to the dis-tance between them. If this is the case, one needs toperform accurate spatial averaging of the count rateover the effective volumes of the source and detector.The plausible spatial inhomogeneities of the neutrinobeam and the detector medium need to be taken intoaccount. In the present section, we limit ourselves tothe simple (but methodologically interesting) casewhen such inhomogeneities are negligible. This impliesthat density functions are independent of within the source and detector volumes and turn to zerooutside the bounds of these volumes. Within theseapproximations, we are interested only in -dependentfactor (with ) in thecomplete expression for the count rate; it is worthreminding that and do not depend on .

The following simplification is also adopted below:the linear size of the detector in the direction of theneutrino beam is assumed to be negligible comparedto the size of the source in the same direction, which,in turn, is small compared to the average source–detector distance. The origin of coordinates is locatedwithin the detector (point in Fig. 27), and axis iscodirectional with unit vector (i.e., directed towarda certain inner point of the source). Let and

; the sought-for integral over the spatial vol-ume of the source may then be written as follows:

(F.1)

Here, is the working volume of the source, isthe solid angle under which this volume is seen frompoint , and and are the distances from to

, ;( )a a af sp x x

LΦ ( ) 2ij Le L Φ = ϕ −! 2( ) ( ) ( )ij ij ijL i L L

@2ij Θij L

dO z−l

sO = Lx l= s dL O O

Ω

Ω

Φ Φ

Ω

≡ = . ) Ω( ) ( )

2V

F

ij ij

Ns s

LL L

ij

L

d e d dLeLx

Vs Ωs

dO Ω

NL Ω

FL dO

PHYSICS O

the near and far boundaries of the source (for simplic-ity, it is assumed to be convex) along unit vector

. It is convenient tointroduce the following measure of distance betweenthe source and detector:

To avoid any misunderstanding, we note that,depending on the angular resolution of the detectorand features of the experimental setup, solid angle may turn out to be smaller than the full solid angleunder which the source is seen. Figure 27 illustratesthis possibility schematically, while equality (F.1)holds true in the general case. Note also that the small-ness of angle is not equivalent to the smallness ofthe source itself; a well-collimated meson beam froman accelerator (neutrino factory) in a long decay chan-nel is an important counterexample. Elementary inte-gration over yields

(F.2a)

and

(F.2b)

These general formulas may be used in processingthe data of short-baseline neutrino experiments,where the distance from the source (e.g., a decaychannel) to the detector is comparable to the longitu-dinal size of the source.

( )= φ θ, ϕ θ, θΩ sin sin cos sin cos

( )Ω

= + .Ω Ω ΩΩ1

2s

F N

s

L d L L

Ωs

Ωs

L

ν

Ω

ν νΩ

ν ν

ν

=π

π π× − − −

+ Δ× − ≠

)

Ω

Ω

2 2

2

4

2 2erf erf2 2

2 ( )exp ( ),

8

s

ijij

F N

ij ij

ij

E Ld

D

iE iEDL DLE L D E L D

E Ei j

D

( )Ω ΩΩ

= = − . ) Ω2Vs s

F Njj

d d L LLx



Fig. 27. Schematic illustration of spatial averaging.

Source

Detector

Os

Od

L

Ω

Ωs

LFΩ

LNΩ

In an ideal experiment with

(F.3)

one can use the following expansion of the error func-tion:

(F.4)

The contributions of order and inthis expansion may be neglected under the assumptionthat and . In the case under consider-ation, the first condition implies that

(F.5)

while the second one is not necessary due to theapproximate cancellation of terms of second order.Using (F.4), we indeed find that

where

Evidently, . Assuming that

(F.6)

we arrive at the following result (valid for any and ):

(F.7)

which could be deduced from the mean-value theo-rem. Our result, however, is supplemented by fairlynontrivial sufficient conditions (F.3), (F.5), and (F.6),which are impossible to obtain from the mean-valuetheorem alone. Volume in (F.7) is estimated (withthe same accuracy) as

If we now assume that working (reference) detectorvolume is sufficiently small compared to (this

( )

( )ΩΩ∈Ω

ΩΩ∈Ω

= −

= − ,

max

and maxs

s

NN

FF

r L L L

r L L L

!

