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

To derive a structure revealing expression of the neutrino oscillation probability in matter with non-unitarity, we formulate a perturbative framework with the expansion parameters, the ratio $\epsilon$ of the solar $\Delta m^2_{sol}$ to atmospheric $\Delta m^2_{atm}$ and the parameters which describe unitarity violation (UV). Using the $\alpha$ parametrization for non-unitary mixing matrix and to first order in our perturbation theory, we show that there is a universal correlation between the $\nu$SM CP phase $\delta$ and the three UV complex $\alpha$ parameter phases. Using a phase convention of the flavor mixing matrix $U_{\text{\tiny MNS}}$ in which $e^{ \pm i \delta }$ is attached to $\sin \theta_{23}$, it is expressed as $e^{- i \delta } \alpha_{\mu e}$, $\alpha_{\tau e}$, and $e^{i \delta} \alpha_{\tau \mu}$, always the same combination in all the oscillation channels. We also show that in a different $U_{\text{\tiny MNS}}$ phase convention with $e^{ \pm i \delta }$ attached to $\sin \theta_{12}$, the $\delta - \alpha$ parameter phase correlation is absent. We discuss the meaning of the phase-convention dependence. Finally, we argue that the three-flavor neutrino evolution has to be unitary in the presence of non-unitary mixing matrix, and discuss how it can be reconciled with the non-unitarity of the whole system.

