FERMILAB-PUB-16-115-TACCEPTED

Generalized mass ordering degeneracy in neutrino oscillation experiments

Pilar Coloma1 and Thomas Schwetz2

1Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA

2Institut fur Kernphysik, Karlsruhe Institute of Technology (KIT), 76021 Karlsruhe, Germany

We consider the impact of neutral-current (NC) non-standard neutrino interactions (NSI) on thedetermination of the neutrino mass ordering. We show that in presence of NSI there is an exactdegeneracy which makes it impossible to determine the neutrino mass ordering and the octant ofthe solar mixing angle θ12 at oscillation experiments. The degeneracy holds at the probability leveland for arbitrary matter density profiles, and hence, solar, atmospheric, reactor, and acceleratorneutrino experiments are affected simultaneously. The degeneracy requires order-one correctionsfrom NSI to the NC electron neutrino–quark interaction and can be tested in electron neutrino NCscattering experiments.

PACS numbers: 14.60.Pq,14.60.St

Keywords: non-standard interactions, oscillations, neutrino mass ordering

I. INTRODUCTION

Neutrino oscillation physics has entered the preci-sion era. Present data determines all three lep-tonic mixing angles and the absolute value of thetwo mass-squared differences with few percent pre-cision [1]. Crucial goals of future oscillation exper-iments are (a) the determination of the neutrinomass ordering and the CP-violating phase δ, and(b) establishing the robustness of three-flavour os-cillations with respect to physics beyond the Stan-dard Model (SM). In the present work we show thatthose two items are intimately related. We considerthe hypothesis that additional interactions affect theneutrino sector, beyond the SM weak interaction [2–4], see [5, 6] for recent reviews. We will show that,for a certain choice of these non-standard interac-tions (NSI), the determination of the neutrino massordering—one of the main goals of upcoming oscil-lation experiments [7–11]—becomes impossible, dueto an exact degeneracy in the evolution equationgoverning neutrino oscillations in matter.

The paper is structured as follows. In Sec. we in-troduce the NSI framework and the notation usedin the rest of the paper. Section shows the ori-gin of the degeneracy and how it can be realizedin both vacuum and matter regimes. In Sec. weexplain how the degeneracy affects neutrino oscil-lation data, while in Sec. we explore the possiblecombination with neutrino scattering data to try toremove the degeneracy. Finally, our conclusions aresummarized in Sec. .

II. NON-STANDARD INTERACTIONS INNEUTRINO PROPAGATION

Three-flavour neutrino evolution in an arbitrarymatter potential is described by the Schroedingerequation

id

dxΨ = H(x)Ψ , (1)

where Ψ is a vector of the flavour amplitudes, Ψ =(ae, aµ, aτ )T , and H(x) = Hvac + Hmat(x). TheHamiltonian describing evolution in vacuum is

Hvac = Udiag(0,∆21,∆31)U† , (2)

with ∆ij = ∆m2ij/(2Eν), where ∆m2

ij ≡ m2i − m2

j

stands for the neutrino mass-squared difference, andEν is the neutrino energy. From neutrino oscillationdata, we know that |∆m2

31| ≈ |∆m232| ≈ 30∆m2

21.The neutrino mass ordering is parametrized by thesign of the larger mass-squared difference, with nor-mal ordering (NO) corresponding to ∆m2

31 > 0 andinverted ordering (IO) to ∆m2

31 < 0. The sign of∆m2

21 by convention is chosen positive. The stan-dard parametrization for the leptonic mixing ma-trix is U = O23U13O12, where Oij (Uij) denotesa real (complex) rotation in the ij sector, withmixing angle θij . Here we find it convenient touse an equivalent parametrization, where we putthe complex phase δ in the 12 rotation, such thatU = O23O13U12. After subtracting a term propor-tional to the unit matrix, the vacuum Hamiltonianbecomes

Hvac = O23O13

(H(2) 0

0 ∆31 − ∆21

2

)OT13O

T23 , (3)

Operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy.

2

with the 12 block given by

H(2) =∆21

2

(− cos 2θ12 sin 2θ12e

iδ

sin 2θ12e−iδ cos 2θ12

). (4)

Let us consider now the presence of neutral-current(NC) NSI in the form of dimension-6 four-fermionoperators, which may contribute to the effective po-tential in matter in Hmat. We follow the notationof [12], for a recent review see e.g. [6]. NSI are de-scribed by the Lagrangian

LNSI = −2√

2GF εfαβ(ναLγ

µνβL)(fγµf) , (5)

where, α, β = e, µ, τ , and f denotes a fermionpresent in the background medium. The parameter

εfαβ parametrizes the strength of the new interactionwith respect to the Fermi constant GF . Hermiticity

requires that εfαβ = (εfβα)∗. Note that we restrictto vector interactions, since we are interested in thecontribution to the effective matter potential. Ingeneric models of new physics NSI parameters areexpected to be small. However, examples of viable

gauge models leading to εu,dαβ ∼ O(1) can be found in

[13, 14] (see also [6] for a discussion of NSI models).

