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IFT-UAM/CSIC-18-0611

Prepared for submission to JHEP

Standard versus Non-Standard CP Phases inNeutrino Oscillation in Matter with Non-Unitarity

Ivan Martinez-Solera,b,c,d Hisakazu Minakataa,e,f

aInstituto Física Teórica, UAM/CSIC, Calle Nicola’s Cabrera 13-15, Cantoblanco E-28049 Madrid,SpainbTheoretical Physics Department, Fermi National Accelerator Laboratory, P.O. Box 500, BataviaIL 60510, USAcDepartment of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USAdColegio de Física Fundamental e Interdisciplinaria de las Américas (COFI), 254 Norzagaraystreet, San Juan, Puerto Rico 00901eCenter for Neutrino Physics, Department of Physics, Virginia Tech, Blacksburg, Virginia 24061,USAfResearch Center for Cosmic Neutrinos, Institute for Cosmic Ray Research, University of Tokyo,Kashiwa, Chiba 277-8582, Japan

E-mail: [email protected], [email protected]

Abstract:To derive a structure revealing expression of the neutrino oscillation probability in

matter with non-unitarity, we formulate a perturbative framework with the expansion pa-rameters, the ratio ε of the solar ∆m2

sol to the atmospheric ∆m2atm and the parameters

which describe unitarity violation (UV). Using the α parametrization for non-unitary mix-ing matrix and to first order in our perturbation theory, we show that there is a universalcorrelation between the νSM CP phase δ and the three UV complex α parameter phases.Using a phase convention of the mixing matrix UMNS in which e±iδ is attached to sin θ23,it is expressed as [e−iδαµe, ατe, eiδατµ], always the same combination in all the oscillationchannels. The generation mechanism of the phase correlation and the reason for its stabilityis elucidated. However, thanks to the UMNS convention dependence, we can eliminate e±iδ

attached to the α parameters in a different convention with e±iδ attached to sin θ12. Whileit casts doubt on physical nature of the correlation, we describe the way of resolving thisissue and the results emerged are briefly reported. Finally, we have clarified how unitaryevolution of three active neutrinos can be reconciled with the non-unitarity of the wholesystem in the presence of non-unitary flavor mixing.

1First version released while the authors were at: Instituto Física Teórica, UAM/CSIC, Calle Nicola’sCabrera 13-15, Cantoblanco E-28049 Madrid, Spain

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Contents

1 Introduction 1

2 Correlation between the CP phase of νSM and the unitarity violation(UV) parameter phases 3

3 Formulating the helio-unitarity violation (UV) perturbation theory 43.1 Unitary evolution of neutrinos in the mass eigenstate basis 53.2 α parametrization of the non-unitary mixing matrix and its convention de-

pendence 63.2.1 Neutrino evolution with general convention of the MNS matrix 63.2.2 The three useful conventions of the MNS matrix 7

3.3 Preliminary step toward perturbation theory: Tilde-basis 83.4 Unperturbed and perturbed Hamiltonian in the tilde basis 93.5 Diagonalization of zeroth-order Hamiltonian and the hat basis 103.6 Flavor basis, the tilde and hat bases, the S and S matrices, and their relations 113.7 Calculation of S matrix 123.8 Recapitulating the leading order S matrix and the first order helio corrections 133.9 Calculation of S(1) and S(1) for the UV part 13

4 Neutrino oscillation probability to first order: νe − νµ sector 144.1 P (νe → νe)

(0+1)helio and P (νµ → νe)

(0+1)helio : the “simple and compact” formulas 15

4.2 P (νe → νe)(1) and P (νµ → νe)

(1): Both intrinsic and extrinsic UV contribu-tions 16

4.3 Symmetry of the oscillation probability 18

5 Unitarity of neutrino evolution with first order UV corrections: νe row 195.1 The oscillation probabilities for proving perturbative unitarity in νe row 205.2 Perturbative unitarity of intrinsic UV contribution: νe row 205.3 No perturbative unitarity of extrinsic UV contribution: νe row 21

6 How accurate are the first order formulas for P (νβ → να)UV? 21

7 Canonical phase combination: stability and phase convention dependence 237.1 Mechanism for generating the canonical phase combination 237.2 Key to the stability and universality of the canonical phase combination 247.3 Phase convention dependence of the canonical phase combination 267.4 Physical interpretation of the phase correlation 27

8 Some additional remarks 278.1 Vacuum limit 278.2 Non-unitarity and Non-standard interactions (NSI) 28

– i –

9 Concluding remarks 28

A Expression of H and Φ matrix elements 30

B Expressions of S(1)UV matrix elements 31

C The oscillation probabilities in νµ − ντ sector 32C.1 P (νµ → νµ)

(0+1)helio and P (νµ → ντ )

(0+1)helio : “the simple and compact” formula 32

C.2 P (νµ → νµ)(1)int-UV and P (νµ → ντ )

(1)int-UV : Intrinsic UV contribution 33

C.3 P (νµ → νµ)(1)ext-UV and P (νµ → ντ )

(1)ext-UV : Extrinsic UV contribution 35

C.4 Perturbative unitarity yes or no of intrinsic and extrinsic UV contributions:νµ row 35

D Identifying the relevant variables 35

1 Introduction

It appears that by now the three flavor lepton mixing [1] is well established after the longterm best endeavor by the experimentalists, which are recognized in an honorable way [2, 3].Though we do not know the value of CP phase δ, the lepton Kobayashi-Maskawa (KM)phase [4], and the neutrino mass ordering, there appeared some hints toward identifyingthese unknowns. That is, the long-baseline (LBL) neutrino experiment T2K sees with acontinuously improving confidence level (CL) that the phase δ is around the value ∼ 3π

2 [5].1

This is the best place for the determination of the mass ordering, as can be seen clearly bythe bi-probability plot introduced in ref. [8]. The preference of the normal mass orderingover the inverted one has been seen in the atmospheric neutrino observation by Super-Kamiokande [9], which is modestly strengthened by the ongoing LBL experiments [5, 6].A recent global analysis [10] showed for the first time that it can be claimed at 3σ CL.Also, a reanalysis of NOνA data seem to confirm the so far dominant result that θ23 is nearmaximal [6].

The apparent convergence of various results from dozens of experiments suggests thatwe may reach a stage of knowing the remaining unknowns at a time earlier than we thought.It will allow us to confirm or reject the important phenomenon of lepton CP violation ina definitive way, for example, by Hyper-K [11], T2HKK [12], and DUNE [13]. Then, itprompts us to think about how to conclude the era of discovery of neutrino mass and thelepton flavor mixing. One of the most important key elements is the paradigm test, thatis, to verify the standard three flavor mixing scheme of neutrinos. As in the quark sector,unitarity test is the most popular, practical way of carrying this out.

1 However, this tendency has not been confirmed by the most recent analysis of NOνA data [6]. Thispoint should be settled in the further progress of the measurements, in particular in T2K II, an extendedrun proposed [7].

– 1 –

A favourable way of performing a leptonic unitarity test is to formulate a model inde-pendent generic framework in which unitarity is violated, and confront it to the experimentaldata. It was attempted in a pioneering work by Antusch et al. [14], which indeed providedsuch a framework in the context of high-scale unitarity violation (UV).2 In low-scale UV,on the other hand, the currently available model is essentially unique, the 3 active plusNs sterile model, see refs. [15, 16] for a partial list of the early references. In the presentcontext, low and high scales imply, typically, energy scales of new physics much lower andhigher than the electroweak scale, respectively. Recently, within the (3 + Ns) model, amodel-independent framework is created to describe neutrino propagation in vacuum [17]and in matter [18] in such a way that the observable quantities are insensitive to details ofthe sterile sector, e.g., its mass spectrum and active-sterile mixing.

In this paper, we construct a perturbative framework by which we can derive a sim-ple expression of the neutrino oscillation probability in matter in the presence of UV. Theframework has the two kind of expansion parameters, the ratio ε ≈ ∆m2

21/∆m231 (precise

definition is in eq. (3.16)), and the UV parameters, hence dubbed as the “helio-UV pertur-bation theory” in this paper. It can be regarded as an extension of the “renormalized helioperturbation theory” in matter to include non-unitarity [19]3, which allows us to discussUV flavor transition of neutrinos with sizeable matter effect. It would be useful for analyz-ing experiments such as Super-K, Hyper-K, T2HKK, DUNE, IceCube-Gen2/PINGU, andKM3NeT-ORCA [9, 11–13, 20–22].

Introduction of non-unitary mixing matrix, which replaces the unitary MNS flavor mix-ing matrix in the neutrino-mass embedded standard model (νSM), brings nine additionalparameters in the neutrino oscillation probability. Unraveling the correlations between theνSM and the UV parameters, as well as among the UV parameter themselves would benecessary to analyse the system with non-unitarity. It turns out that our helio-UV per-turbation theory is extremely structure-revealing. That is, we will see that the lepton KMphase δ and the complex UV parameters come in into the oscillation probability in a cer-tain fixed combination. If we use so called the α parametrization [23], and use a conventionof the flavor mixing matrix UMNS in which e±iδ is attached to sin θ23, it is expressed as[e−iδαµe, ατe, eiδατµ], a universal feature in all the oscillation channels. It will be referredto as the “canonical phase combination” in this paper. We will see, however, the form ofCP phase correlation is UMNS convention dependent. Here, it is appropriate to mention theforegoing studies of δ and UV phase correlation done in the references in [23–25].

A few remarks on high-scale vs. low-scale UV are in order: As recapitulated in [17],the notable differences between them are presence (high-scale) or absence (low-scale) offlavor non-universality and zero-distance flavor transition. In an effort toward formulatingmodel-independent framework for testing low-scale UV the two more criteria are uncoveredfor distinguishing low-scale from high-scale UV. That is, presence of the probability leaking

2 We are aware that in the physics literature UV usually means “ultraviolet”. But, in this paper UV isused as an abbreviation for “unitarity violation” or “unitarity violating”.

3 Having the same zeroth order Hamiltonian as the one in [19] was not expected because the neutralcurrent reaction is involved in the Hamiltonian. But, it came out quite naturally, as will be explained insection 3.

– 2 –

term in the oscillation probability and possible detection of UV perturbative correctionswhich testifies for the low-scale UV [17, 18]. See refs. [26, 27] for the current constraints onunitarity violation in low-scale UV scenario.

High-scale unitarity violation is a well studied subject with many references, only partof which is quoted here [14, 23–25, 27–40]. In the context of the present paper, we wantto remark that the evolution equation of the three flavor active neutrinos in the masseigenstate basis in high-scale UV [27], is the same as the one obtained in the leading orderin W expansion in low-scale UV [18], see eq. (3.2). This property allows us to discuss bothhigh-scale and the leading-order low-scale UV in the same footing.

Finally, in this paper we give a pedagogical discussion to clarify the point of how non-unitarity of the flavor mixing matrix leads to non-unitarity of the observable, the oscillationprobability P (νβ → να). The answer is not totally trivial, because the neutrino evolutionhas to be unitary in high-scale UV, and to our knowledge this point has never been discussedexplicitly in the literature. By integrating out heavy new physics sector at high scale, onlythe three active neutrinos remains as the neutral leptons in low energy effective theory. Thatis, they span the complete state space of neutral leptons at low energies.4 The completenessimplies that neutrino evolution must be unitary, because there is no way to go outside ofthe complete neutral lepton state space during propagation, assuming absence of inelasticscattering, absorption, etc. Then, the question is: how and why the oscillation probabilitydoes not respect unitarity? We will answer these questions in section 3.

In section 2, we start with a pedagogical discussion to argue that the phase correla-tion between νSM and UV variables is quite natural thing to think about in non-unitarysubsystems inside the larger unitary theory. In section 3, we construct our perturbativeframework with UV in matter from scratch in a step-by-step manner, and address its UMNS

convention dependence. In section 4, we compute the neutrino oscillation probabilities inthe νe− νµ sector to first order in the helio-UV expansion. A universal correlation betweenthe complex α parameters and the νSM CP phase is demonstrated. An explicit proof ofunitarity in neutrino evolution is given in sections 5, and accuracy of the helio-UV expansionis examined in section 6. In section 7, the stability and the UMNS convention dependenceof the phase correlation are discussed, which is followed by a brief report on the natureof the correlation. The two miscellaneous topics, the vacuum limit and the relation withNSI, are addressed in section 8 before giving our conclusion in section 9. The oscillationprobabilities in the νµ − ντ sector are calculated in appendix C. Table 1 summarizes theequation numbers of all the oscillation probabilities.