5 Diagonalization of zeroth-order Hamiltonian and the hat basis 9 2.6 Flavor basis, the tilde and hat bases, the S andŜ matrices, and their relations 10 2.7 Calculation ofŜ matrix 11 2.8 Recapitulating the leading orderS matrix and the first order helio corrections 12 2.9 Calculation ofŜ (1) andS (1) for the UV part 12 3 Neutrino oscillation probability to first order: ν e − ν µ sector 13 3.1 P (ν e → ν e ) (0+1) helio and P (ν µ → ν e ) (0+1) helio : the "simple and compact" formulas 14 3.2 P (ν e → ν e ) (1) and P (ν µ → ν e ) (1) : Both intrinsic and extrinsic UV contributions 14 B Expressions ofS (1) UV matrix elements 28 C The oscillation probabilities in ν µ − ν τ sector 29 C.1 P (ν µ → ν µ ) (0+1) helio and P (ν µ → ν τ ) (0+1) helio : "the simple and compact" formula 29 C.2 P (ν µ → ν µ ) (1) int-UV and P (ν µ → ν τ ) (1) int-UV : Intrinsic UV contribution 30 C.3 P (ν µ → ν µ ) (1) ext-UV and P (ν µ → ν τ ) (1) ext-UV : Extrinsic UV contribution 32 C.4 Perturbative unitarity yes or no of intrinsic and extrinsic UV contributions: ν µ row 32 D Identifying the relevant variables 32 1 Introduction It appears that by now the three flavor lepton mixing [1] is well established after the long term 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, which we call the lepton KM phase [4], and the neutrino mass ordering, there appeared some hints toward identifying these unknowns. That is, the long-baseline (LBL) neutrino experiment T2K sees with a continuously 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 by the bi-probability plot introduced in ref. [8]. The preference of the normal mass ordering over 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] shows that it can be claimed at 3σ CL. Also, a reanalysis of NOνA data seem to confirm the so far dominant result that θ 23 is near maximal [6].
The apparent convergence of various results from dozens of experiments suggests that we 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 in a definitive way, for example, by Hyper-K [11], T2HKK [12], and DUNE [13]. Yet, it prompts us to think about how to conclude the era of discovery of neutrino mass and the lepton flavor mixing. One of the most important key elements is the paradigm test, that is, 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.
A favourable way of performing a leptonic unitarity test is to formulate a model independent generic framework in which unitarity is violated, and confront it to the experimental data. It was attempted in a pioneering work by Antusch et al. [14], which indeed provided such a framework in the context of high-scale unitarity violation (UV). 2 In low-scale UV, 1 However, this tendency has not been confirmed by the most recent analysis of NOνA data [6]. This point should be settled in the further progress of the measurements, in particular in T2K II, an extended run proposed [7]. 2 We are aware that in the physics literature UV usually means "ultraviolet". But, in this paper UV is used as an abbreviation for "unitarity violation" or "unitarity violating".
on the other hand, the currently available model is essentially unique, the 3 active plus N s sterile model, see refs. [15,16] for a partial list of the early references. In the present context, low and high scales imply, typically, energy scales of new physics much lower and higher than the electroweak scale, respectively. Recently, within the (3 + N s ) model, a model-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 of the 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 simple expression of the neutrino oscillation probability in matter in the presence of UV. The framework has the two kind of expansion parameters, the ratio ≈ ∆m 2 21 /∆m 2 31 (precise definition is in eq. (2.15)), and the UV parameters, hence dubbed as the "helio-UV perturbation theory" in this paper. It can be regarded as an extension of the "renormalized helio perturbation theory" in matter to include non-unitarity [19] 3 , which allows us to discuss UV flavor transition of neutrinos with sizeable matter effect. It would be useful for analyzing experiments such as Super-K, Hyper-K, T2HKK, DUNE, IceCube-Gen2/PINGU, and KM3NeT-ORCA [9,[11][12][13][20][21][22].
Introduction of non-unitary mixing matrix, which replaces the unitary MNS flavor mixing matrix in the neutrino-mass embedded standard model (νSM), brings nine additional parameters in the neutrino oscillation probability. Unraveling the correlations between the MNS and the UV parameters, as well as among the UV parameter themselves would be necessary to analyse the system with non-unitarity. It turns out that our helio-UV perturbation theory is extremely structure-revealing. That is, we will see that the lepton KM phase δ and the complex UV parameters come in into the oscillation probability in a certain fixed combination. If we use so called the α parametrization [23], and use a phase convention of the flavor mixing matrix U MNS in which e ±iδ is attached to sin θ 23 , it is expressed as e −iδ α µe , α τ e , and e iδ α τ µ , a universal feature in all the oscillation channels. It will be referred to as the "canonical phase combination" in this paper. We will see, however, the form of CP phase correlation is U MNS phase convention dependent.
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) of flavor non-universality and zero-distance flavor transition. In an effort toward formulating model-independent framework for testing low-scale UV the two more criteria are uncovered for distinguishing low-scale from high-scale UV. That is, presence of the probability leaking term in the oscillation probability and possible detection of UV perturbative corrections which testifies for the low-scale UV [17,18]. See refs. [24,25] for the current constraints on unitarity violation in low-scale UV scenario.
High-scale unitarity violation is a well studied subject with many references, only part of which is quoted here [14,23,[25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40]. In the context of the present paper, we want to remark that the evolution equation of the three flavor active neutrinos in the mass eigenstate basis in high-scale UV, see e.g., [25], is the same as the leading order one in low-scale UV, which can be singled out by vanishing limit of active-sterile transition elements [18]. This property allows us to discuss both high-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 nonunitarity of the flavor mixing matrix leads to non-unitarity of the observable, the oscillation probability P (ν β → ν α ). The answer is not totally trivial, because the neutrino evolution has to be unitary in high-scale UV, and to our knowledge this point has never been discussed explicitly in the literature. By integrating out heavy new physics sector at high scale, only the three active neutrinos remains as the neutral leptons in low energy effective theory. That is, they span the complete state space of neutral leptons at low energies. 4 The completeness implies that neutrino evolution must be unitary, because there is no way to go outside of the complete neutral lepton state space during propagation, assuming absence of inelastic scattering, absorption, etc. Then, the question is: how and why the oscillation probability does not respect unitarity? We will answer these questions in the next section.
In section 2, we construct our perturbative framework with UV in matter from scratch in a step-by-step manner, and address its U MNS phase convention dependence. In section 3, we compute the neutrino oscillation probabilities in the ν e − ν µ sector to first order in the helio-UV expansion. A universal correlation between the complex α parameters and the νSM CP phase is demonstrated. An explicit proof of unitarity in neutrino evolution is given in in sections 4, and accuracy of the helio-UV expansion is examined in section 5. The stability and phase convention dependence of the phase correlation are discussed in section 6. The two miscellaneous topics, the vacuum limit and the relation with NSI, are addressed in section 7 before giving our conclusion in section 8. The oscillation probabilities in the ν µ −ν τ sector are calculated in appendix C. Table 1 summarizes the equation numbers of all the oscillation probabilities.
2 Formulating the helio-unitarity violation (UV) perturbation theory Following the observation in ref. [18], we work with the neutrino evolution in 3 × 3 active neutrino space in the vacuum mass eigenstate basis, see eq. (2.2) below. It describes both high-scale UV as well as low-scale UV in the leading (zeroth) order expansion in terms of the active-sterile transition elements (denoted as W ). See, e.g., [25] for the equivalent evolution equation in high-scale UV.

Unitary evolution of neutrinos in the mass eigenstate basis
The three active neutrino evolution in matter in the presence of non-unitary flavor mixing can be described by the Schrödinger equation in the vacuum mass eigenstate basis [18,25] In this paper, we denote the vacuum mass eigenstate basis as the "check basis". In eq. (2.1), N denotes the 3 × 3 non-unitary flavor mixing matrix which relates the flavor neutrino states to the vacuum mass eigenstates as Hereafter, the subscript Greek indices α, β, or γ run over e, µ, τ , and the Latin indices i, j run over the mass eigenstate indices 1, 2, and 3. E is neutrino energy and ∆m 2 ji ≡ m 2 j − m 2 i . The usual phase redefinition of neutrino wave function is done to leave only the mass squared differences.
The functions a(x) and b(x) in eq. (2.1) denote the Wolfenstein matter potential [41] due to charged current (CC) and neutral current (NC) reactions, respectively.
Here, G F is the Fermi constant, N e and N n are the electron and neutron number densities in matter. ρ and Y e denote, respectively, the matter density and number of electrons per nucleon in matter. For simplicity and clarity we will work with the uniform matter density approximation in this paper. But, it is not difficult to extend our treatment to varying matter density case if adiabaticity holds. By writing the evolution equation as in eq. (2.1) with the hermitian Hamiltonian, the neutrino evolution is obviously unitary, which is in agreement with our discussion given at the end of section 1. Then, the answer to the remaining question, "how the effect of nonunitarity comes in into the observables as a consequence of non-unitary mixing matrix" is given in section 2.6.

