FERMILAB-PUB-19-326-T

Eigenvalues: the Rosetta Stone for Neutrino Oscillations in Matter

Peter B. Denton∗

Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA

Stephen J. Parke†

Theoretical Physics Department, Fermi National Accelerator Laboratory, Batavia, IL 60510, USA

Xining Zhang‡

Enrico Fermi Inst. and Dept. of Physics, University of Chicago, Chicago, IL 60637, USA(Dated: May 1, 2020)

We present a new method of exactly calculating neutrino oscillation probabilities in matter. Weleverage the “eigenvector-eigenvalue identity” to show that, given the eigenvalues, all mixing anglesin matter follow surprisingly simply. The CP violating phase in matter can then be determinedfrom the Toshev identity. Then, to avoid the cumbersome expressions for the exact eigenvalues, wehave applied previously derived perturbative, approximate eigenvalues to this scheme and discoveredthem to be even more precise than previously realized. We also find that these eigenvalues convergeat a rate of five orders of magnitude per perturbative order which is the square of the previouslyrealized expectation. Finally, we provide an updated speed versus accuracy plot for oscillationprobabilities in matter, to include the methods of this paper.

I. INTRODUCTION

While the majority of the parameters in the three-neutrino oscillation picture have been measured, mea-surements of the remaining parameters will come byleveraging the matter effect in long-baseline experimentssuch as the currently running T2K and NOvA experi-ments, the now funded and under construction T2HKand DUNE experiments and the proposed T2HKK andESSnuSB experiments, [1–6]. In this context, only afull three-flavor picture including matter effects is ad-equate to probe the remaining parameters. Given thetime and effort that is going into these experiments, itis paramount that we understand neutrino oscillations inmatter as best as we can, both analytically and numer-ically, so as to maximize the oscillation physics outputfrom these major experiments.

The matter effect is the fact that while neutrino prop-agation in vacuum occurs in the mass basis, in mattersince the electron neutrino experiences an additional po-tential, they propagate in a new basis. This effect wasfirst identified in 1978 by Wolfenstein [7]. Exact analyticsolutions for neutrino oscillation probabilities in constantmatter densities are difficult to fully enumerate; a so-lution using Lagrange’s formula appeared in 1980 [8]1

while the full solution was first written down for three fla-vors in 1988 by Zaglauer and Schwarzer (ZS) [10]. Theexact solution requires solving a cubic equation which,in the general case, has the unsightly and impenetrablecos( 1

3 cos−1(· · · )) term present in the eigenvalues which

∗ pdenton@bnl.gov; orcid # 0000-0002-5209-872X† parke@fnal.gov; orcid # 0000-0003-2028-6782‡ xining@uchicago.edu; orcid # 0000-0001-8959-84051 For more on Lagrange’s formula in the context of neutrino oscil-

lations in matter see ref. [9].

are then in nearly every expression involving neutrinooscillations in matter2. Given the eigenvalues, there arethen several choices of how to map this onto the oscilla-tion probabilities. ZS mapped the eigenvalues onto theeffective mixing angles and CP phase in matter; given thephase and the angles, it is then possible to write downthe oscillation probabilities in matter using the vacuumexpressions and the new phase, angles, and eigenvalues.In 2002, Kimura, Takamura, and Yokomakura (KTY)presented a new mapping from the eigenvalues onto theoscillation probabilities by looking at the products of thelepton mixing matrix that actually appear in the proba-bilities [12, 13], see also [14–17]. Another formulation ofthe exact result in the context of the time evolution oper-ator is ref. [18]. Along with these exact expressions, nu-merous approximate expressions have appeared in the lit-erature in various attempts to avoid the cos( 1

3 cos−1(· · · ))term, for a recent review see ref. [19].

In this article we use the eigenvector-eigenvalue iden-tity, that has been recently and extensively surveyed in[20, 21], to write the exact expressions for the mixing an-gles in matter in terms of the eigenvalues of the Hamil-tonian and its principal minors. The benefit of this ap-proach is two-fold. First, it makes the expressions forthe mixing angles in matter, clearer, symmetric, and verysimple. Since our approach for the oscillation probabili-ties in matter is based on the form of the vacuum expres-sions, the intuition that exists for the vacuum still appliesin matter. Second, it allows for a simple replacement ofthe complicated exact eigenvalues with far simpler ap-proximate eigenvalues in a straightforward fashion. Wefind that since this approximate approach only relies on

2 One interesting expression that does not contain thecos( 1

3cos−1(· · · )) term is the Jarlskog invariant in matter [11].

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approximate expressions for the eigenvalues, it is moreaccurate than previous methods, including Denton, Mi-nakata and Parke (DMP) [22], with a comparable levelof simplicity. We also explore the convergence rate ofthe eigenvalues in DMP and find that since all odd-ordercorrections vanish, they converge much faster than ex-pected, at a rate of ∼ 10−5.

II. AN EIGENVALUE BASED EXACTSOLUTION

The technique of ZS is to determine expressions for the

mixing angles and CP-violating phase in matter (θ23, θ13,

θ12, and δ) as a function of the eigenvalues and otherexpressions, while KTY derives the general expressionfor the product of elements of the lepton mixing matrix,UαiU

∗βj . In this section, we describe a technique of using

both approaches.First, we note that, given the eigenvalues, the mixing

angles can be determined from various |Uαi|2 terms. Thisemploys a simpler version of the main result of KTY.Then, to address the CP-violation part of the oscillationprobabilities, we use the Toshev identity [23].