2 Correlation between the CP phase of νSM and the unitarity violation(UV) parameter phases

Focus of our physics discussion in this paper is primarily on the correlation between the CPphase δ of νSM, and the phases associated with the α parameters which will be introduced

4 Since the structure of neutral lepton state space is the same as in the low-scale UV at its leading orderin the W expansion [18], the above and subsequent discussions equally applies to the case of low-scale UVat the leading order as well.

– 3 –

to describe the effect of UV, see eq. (3.6). This section 2 is meant to be an introductionto the topics, aiming at motivating the readers to this relatively unfamiliar subject. Wetry to illuminate, though only intuitively, why the phase correlation is relevant and can becharacteristic in the theories with UV.

To make our discussion transparent, but only for an explanatory purpose in this section,we rely on the particular model of leptonic UV with 3 active and Ns sterile neutrinos, the(3 +Ns) space unitary model. See [17, 18] for an exposition of the properties of the modelin the context of leptonic unitarity test. The model has the (3 + Ns) × (3 + Ns) unitarymixing matrix. For our present discussion, the relevant parts of it are the 3×3 N matrix inthe active neutrino subspace, and 3×Ns W matrix which bridges between the active andthe sterile neutrino subspaces. The active 3× 3 part of the unitarity relation in the wholestate space can be written as

δαβ =3∑j=1

NαjN∗βj +

Ns+3∑J=4

WαJW∗βJ , (2.1)

where j = 1, 2, 3 and the sterile space index J runs over J = 4, 5, ··, Ns+3. The off-diagonalelements of (2.1) are complex numbers and it can be written as polygons with 3 +Ns sides,the unitarity polygons, an extension of the well-known unitarity triangle. Therefore, theCP violating phases associated with N and W are all inter-related with each other. It isthe basic reason why correlation between phases is the relevant point in such models of the(apparent) non-unitarity that is caused in the subsystem even though the whole system isunitary.

In this paper, we deal with the active 3×3 subspace by restricting to the leading orderin the W perturbation theory [18]. Even in this case, the correlation among the phasesin the N and W matrices could leave a part of its memory to the restricted 3 × 3 activeneutrino space. Assuming Ns is large and the magnitudes and phases ofWαJ are random, itis conceivable that cancellation takes place among the terms in the WW † term in eq. (2.1),leaving a small non-unitarity effects. In such cases the major part of the phase correlationoccurs inside the 3× 3 active subspace, which naturally leads to correlations between νSMphase δ and the UV α parameter phases.

3 Formulating the helio-unitarity violation (UV) perturbation theory

In this paper, we investigate the dynamics of active three-flavor neutrino system with non-unitary mixing matrix in matter. In the context of the (3 + Ns) space unitary modelmentioned in the previous section, it is nothing but the leading (zeroth) order term in theW expansion. Following the observation in ref. [18], we work with the neutrino evolutionin 3× 3 active neutrino space in the vacuum mass eigenstate basis, see eq. (3.2) below. Itis conceivable that the system also describes high-mass limit of sterile states, with effectivedecoupling W ≈ 0 of the sterile sector, or more generically the system of high-scale UV.Therefore, it should not come as a surprise that the equivalent evolution equation is usedby Blennow et al [27], for example, to describe high-scale UV.

– 4 –

3.1 Unitary evolution of neutrinos in the mass eigenstate basis

We discuss neutrino evolution in high-scale UV and (leading-order in W ) low-scale UV in asame footing. The three active neutrino evolution in matter in the presence of non-unitaryflavor mixing can be described by the Schrödinger equation in the vacuum mass eigenstatebasis [18, 27]5

id

dxν =

1

2E

0 0 0

0 ∆m221 0

0 0 ∆m231

+N †

a− b 0 0

0 −b 0

0 0 −b

N ν. (3.2)

In this paper, we denote the vacuum mass eigenstate basis as the “check basis”. In eq. (3.2),N denotes the 3 × 3 non-unitary flavor mixing matrix which relates the flavor neutrinostates to the vacuum mass eigenstates as

να = Nαiνi. (3.3)

Hereafter, the subscript Greek indices α, β, or γ run over e, µ, τ , and the Latin indices i, jrun over the mass eigenstate indices 1, 2, and 3. E is neutrino energy and ∆m2

ji ≡ m2j−m2

i .The usual phase redefinition of neutrino wave function is done to leave only the mass squareddifferences.

The functions a(x) and b(x) in eq. (3.2) denote the Wolfenstein matter potential [41]due to charged current (CC) and neutral current (NC) reactions, respectively.

a = 2√

2GFNeE ≈ 1.52× 10−4

(Yeρ

g cm−3

)(E

GeV

)eV2,

b =√

2GFNnE =1

2

(Nn

Ne

)a. (3.4)

Here, GF is the Fermi constant, Ne and Nn are the electron and neutron number densitiesin matter. ρ and Ye denote, respectively, the matter density and number of electrons pernucleon in matter. For simplicity and clarity we will work with the uniform matter densityapproximation in this paper. But, it is not difficult to extend our treatment to varyingmatter density case if adiabaticity holds.

By writing the evolution equation as in eq. (3.2) with the hermitian Hamiltonian, theneutrino evolution is obviously unitary, which is in agreement with our discussion given atthe end of section 1. Then, the answer to the remaining question, “how the effect of non-unitarity comes in into the observables as a consequence of non-unitary mixing matrix” isgiven in section 3.6.

5 We must note that it is a highly nontrivial statement. A naive extension of the unitary system to thenon-unitary one with the Hamiltonian in the flavor basis

H =1

2E

N 0 0 0

0 ∆m221 0

0 0 ∆m231

N† +

a− b 0 0

0 −b 0

0 0 −b

(3.1)

is not equivalent with the one in eq. (3.2) due to non-unitarity of the N matrix [14].

– 5 –

3.2 α parametrization of the non-unitary mixing matrix and its conventiondependence

To parametrize the non-unitary N matrix we use the so-called α parametrization [23],N = (1− α)U , where U ≡ UMNS denotes the νSM 3× 3 unitary flavor mixing matrix.6 Todefine the α matrix, however, we must specify the phase convention by which U matrix isdefined.

We start from the most commonly used form, the Particle Data Group (PDG) [46]convention of the MNS matrix,

UPDG =

1 0 0

0 c23 s23

0 −s23 c23

c13 0 s13e

−iδ

0 1 0

−s13eiδ 0 c13

c12 s12 0

−s12 c12 0

0 0 1

, (3.5)

with the obvious notations sij ≡ sin θij etc. and δ being the CP violating phase. Then, wedefine the non-unitary mixing matrix NPDG as

NPDG = (1− α)UPDG =

1−

αee 0 0

αµe αµµ 0

ατe ατµ αττ

UPDG. (3.6)

By inserting N = NPDG in (3.6) to eq. (3.2), we define the neutrino evolution equation inthe vacuum mass eigenstate basis.

3.2.1 Neutrino evolution with general convention of the MNS matrix

After reducing the standard three-flavor mixing matrix to UPDG, which has four degree offreedom, we still have freedom of phase redefinition

ν →

1 0 0

0 eiβ 0

0 0 eiγ

ν ≡ Γ (β, γ) ν (3.7)

without affecting physics of the system. Then, the evolution equation in the Γ (β, γ) trans-formed basis,

id

dxν =

1

2E

0 0 0

0 ∆m221 0

0 0 ∆m231

ν+

1

2EU(β, γ)† 1− α(β, γ)†

a− b 0 0

0 −b 0

0 0 −b

1− α(β, γ)U(β, γ)ν, (3.8)

6 For early references for parametrizing the UV effect, see e.g., [28, 42–45]. It must be remarked thatthe authors of ref. [27] made an important point in explaining why the α parametrization is more superiorthan their traditional way of using the hermitian η matrix.

– 6 –

describes the same physics. In (3.8), U(β, γ) and α(β, γ) denote, respectively, the Γ (β, γ)

transformed MNS matrix and α matrix:

U(β, γ) ≡

1 0 0

0 e−iβ 0

0 0 e−iγ

UPDG

1 0 0

0 eiβ 0

0 0 eiγ

α(β, γ) ≡

1 0 0

0 e−iβ 0

0 0 e−iγ

α 1 0 0

0 eiβ 0

0 0 eiγ

(3.9)

That is, we can use different convention of the MNS matrix U(β, γ), but then our α matrixhas to be changed accordingly, as in (3.9).

3.2.2 The three useful conventions of the MNS matrix

Among general conventions defined in (3.9), practically, there exist the three useful conven-tions of the MNS matrix. In addition to UPDG in (3.5), they are U(0, δ) and U(δ, δ):

UATM ≡ U(0, δ) =

1 0 0

0 c23 s23eiδ

0 −s23e−iδ c23

c13 0 s13

0 1 0

−s13 0 c13

c12 s12 0

−s12 c12 0

0 0 1

,USOL ≡ U(δ, δ) =

1 0 0

0 c23 s23

0 −s23 c23

c13 0 s13

0 1 0

−s13 0 c13

c12 s12e

iδ 0

−s12e−iδ c12 0

0 0 1

. (3.10)

The reason for our terminology of UATM and USOL in (3.10) is because CP phase δ is attachedto the “atmospheric angle” s23 in UATM, and to the “solar angle” s12 in USOL, respectively.Whereas in UPDG, δ is attached to s13.

Accordingly, we have the three different definition of the α matrix. In addition toNPDG = (1− α)UPDG as in (3.6), we haveNATM = (1− αATM)UATM, andNSOL = (1− αSOL)USOL.The latter two and their relations to α are given by

αATM = α(0, δ) ≡

αee 0 0

αµe αµµ 0

ατe ατµ αττ

=

αee 0 0

αµe αµµ 0

ατee−iδ ατµe

−iδ αττ

,αSOL = α(δ, δ) ≡

αee 0 0

αµe αµµ 0

ατe ατµ αττ

=

αee 0 0

αµee−iδ αµµ 0

ατee−iδ ατµ αττ

. (3.11)

In this paper, for convenience of the calculations, we take the “ATM” convention with UATM

and αATM. But, the translation of our results to the PDG or the “SOL” conventions can bedone easily by using eq. (3.11).

Notice that, because of the structure NATM = (1− αATM)U23U13U12, the α matrix isalways attached to U23. Then, the correlation between the lepton KM phase δ and the UVparameter phases becomes more transparent if e±iδ is attached to U23. This is the reasonwhy we take the MNS matrix convention UATM in (3.10) in our following calculation.

– 7 –

3.3 Preliminary step toward perturbation theory: Tilde-basis

Taking the UATM convention with αATM matrix, we formulate our helio-UV perturbationtheory. We assume that deviation from unitarity is small, so that αβγ 1 hold for allflavor indices β and γ including the diagonal ones. Therefore, we are able to use the twokind of expansion parameters, ε ≈ ∆m2

21/∆m231 (see eq. (3.16) below) and the α parameters

in our helio-UV perturbation theory.We define the following notations for simplicity to be used in the discussions hereafter

in this paper:

∆ji ≡∆m2

ji

2E, ∆a ≡

a

2E, ∆b ≡

b

2E. (3.12)

For convenience in formulating the helio-UV perturbation theory, we move from thecheck basis to an intermediate basis, which we call the “tilde basis”,7 ν = (U13U12)ν, withHamiltonian

H = (U13U12)H(U13U12)† = (U13U12)

0 0 0

0 ∆21 0

0 0 ∆31

(U13U12)†

+ U †23

1−

αee α∗µe α∗τe0 αµµ α∗τµ

0 0 αττ

∆a −∆b 0 0

0 −∆b 0

0 0 −∆b

1−

αee 0 0

αµe αµµ 0

ατe ατµ αττ

U23

= Hvac + H UV . (3.13)

In the last line, we have denoted the first and the second terms in eq. (3.13) as Hvac

and H UV , respectively. The explicit form of the Hvac in a form decomposed into theunperturbed and perturbed parts is given by

H(0)vac(x) = ∆ren

s2

13 0 c13s13

0 0 0

c13s13 0 c213

+ ε

s212 0 0

0 c212 0

0 0 s212

, (3.14)

H(1)vac(x) = εc12s12∆ren

0 c13 0

c13 0 −s13

0 −s13 0

, (3.15)

where

∆ren ≡∆m2

ren2E

, ∆m2ren ≡ ∆m2

31 − s212∆m2

21,

ε ≡ ∆m221

∆m2ren. (3.16)

7 From the flavor basis, the tilde basis is U23 transformed basis, να = (U†23)αβνβ and H = U†23HflavorU23,which is commonly used in various treatments of neutrino propagation in matter.