α parametrization of the non-unitary mixing matrix and its convention dependence
To parametrize the non-unitary N matrix we use the so-called α parametrization [23], N = (1 − α) U , where U ≡ U MNS denotes the νSM 3 × 3 unitary flavor mixing matrix. 5 To define the α matrix, however, we must specify the phase convention by which U matrix is defined. We start from the most commonly used form, the Particle Data Group (PDG) [46] convention of the MNS matrix, with the obvious notations s ij ≡ sin θ ij etc. and δ being the CP violating phase. Then, we define the non-unitary mixing matrix N PDG as (2.5) By inserting N = N PDG in (2.5) to eq. (2.1), we define the neutrino evolution equation in the vacuum mass eigenstate basis.

Neutrino evolution with general convention of the MNS matrix
After reducing the standard three-flavor mixing matrix to U PDG , which has four degree of freedom, we still have freedom of phase redefinitioň without affecting physics of the system. Then, the evolution equation in the Γ (β, γ) transformed basis, describes the same physics. In (2.7), U (β, γ) and α(β, γ) denote, respectively, the Γ (β, γ) transformed MNS matrix andᾱ matrix: That is, we can use different convention of the MNS matrix U (β, γ), but then our α matrix has to be changed accordingly, as in (2.8).

The three useful conventions of the MNS matrix
Among general conventions defined in (2.8), practically, there exist the three useful conventions of the MNS matrix. In addition to U PDG in (2.4), they are U (0, δ) and U (δ, δ): The reason for our naming of U ATM and U SOL in (2.9) is because CP phase δ is attached to the "atmospheric angle" s 23 in U ATM , and to the "solar angle" s 12 in U SOL , respectively. Whereas in U PDG , δ is attached to s 13 . Accordingly, we have the three different definition of the α matrix. In addition to The latter two and their relations toᾱ are given by (2.10) In this paper, for convenience of the calculations, we take the "ATM" convention with U ATM and α ATM . But, the translation of our results to the PDG or the "SOL" conventions can be done easily by using eq. (2.10). Notice that, because of the structure N ATM = (1 − α ATM ) U 23 U 13 U 12 , the α matrix is always attached to U 23 . Then, the correlation between the lepton KM phase δ and the UV parameter phases becomes more transparent if e ±iδ is attached to U 23 . This is the reason why we take the MNS matrix convention U ATM in (2.9) in our following calculation.

Preliminary step toward perturbation theory: Tilde-basis
Taking the U ATM convention with α ATM matrix, we formulate our helio-UV perturbation theory. We assume that deviation from unitarity is small, so that α βγ 1 hold for all flavor indices β and γ including the diagonal ones. Therefore, we are able to use the two kind of expansion parameters, ≈ ∆m 2 21 /∆m 2 31 (see eq. (2.15) 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: For convenience in formulating the helio-UV perturbation theory, we move from the check basis to an intermediate basis, which we call the "tilde basis", 6ν = (U 13 U 12 )ν, with Hamiltoniañ In the last line, we have denoted the first and the second terms in eq. (2.12) asH vac andH UV , respectively. The explicit form of theH vac in a form decomposed into the unperturbed and perturbed parts is given bỹ The superscripts (0) and (1) in eqs. (2.13) and (2.14), respectively, show that they are zeroth and first order in . After transforming to the tilde basis, as we expected, we recover the same Hamiltonian as used in the "renormalized helio perturbation theory" without unitarity violation [19]. An order term is intentionally absorbed into the zeroth-order term inH (0) vac as in eq. (2.13) to make the formulas of the oscillation probabilities simple and compact. We note that the matter termH UV in eq. (2.12) can be decomposed into the zeroth, first and the second order terms in α (orα) matrix elements asH UV =H From the flavor basis, the tilde basis is U23 transformed basis,να = (U † 23 ) αβ ν β andH = U † 23 H flavor U23, which is commonly used in various treatments of neutrino propagation in matter. (2.16) The total Hamiltonian in the tilde basis is, therefore, given byH =H vac +H UV , wherẽ vac .

Unperturbed and perturbed Hamiltonian in the tilde basis
To formulate the helio-UV perturbation theory, we decompose the tilde basis Hamiltonian in the following way:H The unperturbed (zeroth-order) Hamiltonian is given byH matt . We make a phase redefinitioñ which is valid even for non-uniform matter density. Then, the Schrödinger equation forν becomes the form in eq. (2.2) with unperturbed part of the Hamiltonian (H (0) ) as given in (2. 19) namely, without NC matter potential terms. It is evident that the phase redefinition does not affect the physics of flavor change. Hereafter, we omit the prime symbol and use the zeroth-order Hamiltonian eq. (2.19) without NC term. This is nothing but the zeroth order Hamiltonian used in [19], which led to the "simple and compact" formulas of the oscillation probabilities in the standard three-flavor mixing. The perturbed Hamiltonian is then given bỹ where each term in eq. (2.20) is defined in eqs. (2.14) and (2.16). In the actual computation, we drop the second-order term (the last term) in eq. (2.20) because we confine ourselves into the zeroth and first order terms in the UV parameters in this paper.