A. Mixing Angles in Matter

The neutrino oscillation Hamiltonian in matter in theflavor basis is

H =

1

2E

UPMNS

0∆m2

21

∆m231

U†PMNS +

a

00

,

(1)

where we have subtracted out an overallm2

1

2E 1, a ≡2√

2GFneE is the Wolfenstein matter potential [7], andthe PMNS lepton mixing matrix [24, 25] is parameter-ized,

UPMNS =

1c23 s23e

iδ

−s23e−iδ c23

c13 s13

1−s13 c13

c12 s12

−s12 c12

1

,

(2)

where sij = sin θij , cij = cos θij , and we have shifted theCP-violating phase δ from its usual position on s13 to s23

which does not affect any observable. For our numericalstudies we use ∆m2

21 = 7.55 × 10−5 eV2, ∆m231 = 2.5 ×

10−3 eV2, s212 = 0.32, s2

13 = 0.0216, s223 = 0.547, and

δ = 1.32π from [26].Using the eigenvector-eigenvalue identity [20], the

squares of the elements of the lepton mixing matrix in

matter are simple functions of the eigenvalues of theneutrino oscillation Hamiltonian in matter, λi/2E fori ∈ {1, 2, 3}, and new submatrix eigenvalues, ξα/2E andχα/2E for α ∈ {e, µ, τ}. In general, the square of theelements of the mixing matrix are parameterization in-dependent,

|Uαi|2 =(λi − ξα)(λi − χα)

(λi − λj)(λi − λk), (3)

where i, j, and k are all different, and the λi are the exacteigenvalues, see appendix A. This result, eq. 3, can alsobe directly obtained from KTY as shown in appendixB. This equation is valid for every element of the mix-ing matrix, even the µ and τ rows, which are relativelycomplicated in the standard parameterization.

Eq. 3 is one of the primary results of our paper. Giventhe eigenvalues of the Hamiltonian and the eigenvalues ofthe submatrix Hamiltonian, it is possible to write downall nine elements of the mixing matrix in matter, squared.This result is also quite simple and easy to memorizewhich is contrasted with the complicated forms from pre-vious solutions [10, 13, 27].

The submatrix eigenvalues ξα/2E and χα/2E are theeigenvalues of the 2× 2 submatrix of the Hamiltonian,

Hα ≡(Hββ Hβγ

Hγβ Hγγ

), (4)

for α, β, and γ all different. Explicit expressions for theHamiltonian are given in appendix C and the eigenvaluesof the submatrices, which require only the solution toa quadratic, are plotted in fig. 1. We note that whilesolving a quadratic is necessary to evaluate the submatrixeigenvalues, since only the sum and the product of theeigenvalues (that is, the trace and the determinant of thesubmatrix Hamiltonian) appear in eq. 3 whose numeratorcan be rewritten as λ2

i−λi(ξα+χα)+ξαχα, the submatrixeigenvalues do not have to be explicitly calculated. Theexpressions for the sums and products of the eigenvaluesare given in appendix C.

Given the standard parameterization of the leptonmixing matrix, this allows us to write all three mixingangles in matter as simple expressions of the eigenvalues,

s212c213

= |Ue2|2 = − (λ2 − ξe)(λ2 − χe)(λ3 − λ2)(λ2 − λ1)

, (5)

s213

= |Ue3|2 =(λ3 − ξe)(λ3 − χe)(λ3 − λ1)(λ3 − λ2)

, (6)

s223c213

= |Uµ3|2 =(λ3 − ξµ)(λ3 − χµ)

(λ3 − λ1)(λ3 − λ2), (7)

where the hat indicates that it is the mixing angle inmatter. While similar versions of eqs. 5 and 6 have pre-viously appeared in the literature [10], eq. 7 is originalto this manuscript. The general form of eq. 3 allows usto write any term in the lepton mixing matrix, and thusany mixing angle with considerable ease. In addition, as

3

−20 −10 0 10 20E [GeV]

−5.0

−2.5

0.0

2.5

5.0λ,ξ,χ

[10−

3eV

2]

NOρ = 3 g/ccNOρ = 3 g/cc e

λ3

λ2

λ1

ξe, χe

−20 −10 0 10 20E [GeV]

−5.0

−2.5

0.0

2.5

5.0

λ,ξ,χ

[10−

3eV

2]

NOρ = 3 g/ccNOρ = 3 g/cc µ

λ3

λ2

λ1

ξµ, χµ

−20 −10 0 10 20E [GeV]

−5.0

−2.5

0.0

2.5

5.0

λ,ξ,χ

[10−

3eV

2]

NOρ = 3 g/ccNOρ = 3 g/cc τ

λ3

λ2

λ1

ξτ , χτ

FIG. 1. The two submatrix eigenvalues, ξα and χα, as a function of neutrino energy, are shown in the solid blue curveswith α = e, µ, τ in the left, center, and right figures respectively. For comparison, the full matrix eigenvalues λi are shown indashed red, green, and orange in each panel. When a submatrix eigenvalue (solid) overlaps one of the full matrix eigenvalues(dashed) the corresponding |Uαi|2 → 0, as seen from the numerator of eq. 3. Note the Cauchy interlacing condition is satisfied,λ1 ≤ ξα ≤ λ2 ≤ χα ≤ λ3, for each α = (e, µ, τ) and all E, using the convention ξα < χα. See appendix D for further discussion.

we will show in the next section, this also allows us tocalculate the CP violating phase quite easily.

In appendix E we show how to use this method in thevacuum (θ23, δ)-rotated flavor basis. Further extensionsof eq. 3 to an arbitrary number of neutrinos is also dis-cussed in appendix F.