– 8 –

The superscripts (0) and (1) in eqs. (3.14) and (3.15), respectively, show that they are zerothand first order in ε. After transforming to the tilde basis, as we expected, we recover thesame Hamiltonian as used in the “renormalized helio perturbation theory” without unitarityviolation [19]. An order ε term is intentionally absorbed into the zeroth-order term in H(0)

vac

as in eq. (3.14) to make the formulas of the oscillation probabilities simple and compact.We note that the matter term H UV in eq. (3.13) can be decomposed into the zeroth,

first and the second order terms in α (or α) matrix elements as H UV = H(0)matt + H

(1)UV +

H(2)UV :

H(0)matt =

∆a −∆b 0 0

0 −∆b 0

0 0 −∆b

,

H(1)UV = U †23

∆b

2αee

(1− ∆a

∆b

)α∗µe α∗τe

αµe 2αµµ α∗τµατe ατµ 2αττ

U23,

H(2)UV = −U †23

∆b

α2ee

(1− ∆a

∆b

)+ |αµe|2 + |ατe|2 α∗µeαµµ + α∗τeατµ α

∗τeαττ

αµeαµµ + ατeα∗τµ α2

µµ + |ατµ|2 α∗τµαττατeαττ ατµαττ α2

ττ

U23.

(3.17)

The total Hamiltonian in the tilde basis is, therefore, given by H = Hvac + H UV , whereHvac = H

(0)vac + H

(1)vac.

3.4 Unperturbed and perturbed Hamiltonian in the tilde basis

To formulate the helio-UV perturbation theory, we decompose the tilde basis Hamiltonianin the following way:

H = H(0) + H(1). (3.18)

The unperturbed (zeroth-order) Hamiltonian is given by H(0) = H(0)vac + H

(0)matt .

We make a phase redefinition

ν = exp [i

∫ x

dx′∆b(x′)]ν ′ (3.19)

which is valid even for non-uniform matter density. Then, the Schrödinger equation for ν ′

becomes the form in eq. (3.3) with unperturbed part of the Hamiltonian (H(0))′ as given in

(H(0))′ = ∆ren

a(x)∆m2

ren+ s2

13 0 c13s13

0 0 0

c13s13 0 c213

+ ε

s212 0 0

0 c212 0

0 0 s212

. (3.20)

namely, without NC matter potential terms. It is evident that the phase redefinition doesnot affect the physics of flavor change. Hereafter, we omit the prime symbol and use the

– 9 –

zeroth-order Hamiltonian eq. (3.20) without NC term. This is nothing but the zeroth orderHamiltonian used in [19], which led to the “simple and compact” formulas of the oscillationprobabilities in the standard three-flavor mixing.

The perturbed Hamiltonian is then given by

H(1) = H(1)vac + H

(1)UV + H

(2)UV (3.21)

where each term in eq. (3.21) is defined in eqs. (3.15) and (3.17). In the actual computation,we drop the second-order term (the last term) in eq. (3.21) because we confine ourselvesinto the zeroth and first order terms in the UV parameters in this paper.

3.5 Diagonalization of zeroth-order Hamiltonian and the hat basis

To carry out perturbative calculation, it is convenient to transform to a basis which diago-nalizes H(0), which we call the “hat basis”. H(0) is diagonalized by the unitary transforma-tion as follows:

H0 = U †φH0Uφ =

h1 0 0

0 h2 0

0 0 h3

, (3.22)

where the eigenvalues hi are given by

h1 =1

2

[(∆ren + ∆a)− sign(∆m2

ren)

√(∆ren −∆a)

2 + 4s213∆ren∆a

]+ ε∆rens

212,

h2 = c212 ε ∆ren, (3.23)

h3 =1

2

[(∆ren + ∆a) + sign(∆m2

ren)

√(∆ren −∆a)

2 + 4s213∆ren∆a

]+ ε∆rens

212.

See eqs. (3.12) and (3.16) for the definitions of ∆ren, ∆a etc. By the convention withsign(∆m2

ren), we can treat the normal and the inverted mass orderings in a unified way.The foregoing and the following treatment of the system without the UV α parameters inthis section, which recapitulates the one in ref. [19], is to make description in this paperself-contained.

Uφ is parametrized as

Uφ =

cosφ 0 sinφ

0 1 0

− sinφ 0 cosφ

. (3.24)

where φ is nothing but the mixing angle θ13 in matter. With the definitions of the eigen-values eq. (3.23), the following mass-ordering independent expressions for cosine and sine2φ are obtained:

cos 2φ =∆ren cos 2θ13 −∆a

h3 − h1,

sin 2φ =∆ren sin 2θ13

h3 − h1. (3.25)

– 10 –

The perturbing Hamiltonian in vacuum in the tilde basis, H(1)vac, has a simple form such

that the positions of “zeros” are kept after transformed into the hat basis:

H(1)vac = U †φH

(1)vacUφ

= εc12s12∆ren

0 cos (φ− θ13) 0

cos (φ− θ13) 0 sin (φ− θ13)

0 sin (φ− θ13) 0

. (3.26)

In fact, H1 is identical to H1 with θ13 replaced by (θ13−φ). However, the form of H(1)UV is

somewhat complicated,

H(1)UV = U †φH

(1)UV Uφ ≡ ∆bU

†φHUφ (3.27)

where we have defined H matrix

H ≡

H11 H12 H13

H21 H22 H23

H31 H32 H33

= U †23

2αee

(1− ∆a

∆b

)α∗µe α∗τe

αµe 2αµµ α∗τµατe ατµ 2αττ

U23. (3.28)

The explicit expressions of the elements Hij are given in appendix A.

3.6 Flavor basis, the tilde and hat bases, the S and S matrices, and theirrelations

We summarize the relationship between the flavor basis, the check (vacuum mass eigenstate)basis, the tilde, and the hat (zeroth order diagonalized hamiltonian) basis.

Only the unitary transformations are involved in changing from the hat basis to thetilde basis, and from the tilde basis to the check basis:

H = U †φHUφ, or H = UφHU†φ,

H = (U13U12)H(U13U12)†, or H = (U13U12)†H(U13U12). (3.29)

The non-unitary transformation is involved from the check basis to the flavor basis:

να = Nαiνi = (1− α)Uαi νi. (3.30)

The relationship between the flavor basis Hamiltonian Hflavor and the hat basis one H is

Hflavor = (1− α)U H (1− α)U†

= (1− α)U23UφHU†φU†23(1− α)†. (3.31)

Then, the flavor basis S matrix is related to S and S matrices as

S = (1− α)U23UφSU†φU†23(1− α)† = (1− α)U23SU

†23(1− α)†. (3.32)

Notice that both S and S are unitary, but S is not because of non-unitarity of the (1− α)

matrix.

– 11 –

This is the answer to the question we posed in section 1. Namely, the non-unitarityof S matrix in the flavor basis, whose square is the observable, comes from the initialprojection from the flavor- to mass-basis and the final projection back from the mass- toflavor-eigenstate. Notice that there is no other way, because neutrino evolution has to beunitary, as discussed in section 1.8

Because of the reasoning above, we denote U23SU†23 as the “propagation-S matrix”. For

convenience, we write down explicitly all the pieces in eq. (3.32), the propagation-S matrixin terms of the S elements [19]:

(U23SU†23)ee = See,

(U23SU†23)eµ = c23Seµ + s23e

−iδSeτ ,

(U23SU†23)eτ = c23Seτ − s23e

iδSeµ,

(U23SU†23)µe = c23Sµe + s23e

iδSτe

(U23SU†23)µµ = c2

23Sµµ + s223Sττ + c23s23(e−iδSµτ + eiδSτµ),

(U23SU†23)µτ = c2

23Sµτ − s223e

2iδSτµ + c23s23eiδ(Sττ − Sµµ),

(U23SU†23)τe = c23Sτe − s23e

−iδSµe

(U23SU†23)τµ = c2

23Sτµ − s223e−2iδSµτ + c23s23e

−iδ(Sττ − Sµµ)

(U23SU†23)ττ = s2

23Sµµ + c223Sττ − c23s23(e−iδSµτ + eiδSτµ). (3.33)

We note that S, which can be expanded by the small parameters, ε and α, as

S = S(0) + S(1)helio + S

(1)UV (3.34)

To first order in these small parameters, we obtain

S = U23S(0)U †23 + U23

(S

(1)helio + S

(1)UV

)U †23 − αU23S

(0)U †23 − U23S(0)U †23α

†. (3.35)

We shall call the S(1)UV piece in the second term “intrinsic” UV contribution, and the last

two terms in eq. (3.35) as “extrinsic” UV contribution. The intrinsic UV contribution is inunitary part, and the extrinsic UV contribution represents non-unitary effect. Finally, theoscillation probabilities are simply given by

P (νβ → να;x) = |Sαβ(x)|2. (3.36)

3.7 Calculation of S matrix

To calculate S(x) we define Ω(x) as

Ω(x) = eiH0xS(x). (3.37)8 We do not assume that our discussion affects the formulas used so far in the treatment of high-scale

UV, and it is perfectly consistent with that in ref. [27], for example. Our discussion just aims at serving fora transparent understanding of the point, how neutrino’s unitary evolution is reconciled with non-unitarynature of the observable P (νβ → να). We will see below and in the next section that at first order in theUV parameter expansion a clear separation between the unitary and non-unitaly part of the oscillationprobability occurs.

– 12 –

Then, Ω(x) obeys the evolution equation

id

dxΩ(x) = H1Ω(x) (3.38)

where

H1 ≡ eiH0xH1e−iH0x. (3.39)

where H1 = H(1)vac + H

(1)UV . See eqs. (3.26) and (3.27). Then, Ω(x) can be computed

perturbatively as

Ω(x) = 1 + (−i)∫ x

0dx′H1(x′) + (−i)2

∫ x

0dx′H1(x′)

∫ x′

0dx′′H1(x′′) + · · ·, (3.40)

and the S matrix is given by

S(x) = e−iH0xΩ(x). (3.41)

3.8 Recapitulating the leading order S matrix and the first order helio correc-tions

Since all the relevant quantities are computed for the leading order and the helio correctionsin ref. [19], we just recapitulate them in below. The zeroth order result of S matrix is givenby

S(0) = Uφe−iH0xU †φ =

c2φe−ih1x + s2

φe−ih3x 0 cφsφ

(e−ih3x − e−ih1x

)0 e−ih2x 0

cφsφ(e−ih3x − e−ih1x

)0 c2

φe−ih3x + s2

φe−ih1x

(3.42)

where cφ ≡ cosφ and sφ ≡ sinφ. The non-vanishing first order helio corrections (order ∼ ε)to S matrix are given by(S

(1)helio

)eµ

=(S

(1)helio

)µe

= ε∆renc12s12

[cφc(φ−θ13)

e−ih2x − e−ih1x

h2 − h1+ sφs(φ−θ13)


h3 − h2

],(

S(1)helio

)µτ

=(S

(1)helio

)τµ

= ε∆renc12s12

[−sφc(φ−θ13)


h2 − h1+ cφs(φ−θ13)


h3 − h2

],

(3.43)

and all the other elements vanish. The elements of the propagation-S matrix U23SU†23 can

be obtained from S matrix elements by using eq. (3.33).

3.9 Calculation of S(1) and S(1) for the UV part

In this paper, we restrict ourselves to the perturbative calculation to first order in ε ≡∆m2

21/∆m2ren ≈ ∆m2

21/∆m231 and to first order in the UV parameters αβγ . Then, the form

of S matrix and the oscillation probability in zeroth and the first-order helio correctionsare identical with those computed in ref. [19]. Therefore, we only calculate, in the rest ofthis section, the matter part which produces the UV contributions.