Diagonalization of zeroth-order Hamiltonian and the hat basis
To carry out perturbative calculation, it is convenient to transform to a basis which diago-nalizesH (0) , which we call the "hat basis".H (0) is diagonalized by the unitary transformation as follows:Ĥ where the eigenvalues h i are given by See eqs. (2.11) and (2.15) for the definitions of ∆ ren , ∆ a etc. By the convention with sign(∆m 2 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 in this section, which recapitulates the one in ref. [19], is to make description in this paper self-contained.
U φ is parametrized as where φ is nothing but the mixing angle θ 13 in matter. With the definitions of the eigenvalues eq. (2.22), the following mass-ordering independent expressions for cosine and sine 2φ are obtained: The perturbing Hamiltonian in vacuum in the tilde basis,H vac , has a simple form such that the positions of "zeros" are kept after transformed into the hat basis: In fact,Ĥ 1 is identical toH 1 with θ 13 replaced by (θ 13 − φ). However, the form ofĤ The explicit expressions of the elements H ij are given in appendix A.

Flavor basis, the tilde and hat bases, the S andŜ matrices, and their relations
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 the tilde basis, and from the tilde basis to the check basis: The non-unitary transformation is involved from the check basis to the flavor basis: The relationship between the flavor basis Hamiltonian H flavor and the hat basis oneĤ is Then, the flavor basis S matrix is related toŜ andS matrices as Notice that bothŜ andS are unitary, but S is not because of non-unitarity of the (1 − α) matrix.
This is the answer to the question we posed in section 1. Namely, the non-unitarity of S matrix in the flavor basis, whose square is the observable, comes from the initial projection from the flavor-to mass-basis and the final projection back from the mass-to flavor-eigenstate. Notice that there is no other way, because neutrino evolution has to be unitary, as discussed in section 1. 7 7 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. [25], for example. Our discussion just aims at serving for a transparent understanding of the point, how neutrino's unitary evolution is reconciled with non-unitary nature of the observable P (ν β → να). We will see below and in the next section that at first order in the UV parameter expansion a clear separation between the unitary and non-unitaly part of the oscillation probability occurs.
Because of the reasoning above, we denote U 23S U † 23 as the "propagation-S matrix". For convenience, we write down explicitly all the pieces in eq. (2.31), the propagation-S matrix in terms of theS elements [19]: We note thatS, which can be expanded by the small parameters, and α, as To first order in these small parameters, we obtain We shall call theS (1) UV piece in the second term "intrinsic" UV contribution, and the last two terms in eq. (2.34) as "extrinsic" UV contribution. The intrinsic UV contribution is in unitary part, and the extrinsic UV contribution represents non-unitary effect. Finally, the oscillation probabilities are simply given by (2.35)

Recapitulating the leading orderS matrix and the first order helio corrections
Since all the relevant quantities are computed for the leading order and the helio corrections in ref. [19], we just recapitulate them in below. The zeroth order result ofS matrix is given byS where c φ ≡ cos φ and s φ ≡ sin φ. The non-vanishing first order helio corrections (order ∼ ) toS matrix are given by and all the other elements vanish. The elements of the propagation-S matrix U 23S U † 23 can be obtained fromS matrix elements by using eq. (2.32). (1) andS (1) for the UV part