B. CP-Violating Phase in Matter

In order to determine the CP-violating phase in mat-

ter, we note that cos δ can be determined, given the othermixing angles in matter, from |Uµ1|2 (or |Uµ2|2, |Uτ1|2,

or |Uτ2|2) from eq. 3. The sign of δ needs to be sepa-

rately determined. We note that the sign of δ must bethe same as the sign of δ. To see this, we employ theNaumov-Harrison-Scott (NHS) identity [28, 29],

J =∆m2

21∆m231∆m2

32

∆m221∆m2

31∆m232

J , (8)

where J = =[Ue1Uµ2U∗e2U

∗µ1] = s23c23s13c

213s12c12 sin δ is

the Jarlskog invariant [30]. We note that the numeratorand denominator in eq. 8 always have the same sign, so

sin δ has the same sign as sin δ. That is, the eigenvaluesin matter never cross.

In practice, it is simpler to determine sin δ from theToshev identity [23],

sin δ =sin 2θ23

sin 2θ23

sin δ , (9)

and use θ23 determined in eq. 7. An alternative methodfor determining the CP violating phase is given in ap-pendix E.

C. Oscillation Probabilities in Matter

Finally, these can all be combined into any oscilla-tion probability. For the primary appearance channel

at NOVA, T2K, DUNE, T2HK(K), ESSnuSB [1–6], orany other long-baseline neutrino experiment, the oscilla-tion probability can be written in the following compactform,

P (νµ → νe) =∣∣A31 + e±i∆32A21

∣∣2 , (10)

where the upper (lower) sign is for neutrinos (anti-neutrinos) and

A31 = 2s13c13s23 sin ∆31 , (11)

A21 = 2s12c13(c12c23e−iδ − s12s13s23) sin ∆21 , (12)

∆ij =(λi − λj)L

4E. (13)

The above determination of the mixing angles and CPviolating phase in matter allow for the simple determi-nation of the oscillation probability in matter for theνµ → νe appearance channel, or any other channel via thevacuum oscillation probabilities. Therefore, any physicsintuition already obtained for vacuum oscillation proba-bilities is easily transferred to oscillation probabilities inmatter.

In addition to all the appearance channels this ap-proach also works in a straightforward fashion for thedisappearance channels as well. For disappearance theoscillation probabilities in matter can be written as,

P (να → να) = 1− 4∑

i<j

|Uαi|2|Uαj |2 sin2 ∆ij . (14)

Thus the coefficients, |Uαi|2|Uαj |2, can be read off as sim-ple functions of the eigenvalues and the submatrix eigen-values, eq. 3, without any need to even convert to themixing angles in matter.

III. APPROXIMATE EIGENVALUES

While the form of the mixing angles in matter pre-sented above is exact, it still relies on the complicated

4

−20 −10 0 10 20E [GeV]

10−14

10−10

10−6

10−2|∆λ

exac

t−

∆λ

app

rox|/∆

λex

act

NOρ = 3 g/cc

∆λ21

∆λ31

∆λ32

FIG. 2. The fractional precision of the zeroth order and thesecond order DMP eigenvalues are shown in solid and dashedcurves respectively. We plot the different in eigenvalues sothat we are insensitive to an overall shift in the eigenvalues.Continuing to higher order in the eigenvalues continues toincrease the precision by comparable levels since all odd ordercorrections to the eigenvalues in DMP are zero (see appendixG).

expression of the eigenvalues. It has been previouslyshown, however, that the eigenvalues can be extremelywell approximated via a mechanism of changing bases asdemonstrated by Denton, Minakata, and Parke (DMP)[22], see also refs. [31–33]. While expressions for the dif-ferences of eigenvalues in DMP are quite compact [34],the expressions in eqs. 5-7 require the individual eigen-values so we list those here as well. Beyond the zerothorder expressions, it is possible to derive higher orderterms through perturbation theory [22] or through fur-ther rotations [33]. This approach leads to a smallness

parameter that is no larger that c12s12∆m2

21

∆m231∼ 1.5% and

is zero in vacuum confirming that the exact solution isrestored at zeroth order in vacuum, see eq. 23 below.

A. Zeroth Order Eigenvalues

The zeroth-order eigenvalues are extremely precisewith a fractional error in the difference of the eigenvaluesof < 10−5 at DUNE. We define eigenvalues of two inter-mediate steps. First, the eigenvalues of the un-rotatedHamiltonian, after a constant U23(θ23, δ) rotation3,

λa = a+ s213∆m2

ee + s212∆m2

21 , (15)

λb = c212∆m221 , (16)

λc = c213∆m2ee + s2

12∆m221 . (17)

3 A term (s212∆m221/2E)1 could be subtract from the Hamiltonian,

eq. 1, to simplify the following expressions.

Next, after an O13 rotation, we have

λ± =1

2[λa + λc

±sign(∆m2ee)√

(λa − λc)2 + (2s13c13∆m2ee)

2],

λ0 = λb , (18)

and

sin2 φ =λ+ − λcλ+ − λ−

. (19)

Finally, after an O12 rotation, the eigenvalues throughzeroth order are

λ1,2 =1

2[λ0 + λ− (20)

∓√

(λ0 − λ−)2 + (2 cos(φ− θ13)s12c12∆m221)2

],

λ3 = λ+ , (21)

and

sin2 ψ =λ2 − λ0

λ2 − λ1

. (22)

Here x represents an approximate expressions for thequantity x in matter. At this order, θ23 and δ, are un-changed from their vacuum values. Eqs. 18 to 22 definethe zeroth order approximation.