– 13 –

By inserting U †φUφ, H1 (hereafter the matter part only) can be written as

H1 ≡ ∆bU†φUφe

iH0xU †φHUφe−iH0xU †φUφ

= ∆bU†φ

c2φeih1x + s2

φeih3x 0 cφsφ

(eih3x − eih1x

)0 eih2x 0

cφsφ(eih3x − eih1x

)0 c2

φeih3x + s2

φeih1x

H11 H12 H13

H21 H22 H23

H31 H32 H33

×

c2φe−ih1x + s2

φe−ih3x 0 cφsφ


)0 e−ih2x 0


)0 c2

φe−ih3x + s2

φe−ih1x

Uφ≡ ∆bU

†φ

Φ11 Φ12 Φ13

Φ21 Φ22 Φ23

Φ31 Φ32 Φ33

Uφ ≡ ∆bU†φΦUφ, (3.44)

where we have introduced another simplifying matrix notation Φ and its elements Φij . Theexplicit expressions of Φij are given in appendix A.

Assuming the uniform matter density, we obtain

S(x)(1)matt = e−iH0xΩ(x)

(1)matt = ∆bU

†φUφe

−iH0xU †φ

[(−i)

∫ x

0dx′Φ(x′)

]Uφ

= ∆bU†φ

c2φe−ih1x + s2

φe−ih3x 0 cφsφ


)0 e−ih2x 0


)0 c2

φe−ih3x + s2

φe−ih1x

× (−i)

∫ x

0dx′

Φ11(x′) Φ12(x′) Φ13(x′)

Φ21(x′) Φ22(x′) Φ23(x′)

Φ31(x′) Φ32(x′) Φ33(x′)

Uφ (3.45)

Since S(1)UV = UφS

(1)mattU

†φ, one can obtain S(1)

UV by removing U †φ and Uφ from eq. (3.45). Theirexplicit forms are given in appendix B. Then, the first order UV contribution to the flavorbasis S matrix can be readily calculated as

S(1)UV = U23S

(1)UVU

†23 (3.46)

because the extrinsic factors (1− α) and/or (1− α)† only yield higher order terms.

4 Neutrino oscillation probability to first order: νe − νµ sector

In this section, we calculate the expressions of the oscillation probabilities. For clarity, weconcentrate on νe → νe and νµ → νe channels. The other oscillation probabilities whichare required to discuss unitarity in the νe row will be obtained in section 5. The oscillationprobabilities in the νµ−ντ sector are given in appendix C. Table 1 at the end of this sectionsummarizes the locations and the equation numbers of all the probability formulas.

We have obtained S matrix elements in the zeroth order and first order helio collec-tions using eq. (3.33) with the S(0) and S

(1)helio matrix elements in eqs. (3.42) and (3.43),

– 14 –

respectively, and the first order matter correction S(1)UV in eq. (3.46). Therefore, we know

the whole S matrix to first order in the helio and the UV parameters

S = S(0) + S(1)helio + S

(1)UV − αS

(0) − S(0)α†. (4.1)

Then, we are ready to calculate the expressions of the oscillation probabilities using theformula P (νβ → να;x) = |Sαβ|2 to first order in the expansion parameters. Since allthe building elements are known, we just present the final expressions of the oscillationprobabilities.

We categorize P (νβ → να) into the three types of terms:

P (νβ → να) = P (νβ → να)(0+1)helio + P (νβ → να)

(1)int-UV + P (νβ → να)

(1)ext-UV , (4.2)

where

P (νβ → να)(0+1)helio = |S(0)

αβ |2 + 2Re

[(S

(0)αβ

)∗ (S

(1)helio

)αβ

],

P (νβ → να)(1)int-UV = 2Re

[(S

(0)αβ

)∗ (S

(1)UV

)αβ

],

P (νβ → να)(1)ext-UV = −2Re

[(S

(0)αβ

)∗ (αS(0) + S(0)α†

)αβ

]. (4.3)

The first term in eq.(4.2), P (νβ → να)(0+1)helio , is nothing but the “simple and compact”

formulas for the probability derived in ref. [19] which is based on the standard unitarythree-flavor mixing and is valid to first order in ε.

4.1 P (νe → νe)(0+1)helio and P (νµ → νe)

(0+1)helio: the “simple and compact” formulas

Since all the calculations for P (νe → νe)(0+1)helio and P (νµ → νe)

(0+1)helio are done in [19] and

described in detail in this reference we just present here the result:

P (νe → νe)(0+1)helio = 1− sin2 2φ sin2 (h3 − h1)x

2, (4.4)

P (νµ → νe)(0+1)helio

=

[s2

23 sin2 2θ13 + 4εJr cos δ

(h3 − h1)− (∆ren −∆a)

(h3 − h2)

](∆ren

h3 − h1

)2

sin2 (h3 − h1)x

2

+ 8εJr(∆ren)3

(h3 − h1)(h3 − h2)(h2 − h1)sin

(h3 − h1)x

2sin

(h2 − h1)x

2cos

(δ +

(h3 − h2)x

2

),

(4.5)

where x is the baseline and Jr, the reduced Jarlskog factor [47], is defined as

Jr ≡ c12s12c23s23c213s13. (4.6)

For simplified notations such as ∆ren ≡ ∆m2ren

2E , and ∆a ≡ a2E , see sections 3.1 and 3.3. hi

(i = 1, 2, 3) denote the eigenvalues of H(0) as defined in eq. (3.22).Because the matter potential due to the NC interaction is removed from the zeroth-

order Hamiltonian H(0) by the phase redefinition (see section 3.4), the unitary part P (νβ →να)

(0+1)helio is free from NC matter potential ∆b ≡ b

2E .

– 15 –

4.2 P (νe → νe)(1) and P (νµ → νe)

(1): Both intrinsic and extrinsic UV contribu-tions

The first order intrinsic and extrinsic UV contributions to P (νe → νe) read

P (νe → νe)(1)int-UV = sin2 2φ

cos 2φ

[αee

(1− ∆a

∆b

)− s2

23αµµ − c223αττ − c23s23Re

(eiδατµ

)]− sin 2φ

[s23Re

(e−iδαµe

)+ c23Re (ατe)

](∆bx) sin(h3 − h1)x

− 4 cos 2φ sin 2φ

sin 2φ

[αee

(1− ∆a

∆b

)− s2


(eiδατµ

)]+ cos 2φ

[s23Re

(e−iδαµe

)+ c23Re (ατe)

] ∆b

h3 − h1sin2 (h3 − h1)x

2, (4.7)

P (νe → νe)(1)ext-UV = −4αee

[1− sin2 2φ sin2 (h3 − h1)x

2

]. (4.8)

– 16 –

Similarly, the first order intrinsic and extrinsic UV contributions to P (νµ → νe) are givenby

P (νµ → νe)(1)int-UV = 2Re

[(S(0)eµ

)∗ (S

(1)UV

)eµ

]= −s2

23 sin2 2φ cos 2φ

[αee

(1− ∆a

∆b

)− s2


(eiδατµ

)](∆bx) sin(h3 − h1)x

+ s223 sin3 2φ

[s23Re

(e−iδαµe

)+ c23Re (ατe)


+ sin 2θ23 sin 2φ

×c2φ

[c23Re

(e−iδαµe

)− s23Re (ατe)

]− cos 2θ23cφsφRe

(eiδατµ

)− sin 2θ23cφsφ(αµµ − αττ )

× ∆b

(h2 − h1)

− sin2 (h3 − h2)x

2+ sin2 (h3 − h1)x

2+ sin2 (h2 − h1)x

2

+ sin 2θ23 sin 2φ

×s2φ

[c23Re

(e−iδαµe

)− s23Re (ατe)

]+ cos 2θ23cφsφRe

(eiδατµ

)+ sin 2θ23cφsφ(αµµ − αττ )

× ∆b

(h3 − h2)

− sin2 (h2 − h1)x

2+ sin2 (h3 − h2)x

2+ sin2 (h3 − h1)x

2

+ 4s2

23 sin 2φ cos 2φ

sin 2φ

[αee

(1− ∆a

∆b

)− s2


(eiδατµ

)]+ cos 2φ

[s23Re

(e−iδαµe

)+ c23Re (ατe)

] ∆b

h3 − h1sin2 (h3 − h1)x

2

+ 2 sin 2θ23 sin 2φ

c2φ

[c23Im

(e−iδαµe

)− s23Im (ατe)

]+ cφsφIm

(eiδατµ

)× ∆b

(h2 − h1)sin

(h3 − h1)x

2sin

(h1 − h2)x

2sin

(h2 − h3)x

2

− 2 sin 2θ23 sin 2φ

s2φ

[c23Im

(e−iδαµe

)− s23Im (ατe)

]− cφsφIm

(eiδατµ

)× ∆b

(h3 − h2)sin

(h3 − h1)x

2sin

(h1 − h2)x

2sin

(h2 − h3)x

2, (4.9)

P (νµ → νe)(1)ext-UV = 2Re

[(S(0)eµ

)∗S(1)eµ |ext

]= 2s23 sin 2φ

[cos 2φRe

(e−iδαµe

)− s23 sin 2φ (αee + αµµ)

]sin2 (h3 − h1)x

2

− s23 sin 2φIm(e−iδαµe

)sin(h3 − h1)x. (4.10)

The expressions of the oscillation probabilities in eqs. (4.7), (4.8), (4.9), and (4.10) arethe first explicit demonstration of the canonical phase combination [e−iδαµe, ατe, eiδατµ] inthis paper. Non-association of e±iδ to ατe must be understood as a particular “correlation”,and naturally there is no association of δ in the diagonal α parameters, αββ (β = e, µ, τ). Wewill see in the rest of this paper that the canonical phase combination is always realized in

– 17 –

both the first order intrinsic and extrinsic UV correction terms in the oscillation probabilitiesin all the channels.9

We should mention here that the phase correlation e−iδαµe, a part of our canonicalphase combination, has been observed in the foregoing studies including refs. [23–25]. It isnice that we can extend the notion of phase correlation to include all the α parameters.10

We defer our presentation of the oscillation probabilities in the νµ − ντ sector to ap-pendix C. The generic features of them are very similar to the ones presented in this section,with slightly more complicated expressions. Importantly, the canonical phase combinationprevails in the oscillation probabilities in the νµ−ντ sector, as we will see in appendix C. Apossibility of reducing the number of parameters by expanding in another small parameteris briefly discussed in appendix D.

The readers may ask “Why is such canonical phase combination realized so univer-sally?”. We plan to give an organized discussion of the mechanism for generating thecanonical phase combination and its stability in section 7.

The expressions of the oscillation probabilities are scattered into various places in thispaper. Therefore, for the readers’ convenience, we tabulate in table 1 the equation numbersfor P (νβ → να)

(1)int-UV and P (νβ → να)

(1)ext-UV in various channels.

Table 1. The equation numbers for P (νβ → να)(1)int-UV and P (νβ → να)

(1)ext-UV (see eq. (4.3) for

definitions) are summarized.

channel P (νβ → να)(1)int-UV P (νβ → να)

(1)ext-UV

νµ → νe eq. (4.9) in section 4.2 eq. (4.10) in section 4.2νe → νe eq. (4.7) in section 4.2 eq. (4.8) in section 4.2νe → νµ T transformation from eq. (4.9) eq. (5.5) in section 5.2νe → ντ eq. (5.3) in section 5.2 eq. (5.6) in section 5.3νµ → νµ eq. (C.3) in appendix C.2 eq. (C.5) in appendix C.3νµ → ντ eq. (C.4) in appendix C.2 eq. (C.6) in appendix C.3

4.3 Symmetry of the oscillation probability

We have pointed out in ref. [52] that the unitary part of our present system possesses thesymmetry under the transformations involving angle φ.11 Here, we give a more completediscussion including the UV effect.

One can observe that the oscillation probability given in this section, P (νe → νe)(1)

and P (νµ → νe)(1), including both the intrinsic and extrinsic UV parts are invariant under

9 A perturbative treatment using the similar expansion parameters is presented in ref. [48] within theframework of 3 + 3 model, in which the calculation of the oscillation probabilities of the first order arecarried out. However, due to different implementation of UV, it is essentially impossible to compare ourformulas to theirs. As a consequence, none of the points of our emphasis, the canonical phase combinationand unitarity of neutrino propagation in matter is not reached in their paper.