Calculation ofŜ
In this paper, we restrict ourselves to the perturbative calculation to first order in ≡ ∆m 2 21 /∆m 2 ren ≈ ∆m 2 21 /∆m 2 31 and to first order in the UV parameters α βγ . Then, the form of S matrix and the oscillation probability in zeroth and the first-order helio corrections are identical with those computed in ref. [19]. Therefore, we only calculate, in the rest of this section, the matter part which produces the UV contributions.
By inserting U † φ U φ , H 1 (hereafter the matter part only) can be written as where we have introduced another simplifying matrix notation Φ and its elements Φ ij . The explicit expressions of Φ ij are given in appendix A.
Assuming the uniform matter density, we obtain 3 Neutrino oscillation probability to first order: ν e − ν µ sector In this section, we calculate the expressions of the oscillation probabilities. For clarity, we concentrate on ν e → ν e and ν µ → ν e channels. The other oscillation probabilities which are required to discuss unitarity in the ν e row will be obtained in section 4. The oscillation probabilities in the ν µ − ν τ sector are given in appendix C. Table 1 at the end of this section summarizes 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 collections using eq. (2.32) with theS (0) andS (1) helio matrix elements in eqs. (2.41) and (2.42), respectively, and the first order matter correction S (1) UV in eq. (2.45). Therefore, we know the whole S matrix to first order in the helio and the UV parameters Then, we are ready to calculate the expressions of the oscillation probabilities using the formula P (ν β → ν α ; x) = |S αβ | 2 to first order in the expansion parameters. Since all the building elements are known, we just present the final expressions of the oscillation probabilities. We categorize P (ν β → ν α ) into the three types of terms: The first term in eq.(3.2), P (ν β → ν α ) (0+1) helio , is nothing but the "simple and compact" formulas for the probability derived in ref. [19] which is based on the standard unitary three-flavor mixing and is valid to first order in .
3.1 P (ν e → ν e ) (0+1) helio and P (ν µ → ν e ) (0+1) helio : the "simple and compact" formulas Since all the calculations for P (ν e → ν e ) (0+1) helio and P (ν µ → ν e ) (0+1) helio are done in [19] and described in detail in this reference we just present here the result: where x is the baseline and J r , the reduced Jarlskog factor [47], is defined as Because the matter potential due to the NC interaction is removed from the zerothorder HamiltonianH (0) by the phase redefinition (see section 2.4), the unitary part P (ν β → ν α ) 3.2 P (ν e → ν e ) (1) and P (ν µ → ν e ) (1) : Both intrinsic and extrinsic UV contributions The first order intrinsic and extrinsic UV contributions to P (ν e → ν e ) read int-UV = sin 2 2φ cos 2φ α ee 1 − Similarly, the first order intrinsic and extrinsic UV contributions to P (ν µ → ν e ) are given by The expressions of the oscillation probabilities in eqs. (3.7), (3.8), (3.9), and (3.10) are the first explicit demonstration of the canonical phase combination e −iδ α µe , α τ e , and e iδ α τ µ in this paper. Non-association of e ±iδ to α τ e must be understood as a particular "correlation", and naturally there is no association of δ in the diagonal α parameters, α ββ (β = e, µ, τ ). We will see in the rest of this paper that the canonical phase combination is always realized in both the first order intrinsic and extrinsic UV correction terms in the oscillation probabilities in all the channels. 8 We should mention that the phase correlation e −iδ α µe , a part of our canonical phase combination, has been observed in ref. [23] but only in vacuum, and in ref. [39] in matter but only as an outcome of numerical study in the particular channels. 9 We defer our presentation of the oscillation probabilities in the ν µ − ν τ sector to appendix 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 combination prevails in the oscillation probabilities in the ν µ − ν τ sector, as we will see in appendix C. A possibility of reducing the number of parameters by expanding in another small parameter is briefly discussed in appendix D.
The expressions of the oscillation probabilities are scattered into various places in this paper. Therefore, for the readers' convenience, we tabulate in table 1 the equation numbers for P (ν β → ν α ) (1) int-UV and P (ν β → ν α ) (1) ext-UV in various channels.  In section 2 we have shown that, to first order in the helio-UV perturbation theory, the oscillation probability P (ν β → ν α ) can be decomposed into the two parts, the unitary part denoted as P (ν β → ν α ) int-UV , and the non-unitary part P (ν β → ν α ) (1) ext-UV . It is the unique form of the probability which is consistent with the unitarity of the propagation-S matrix U 23S U † 23 before the initial and final projection to and from the mass eigenstate basis, respectively, are applied. As discussed in section 2.6, 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. 8 A perturbative treatment using the similar expansion parameters is presented in ref. [48] within the framework of 3 + 3 model, in which the calculation of the oscillation probabilities of the first order are carried out. However, due to different implementation of UV, it is essentially impossible to compare our formulas to theirs. As a consequence, none of the points of our emphasis, the canonical phase combination and unitarity of neutrino propagation in matter is not reached in their paper. 9 Our result is consistent with theirs if the authors of refs. [23] and [39] have used the UPDG or UATM phase conventions because the correlation e −iδ αµe holds in the both conventions. See section 6.2.
In this section, we give an explicit proof of unitarity of P (ν e → ν α ) (0+1) helio + P (ν e → ν α ) (1) int-UV in ν e row. The similar explicit proof of unitarity in ν µ row will be given in appendix C. To our knowledge it is the first explicit proof at the probability level that neutrino propagation is unitary in the presence of non-unitary mixing matrix. Since we already know that the oscillation probability to first-order helio corrections is unitary [19], to prove perturbative unitarity. In (4.2) we have assumed that all α βγ ∼ .

Perturbative unitarity of intrinsic UV contribution: ν e row
To examine unitarity of intrinsic UV contribution we compute P (ν e → ν τ ) (1) int-UV with the result Given the expressions of the oscillation probabilities in ν e row, one can readily prove perturbative unitarity for neutrino evolution without extrinsic UV corrections 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 ν e row of the intrinsic UV contributions to first order in the α parameters. The similar result will be shown to hold in ν µ row in appendix C.4.