We note that φ and ψ are an excellent approximation

for θ13 and θ12 respectively, [22], see ref. [33] for the ex-plicit higher order correction terms. These are effective

two-flavor approximations to θ12 and θ13 while eqs. 5 and6 are the full three-flavor exact expressions. We furtherdiscuss the similarity in these expressions in sec. F.

B. Second Order Eigenvalues

After performing the rotations that lead to the eigen-values in eqs. 20 and 21, the smallness parameter is

ε′ ≡ sin(φ− θ13)s12c12∆m221/∆m

2ee < 1.5% (23)

and is zero in vacuum since φ = θ13 in vacuum. Becauseof the nature of the DMP approximation, the zeroth or-der eigenvalues in eqs. 20 and 21 already contain the firstorder in ε′ corrections. That is, the first order correctionsare just the diagonal elements in the perturbing Hamil-tonian which are all zero by construction. The secondorder corrections are simply,

λ(2)1 = −(ε′∆m2

ee)2

s2ψ

λ3 − λ1

, (24)

λ(2)2 = −(ε′∆m2

ee)2

c2ψ

λ3 − λ2

, (25)

λ(2)3 = −λ(2)

1 − λ(2)2 . (26)

5

It is useful to note that λ1 and λ2 are related by the 1-2interchange symmetry [22]. The 1-2 interchange sym-metry says that all oscillation observables in matter areindependent of the following transformations,

λ1 ↔ λ2 , c2ψ ↔ s2ψ , and cψsψ → −cψsψ . (27)

It is clear that λ(2)3 is invariant under this interchange,

and λ(2)2 follows directly from λ

(2)1 and the interchange.

As with φ in eq. 19, ψ is an excellent approximation

for θ12. The fractional precision of the eigenvalues at ze-roth and second order are shown in fig. 2. Since φ and ψ

are good approximations for θ13 and θ12 respectively, and

since θ23 and δ don’t vary very much in matter, one couldimagine using the vacuum probabilities with the approx-imate eigenvalues and replacing only θ13 and θ12 with φand ψ respectively. This is exactly the DMP approach atzeroth order. Thus one way to quantify the improvementof this approach over DMP is to compare the precision

with which we can approximate θ13 and θ12 with either φand ψ which result from a two-flavor rotation (see eqs. 19and 22) or with eqs. 3, 6, and 5. We have numericallyverified that the full three-flavor approach to calculatingthe mixing angles improves the precision on the mixingangles in matter (and thus the oscillation probabilities)compared with the two-flavor approach that leads to φand ψ.

Next, we note that for similar reasons that the first

order corrections vanish, λ(1)i = 0, all the odd corrections

vanish within the DMP framework4. That is, λ(k)i = 0

for all i ∈ {1, 2, 3} and any k odd, see appendix G. Whileit would appear that, given a perturbing Hamiltonian∝ ε′ that the precision would converge as ε′, this showsthat, in fact, the precision converges considerably fasterat ε′2. This result had not been previously identified inthe literature.

We now compare the precision of sine of the mixingangles and CP violating phase in matter using the ap-proximate eigenvalues through zeroth order and secondorder to the exact expressions in fig. 3. Using the ze-roth order eigenvalues to evaluate the angles and thephase is quite precise even at zeroth order, at the 1%level or much better. Adding in the second order correc-tions dramatically increases the precision by about fourorders of magnitude for neutrinos and six orders of mag-nitude for anti-neutrinos, consistent with the fact thatε′ is ∼ 10−2 in the limit as E → ∞ and ∼ 10−3 in thelimit as E → −∞. We also see that we recover the ex-act answers in vacuum, a trait that many approximationschemes do not share [19].

4 In fact, the conditions for the odd corrections to vanish canbe generally achieved in an arbitrary three or four dimensionalHamiltonian, but not for general higher dimensional Hamiltoni-ans, see the end of appendix G.

Next, we show the precision of the appearance oscilla-tion probability for DUNE in fig. 45. The scaling law ofthe precision remains the same as previously shown andwe have verified that it continues at the same rate to evenhigher orders. In fact, as we continue to higher orders wefind that all the odd corrections to the eigenvalues vanish,see appendix G.

Finally, in an effort to roughly quantify the “simplic-ity” of our results, we computed the speed with whichwe can calculate one oscillation probability as shown infig. 5. For comparison we have included many other ap-proximate and exact expressions as previously exploredin [19]. For the sake of openness, the nu-pert-comparecode used for each of these is publicly available https://github.com/PeterDenton/Nu-Pert-Compare [35]. Wenote that while our new results using DMP eigenvaluesare not as fast as others, adding in higher order cor-rections is extremely simple, as indicated in eqs. 24-26which give rise to an impressive six orders of magnitudeimprovement in precision for almost no additional com-plexity. All points use δ = −0.4π except for OS and Expwhere δ = 0, for a detailed discussion see ref. [19].

Computational speed is a useful metric not only forsimplicity but also for long-baseline experiments whichmust compute oscillation probabilities many times whenmarginalizing over a large number of systematics andstandard oscillation parameters. In addition, performingthe Feldman-Cousins method of parameter estimation isknown to be extremely computationally expensive [36].

IV. CONCLUSIONS

In this article we have used the eigenvector-eigenvalueidentity to develop a new way to write neutrino oscilla-tion probabilities in matter, both exactly and with sim-pler approximate expressions. The primary new resultinvolves determining the mixing angles in matter whichhas the benefit in that intuition gained about vacuumoscillations still applies to oscillations in matter. TheCP violating phase in matter is then determined in a

straightforward fashion from θ23 and the Toshev iden-tity [23]. Given the mixing angles and the CP violatingphase in matter, writing down the oscillation probabil-ity in matter follows directly from the simple vacuumexpression.