10 Our result is consistent with theirs if the authors of refs. [23] and [24] have used the UPDG or UATM

conventions because the correlation e−iδαµe holds in the both conventions. See section 7.3.11 For more transparent expressions of the oscillation probability for this purpose, go to appendix B in

ref. [19].

– 18 –

the transformation

φ→ φ+π

2. (4.11)

It implies the simultaneous transformations

cφ → −sφ, sφ → cφ, cos 2ϕ→ − cos 2ϕ, sin 2ϕ→ − sin 2ϕ,

h1 → h3, h3 → h1, (4.12)

whose last two, exchanging eigenvalues h1 ↔ h3, is enforced by (4.11). It can be understoodby noticing that cos(φ−θ12)→ − sin(φ−θ12) and sin(φ−θ12)→ cos(φ−θ12) under (4.11),together with an alternative expressions of the eigenvalues

h1 = sin2 (φ− θ13) ∆ren + c2φ∆a + ε∆rens

212,

h3 = cos2 (φ− θ13) ∆ren + s2φ∆a + ε∆rens

212. (4.13)

It is also easy to verify that the invariance under (4.11) holds in all the oscillation proba-bilities listed in table 1.

It is interesting to note that in both cases of the “solar resonance” perturbation theory[52, 53] and the helio-UV perturbation theory in this paper, the respective matter enhancedangles are involved. In the former it is θ12 and in the latter θ13.

5 Unitarity of neutrino evolution with first order UV corrections: νe row

In section 3 we have shown that, to first order in the helio-UV perturbation theory, theoscillation probability P (νβ → να) can be decomposed into the two parts, the unitarypart denoted as P (νβ → να)

(0+1)helio + P (νβ → να)

(1)int-UV , and the non-unitary part P (νβ →

να)(1)ext-UV . It is the unique form of the probability which is consistent with the unitarity of

the propagation-S matrix U23SU†23. As discussed in section 3.6, the property is in complete

harmony with the reasonings for unitary evolution even with non-unitary mixing matrix,which is spelled out in section 1.

In this section, we give an explicit proof of unitarity of P (νe → να)(0+1)helio + P (νe →

να)(1)int-UV in νe row. The similar explicit proof of unitarity in νµ row will be given in

appendix C. To our knowledge it is the first explicit proof at the probability level thatneutrino propagation is unitary in the presence of non-unitary mixing matrix. Since wealready know that the oscillation probability to first-order helio corrections is unitary [19],∑

α=e,µ,τ

P (νβ → να)(0+1)helio = 1 +O(ε2), (5.1)

it is sufficient to show ∑α=e,µ,τ

P (νβ → να)(1)int-UV = O(ε2) (5.2)

to prove perturbative unitarity. In (5.2) we have assumed that all αβγ ∼ ε.

– 19 –

5.1 The oscillation probabilities for proving perturbative unitarity in νe row

To discuss perturbative unitarity in νe row, we need to prepare the following three oscil-lation probabilities at first order, P (νe → νe), P (νe → νµ), and P (νe → ντ ). P (νe →νe)

(1)int-UV and P (νe → νe)

(1)ext-UV are given in eqs. (4.7) and (4.8), respectively. P (νe →

νµ)(1)int-UV and P (νe → νµ)

(1)ext-UV , can be obtained by generalized T transformation [18]

of eqs. (4.9) and (4.10), respectively. More generally, the transformation of the probabil-ities from νβ → να to να → νβ channels can be done by taking the complex conjugateof all the complex parameters, that is, to flip the sign of the imaginary part of e±iδ andthe UV parameters. Therefore, what we need to compute here are P (νe → ντ )

(1)int-UV and

P (νe → ντ )(1)ext-UV .

5.2 Perturbative unitarity of intrinsic UV contribution: νe row

To examine unitarity of intrinsic UV contribution we compute P (νe → ντ )(1)int-UV with the

result

P (νe → ντ )(1)int-UV

= −c223 sin2 2φ cos 2φ

[αee

(1− ∆a

∆b

)− s2


(eiδατµ

)](∆bx) sin(h3 − h1)x

+ c223 sin3 2φ

[s23Re

(e−iδαµe

)+ c23Re (ατe)


+ 4c223 sin 2φ cos 2φ

sin 2φ

[αee

(1− ∆a

∆b

)− s2


(eiδατµ

)]+ cos 2φ

[s23Re

(e−iδαµe

)+ c23Re (ατe)

] ∆b

h3 − h1sin2 (h3 − h1)x

2

− sin 2θ23 sin 2φ

×c2φ

[c23Re

(e−iδαµe

)− s23Re (ατe)

]− cφsφ

[sin 2θ23(αµµ − αττ ) + cos 2θ23Re

(eiδατµ

)]× ∆b

(h2 − h1)

− sin2 (h3 − h2)x

2+ sin2 (h3 − h1)x

2+ sin2 (h2 − h1)x

2

− sin 2θ23 sin 2φ

×s2φ

[c23Re

(e−iδαµe

)− s23Re (ατe)

]+ cφsφ

[sin 2θ23(αµµ − αττ ) + cos 2θ23Re

(eiδατµ

)]× ∆b

(h3 − h2)

− sin2 (h2 − h1)x

2+ sin2 (h3 − h2)x

2+ sin2 (h3 − h1)x

2

+ 2 sin 2θ23 sin 2φ

c2φ

[c23Im

(e−iδαµe

)− s23Im (ατe)

]+ cφsφIm

(eiδατµ

)× ∆b

(h2 − h1)sin

(h3 − h1)x

2sin

(h1 − h2)x

2sin

(h2 − h3)x

2


s2φ

[c23Im

(e−iδαµe

)− s23Im (ατe)

]− cφsφIm

(eiδατµ

)× ∆b

(h3 − h2)sin

(h3 − h1)x

2sin

(h1 − h2)x

2sin

(h2 − h3)x

2. (5.3)

– 20 –

Given the expressions of the oscillation probabilities in νe row, one can readily proveperturbative unitarity for neutrino evolution without extrinsic UV corrections

P (νe → νe)(1)int-UV + P (νe → νµ)

(1)int-UV + P (νe → ντ )

(1)int-UV = 0. (5.4)

where “0” in the right-hand side implies absence of first order terms in the UV parameters.This completes our proof of perturbative unitarity in νe row of the intrinsic UV contributionsto first order in the α parameters. The similar result will be shown to hold in νµ row inappendix C.4.

5.3 No perturbative unitarity of extrinsic UV contribution: νe row

For completeness, we explicitly verify that P (νβ → να)(1)ext-UV gives raise to non-unitary

contribution. Among the relevant three probabilities, P (νe → νe)(1)ext-UV is given in eq. (4.8)

in section 4.2. P (νe → νµ)(1)ext-UV can be obtained by generalized T transformation of

eq. (4.10)

P (νe → νµ)(1)ext-UV = 2s23 sin 2φ

[cos 2φRe

(e−iδαµe

)− s23 sin 2φ (αee + αµµ)

]sin2 (h3 − h1)x

2

+ s23 sin 2φIm(e−iδαµe

)sin(h3 − h1)x. (5.5)

Finally, P (νe → ντ )(1)ext-UV can be easily computed as

P (νe → ντ )(1)ext-UV

= −2c23 sin 2φ

sin 2φ

[s23Re

(eiδατµ

)+ c23 (αee + αττ )

]− cos 2φRe (ατe)

sin2 (h3 − h1)x

2

+ c23 sin 2φIm (ατe) sin(h3 − h1)x. (5.6)

It is evident that they do not add up to zero, as there is no way for imaginary partcancels when P (νe → νµ)

(1)ext-UV and P (νe → ντ )

(1)ext-UV are added up. We find no indication

of even partial cancellation between the various terms. Therefore, there is no perturbativeunitarity of extrinsic UV contribution in νe row, as expected. We will see the same resultin the νµ row in section C.4.

6 How accurate are the first order formulas for P (νβ → να)UV?

The principal objective of constructing our helio-UV perturbation theory is to understandthe qualitative features of oscillation probability with UV. Yet, it may be better to havean idea of how good is the approximation it can offer. In particular, we are interested inthe UV part, P (νβ → να)

(1)UV ≡ P (νβ → να)

(1)int-UV + P (νβ → να)

(1)ext-UV , because the

accuracies of P (νβ → να)(0+1)helio have been examined in [19]. Notice that it corresponds to

the quantity

∆P (νβ → να) ≡ P (νβ → να)standard − P (νβ → να)(0)non-unitary (6.1)

– 21 –

Figure 1. The iso-contour of −P (νµ → να)(1)UV ≡ −[P (νµ → να)

(1)int-UV + P (νµ → να)

(1)ext-UV ] is

presented in space of neutrino energy E and baseline L for α = e (upper panel) and α = µ (lowerpanel). It corresponds to the difference ∆P (νµ → να) ≡ P (νµ → να)standard−P (νµ → να)

(0)non-unitary

plotted in the lower panel of figures 1 (α = e) and 3 (α = µ) of ref. [18]. In this calculation, thesame values for the standard mixing parameters as well as the UV α parameters as in [18] are used:αee = 0.01, αµe = 0.0141, αµµ = 0.005, ατe = 0.0445, ατµ = 0.0316, αττ = 0.051. The matterdensity is taken to be ρ = 3.2 g cm−3 over the entire baseline.

– 22 –

which is numerically computed with high precision and is plotted in the lower panels offigures 1-3 in [18] for (βα) = µe, µτ , and µµ. Then, we confront our first order formulas ofP (νβ → να)

(1)UV to ∆P (νβ → να) in [18].

In figure 1, plotted are the iso-contours of −P (νµ → να)(1)UV as a function of energy E

and baseline L for α = e (upper panel) and α = µ (lower panel). We have used the samevalues for the νSM mixing parameters as well as the UV α parameters as in ref. [18]. We seeoverall agreement, not only qualitatively but also quantitatively to a certain level, betweenthe iso-contours in the upper and lower panels in figure 1 and the ones given in figures 1(for νµ → νe) and 3 (for νµ → νµ) in ref. [18], respectively.12 If the numerical accuracyof the first order formula is affected by the treatment of the solar level crossing [19], itwould be worthwhile to re-examine this problem with a different framework developed inrefs. [50, 51].

7 Canonical phase combination: stability and phase convention depen-dence

Knowing the universal phase correlation between the νSM CP phase and the ones associatedwith the UV parameters may help interpreting correctly the results e.g., in the analysesto constrain non-unitarity. Therefore, it may be important to understand how the phasecorrelation comes about and why it is so stable. In this section we clarify some aspects ofthe canonical phase combination, the specific way how the α parameter phases appear in thespecial combination with the νSM CP phase. In particular, we answer the following threequestions on the canonical phase combination: (1) What is the mechanism for generationof such structure? (2) Why is it so stabile over the oscillation channels as well as both theintrinsic and and extrinsic contributions with quite different nature?, and (3) How does itdepend on the convention of the MNS matrix?

7.1 Mechanism for generating the canonical phase combination

To illuminate the point, we focus on the particular example, the intrinsic UV contributionto the oscillation probability in the νµ → νe channel. It is the interference term between thezeroth-order helio amplitude S(0)

eµ = c23S(0)12 + s23e

−iδS(0)13 = s23e

−iδcφsφ(e−ih3x − e−ih1x

)(see (3.33) or (7.4)), and the first order intrinsic UV amplitude(

S(1)UV

)eµ

= c23

(S

(1)UV

)12

+ s23e−iδ(S

(1)UV

)13, (7.1)

where the expressions of the two terms in the right-hand-side of (7.1) are in appendix (B).Then, the first order contribution to P (νµ → νe)

(1)int-UV is given by twice the real part of

12 The only exception might be in a relatively small region with shape of oblique ellipse centered aroundL = 2000 km and E = 100 MeV, which extends to a few 100 MeV, the region of solar MSW enhancement [41,49]. But, it was understood that the disagreement is largely due to a difference in mesh between ourfigure 1 and figures 1 and 3 in ref. [18]. Notice that an extremely fine mesh is required to display thecontours accurately, given the feature of the probability in this region due to superimposed high-frequencyatmospheric-scale oscillations on long-wavelength solar-scale oscillations. But, we did not try to elaboratethis point, because a numerical accuracy of the first order formula is not the main point of this paper.