!

−

+ δ ≈δ + − δ + − δ + .

π

2 2 2

erf( ) erf( )2 21 (2 1)

3z

z z

e z z …

δ22( ) δ2

2 2( )z

δ ! 1 δ ! 1z

,

ν

π,!

21N F

ij

DrE L

( ) Φ

Ω

Ων

≈ −

π π× − Δ − ,

) Ω ΩΩ( )

22 21

ij

s

LF Nij

ij ij

d L L e

i L DLL E L

Ω + − −Δ = − = − .

Ω Ω Ω Ω2 12

N F F NL L L L L LL L L

ΩΔ ! 1/

Ω∈Ων

π πΔ + ,

Ω

!

1 24 22 2max 1

s ij ij

DL LE L L

i j

( )Φ

Φ

Ω

≈ − ≈ ,) Ω ΩΩ

( )( )

2Vij

ij

s

LL F N

ij see d L L

L

Vs

( ) ( )Ω

Ω

= = −

≈ − .

Ω Ω

Ω Ω

Ω

Ω

3 3

V2

1V3

( )s s

s

F Ns

F N

d d L L

L d L L

x

Vd Vs


condition is satisfied in a typical neutrino experiment)and that the detector geometry is not too fancy, theintegration over becomes trivial and yields

(F.8)

where still denotes the conventional measure of dis-tance between the source and detector.

In order to illustrate the importance and nontrivialnature of conditions (F.3), (F.5), and (F.6), we con-sider the simplest case of a spheroidal source withradius . Its angular size does not exceed the angu-lar resolution of the detector. It follows from simplegeometric considerations that and,naturally, . Therefore,

(F.9)

To simplify the scenario further, we assume that (this inequality is always fulfilled for solar

and astrophysical neutrinos detected on the Earth)and (this is not always true forremote astrophysical sources, but is valid for the Sun).

y

Φ Φ≈ ,

( ) ( )

2 2V V

V Vij ij

d s

L L

s de ed d

L Ly x

L

r θs

Δ = − θΩ 2(1 cos )= =N Fr r r

( ) ( )∈ΩΔ = Δ = − θ ≈ θ ≈ .Ω Ω

Ω

22max 2 1 coss

ss s r L

π @2 ijL L

νπ &2 ijL L E D

1 2020


Condition (F.5) is then satisfied automatically, whilecondition (F.6) takes the form

(F.10)

This inequality is definitely not fulfilled for the Sunand the currently adopted value of . The regionsof efficient neutrino production in the solar core arerelatively thin concentric layers with typical radii vary-ing from for 8B, 7Be, and CNO neutrinos to

for , , and neutrinos71 ( is the solarradius). The left-hand side of (F.10) may then be esti-mated as

which demonstrates (and this is no news) that approx-imation (F.8) is definitely not applicable in the analy-sis of oscillations of solar neutrinos.

Summarizing the results, we note the following:although the above analysis was simplified in manyrespects72, it demonstrates that the finite size of thesource needs to be taken into account to process datacorrectly in both short-baseline oscillation experi-ments and very-long-baseline experiments (includingthe ones with on the order of an astronomical unit,which is the case with experiments involving solarneutrinos).

APPENDIX G COMPLEX ERROR FUNCTION AND ASSOCIATED FORMULAS

The error function and the complementary errorfunction of complex argument have already been stud-ied extensively (see, e.g., [146–150] and referencestherein). Let us list here some results that were used inSection 11 to analyze the decoherence function. Forconvenience, we first write the following well-knownformulas [103]:

(G.1)

(G.2)

71See [142, 143] for details.72A more accurate analysis should cover the spatial distributions

of colliding and/or decaying particles in the source and theeffects of interaction of virtual neutrinos with the sourcemedium (namely, coherent scattering of neutrinos off electronsin the medium, or the MSW effect [144, 145]), which are cru-cial for astrophysical applications such as experiments withsolar neutrinos.