This technique benefits from the simplicity of usingthe expression for the oscillation probability in vacuum,and the clear, compact, expressions for the mixing anglesin matter, and can be applied to any oscillation channel.

5 We also compared calculating the oscillation probabilities withthe approximate eigenvalues using the Toshev identity to deter-mine δ and the NHS identity to determine δ through the Jarlskog,and found that the Toshev identity performs better. This is dueto the fact that the δ and θ23 both don’t vary very much inmatter while all the other parameters do.

6

−20 −10 0 10 20E [GeV]

10−16

10−12

10−8

10−4

100

|∆x|/x

NOρ = 3 g/cc

s13 s12

−20 −10 0 10 20E [GeV]

10−16

10−12

10−8

10−4

100

|∆x|/x

NOρ = 3 g/cc

s23 sδ

FIG. 3. The fractional precision of sine of the mixing angles and the CP violating phase in matter in eqs. 5-9. The precisionusing the zeroth order DMP eigenvalues is shown with solid curves and with the second order eigenvalues with dashed curves.

There is also the fact that by explicitly writing the mixingangles as simple functions of the eigenvalues, they can bereplaced with simple approximate expressions, such asthose derived by Denton, Minakata, and Parke (DMP)[22]. This new technique presented here is more precisethan that in DMP, order by order, since this result iseffectively complete to all orders in the eigenvectors andonly requires correction to the eigenvalues.

The primary new results of this article are,

• Eq. 3, reproduced here,

|Uαi|2 =(λi − ξα)(λi − χα)

(λi − λj)(λi − λk),

which presents a simple, clear and easy to remem-ber way to determine the norm of the elementsof the mixing matrix and hence the mixing angles

0.0

0.1

Pµe NO

ρ = 3 g/ccL = 1300 km

DU

NE

DU

NE

−20 −10 0 10 20E [GeV]

10−14

10−12

10−10

10−8

10−6

10−4

10−2

|∆Pµe|/Pµe

Zeroth

Second

FIG. 4. Top: The oscillation probability P (νµ → νe) at L =1300 km in the NO. Bottom: The fractional precision of theprobability using the zeroth (second) order DMP eigenvaluesin blue (orange). The vertical red bands show DUNE’s regionof interest. The precision is comparable in the IO.

in matter given the eigenvalues. Then the oscilla-tion probabilities can be calculated in a straightfor-ward fashion using the CP-violating phase in mat-ter from the Toshev identity.

• The form of eq. 3 allows for the direct substitu-tion of approximate eigenvalues, such as those fromDMP. As shown here for the first time, the DMPeigenvalues converge extremely quickly, ∼ 10−5 perstep since all the odd order corrections to the eigen-

0.02 0.1 1t [µs]

10−12

10−10

10−8

10−6

10−4

10−2

|∆P|/P

DMP0

DMP1

MP AM2

AM5/2MFAKT

DPZ0

DPZ2

ZS DiagOS ExpBEST

WORST

Nu-Pert-Compare v1.2

FIG. 5. We have plotted the fractional precision at the firstoscillation maximum for DUNE at δ = −0.4π versus the timeto compute one oscillation probability on a single core. Ourresults are labeled DPZ and are in orange. ZS [10] and Diagare two exact solutions where Diag represents an off-the-shelflinear algebra diagonalization package. Two other exact so-lutions from [18] are labeled OS (using the Cayley-Hamiltonmethod) and Exp (exponentiating the Hamiltonian) which donot account for CP violation (δ = 0). We only plot expres-sions that reach at least 1% precision at the first oscillationmaximum. The remaining expressions are: MP [32], AM [37],MF [38], AKT [31], and DMP [22]. For a detailed discussionsee ref. [19].

7

values vanish.

• The form of eq. 3 is trivially generalizable to anynumber of neutrinos.

Given the formalism presented here, we have a clear andsimple mechanism for calculating the oscillation proba-bilities in matter either exactly or approximately. Thismethod hinges on the eigenvalues, therefore the eigen-values are the Rosetta Stone for neutrino oscillations inmatter.

ACKNOWLEDGMENTS

We want to thank Terence Tao for many useful and in-teresting discussions on the eigenvector-eigenvalue iden-tity. PBD acknowledges the United States Departmentof Energy under Grant Contract desc0012704 and theNeutrino Physics Center. This manuscript has been au-thored by Fermi Research Alliance, LLC under ContractNo. DE-AC02-07CH11359 with the U.S. Department ofEnergy, Office of Science, Office of High Energy Physics.SP received funding/support from the European UnionsHorizon 2020 research and innovation programme underthe Marie Sklodowska-Curie grant agreement No 690575and No 674896.

Appendix A: The Exact Eigenvalues in Matter

From refs. [8, 10, 39], the exact eigenvalues in matterare,

λ1 =1

3A− 1

3

√A2 − 3B

(S +√

3√

1− S2),

λ2 =1

3A− 1

3

√A2 − 3B

(S −√

3√

1− S2),

λ3 =1

3A+

2

3

√A2 − 3B S . (A1)

The terms A, B, and C are the sum of the eigenval-ues, the sum of the products of the eigenvalues, and thetriple product of the eigenvalues, while S contains thecos( 1

3 cos−1(· · · )) terms,

A = ∆m221 + ∆m2

31 + a , (A2)

B = ∆m221∆m2

31 + a[∆m2

31c213 + ∆m2

21(1− c213s212)],

(A3)

C = a∆m221∆m2

31c213c

212 , (A4)

S = cos

{1

3cos−1

[2A3 − 9AB + 27C

2(A2 − 3B)3/2

]}, (A5)

where a ≡ 2E√

2GFne is the matter potential, E is theneutrino energy, GF is Fermi’s constant, and ne is theelectron number density.