– 23 –

the following interference term (see eq. (A.1) for the expression of Hij)(S(0)eµ

)∗ (S

(1)UV

)eµ

= c23s23eiδcφsφ

(eih3x − eih1x

)×[

∆b

h2 − h1


) (c2φH12 − cφsφH32

)+

∆b

h3 − h2


) (s2φH12 + cφsφH32

)]+ s2

23c2φs

2φ(eiδe−iδ)

(eih3x − eih1x

)×[(−i∆bx)


)H33 +

(s2φe−ih3x − c2

φe−ih1x

)(H11 −H33)

+ cφsφ

(e−ih3x + e−ih1x

)(H13 +H31)

+

∆b

h3 − h1


)cφsφ cos 2φ (H11 −H33) +H13 − 2c2

φs2φ (H13 +H31)

]. (7.2)

That is, eiδ from(S

(0)eµ

)∗cancels unwanted e−iδ from H12 and H32 (lozenge structure,

see below) as well as explicit e−iδ factor from δ-independent diagonal Hii (i = 1, 2, 3)and H13 terms, leaving no explicit e±iδ apart from the ones hidden as the canonical phasecombination with α’s inHij . It is straightforward to repeat this analysis in all the remainingoscillation channels.

7.2 Key to the stability and universality of the canonical phase combination

The above example suggests that the particular way e±iδ is distributed in various quantities,Hamiltonian as well as the S matrix, is the key to understand occurence of the canonicalphase combination. Let us introduce the terminology “lozenge position e±iδ matrix” orsimply “lozenge matrix”, or “lozenge structure” for short. We define Y as lozenge positione±iδ matrix if Y has the following form

Y ≡

Yee Yeµe−iδ Yeτ

Yµeeiδ Yµµ Yµτe

iδ

Yτe Yτµe−iδ Yττ

, (7.3)

where Yαβ may contain δ and the α parameter UV phases but only in the form of canonicalphase combination, [e−iδαµe, ατe, eiδατµ]. The simplest example of the lozenge matrixis the H matrix defined in (3.28) whose explicit expressions of the elements are given ineq. (A.1).

If the total S matrix is written in the form of lozenge matrix, it automatically guaranteesthat the νSM phase δ and UV phases enter into the oscillation probability in the form ofcanonical phase combination. The leading (zeroth) order helio amplitude

S(0) = U23S(0)U †23

=

c2φe−ih1x + s2

φe−ih3x s23e

−iδcφsφ(e−ih3x − e−ih1x

)c23cφsφ


)s23e

iδcφsφ(e−ih3x − e−ih1x

)c2

23e−ih2x + s2

23

(c2φe−ih3x + s2

φe−ih1x

)c23s23e

iδ(c2φe−ih3x + s2

φe−ih1x − e−ih2x

)c23cφsφ


)c23s23e

−iδ(c2φe−ih3x + s2

φe−ih1x − e−ih2x

)s2

23e−ih2x + c2

23

(c2φe−ih3x + s2

φe−ih1x

)

(7.4)

– 24 –

is indeed has the lozenge structure, but in a very special manner such that there is no phasedependence inside the Y elements.

Now, we discuss the first order intrinsic UV contribution to the probability. In this case,the first order contribution is of the form (Yαβ)∗ Zαβ where Z is the matrix representation ofthe first order intrinsic UV amplitude. If Z also has the form of lozenge matrix it is obviousthat the canonical phase combination is realized in the oscillation probability. We knowfrom appendix B that the tilde basis S(1)

UV matrix has the lozenge structure, which essentiallycomes from the same property of the H matrix. One can easily show that if a matrix X isa lozenge position e±iδ matrix, U23XU

†23 is also a a lozenge position matrix.13 Therefore,

the first order intrinsic UV amplitude S(1)UV = U23S

(1)UVU

†23 has the lozenge structure. This

is to prove that the first order intrinsic UV contribution to the probability in our helio-UVperturbation theory respects the canonical phase combination in all the flavor oscillationchannels.

Then, one may ask “What about the extrinsic UV contribution?”. One must the noticethat the α matrix has a desirable property that αY and Y α† are both both lozenge positionmatrices:

αY =

[αeeYee] [αeeYeµ] e−iδ [αeeYeτ ][e−iδαµeYee + αµµYµe

]eiδ

[e−iδαµeYeµ + αµµYµµ

] [e−iδαµeYeτ + αµµYµτ

]eiδ[

ατeYee + eiδατµYµe + αττYτe] [ατeYeµ + eiδατµYµµ + αττYτµ

]e−iδ

[ατeYeτ + eiδατµYµτ + αττYττ

](7.5)

Y α† =

[αeeYee]

[(e−iδαµe

)∗Yee + αµµYeµ

]e−iδ

[α∗τeYee +

(eiδατµ

)∗Yeµ + αττYeτ

][αeeYµe] e

iδ[(e−iδαµe

)∗Yµe + αµµYµµ

] [α∗τeYµe +

(eiδατµ

)∗Yµµ + αττYµτ

]eiδ

[αeeYτe][(e−iδαµe

)∗Yτe + αµµYτµ

]e−iδ

[α∗τeYτe +

(eiδατµ

)∗Yτµ + αττYττ

]

(7.6)

In eqs. (7.5) and (7.6), the square parentheses imply that inside which only the canonicalphase combination is contained. Therefore, given that S(0) has the lozenge structure itguarantees that the first order extrinsic UV correction terms in the oscillation probabilityrespect the canonical phase combination.

The only exception which does not respect the canonical phase combination to firstorder in our expansion is the interference term between the zeroth and the first orderhelio amplitudes, the second term in the first line in eq. (4.3). It is because the latter,S

(1)helio (see (3.43)) and hence S(1)

helio, is not the lozenge position e±iδ matrix. But, sincethe interference term does not contain UV α parameters, it does not affect if the canonicalphase combination is realized or not.

Thus, the reason why such canonical phase combination appears systematically in bothP (νµ → νe)

(1)int-UV and P (νµ → νe)

(1)ext-UV simultaneously, and throughout all the oscillation

channels is due to a remarkable stability of the lozenge structure, as we saw in the above. Itis not difficult to extend this argument to show that arbitrary order purely intrinsic UV or

13 In fact, U23X and XU†23 are both lozenge position matrices.

– 25 –

purely extrinsic UV contributions as well as any intrinsic - extrinsic mixed UV contributionsto the oscillation probability respect the canonical phase combination to all orders in thehelio-UV parturbation theory.

7.3 Phase convention dependence of the canonical phase combination

It must be obvious from our discussion in section 3.2 that the α matrix, and hence the αβγparameters, depend on the phase convention of the MNS matrix. Then, the form of thecanonical phase combination also depends on the UMNS phase convention. But, since therelationship between the α parameters belonging to the three different phase conventions isexplicitly given in eq. (3.11), it is straightforward to translate the form of canonical phasecombination from one convention to another.

We have obtained the canonical phase combination with the UATM phase convention as

e−iδαµe, ατe, eiδατµ. (7.7)

It can be translated into the one with the UPDG phase convention

e−iδαµe, e−iδατe, ατµ, (7.8)

for the α parameters and the one with the USOL phase convention

αµe, ατe, ατµ, (7.9)

for the α parameters. See eqs. (3.6) and (3.11) for the definitions of α and α parameters,and their relationship with the UATM convention α parameters. That is, under the USOL

phase convention, no correlation between νSM phase δ and the UV α parameter phasesexists. Conversely, one can easily show that the USOL phase convention is the unique casewithout phase correlations.

We should note that there is no preference between the three phase conventions ofUMNS. Suppose we do analysis by taking the UATM convention with defining magnitudesand phases of the complex α parameters as

αµe = |αµe|eiφµe , ατe = |ατe|eiφτe , ατµ = |ατµ|eiφτµ , (7.10)

and obtained the best fit values of the magnitudes and phases. For the sake of discussion weassume temporarily that UV is favoured by the data. Then, there is a simple translation rulebetween the phases one obtains as a result of fitting the data. If we denote the magnitudeand phases of α (UPDG convention) and α (USOL convention) parameters as analogously in(7.10) the translation rule reads

φµe = φµe = φµe + δ, φτe = φτe − δ = φτe, φτµ = φτµ − δ = φτµ − δ.(7.11)

Therefore, one can use any conventions of UMNS in doing analysis. The only thing one mustremember is that the best fit value of φαβ is UMNS convention dependent.

To summarize, we have observed that, to first-order in the helio-UV perturbation the-ory, the UV α parameter phases come in into the oscillation probabilities in a fixed com-bination of νSM CP phase δ. However, the relation is UMNS phase-convention dependent,

– 26 –

as we saw in eqs. (7.7), (7.8), and (7.9). The convention dependence is perfectly legitimatebecause a change in phase convention for the UMNS matrix translates into the one of theα parameters, as shown in section 3.2. The simple translation rule exists for the best fitvalues of α parameter phases.

7.4 Physical interpretation of the phase correlation

As we discussed in section 2, the correlation between the νSM CP phase δ and phases ofthe complex α parameters can be viewed as reflection of property of non-unitarity sub-system inside the larger unitary theory. In this view, the phase correlation can naturallybe physical. On the contrary, some of the readers may argue that existence of UMNS phase-convention (called the SOL convention) in which the δ − α parameter phase correlation isapparently absent testifies for the superficial nature of the correlation. Probably, we shouldconsider the both views as a candidate for valid interpretation.

We report below our current understanding of the interpretation of the phase correla-tion. In fact, there is a simple consistency check of the second view, no phase correlation inthe USOL convention of UMNS. One can perform the similar exercise as in the present paperto examine whether absence of phase correlation in the USOL convention prevails outsidethe region of validity of the helio-UV perturbation theory.

A natural candidate for such perturbative framework is the solar-resonance pertur-bation theory recently formulated in ref. [52], which has a validity in region around thesolar-scale ∆m2 enhanced oscillations. The result of this calculation, which will be re-ported elsewhere [53], shows that there exists a δ − α parameter phase correlation in theUSOL convention. In fact, notion of phase correlation is somewhat enlarged, it looks morelike the correlation between δ and the certain blobs of α parameters. Therefore, it appearsthat the correlation between the νSM CP phase δ and phases of the complex α parameterswe saw in the present paper is physical.

8 Some additional remarks

8.1 Vacuum limit

The vacuum limit in our helio-UV perturbation theory can be taken in a straightforwardmanner. With vanishing matter potentials, the Hamiltonian in eq. (3.2) in the vacuum masseigenstate basis reduces to the free Hamiltonian. Then, all the intrinsic UV contributionsP (νβ → να)

(1)int-UV vanish, and the neutrino oscillation probability coincides with the vacuum

limit of P (νβ → να)(0+1)helio + P (νβ → να)

(1)ext-UV to first order in the α parameters. Notice

that the vacuum limit of the probabilities implies to take the following limits:

φ→ θ13, h1 → 0, h2 →∆m2

21

2E, h3 →

∆m231

2E. (8.1)

See eq. (3.25) to understand the first one. Since it is straightforward to take the vacuumlimit in the expressions of the helio and the extrinsic UV contributions, P (νβ → να)

(0+1)helio +

P (νβ → να)(1)ext-UV, we do not write the explicit forms of the oscillation probabilities in

vacuum.

– 27 –

8.2 Non-unitarity and Non-standard interactions (NSI)

A question is often raised: What is the relationship between non-unitarity and non-standardinteractions (NSI) [54–56]? A short answer is that starting from a generic situation whichinclude not only NSI in propagation but also the ones in production and detection (total 27parameters), our framework could be reproduced by placing appropriate relations betweenthe three kind of NSI parameters. Notice that the latter two introduce non-unitarity [57].

In looking this issue in closer detail, correspondence between NSI and UV approachesis more tenable in propagation part. Since neutrino propagation with NSI is usually formu-lated by implementing unitarity (see e.g., [58, 59]), and the intrinsic UV part of neutrinoevolution is unitary, it is possible to establish one to one correspondence between our αparameters and the propagation NSI elements εαβ (α, β = e, µ, τ) assuming Ne = rNn (ris a constant), which was done in [27] for the case of r = 1. However, one must be awarethat even in this case, the bound on the NSI εβγ parameters obtained by analysis onlywith the propagation NSI cannot be directly translated to the αβγ parameter bound usingthe translation rule described in [27]. It is because the NSI model equivalent to the UV α

system is the one added with the production and detection NSI in the appropriate manneras described in ref. [27]. One should also keep in mind that the condition of constant ratioNe/Nn = r does not hold in the sun, and is broken in deep interior of the earth.