The matter part of the Hamiltonian is then obtainedas

Hmat =√

2GFNe(x)

1 + εee εeµ εeτε∗eµ εµµ εµτε∗eτ ε∗µτ εττ

, (6)

εαβ =∑

f=e,u,d

Yf (x)εfαβ , (7)

with Yf (x) ≡ Nf (x)/Ne(x), Nf (x) being the densityof fermion f along the neutrino path. This impliesthat the effective NSI parameters εαβ may dependon x. The “1” in the ee entry in eq. (6) correspondsto the standard matter potential [2, 15]. For neutralmatter, the densities of electrons and protons areequal. Thus, the relative densities of up and downquarks are

Yu(x) = 2 + Yn(x) , Yd(x) = 1 + 2Yn(x) , (8)

where Yn(x) is the relative neutron density alongthe neutrino path. Below we will use the notationε⊕αβ and ε�αβ to indicate when the εαβ refer to thespecific matter composition of the Earth or the Sun,respectively.

III. THE GENERALIZED MASSORDERING DEGENERACY

Let us consider first the vacuum part of the Hamil-tonian, Hvac defined in eqs. (3) and (4). It is easy

to show that the transformation

∆m231 → −∆m2

31 + ∆m221 = −∆m2

32 ,sin θ12 ↔ cos θ12 ,δ → π − δ

(9)

implies that Hvac → −H∗vac. Inserting this intoeq. (1) and taking the complex conjugate we re-cover exactly the same evolution equation, when wetake into account that complex conjugation of theamplitudes (Ψ → Ψ∗) is irrelevant, as only moduliof flavour amplitudes are observable.1 This provesthat the transformation (9) leaves the three-flavourevolution in vacuum invariant.

Note that this transformation corresponds to a com-plete inversion of the neutrino mass spectrum. Thetransformation ∆m2

31 ↔ −∆m232 exchanges NO and

IO, while changing the octant of θ12 exchanges theamount of νe present in ν1 and ν2. We denote the ef-fect of the transformation (9) as “flipping” the massspectrum. The corresponding degeneracy is knownin limiting cases, for instance, the so-called massordering degeneracy in the context of long-baselineexperiments [17]. It is manifest also in the exact ex-pression for the three-flavour νe survival-probabilityPee in vacuum, relevant for medium-baseline reactorexperiments [18].

It is clear that for a non-zero standard matter ef-fect, eq. (6) with εαβ = 0, the transformation (9) nolonger leaves the evolution invariant, since Hmat re-mains constant. The matter effect in the 13-sector isthe basis of the mass ordering determination in long-baseline accelerator [7, 8] and atmospheric neutrino[10, 11] experiments. Moreover, the observation ofthe MSW [2, 15] matter resonance in the Sun re-quires that θ12 < 45◦, which forbids the transfor-mation in the second line of eq. (9). This allows, inprinciple, for the determination of the mass order-ing via a precise measurement of Pee in vacuum [19],as intended for instance by the JUNO collaboration[9].

However, if in addition to the transformation (9), itis also possible to transform Hmat → −H∗mat, thenthe full Hamiltonian including matter would trans-form as H → −H∗, leaving the evolution equationinvariant. This can be achieved in presence of NSI,supplementing the transformation (9) with [18]

εee → −εee − 2 ,εαβ → −ε∗αβ (αβ 6= ee) .

(10)

1 The invariance of the evolution under the transformationH → −H∗ is a consequence of CPT invariance. It hasbeen noted in the context of NSI in [16] and applied insome limiting cases, see also [12].

3

The transformation of εee is crucial to change thesign of the ee element of Hmat including the stan-dard matter effect. Note that eq. (10) depends onthe parametrization used for Hvac in eq. (3). Ifthe standard parametrization with U = O23U13O12

was used instead, then we would obtain εeµ → ε∗eµ,εeτ → ε∗eτ , and all other εαβ transforming as ineq. (10).

Since in general the εαβ are dependent on the neu-tron density, the degeneracy can be broken in prin-ciple by comparing experiments in matter with dif-ferent neutron densities, or in configurations wherethe neutron density changes significantly along theneutrino path. However, one can choose couplingssuch that NSI with neutrons are zero and take placeonly with protons and/or electrons, by choosingεqαβ proportional to the quark electric charge, i.e.,

εuαβ = −2εdαβ . In this situation the εαβ are alwaysindependent of x, the degeneracy is complete andcannot be broken by any combination of neutrinooscillation experiments.