As an example of the analytic impenetrability of S,setting a = 0 and recovering the vacuum values for theeigenvalues, (0,∆m2

21,∆m231), is a highly non-trivial ex-

ercise.Appendix B: Derivation From KTY

Since only the product of elements of the PMNS matrixare necessary to write down the oscillation probabilities,we start with the definition of the product of two ele-ments of the lepton mixing matrix in matter from eq. 39in KTY, ref. [13],

UαiU∗βi =

pαβλi + qαβ − δαβλi(λj + λk)

(λj − λi)(λk − λi), (B1)

where x is the quantity x evaluated in matter and theλi’s are the exact eigenvalues in matter6, see appendixA. We note that a similar approach was used in [27].

The matrix p is just the Hamiltonian in matter, pαβ =

(2E)Hαβ =∑i λiUαiU

∗βi, see appendix C. The other

term, q, is given by qαβ =∑i<j λiλjUαkU

∗βk for k 6= i, j.

It is also equivalent to qαβ = (2E)2(HγβHαγ −HαβHγγ)for γ 6= i, j.

We evaluate eq. B1 in the case of α = β.

|Uαi|2 =λ2i − λi(2E)(Hββ +Hγγ) + qαα

(λj − λi)(λk − λi), (B2)

where α, β, and γ are all different, as are i, j, and k. Wecan then write the numerator as (λi−ξα)(λi−χα) whereξα and χα satisfy

ξα + χα = (2E) (Hββ +Hγγ) , (B3)

ξαχα = (2E)2 (HββHγγ −HβγHγβ) . (B4)

That is, ξα and χα are the eigenvalues of Hα, the 2 × 2submatrix of the Hamiltonian,

We also note that eq. 3 leads to the following identityalso presented in ref. [13],

(λ2 − λ1)(λ3 − λ1)(λ3 − λ2)s12c12s13c213

= ∆m221∆m2

31∆m232s12c12s13c

213 , (B5)

which is the NHS identity [28, 29] divided by the Toshevidentity [23]. That is, this quantity

(λ2 − λ1)(λ3 − λ1)(λ3 − λ2)|Ue1||Ue2||Ue3| , (B6)

is independent of the matter potential.

Appendix C: The Hamiltonian

Here we multiply out the Hamiltonian in matter foruse in the above expressions. First, we define (2E)H =

O13(θ13)O12(θ12)M2O†12(θ12)O†13(θ13) + diag(a, 0, 0), areal matrix, which is,

8

H =1

2E

a+ ∆m2

ees213 + ∆m2

21s212 c13s12c12∆m2

21 s13c13∆m2ee

· ∆m221c

212 −s13s12c12∆m2

21

· · ∆m2eec

213 + ∆m2

21s212

, (C1)

where Hαβ = Hβα and ∆m2ee ≡ c212∆m2

31 + s212∆m2

31 [40]. Then the Hamiltonian in the flavor basis is H =

U23(θ23, δ)HU†23(θ23, δ) which is,

H =

Hee s23e−iδHeτ + c23Heµ c23Heτ − s23e

iδHeµ

· c223Hµµ + s223Hττ+

2s23c23 cos δHµτeiδ [s23c23(Hττ −Hµµ)+(c223e

−iδ − s223e

iδ)Hµτ]

· · c223Hττ + s223Hµµ−

2s23c23 cos δHµτ

, (C2)

where Hαβ = H∗βα.Then the eigenvalues of the submatrices, ξα and χα are given by

ξe + χe = (2E) (Hµµ +Hττ ) , (C3)

ξeχe = (2E)2(HµµHττ −H2

µτ

), (C4)

ξµ + χµ = (2E)(Hee + c223Hττ + s2

23Hµµ − 2s23c23 cos δHµτ), (C5)

ξµχµ = (2E)2[Hee

(c223Hττ + s2

23Hµµ − 2s23c23 cos δHµτ)−∣∣c23Heτ − s23e

−iδHeµ∣∣2]. (C6)

The ξτ and χτ eigenvalues are the same as ξµ and χµ under the interchange s223 ↔ c223 and s23c23 → −s23c23. Note

that the complicated Hµτ term does not appear in the ξe and χe terms since the eigenvalues of the 2× 2 submatrixHe are the same as those of He.

For illustration, we write down the electron submatrixeigenvalues, although we note that explicit calculationof the submatrix eigenvalue is not necessary since eq. 3depends only on the sum and product of the eigenvalueswhich are directly given in eqs. C3-C6,

ξe, χe =∆m2

ee

2

[c213 + ε±

√(c213 − ε)2 + (2s13s12c12ε)2

],

(C7)

where ε ≡ ∆m221/∆m

2ee. (This ε is different from

ε′ = sin(φ − θ13)s12c12∆m221/∆m

2ee used as our pertur-

bative expansion parameter, see eq. 23.)