9 Concluding remarks

In this paper, we have formulated a perturbative framework which is called the “helio-UV(unitarity violation) perturbation theory”. It utilizes the two kind of expansion parameters,ε ≈ ∆m2

21/∆m231 and the UV α parameters. To our knowledge it is one of the first trials to

formulate perturbation theory of neutrino oscillation with UV in matter. As an outcome offirst-order computation of the oscillation probability, we were able to obtain the followinginteresting results:

• The phases of the complex UV parameters always come in into the observable in theparticular combination with the νSM CP phase δ, [e−iδαµe, ατe, and eiδατµ], underthe ATM convention of UMNS in which e±iδ is attached to s23. However, we also foundthat the form of phase correlation is UMNS convention dependent, and takes a differentform in the UPDG and USOL conventions. In the latter, in which e±iδ is attached tos12, no phase correlation is observed.

• We have given a clarifying discussion on how (must be) unitary nature of neutrinoevolution in high-scale UV is reconciled with non-unitarity of the whole system. For-tunately, it is built-in in our framework.

While the two conflicting views are mentioned in section 7.4 on how to interpret appar-ent absence of correlation between δ and the UV α parameters in the USOL phase convention,the result obtained in a forthcoming paper [53] indicates that nature of the phenomenonof phase correlation is physical, not an artifact of UMNS convention choice. We want to

– 28 –

emphasize that correlations between νSM CP phase and the phases originated UV newphysics is an entirely natural thing to think about, as we have argued in section 2.

We also note a practical utility of knowing the correlation between δ and the α param-eter phases. Simply, analytic demonstration of parameter correlation makes interpretationof the results of data analysis transparent [23–25]. It is also what happened in the obser-vation of sin θ13-NSI parameter confusion [60], and its analytic understanding in terms ofthe sin θ13 − εeµ − εeτ combined variable uncovered in ref. [58].

It may be worthwhile to note that beyond the phase correlation much discussed in thispaper we have observed a clustering of α parameters in the expressions of the oscillationprobabilities. UsingH matrix elements (see appendix A for explicit expressions) the cluster-ing of α parameters take the form of 2 pairs: (c2

φH12−cφsφH32)-and-(s2φH12+cφsφH32), and

(cφsφH12−s2φH32)-and-(cφsφH12+c2

φH32).14 In each pairs the two variables are transformedto each other by the symmetry transformation (4.11), so that they are not independent witheach other. Therefore, the independent cluster variables of α parameters are only two, onefrom each pair. We do not know physical interpretation of such cluster of α parameters.

Unitarity of neutrino evolution must hold even with the non-unitary mixing matrix,given the Schrödinger equation (3.2) in the mass eigenstate basis with hermitian Hamilto-nian. Our formulation indicates explicitly that the propagation in matter of three flavoractive neutrinos is indeed unitary with the propagation-S matrix U23SU

†23. Then, non-

unitarity of the S matrix, or of the oscillation probability, occurs only when initial andfinal projections of flavor states from/onto the mass eigenstates come into play. Whenunderstood in this way, the structure of unitary propagation-S matrix sandwiched by thenon-unitary α matrix holds to all orders in our helio-UV perturbation theory.

As being a consistent framework, perturbation theory is often useful for finding answersto such qualitative questions as the one discussed above, even though low order calculationsmay not be so accurate numerically. Yet, we have observed that our formulas for the firstorder UV corrections agree reasonably well with the exact results. Utility of first orderformula must increase even more in the precision measurement era in which constraints onUV would reach to |α| <∼ 10−3.

Acknowledgments

We thank Enrique Fernandez-Martinez for illuminating discussions about theories withnon-unitarity, and Hiroshi Nunokawa for raising the issue discussed in appendix D. H.M.expresses a deep gratitude to Instituto Física Teórica, UAM/CSIC in Madrid, for its supportvia “Theoretical challenges of new high energy, astro and cosmo experimental data” project,Ref: 201650E082. I.M.S. acknowledges support from the Spanish grant FPA2015-65929-P(MINECO/FEDER, UE) and the Spanish Research Agency (“Agencia Estatal de Investiga-cion”) grants IFT “Centro de Excelencia Severo Ochoa” SEV2012-0249 and SEV-2016-0597.

14 The diagonal Hjj falls into the form of difference such as (H11 −H33) owing to the re-phasing degreeof freedom. It should be noticed that (c2φH21 − cφsφH23)-and-(s2φH21 + cφsφH23) and (cφsφH21 − s2φH23)-and-(cφsφH21 + c2φH23) are complex conjugate of the two pairs described above. Therefore, they are notindependent variables.

– 29 –

This work has received funding/support from the European Union’s Horizon 2020 researchand innovation programme under the Marie Sklodowska-Curie grant agreement No 674896and No 690575.

A Expression of H and Φ matrix elements

The expressions of the elements Hij defined in eq. (3.28) are given by

H11 = 2αee

(1− ∆a

∆b

),

H12 = e−iδc23

(e−iδαµe

)∗− s23α

∗τe

,

H13 = s23

(e−iδαµe

)∗+ c23α

∗τe,

H21 = eiδ(c23e

−iδαµe − s23ατe

),

H22 = 2[c2

23αµµ + s223αττ − c23s23Re

(eiδατµ

)],

H23 = eiδ[2c23s23(αµµ − αττ ) + c2

23

(eiδατµ

)∗− s2

23eiδατµ

],

H31 = s23e−iδαµe + c23ατe,

H32 = e−iδ[2c23s23(αµµ − αττ ) + c2

23eiδατµ − s2

23

(eiδατµ

)∗],

H33 = 2[s2

23αµµ + c223αττ + c23s23Re

(eiδατµ

)]. (A.1)

The expressions of Φij defined by eq. (3.44) in section 3.9 is given by

Φ11 =H11 − 2c2

φs2φ (H11 −H33)− cφsφ cos 2φ (H13 +H31)

+ ei(h3−h1)x

c2φs

2φ (H11 −H33) + cφsφ

(−s2

φH13 + c2φH31

)+ e−i(h3−h1)x

c2φs

2φ (H11 −H33) + cφsφ

(c2φH13 − s2

φH31

),

Φ12 = e−i(h2−h1)x(c2φH12 − cφsφH32

)+ ei(h3−h2)x

(s2φH12 + cφsφH32

),

Φ13 =−cφsφ cos 2φ (H11 −H33) + 2c2

φs2φ (H31 +H13)

+ ei(h3−h1)x

−cφs3

φ (H11 −H33) + s4φH13 − c2

φs2φH31

+ e−i(h3−h1)x

c3φsφ (H11 −H33) + c4

φH13 − c2φs

2φH31

,

Φ21 = ei(h2−h1)x(c2φH21 − cφsφH23

)+ e−i(h3−h2)x


),

Φ22 = H22,

Φ23 = ei(h2−h1)x−cφsφH21 + s2

φH23

+ e−i(h3−h2)x

cφsφH21 + c2

φH23

,

Φ31 =−cφsφ cos 2φ (H11 −H33) + 2c2

φs2φ (H13 +H31)

+ ei(h3−h1)x

c3φsφ (H11 −H33)− c2

φs2φH13 + c4

φH31

+ e−i(h3−h1)x

−cφs3

φ (H11 −H33)− c2φs

2φH13 + s4

φH31

,

Φ32 = e−i(h2−h1)x−cφsφH12 + s2

φH32

+ ei(h3−h2)x

cφsφH12 + c2

φH32

,

Φ33 =H33 + 2c2

φs2φ (H11 −H33) + cφsφ cos 2φ (H13 +H31)

+ ei(h3−h1)x

−c2

φs2φ (H11 −H33) + cφs

3φH13 − c3

φsφH31

+ e−i(h3−h1)x

−c2

φs2φ (H11 −H33)− c3

φsφH13 + cφs3φH31

. (A.2)

– 30 –

B Expressions of S(1)UV matrix elements

The expressions of S(1)UV matrix elements computed with S(1)

UV = UφS(1)mattU

†φ where S(1)

matt isdefined by eq. (3.45) in section 3.9 is given by(

S(1)UV

)11

= (−i)(∆bx)

[(s2φe−ih3x + c2

φe−ih1x

)H11 − c2

φs2φ


)(H11 −H33)

+ cφsφ


φe−ih1x

)(H13 +H31)

]+

∆b

h3 − h1


)[2c2φs

2φ (H11 −H33) + cφsφ cos 2φ (H13 +H31)

]. (B.1)

(S

(1)UV

)21

= ∆b

[e−ih2x − e−ih1x

(h2 − h1)

(c2φH21 − cφsφH23

)+e−ih3x − e−ih2x

(h3 − h2)


)].

(B.2)

(S

(1)UV

)31

= (−i)(∆bx)

[cφsφ


)H11 − cφsφ

(c2φe−ih3x − s2

φe−ih1x

)(H11 −H33)

+ c2φs

2φ


)(H13 +H31)

]+

∆b

h3 − h1


)[cφsφ cos 2φ (H11 −H33) +H31 − 2c2

φs2φ (H13 +H31)

]. (B.3)

(S

(1)UV

)12

= ∆b

[e−ih2x − e−ih1x

(h2 − h1)

(c2φH12 − cφsφH32


(h3 − h2)


)].

(B.4)

(S

(1)UV

)22

= (−i)(∆bx)e−ih2xH22. (B.5)

(S

(1)UV

)32

= ∆b

[−e−ih2x − e−ih1x

(h2 − h1)

(cφsφH12 − s2

φH32


(h3 − h2)

(cφsφH12 + c2

φH32

)].

(B.6)

(S

(1)UV

)13

= (−i)(∆bx)

[cφsφ


)H33 + cφsφ


φe−ih1x

)(H11 −H33)

+ c2φs

2φ


)(H13 +H31)

]+

∆b

h3 − h1


)[cφsφ cos 2φ (H11 −H33) +H13 − 2c2

φs2φ (H13 +H31)

]. (B.7)

(S

(1)UV

)23

= ∆b

[−e−ih2x − e−ih1x

(h2 − h1)

(cφsφH21 − s2

φH23


(h3 − h2)

(cφsφH21 + c2

φH23

)].

(B.8)

– 31 –

(S

(1)UV

)33

= (−i)(∆bx)

[(c2φe−ih1x + s2

φe−ih3x

)H33 + c2

φs2φ


)(H11 −H33)

+ cφsφ

(c2φe−ih3x − s2

φe−ih1x

)(H13 +H31)

]− ∆b

h3 − h1


)[2c2φs

2φ (H11 −H33) + cφsφ cos 2φ (H13 +H31)

]. (B.9)

C The oscillation probabilities in νµ − ντ sector

In this section, we discuss νµ → νµ and νµ → ντ channels in parallel. Using the notationsdefined in eqs. (4.2) with (4.3), we present the oscillation probabilities in νµ − ντ sector.15

We start from the zeroth-order and the “helio contributions”, by just copying the “simpleand compact” formula in [19] for self-containedeness.

C.1 P (νµ → νµ)(0+1)helio and P (νµ → ντ )

(0+1)helio: “the simple and compact” formula

There are many ways to write P (νµ → νµ)(0+1)helio and P (νµ → ντ )

(0+1)helio . We present here the

ones which may be convenient to verify unitarity:16

P (νµ → νµ)(0+1)helio

= 1−[s4

23 sin2 2φ+ 8εJr cos δ s223

(∆ren)2 (h3 − h1)− (∆ren −∆a)(h3 − h1)2(h3 − h2)

]sin2 (h3 − h1)x

2

−[sin2 2θ23c

2φ − 4ε

(Jr cos δ/c2

13

)cos 2θ23

∆ren (h3 − h1)− (∆ren + ∆a)(h3 − h1)(h3 − h2)

]sin2 (h3 − h2)x

2

−[sin2 2θ23s

2φ − 4ε

(Jr cos δ/c2

13

)cos 2θ23

∆ren (h3 − h1) + (∆ren + ∆a)(h3 − h1)(h1 − h2)

]sin2 (h1 − h2)x

2

− 16εJr cos δ s223

(∆ren)3

(h3 − h1)(h3 − h2)(h1 − h2)sin

(h3 − h1)x

2sin

(h1 − h2)x

2cos

(h3 − h2)x

2.