Let us illustrate the degeneracy by the following ex-ample: assume that there are no NSI in Nature.Then we can fit data from any neutrino oscillationexperiment either with standard oscillations and thecorrect spectrum, or equally well with a flippedspectrum and εee = −2. For

εuee = −4/3 , εdee = 2/3 (11)

we obtain εee = −2 independent of the neutron den-sity, and hence the degeneracy will be perfect, irre-spective of the matter environment.

IV. IMPACT OF THE DEGENERACY ATOSCILLATION EXPERIMENTS

A manifestations of this result is the so-called LMA-dark solution for solar neutrinos [20], which corre-sponds to a fit to solar neutrino data with θ12 > 45◦

(“dark octant”) and values of εu,dee ' −1. In the Sunthe neutron fraction Yn drops from about 1/2 in thecentre to about 1/6 at the border of the solar core.From eqs. (8) and (7) follows, that for εu,dee ' −1we obtain ε�ee ' −2, close to the value needed forthe generalized mass ordering degeneracy. In [12]a recent analysis of solar neutrino data has beenperformed, assuming either NSI with up or downquarks. In this case Yn does not drop out of εαβdefined in eq. (7), and hence the condition εee = −2cannot be fulfilled along the whole neutrino pathin the Sun. Therefore, the degeneracy is not per-fect. In [12] the ∆χ2 of the LMA-dark solution isnearly zero for NSI on up quarks and . 2 for down

quarks. While the sign of ∆m231 is irrelevant for so-

lar neutrino phenomenology, it has been realised in[18], that the sin θ12 ↔ cos θ12 ambiguity introducedby the LMA-dark solution leads to a mass orderingambiguity in the planned reactor experiment JUNO[9]. This is a manifestation of the generalized degen-eracy discussed above.

As another example, we will now demonstrate theimpact of the generalized degeneracy on the sensi-tivity of the long-baseline Deep Underground Neu-trino Experiment (DUNE) [8] to the mass ordering.In absence of NSI, the DUNE experiment would beable to reject the wrong hypothesis for the massordering with a significance above ∼ 5σ regardlessof the true value of δ [8]. We calculate expecteddata for NO, δ = 40◦, sin2 θ12 = 0.3, and no NSI.The simulation is performed using GLoBES [21]; thesimulation details are the same as in Ref. [22]. Thenthese artificial data are fitted by allowing for the si-multaneous presence of εee and εeτ , while all otherNSI parameters are set to zero, for simplicity. Re-sults are shown by the shaded regions in Fig. 1. Thelower panel confirms the perfect degeneracy of theflipped mass spectrum at ε⊕ee = −2 and εeτ = 0, with∆χ2 = 0 with respect to the true best fit point. Inboth panels of Fig. 1 we observe also a strong corre-lation between εee and εeτ , see [22]. Therefore, whilethe degeneracy is exact for (εee, εeτ ) = (−2, 0), it isrecovered to a good accuracy for nonzero values ofεeτ as long as |εeτ | ' 0.2|εee + 2|. The importanceof εee and εeτ for the mass ordering determinationin DUNE has been pointed out recently in [23].

V. COMBINATION WITH NEUTRINOSCATTERING DATA

Since the generalized degeneracy is exact and holdsfor any oscillation experiment, the only way to breakit are non-oscillation experiments. Indeed, opera-tors of the type in eq. (5) contribute to the neutralcurrent (NC) neutrino scattering cross section. Un-fortunately, data on electron neutrino NC scatteringis scarce. A relevant constraint on the parametersof interest to us comes from the historical CHARMexperiment [24], which has measured the quantityRe = 0.406 ± 0.140, where Re is ratio of the elec-tron neutrino plus antineutrino NC cross sections tothe corresponding charged current ones. In presenceof NSI we have Re = g2

L + g2R, where [25]

g2P =

∑q=u,d

[(gqP +

εqee2

)2

+|εqeµ|2 + |εqeτ |2

4

], (12)

4

Fit to NO

●

DUNE + Osc. + CHARM

DUNE only

1σ, 2σ, 3σ regions

-3 -2 -1 0 10.0

0.1

0.2

0.3

0.4

0.5

ϵ��⊕

|ϵ�τ⊕|

Fit to IO

1σ, 2σ, 3σ regions

-3 -2 -1 0 10.0

0.1

0.2

0.3

0.4

0.5

ϵ��⊕

|ϵ�τ⊕|

FIG. 1: Results from a fit to simulated data for DUNE.We assume a true NO and no NSI, and perform a fitallowing for non-zero values of εee and εeτ . In the upperpanel we fit with the correct mass spectrum, while in thelower panel we adopt IO and exchange sin θ12 ↔ cos θ12.The shaded regions correspond to DUNE alone, whereasthe contour curves include the constraints from globaloscillation data [12] and from the CHARM experiment[24] on the NC cross section. We marginalize over ∆m2

31,δ, θ23 and the complex phase of εeτ .

with P = L,R, and gqP are the SM NC couplings.We have included only the vector-like NSI parame-ters. Note that the CHARM constraint is somewhatmodel dependent, since it would not apply if themediator particle responsible for the NSI is muchlighter than the momentum transfer in CHARM(typically of several tens of GeV) [13].