Appendix D: Asymptotics of Submatrix Eigenvalues

For convenience we define ξα < χα. Then we note thatthe submatrix eigenvalues are asymptotically the same as

certain full eigenvalues.

limE→−∞

λ2 = limE→∞

λ1 = ξe , (D1)

limE→−∞

λ3 = limE→∞

λ2 = χe , (D2)

limE→−∞

λ1 = limE→−∞

ξµ = limE→−∞

ξτ , (D3)

limE→∞

λ3 = limE→∞

χµ = limE→∞

χτ , (D4)

limE→−∞

χµ = limE→∞

ξµ = m2τ , (D5)

limE→−∞

χτ = limE→∞

ξτ = m2µ , (D6)

where

m2α =

∑

i

m2i |Uαi|2 . (D7)

Since m2µ ≈ c213s

223∆m2

31 and m2τ ≈ c213c

223∆m2

31, chang-

ing the octant changes which of m2µ and m2

τ are larger.This in turn swaps the ordering of the µ and τ submatrixeigenvalues.

Furthermore, the eigenvalues of H and its principalminors, λi’s and ξα’s, χα’s (ξα < χα), satisfy the Cauchyinterlacing identity, λ1 ≤ ξα ≤ λ2 ≤ χα ≤ λ3 for α =(e, µ, τ), for all values of the matter potential.

6 Note that the eigenvalues in matter can also be expressed in a

longer notation as λi = m2i.

9

ξ′ + χ′ ξ′χ′

e ∆m231c

213 + ∆m2

21(1− c213s212) (∆m231c

213)(∆m2

21c212)

µ a+ ∆m231 + ∆m2

21s212 a(∆m2

31c213 + ∆m2

21s212s

213) + ∆m2

31∆m221s

212

τ a+ ∆m231s

213 + ∆m2

21(1− s213s212) a∆m221c

212 + (∆m2

31s213)(∆m2

21c212)

Sum 2(a+ ∆m231 + ∆m2

21) = 2A a(∆m231c

213 + ∆m2

21(1− c213s212)) + ∆m231∆m2

21 = B

TABLE I. The sum and product of the eigenvalues of the principal minors of the rotated Hamiltonian, eq. E2 and theirrelationship to eigenvalues of the full Hamiltonian,

∑j λj(H) = A and

∑j>k λj(H)λk(H) = B given in appendix A.

Appendix E: Using the (θ23, δ)-Rotated Flavor Basis

In this appendix, we use the eigenvector-eigenvalue identity in the vacuum (θ23, δ)-rotated flavor basis and recoverthe full PMNS matrix in matter by performing the vacuum (θ23, δ)-rotated at the end. The (θ23, δ)-rotated flavorbasis is defined as

U†23(θ23, δ)

νeνµντ

with U23(θ23, δ) =

1

c23 s23eiδ

−s23e−iδ c23

. (E1)

In this basis the Hamiltonian, H, is given as

H =1

2E

a+ ∆m2ees

213 + ∆m2

21s212 c13s12c12∆m2

21 s13c13∆m2ee

· ∆m221c

212 −s13s12c12∆m2

21

· · ∆m2eec

213 + ∆m2

21s212

. (E2)

Note, it is now real and independent of θ23 and δ and the same as eq. C1.

This Hamiltonian can be diagonalized by the followingunitary matrix

V ≡ V23(α) U13 U12 , (E3)

with only real entries, with7

s213

= |Ve3|2 =(λ3 − ξ′e)(λ3 − χ′e)(λ3 − λ1)(λ3 − λ2)

,

s212c213

= |Ve2|2 = − (λ2 − ξ′e)(λ2 − χ′e)(λ2 − λ1)(λ3 − λ2)

,

s2αc

213

= |Vµ3|2 =(λ3 − ξ′µ)(λ3 − χ′µ)

(λ3 − λ1)(λ3 − λ2), (E4)

where the sum and the product of ξ′’s and χ′’s are givenin Table I and are obtained from the trace and determi-nant of the principal minors of H, eq. E2. The expres-

sions for θ12 and θ13 are the same as eqs. 5 and 6, sinceξ′e = ξe and χ′e = χe, eqs. C3 and C4. |α| is tiny (< 0.01)and is zero in vacuum.

7 The relationship between this work and ref. [10] is sin2 α =F 2/(E2 + F 2) and F 2 = (λ3 − ξ′e)(λ3 − χ′e)(λ3 − ξ′µ)(λ3 − χ′µ)

and E2 = (λ3 − ξ′e)(λ3 − χ′e)(λ3 − ξ′τ )(λ3 − χ′τ ). The character-istic equation, using λ3 as the solution, is needed to prove thisequivalence.

For the full PMNS matrix in matter, we combine

U23(θ23, δ) with V23(α) into U23(θ23, δ), as follows(

c23 s23eiδ

−s23e−iδ c23

) (cα sα−sα cα

)

=

(eiρ

eiσ

)(c23 s23e

iδ

−s23e−iδ c23

). (E5)

The solution is σ = −ρ with

c23cα − s23sαeiδ = c23e

iρ ,

and sαc23 + cαs23eiδ = s23e

i(ρ+δ) .

Therefore

s223

= c2αs223 + s2

αc223 + 2sαcαs23c23 cos δ , (E6)

cos δ = (cos δ s23c23(c2α − s2α) + (c223 − s2

23)sαcα)/(s23c23) ,

sin δ = (sin δ s23c23)/(s23c23) . (E7)

The equation for sin δ is exactly the Toshev identity [23].The full PMNS matrix in matter is then by

UPMNS = U23(θ23, δ) U13(θ13) U12(θ12) . (E8)

with θ23, δ, θ12 and θ13 given by eqs. E6, E7, 5, and 6,respectively, in total agreement with what was obtainedin ref. [10], using a different method.