(C.1)

15 Similarly, if necessary, one can compute P (ντ → ντ )(1)int-UV and P (ντ → ντ )

(1)ext-UV in the same way.

The former can be used to verify unitarity in ντ row with the other probabilities P (ντ → νe)(1)int-UV and

P (ντ → νµ)(1)int-UV which can be obtained by generalized T transformation from eqs. (5.3) and (C.4),

respectively.16 The formulas written here may be more reader friendly compared to the ones in ref. [19] which are

presented in a condensed and abstract fashion.

– 32 –

P (νµ → ντ )(0+1)helio

= −[c2

23s223 sin2 2φ+ 4εJr cos δ cos 2θ23

(∆ren)2 (h3 − h1)− (∆ren −∆a)(h3 − h1)2(h3 − h2)

]sin2 (h3 − h1)x

2

+

[sin2 2θ23c

2φ − 4ε

(Jr cos δ/c2

13

)cos 2θ23

∆ren (h3 − h1)− (∆ren + ∆a)(h3 − h1)(h3 − h2)

]sin2 (h3 − h2)x

2

+

[sin2 2θ23s

2φ − 4ε

(Jr cos δ/c2

13

)cos 2θ23

∆ren (h3 − h1) + (∆ren + ∆a)(h3 − h1)(h1 − h2)

]sin2 (h1 − h2)x

2

− 8εJr(∆ren)3

(h3 − h1)(h3 − h2)(h1 − h2)sin

(h3 − h1)x

2sin

(h1 − h2)x

2

×[cos 2θ23 cos δ cos

(h3 − h2)x

2− sin δ sin

(h3 − h2)x

2

]. (C.2)

C.2 P (νµ → νµ)(1)int-UV and P (νµ → ντ )

(1)int-UV : Intrinsic UV contribution

The first order intrinsic UV correction to the oscillation probability P (νµ → νµ) reads

P (νµ → νµ)(1)int-UV

= − sin2 2θ23

[cos 2θ23 (αττ − αµµ) + sin 2θ23Re

(eiδατµ

)](∆bx)

c2φ sin(h3 − h2)x− s2

φ sin(h2 − h1)x

− s2

23 sin2 2φ

[αee

(1− ∆a

∆b

)− s2


(eiδατµ

)]× (∆bx)

c2

23 sin(h3 − h2)x− c223 sin(h2 − h1)x− s2

23 cos 2φ sin(h3 − h1)x

− 2s2

23 sin 2φ[s23Re

(e−iδαµe

)+ c23Re (ατe)

]× (∆bx)

c2

23c2φ sin(h3 − h2)x+ c2

23s2φ sin(h2 − h1)x+ 2s2

23c2φs

2φ sin(h3 − h1)x

− 4s2

23 sin 2φ

sin 2φ

[αee

(1− ∆a

∆b

)− s2


(eiδατµ

)]+ cos 2φ

[s23Re

(e−iδαµe

)+ c23Re (ατe)

]× ∆b

h3 − h1

s2

23 cos 2φ sin2 (h3 − h1)x

2− c2

23 sin2 (h3 − h2)x

2+ c2

23 sin2 (h2 − h1)x

2

− 4 sin 2θ23

cφsφ

[c23Re

(e−iδαµe

)− s23Re (ατe)

]− cos 2θ23s

2φRe

(eiδατµ

)− sin 2θ23s

2φ(αµµ − αττ )

× ∆b

(h2 − h1)

s2

23c2φ sin2 (h3 − h1)x

2− s2

23c2φ sin2 (h3 − h2)x

2+(c2

23 − s223s

2φ

)sin2 (h2 − h1)x

2

− 4 sin 2θ23

cφsφ

[c23Re

(e−iδαµe

)− s23Re (ατe)

]+ cos 2θ23c

2φRe

(eiδατµ

)+ sin 2θ23c


× ∆b

(h3 − h2)

s2

23s2φ sin2 (h3 − h1)x

2+(c2

23 − s223c

2φ

)sin2 (h3 − h2)x

2− s2

23s2φ sin2 (h2 − h1)x

2

. (C.3)

– 33 –

While the first order intrinsic UV correction to the appearance oscillation probabilityP (νµ → ντ ) reads

P (νµ → ντ )(1)int-UV

= sin2 2θ23

[cos 2θ23 (αττ − αµµ) + sin 2θ23Re

(eiδατµ

)](∆bx)

c2φ sin(h3 − h2)x− s2

φ sin(h2 − h1)x

+ sin2 2θ23c

2φs

2φ

[αee

(1− ∆a

∆b

)− s2


(eiδατµ

)]× (∆bx)

sin(h3 − h2)x− sin(h2 − h1)x+ cos 2φ sin(h3 − h1)x

+ sin2 2θ23cφsφ

[s23Re

(e−iδαµe

)+ c23Re (ατe)

]× (∆bx)

c2φ sin(h3 − h2)x+ s2

φ sin(h2 − h1)x− 2c2φs

2φ sin(h3 − h1)x

− 2 sin 2θ23 cos 2θ23

×cφsφ

[c23Re

(e−iδαµe

)− s23Re (ατe)

]− cos 2θ23s

2φRe

(eiδατµ

)− sin 2θ23s


× ∆b

h2 − h1

−c2

φ sin2 (h3 − h2)x

2+ c2


2− (1 + s2

φ) sin2 (h2 − h1)x

2

+ 2 sin 2θ23 cos 2θ23

×cφsφ

[c23Re

(e−iδαµe

)− s23Re (ατe)

]+ cos 2θ23c

2φRe

(eiδατµ

)+ sin 2θ23c


× ∆b

h3 − h2

(1 + c2

φ) sin2 (h3 − h2)x

2− s2


2+ s2


2

− sin2 2θ23 sin 2φ

sin 2φ

[αee

(1− ∆a

∆b

)− s2


(eiδατµ

)]+ cos 2φ

[s23Re

(e−iδαµe

)+ c23Re (ατe)

]× ∆b

h3 − h1

sin2 (h3 − h2)x

2+ (c2

φ − s2φ) sin2 (h3 − h1)x

2− sin2 (h2 − h1)x

2

− 4 sin 2θ23c

2φ

cφsφ

[c23Im

(e−iδαµe

)− s23Im (ατe)

]+ s2

φIm(eiδατµ

)× ∆b

h2 − h1sin

(h3 − h1)x

2sin

(h1 − h2)x

2sin

(h2 − h3)x

2

+ 4 sin 2θ23s2φ

cφsφ

[c23Im

(e−iδαµe

)− s23Im (ατe)

]− c2

φIm(eiδατµ

)× ∆b

h3 − h2sin

(h3 − h1)x

2sin

(h1 − h2)x

2sin

(h2 − h3)x

2(C.4)

The expressions of P (νµ → νµ)(1)int-UV in eq. (C.3), and P (νµ → ντ )

(1)int-UV in eq. (C.4) are

another explicit demonstration of the canonical phase combination, [e−iδαµe, ατe, eiδατµ],with no correlation between δ and the diagonal α parameters. The exposition of the mech-anism which leads to the canonical phase combination is given in section 7.1.

– 34 –

C.3 P (νµ → νµ)(1)ext-UV and P (νµ → ντ )

(1)ext-UV : Extrinsic UV contribution

The first order extrinsic UV contributions P (νµ → νµ)(1)ext-UV and P (νµ → ντ )

(1)ext-UV are

given by

P (νµ → νµ)(1)ext-UV

= 4s23 sin 2φRe(e−iδαµe

)[−s2

23 cos 2φ sin2 (h3 − h1)x

2+ c2

23

sin2 (h3 − h2)x

2− sin2 (h2 − h1)x

2

]+ 4αµµ

[−1 + s4

23 sin2 2φ sin2 (h3 − h1)x

2+ sin2 2θ23

c2φ sin2 (h3 − h2)x

2+ s2


2

],(C.5)

P (νµ → ντ )(1)ext-UV

= − sin 2θ23 sin 2φ[c23Re

(e−iδαµe

)+ s23Re (ατe)

]×

cos 2φ sin2 (h3 − h1)x

2+ sin2 (h3 − h2)x

2− sin2 (h2 − h1)x

2

+ sin 2θ23 sin2 2φ

[s2

23Re(eiδατµ

)+ c23s23(αµµ + αττ )

]sin2 (h3 − h1)x

2

+ 2 sin 2θ23

[cos 2θ23Re

(eiδατµ

)− sin 2θ23(αµµ + αττ )

]c2φ sin2 (h3 − h2)x

2+ s2


2


[c23Im

(e−iδαµe

)− s23Im (ατe)

]sin

(h3 − h1)x

2sin

(h1 − h2)x

2sin

(h2 − h3)x

2

+ sin 2θ23Im(eiδατµ

)c2φ sin(h3 − h2)x− s2

φ sin(h2 − h1)x

. (C.6)

C.4 Perturbative unitarity yes or no of intrinsic and extrinsic UV contribu-tions: νµ row

Given the expressions of the oscillation probabilities in νµ row in eqs. (4.9), (C.3), and (C.4),it is straightforward to prove perturbative unitarity for neutrino evolution only with intrinsicUV corrections to first order in α’s

P (νµ → νe)(1)int-UV + P (νµ → νµ)

(1)int-UV + P (νµ → ντ )

(1)int-UV = 0. (C.7)

On the other hand, the extrinsic UV corrections in first order in α parameters in theoscillation probabilities in νµ row, eqs. (4.10), (C.5), and (C.6), do not cancel out as in thecase of νe row, giving no indication even of a partial cancellation. Therefore, clearly theextrinsic UV corrections do not respect unitarity.

D Identifying the relevant variables

When non-unitarity is introduced the number of parameters increases from six (νSM) tofifteen (adding nine α parameters), a growth by a factor of 2.5. We look for a possibilityof reducing the number of parameters by finding an extra small parameter by which theoscillation probability can be expanded. By the estimate a/∆m2

ren <∼ 0.1 (assuming Ye =

0.5) for E = 1 GeV and ρ = 3 g/cm3, sinφ can be approximated as s13. Since the measured

– 35 –

value of θ13 is small, s13 = 0.148, which is the one from the largest statistics measurement[61],17 it can be used as another expansion parameter. Then, we can expand the probabilityformulas in terms of sφ ≡ sinφ ' s13 to first order, assuming ρE 10 GeV g/cm3.18

Given the oscillation probability formulas tabulated in table 2, it is easy to expandP (νβ → να) to first order in sφ. Then, we count the α parameters that remain in thezeroth- and the first-order formulas. The results of this exercise are presented in table 2.

Table 2. The UV α parameters which are present in P (νβ → να)(1)UV to zeroth (second column)

and to the first order (third column) in sinφ. The results for anti-neutrino channels are the sameas the corresponding neutrino channels.

channel parameters in P (νβ → να)(1)UV parameters in P (νβ → να)

(1)UV

in zeroth order in sφ to first order in sφνe → νe αee left col. plus Re

(e−iδαµe

), Re (ατe)

νe → νµ, νµ → νe does not apply Re(e−iδαµe

), Im

(e−iδαµe

),

νe → ντ , ντ → νe Re (ατe), Im (ατe)

νµ → νµ αµµ, αττ , Re(eiδατµ

)left col. plus Re

(e−iδαµe

), Re (ατe)

νµ → ντ , ντ → νµ αµµ, αττ , Re(eiδατµ

), Im

(eiδατµ

)left col. plus Re

(e−iδαµe

),

Im(e−iδαµe

), Re (ατe), Im (ατe)

A few remarks are in order: First of all, we should note that in the appearance channels,νµ → νe and νµ → ντ , all the nine UV parameters come in in propagation in matter if we donot expand in terms of sinφ. When expended by sinφ to first order, reduction of numberof parameters is effective for νµ → νe and νe → ντ channels, only four parameters out ofnine. On the other hand, reduction of number of parameters to first order in sinφ is not soeffective for νµ → νµ and νµ → ντ channels, missing only a single parameter αee.

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