No perturbative unitarity of extrinsic UV contribution: ν e row
For completeness, we explicitly verify that P (ν β → ν α ) (1) ext-UV gives raise to non-unitary contribution. Among the relevant three probabilities, P (ν e → ν e ) (1) ext-UV is given in eq. (3.8) in section 3.2. P (ν e → ν µ ) (1) ext-UV can be obtained by generalized T transformation of eq. (3.10) P (ν e → ν µ ) (1) ext-UV = 2s 23 sin 2φ cos 2φRe e −iδ α µe − s 23 sin 2φ (α ee + α µµ ) sin 2 Finally, P (ν e → ν τ ) (1) ext-UV can be easily computed as It is evident that they do not add up to zero, as there is no way for imaginary part cancels when P (ν e → ν µ ) (1) ext-UV and P (ν e → ν τ ) (1) ext-UV are added up. We find no indication of even partial cancellation between the various terms. Therefore, there is no perturbative unitarity of extrinsic UV contribution in ν e row, as expected. We will see the same result in the ν µ row in section C.4.
5 How accurate are the first order formulas for P (ν β → ν α ) UV ?
The principal objective of constructing our helio-UV perturbation theory is to understand the qualitative features of oscillation probability with UV. Yet, it may be better to have an idea of how good is the approximation it can offer. In particular, we are interested in the UV part, P (ν β → ν α ) ext-UV , because the accuracies of P (ν β → ν α ) (0+1) helio have been examined in [19]. Notice that it corresponds to the quantity which is numerically computed with high precision and is plotted in the lower panels of figures 1-3 in [18] for (βα) = µe, µτ , and µµ. Then, we confront our first order formulas of P (ν β → ν α ) (1) UV to ∆P (ν β → ν α ) in [18]. In figure 1, plotted are the iso-contours of −P (ν µ → ν α ) (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 same values for the νSM mixing parameters as well as the UV α parameters as in ref. [18]. We see overall agreement, not only qualitatively but also quantitatively to a certain level, between the iso-contours in the upper and lower panels in figure 1 and the ones given in figures 1 (for ν µ → ν e ) and 3 (for ν µ → ν µ ) in ref. [18], respectively. 10 If the numerical accuracy 10 The only exception might be in a relatively small region with shape of oblique ellipse centered around L = 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 our figure 1 and figures 1 and 3 in ref. [18]. Notice that an extremely fine mesh is required to display the contours accurately, given the feature of the probability in this region due to superimposed high-frequency atmospheric-scale oscillations on long-wavelength solar-scale oscillations. But, we did not try to elaborate this point, because a numerical accuracy of the first order formula is not the main point of this paper.
of the first order formula is affected by the treatment of the solar level crossing [19], it would be worthwhile to re-examine this problem with a different framework developed in refs. [50,51].
6 Canonical phase combination: Stability and phase convention dependence In this section we clarify the two aspects of the canonical phase combination, the particular way how the α parameter phases come in in the special combination with the νSM CP phase. They are to answer the questions on the canonical phase combination: (1) Why so stabile over the oscillation channels as well as quite different nature of the intrinsic and and extrinsic contributions, and (2) Are they independent of phase convention of the MNS matrix?

Mechanism for generating the canonical phase combination
Knowing the universal phase correlation between the lepton KM phase and the ones associated with the UV parameters may simplify the analyses, e.g., to constrain non-unitarity. Therefore, it is important to understand how the phase correlation comes about and why it is so stable.
To make the discussion concrete, let us ask a question: Observe that the phase factor e ±iδ is distributed in the S matrix elements in a quite nontrivial fashion (as will be diagnosed below) which is inherited from those of H ij in eq. (A.1) in the first order amplitudes. The first order oscillation probability is given by the interference between the two different amplitudes. Then, what is the reason why such canonical phase combination appears systematically in both P (ν µ → ν e ) (1) int-UV and P (ν µ → ν e ) (1) ext-UV simultaneously, and throughout all the oscillation channels? In this section, we answer this question.
Toward the goal we first note that the flavor basis S matrix at zeroth order has a characteristic form of e ±iδ , in a lozenge positions, as which can be written in an abbreviated form as 11 where X αβ is independent of any CP phases. 11 Here, we refer S matrices in flavor basis by the notations X and Y not to trigger confusion with S matrix elements S αβ which describe the neutrino flavor transformation.
Then, the obvious (and probably unique) possibility to realize the canonical phase combination is that the interfering amplitude, S (1) int-UV and S (1) ext-UV , has the same structure where Y αβ contain the lepton KM and the UV phases, but in the form of canonical phase combination, e −iδ α µe , α τ e , and e iδ α τ µ . It is obvious that the extra phase factors e ±iδ cancel out in P (ν β → ν α ) ∝ (X αβ ) * Y αβ , leaving the canonical phase combination in the oscillation probabilities. The rest of the task that remains needed to answer the question we posed above is to show that the both S ext-UV . The last two terms of the S matrix in eq. (3.2) are the form αX and Xα † , respectively. For generality and possible use in wider context, we use Y , instead of X: In eqs. (6.4) and (6.5), the square parentheses imply that inside them only the canonical phase combination is contained. Therefore, the canonical phase structure of S ext-UV matrix, e ±iδ located in lozenge positions attached to functions only with CP phases with the canonical phase combination, is maintained in both αS (0) and S (0) α † (or, more generically for αY and Y α † ). It guarantees that the first order extrinsic UV correction terms in the oscillation probability respect the canonical phase combination.
We now examine the structure of S UV U † 23 , as the final task to understand the canonical phase structure. A close examination of the expressions ofS (1) UV matrix elements given in appendix B reveals that they possess the canonical phase structure, the form in eq. (6.3). This structure can be recognized in the process of computing the S matrix and the oscillation probability, which is left as an exercise for the readers.