10

Appendix F: Extension to an Arbitrary Number ofNeutrinos

Equation 3 can be generalized in a straightforwardfashion to an arbitrary number of neutrinos. As an ini-tial illustrative example, for two flavors in matter we havethat the elements of the diagonalized mixing matrix inmatter are

|Uαi|2 =λi − ξαλi − λj

. (F1)

(This two-flavor approach was exploited in a three-flavorcontext in DMP via two two-flavor rotations.)

To evaluate this, we find the eigenvalues of the Hamil-tonian and its submatrix. The Hamiltonian is

H =∆m2

4E

(a/∆m2 − cos 2θ sin 2θ

sin 2θ cos 2θ − a/∆m2

). (F2)

The eigenvalues, λ1,2/2E, of this 2× 2 system are,

λ1,2 = ∓1

2

√(a−∆m2 cos 2θ)2 + (∆m2 sin 2θ)2 , (F3)

and the submatrix eigenvalues, ξe,µ/2E are trivially,

ξe =1

2

(∆m2 cos 2θ − a

), ξµ =

1

2

(a−∆m2 cos 2θ

).

(F4)Then we can write down the mixing matrix in mat-ter, where we note that the off-diagonal term squared

is sin2 θ,

sin2 θ = |Ue2|2 =λ2 − ξeλ2 − λ1

(F5)

=1

2

(1− ∆m2 cos 2θ − a√

(∆m2 cos 2θ − a)2 + (∆m2 sin 2θ)2

),

which agrees with the standard two-flavor expression [41].What is powerful about this method is that it can ex-

tend to more neutrinos as well. For four neutrinos, wecan write down the elements of the mixing matrix in mat-ter in the related, easy to remember form as the two orthree-flavor cases,

|Uαi|2 =(λi − ξα)(λi − χα)(λi − ζα)

(λi − λj)(λi − λk)(λi − λ`), (F6)

for i, j, k, ` all different and where ξα, χα, and ζα arethe three eigenvalues of the associated submatrix. Thismethod can be extended in a straightforward fashion toan arbitrary number of neutrinos.

The general form of eq. 3 for any n × n matrix withpossibly degenerate eigenvalues is,

|Uαi|2n∏

k=1,k 6=i

(λi − λk) =

n−1∏

k=1

(λi − ξα,k) , (F7)

where the k in ξα,k covers the n− 1 eigenvalues of the αsubmatrix, see [20].

Appendix G: Higher Order Eigenvalues

As mentioned in the text, the first order corrections tothe eigenvalues are zero. The second order correctionsare given in eqs. 24-26. We also find that the third ordercorrections are zero.

Given an 3× 3 Hamiltonian

H = H0 + V , (G1)

where H0 = diag(λ1, λ2, λ3) is a non-degenerate zerothorder diagonal matrix and V , which is Hermitian, is theperturbing part of the Hamiltonian in DMP,

V = ε′∆m2

ee

2E

−sψcψ

−sψ cψ

. (G2)

Because all diagonal elements of V vanish, it is straight-forward to see that the first order eigenvalue correctionsare zero. Next we calculate the third order correctionsto eigenvalues

λ(3)i =

∑

j,k 6=i

VijVjkVki(λj − λi)(λk − λi)

−∑

j 6=i

ViiVijVji(λj − λi)2

, (G3)

where Vij = 〈i|V |j〉. The rightmost term is clearly zerosince Vii = 0. In the first summation if j = k, Vjk = 0for the same reason. If not then the numerator containsthe product of all three off-diagonal terms. Since one of

these terms is zero in DMP (V12) we have that λ(3)i is

zero.In fact, all λ

(m)i = 0 for m odd. A brief proof is that

when m is odd λ(m)i is a summation of terms propor-

tional either to a diagonal element of V or the productV12V23V31.

The above conclusion is a special case of a more generalstatement. Let’s consider an n × n perturbing Hamilto-nian Vn×n for which there is an r such that 0 ≤ r ≤ nwherein

Vij 6= 0 only if i ≤ r < j or i > r ≥ j , (G4)

i.e. Vn×n has the block form

Vn×n =

(0r×r Vr×(n−r)

V(n−r)×r 0(n−r)×(n−r)

). (G5)

If the above condition is satisfied, the mth order eigen-

value corrections λ(m)i = 0 for m odd. DMP is the case

of n = 3, r = 2.We list the even order corrections through 6th order.

We note that we only need to write down the λ(n)1 correc-

tions since λ(n)2 is related to λ

(n)1 by the 1-2 interchange

symmetry [22] as given by eq. 27 and∑i λ

(n)i = 0 which

allows for the determination of λ(n)3 .

11

λ(2)1

(ε′∆m2ee)

2= −

s2ψ

∆λ31, (G6)

λ(4)1

(ε′∆m2ee)

4=

s2ψ

∆λ21(∆λ31)3

(−c2ψ∆λ31 + s2

ψ∆λ21

), (G7)

λ(6)1

(ε′∆m2ee)

6=

s2ψ

(∆λ21)2(∆λ31)5

[−c4ψ(∆λ31)2 + c2ψs

2ψ(3∆λ21 + ∆λ31)∆λ31 − 2s4

ψ(∆λ21)2]. (G8)

For three neutrinos, the Jacobi method ensures that theconditions that the odd corrections to the eigenvaluescan always be met by rotating one off-diagonal elementof the perturbing Hamiltonian to zero. The necessary

conditions can also be met for four neutrinos by rotatingtwo off-diagonal elements in disconnected sectors (say,U12 and U34). For general matrices this condition cannotbe met for more than four neutrinos.

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