π .!

22 1ij

rL L

Δ 212m

. 0 1R. 0 3R pp pep hep R

−ν

Δπ ≈ , . ×

2 22125 2

12

1 MeV2 260 2 8 10 eV

mr rL L R E

L

∞ ∞+ +−

= =

−= = ,+ ! + !!π π

22 1 2 1

0 0

( 1)2 2 2erf( )(2 1) (2 1)

n n n nz

n n

z zz en n n

( )

∞−

=

− − !!+ π π→ ∞, < .

∼

2

21

( 1) (2 1)erfc( ) 1(2 )

3arg4

nz

nn

nezz z

z z

PHYSICS O

The following expansions for function ,which was introduced in the main text, are derivedfrom (306) and (G.1):

(G.3a)

(G.3b)

These expansions are useful for small and interme-diate values of 73, respectively. In order to determinethe asymptotics of functions and atlarge , one needs to use (G.2) for and thenapply the following rule:

The result derived from (306) and (G.2) is

(G.4)

where the upper (lower) sign should be taken for ( ).

The formulas given below are helpful in high-accu-racy numerical calculations of the error function. Theyare based on the following integral representation ofthe complementary error function (see, e.g.,[146, 149]):

From this it follows that (cf. the result in [147])

where

73In practice, (G.3a) and (G.3b) work fine at and, respectively.

Ierf( )z

= + − + − + , π2

4 6 82 101Ierf( ) 1 ( )

6 30 168z z zz z z

− = + + + + + . π2

2 4 6 82 104 8 161 2 ( )

3 15 105

ze z z zz z

z

& 1z.& &1 4 5z

erfc( )z Ierf( )zz −erfc( )z

= − − .erfc( ) 2 erfc( )z z

( )

−± +

π × − + − + → ∞ ,

∼

2

2

2

2 4 6 8

Ierf( )2

3 15 105 112 4 8

zez zz

zz z z z

< πarg 3 4z > πarg 3 4z

∞ − += .π +

2 2( )

2 20

2erfc( )t zz dtez

t z

[ ]

[ ]+ −

+ −

+ = + − ω π × , + , ,

+ = − − ω π × , + , ,

( (

( (

2

20 2

2

20 2

Re erfc( ) exp cos(2 )

cos( ) ( ) cos( ) ( )

Im erfc( ) exp cos(2 )

sin( ) ( ) sin( ) ( )

ra ib r

r w a b w a b

ra ib r

r w a b w a b

∞ −

−∞∞ −

−∞

±

, =+ − +

= , + ω + ω

= + ω, = + ,ω = , ω =

2

2

2 2 2 2 2 2

2 22 2 2

2 2

( )( ) 4

cos(2 ) sin(2 )

2cos sin .

n t

n

n t

dtt ea bt a b a b

dtt e

t r r

w ab r a ba r b r

(



All the quantities found here are real. Note that theintegrands in integrals are positive, nonsingu-lar (with the exception of the trivial case of ),and decrease rapidly at large . These properties arehelpful in numerical integration based on the standardquadrature formulas.

ACKNOWLEDGMENTS

We would like to thank E.K. Akhmedov, V.A. Bednya-kov, S.M. Bilenky, Z.G. Berezhiani, F. Vissani, M.I. Vysotsky,A.Z. Gazizov, M.O. Gonchar, D.S. Gorbunov, C. Giunti,I.P. Ivanov, I.D. Kakorin, A.E. Kaloshin, D.I. Kazakov,S.E. Korenblit, K.S. Kuzmin, V.A. Li, W. Potzel, V.A. Rubakov,D.V. Taichenachev, O.V. Teryaev, K.A. Treskov, A.I. Frank,A.S. Sheshukov, M.I. Shirokov, and D.S. Shkirmanov forstimulating discussions.

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Translated by D. Safin


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Quantum Field Theory of Neutrino Oscillations