Phase convention dependence of the canonical phase combination
It must be obvious from our discussion in section 2.2 that the α matrix, and hence the α βγ parameters, depend on the phase convention of the MNS matrix. Then, the form of the canonical phase combination also depends on the phase convention. But, since the relationship between the α parameters belonging to the three different phase conventions is explicitly given in eq. (2.10), it is straightforward to translate the form of canonical phase combination from one convention to another.
We have obtained the canonical phase combination with the U ATM phase convention as e −iδ α µe , α τ e , e iδ α τ µ . (6.6) It can be translated into the one with the U PDG phase convention e −iδᾱ µe , e −iδᾱ τ e ,ᾱ τ µ , (6.7) for theᾱ parameters and the one with the U SOL phase conventioñ α µe ,α τ e ,α τ µ , (6.8) for theα parameters. See eqs. (2.5) and (2.10) for the definitions ofᾱ andα parameters, and their relationship with the U ATM convention α parameters. That is, under the U SOL phase convention, no correlation between νSM phase δ and the UVα parameter phases exists. Conversely, one can easily show that the U SOL phase convention is the unique case without phase correlations.

Meaning of convention dependent phase correlation
To summarize, we have observed that, to first-order in the helio-UV perturbation theory, that the UV α parameter phases come in into the oscillation probabilities in a fixed combination of νSM CP phase δ. However, the relation is U MNS phase-convention dependent, as we saw in eqs. (6.6), (6.7), and (6.8). The convention dependence is perfectly legitimate because a change in phase convention for the U MNS matrix translates into the one of the α parameters, as shown in section 2.2. What would be the consequence and the interpretation of these features? The clearest message we can convey to the readers is: • A natural suggestion in analyzing data, e.g. to place constraints on UV, is to utilize the U SOL convention (2.9) with theα parameters (2.10). In this way, one can avoid unwanted correlations between the physical parameters, νSM CP phase δ and theα parameter phases.
Notice that this prescription is independent on our interpretation below of the phase correlation at a "deeper level". It appears to us that the following two conflicting views are possible: • The δ − α parameter phases correlation in the U ATM and U PDG conventions suggests that the way how UV new physics effect is implemented into the low energy effective theory is dictated by the framework of UV itself.
• The fact that the δ − α parameter phases correlation is absent by taking the U SOL convention implies that the phase correlation is artificial without physical significance.
Despite temptation for the latter view, we note that the phase correlation vanishes only at the fine tuned U SOL convention. In every other phase convention, there is δ − α's phase correlation which is universal in all the oscillation channels. That is, existence of the phase correlation is generic. It would imply that a certain consistency condition must be met when the effect of new physics is introduced into the low energy effective theory, νSM.
At this moment, we are unable to make a definitive choice from the two alternative views above, partly because our discussion is based on a particular perturbative framework, whose region of validity is quite limited.
7 Some additional remarks

Vacuum limit
The vacuum limit in our helio-UV perturbation theory can be taken in a straightforward manner. With vanishing matter potentials, the Hamiltonian in eq. (2.1) in the vacuum mass eigenstate basis reduces to the free Hamiltonian. Then, all the intrinsic UV contributions int-UV vanish, and the neutrino oscillation probability coincides with the vacuum limit of P (ν β → ν α ) ext-UV to first order in the α parameters. Notice that the vacuum limit of the probabilities implies to take the following limits: See eq. (2.24) to understand the first one. Since it is straightforward to take the vacuum limit in the expressions of the helio and the extrinsic UV contributions, P (ν β → ν α ) ext-UV , we do not write the explicit forms of the oscillation probabilities in vacuum.

Non-unitarity and Non-standard interactions (NSI)
A question is often raised: What is the relationship between non-unitarity and non-standard interactions (NSI) [52][53][54]? A short answer is that starting from a generic situation which include not only NSI in propagation, but also the ones in production and detection our framework could be reproduced by placing appropriate relations between the propagation, production, and detection NSI. Notice that the latter two introduce non-unitarity [55]. However, it implies a huge reduction of number of parameters, 27 to 9, excluding νSM ones. In addition, the statement is true only if the ratio of neutron number density to electron number density is constant over the entire environment we deal with. Clearly, the condition is not valid in the sun, and is broken even inside the Earth. Assuming N e = rN n (r is a constant) a more detailed correspondence may be established for propagation NSI. Notice that neutrino propagation with NSI is usually formulated by implementing unitarity (see e.g., [56,57]). Since our intrinsic UV part of neutrino evolution is unitary, it is possible to establish one to one correspondence between our α parameters and the propagation NSI elements αβ (α, β = e, µ, τ ), as shown in [25] for the case of r = 1. 12