Assuming that the CHARM bound applies, we fol-low Ref. [25] and we show in Fig. 2 the allowed re-gion for εuee and εdee from the CHARM data. Thepoint from eq. (11), corresponding to perfect de-

-2 -1 0 1 2-2

-1

0

1

2

✏uee

✏d ee

✏ �ee =

�2

✏ �ee = �

2CHARM

1�, 2�, 3�

FIG. 2: Allowed region at 1, 2, 3σ (2 d.o.f.) fromCHARM [24] (assuming NSI from a heavy mediator) inthe plane of εuee and εdee, for εu,deµ = εu,deτ = 0. The crosscorresponds to the point of exact degeneracy, eq. (11).The diagonal lines indicate the parameters for whichεee = −2 in Earth and solar matter.

generacy for any matter profile, is indicated bythe cross in the figure. We observe that it is ex-cluded by CHARM data: for this point we predictRe ≈ 0.956, which disagrees with the CHARM ex-perimental value at 3.9σ. The diagonal lines in thefigure indicate the parameters for which εee = −2in Earth and solar matter. We use that in Earthmatter, Yn ≈ 1.05, and for the Sun we show thespread induced by Yn = 1/2 → 1/6. For neutrinotrajectories in the Earth, the generalized degener-acy holds along the line indicated in the plot. How-ever, since the degeneracy for the Sun appears forslightly different values of εu,dee there is the poten-tial to break it by the combination. Indeed, for theLMA-dark solution for NSI either on up or downquarks we have εu,dee ≈ −1 [12]. From Fig. 2 we seethat (εuee, ε

dee) ≈ (0,−1) is in strong disagreement

with CHARM, while (εuee, εdee) ≈ (−1, 0) is within

the 1σ region.

Let us therefore adopt the hypothesis of NSI withup quarks only and check whether solar neutrinoand CHARM data could break the mass orderingdegeneracy for DUNE. The contour curves in Fig. 1show the combined analysis, where we include globaloscillation data (including the solar neutrino “SNO-POLY” analysis) from [12], assuming that εuee andεueτ are approximately uncorrelated. If we restrictεeτ = 0, the degeneracy is broken, since εuee = −1(as required by solar data) implies ε⊕ee ≈ −3, whichcan be excluded at high confidence level by DUNE

5

for εeτ = 0. However, if we allow for non-zero εeτ ,we see that a large region with the flipped mass spec-trum remains below the 2σ level around ε⊕ee ≈ −3and |ε⊕eτ | ≈ 0.2. Hence, we conclude that includingpresent constraints from oscillation and scatteringdata, the degeneracy will severely affect the massordering sensitivity of DUNE. Let us note that ifmore NSI parameters are allowed to vary, the fitwith the flipped spectrum may even improve fur-ther.

VI. CONCLUSIONS

We have demonstrated that the so-called LMA-darksolution is a manifestation of an exact degeneracyat the level of the neutrino evolution equation. Thisdegeneracy makes it impossible to determine theneutrino mass ordering by neutrino oscillation ex-periments. It requires |εu,dee | ' 1, i.e., NSI of elec-tron neutrinos comparable in strength to weak in-teractions. The only way to break the degeneracy

is via non-oscillation experiments. We have shownthat taking into account current data on the νeNC cross section excludes NSI needed for the ex-act generalized degeneracy (subject to some modeldependence); however, the degeneracy remains tobe present at an approximate level, still destroy-ing the mass ordering sensitivity of planned exper-iments. In order to break the degeneracy at highconfidence level, improved data on νe NC interac-tions is mandatory. These may be provided, forinstance, by coherent neutrino–nucleus interactionexperiments [26–29].

Acknowledgments. We thank Stefan Vogl and DavidV. Forero for useful discussions. Fermilab is oper-ated by Fermi Research Alliance, LLC under Con-tract No. De-AC02-07CH11359 with the UnitedStates Department of Energy. This project has re-ceived funding from the European Union’s Hori-zon 2020 research and innovation programme un-der the Marie Sklodowska-Curie grant agreementNo 674896.

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