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, ≈ ∆m 2 21 /∆m 2 31 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 of first-order computation of the oscillation probability, we were able to obtain the following interesting results: • The phases of the complex UV parameters always come in into the observable in the particular combination with the νSM CP phase δ, [e −iδ α µe , α τ e , and e iδ α τ µ ], under the phase convention of U MNS in which e ±iδ is attached to s 23 .
• We have also observed that the way the complex α parameters are correlated with δ is U MNS convention dependent, which stems from convention dependence of the α matrix. It is [e −iδ α µe , e −iδ α τ e , α τ µ ] in the PDG convention, and no correlation between α parameters and δ in the U MNS phase convention called U SOL with e ±iδ attached to s 12 .
We would like to emphasize that this is a rare occasion in which the correlation between νSM and the new physics parameters is explicitly discussed and elucidated. From these results, the most importance message, which could be relevant to the readers, is that usage of the U SOL phase convention and the associatedα parameters (see (2.10) for definition) may be preferable for a merit of avoiding unwanted correlations between the physically different two groups of phases. It would simplify analyses of data to constrain UV with clearer interpretation of the results, and makes discussions of the parameter correlation and degeneracy more transparent. Now, the important question is: "What do the above features of the phase correlation mean? Our particular concern is about the U MNS phase convention dependence uncovered in our study. One can argue that existence of the phase convention in which the UV α parameters do not have correlation with δ implies that it is of superficial nature. However, the correlation exists in all the phase convention except for U SOL . The correlation is universal, i.e., the identical combinations in all the channels for a given U MNS phase convention. Then, an alternative interpretation which is natural in this line of thought is that the way νSM CP phase δ couples with the complex UV parameters is dictated by the framework of UV itself. In this paper, we are not able to make a definite choice from these two interpretations. One of the key obstacles is that nothing is known about whether the similar phase correlation 12 The structure corresponding to αα − ββ for diagonal NSI elements, which is due to re-phasing freedom, is not visible in our oscillation probability formulas which are written by the α parameters. But, it must exist at the level of elements Hij defined in eq. (2.27). For an explicit demonstration of the former structure for NSI, see e.g., arXiv version 1 of ref. [56]. exists in regions outside validity of our perturbative framework. Furthermore, even within the current framework we must be able to give an all-order proof of the canonical phase combination for a firmer statement. To carry it out, however, one has to deal with the situation where "helio" and "UV" amplitudes interfere in a fully mixed way. We hope that we can return to these issues in the future.
As being a consistent framework, perturbation theory is often useful for finding answers to such qualitative questions as above, even though low order calculations may not be so accurate numerically. Yet, we have observed that our formulas for the first order UV corrections agree reasonably well with the exact results. Utility of first order formula must increase even more in the precision measurement era in which constraints on UV would reach to |α| < ∼ 10 −3 .
We have given a clarifying discussion on how (must be) unitary nature of neutrino evolution in high-scale UV is reconciled with non-unitarity of the whole system. Unitarity of neutrino evolution must be true even with the non-unitary mixing matrix, given the Schrödinger equation (2.1) in the mass eigenstate basis with hermitian Hamiltonian. Our formulation indicates explicitly that the propagation in matter of three flavor active neutrinos is indeed unitary with the propagation-S matrix U 23S U † 23 . Then, non-unitarity of the S matrix, or of the oscillation probability, occurs only when initial and final projections of flavor states from/onto the mass eigenstates come into play. When understood in this way, the property holds to all orders in our helio-UV perturbation theory.
14 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.
value of θ 13 is small, s 13 = 0.148, which is the one from the largest statistics measurement [58], 15 it can be used as another expansion parameter. Then, we can expand the probability formulas in terms of s φ ≡ sin φ s 13 to first order, assuming ρE 10 GeV g/cm 3 . 16 Given the oscillation probability formulas tabulated in table 2, it is easy to expand P (ν β → ν α ) to first order in s φ . Then, we count the α parameters that remain in the zeroth-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 (ν β → ν α ) (1) UV to zeroth (second column) and to the first order (third column) in sin φ. The results for anti-neutrino channels are the same as the corresponding neutrino channels. channel parameters in P (ν β → ν α ) (1) UV parameters in P (ν β → ν α ) (1) UV in zeroth order in s φ to first order in s φ ν e → ν e α ee left col. plus Re e −iδ α µe , Re (α τ e ) ν e → ν µ , ν µ → ν e does not apply Re e −iδ α µe , Im e −iδ α µe , ν e → ν τ , ν τ → ν e Re (α τ e ), Im (α τ e ) ν µ → ν µ α µµ , α τ τ , Re e iδ α τ µ left col. plus Re e −iδ α µe , Re (α τ e ) ν µ → ν τ , ν τ → ν µ α µµ , α τ τ , Re e iδ α τ µ , Im e iδ α τ µ left col. plus Re e −iδ α µe , Im e −iδ α µe , Re (α τ e ), Im (α τ e ) A few remarks are in order: First of all, we should note that in the appearance channels, ν µ → ν e and ν µ → ν τ , all the nine UV parameters come in in propagation in matter if we do not expand in terms of sin φ. When expended by sin φ to first order, reduction of number of parameters is effective for ν µ → ν e and ν e → ν τ channels, only four parameters out of nine. On the other hand, reduction of number of parameters to first order in sin φ is not so effective for ν µ → ν µ and ν µ → ν τ channels, missing only a single parameter α ee .