Physics of parameter correlations around the solar-scale enhancement in neutrino theory with unitarity violation

...................................................................................................................Wediscussthephysicsofthethreeneutrinoﬂavortransformationwithnon-unitarymixing matrix,withparticularattentiontothecorrelationbetweenthe ν SM-and the α parameters which represent the effect of unitarity-violating (UV) new physics. Towards this goal, a new perturbative framework is created to illuminate the effect of non-unitarity in the region of the solar-scale enhanced oscillations. We refute the skepticism about the physical reality of the ν Standard Model CP phase δ – α parameter phase correlation by analysis with the SOL convention of U MNS , in which e ± i δ is attached to s 12 . Then, a comparative study between the solar-and atmospheric-scale oscillation regions allowed by the framework reveals a dynamical δ –(blobs of the α parameters) correlation in the solar oscillation region, in sharp contrast to the “chiral”-type phase correlation [ e − i δ ¯ α μ e , e − i δ ¯ α τ e , ¯ α τμ ] in the Particle Data Group convention seen in the atmospheric oscillation region. An explicit perturbative calculation to the ﬁrst order in the ν μ → ν e channel allows us to decompose the UV related part of the probability into the unitary evolution part and the genuine non-unitary part. We observe that the effect of non-unitarity tends to cancel between these two parts, as well as between the different α βγ parameters. ...................................................................................................................


Introduction
The discovery of neutrino oscillation and hence neutrino mass [1,2] under the framework of threegeneration lepton flavor mixing [3] created a new field of research in particle physics.It led to the construction of the next-generation accelerator and underground experiments with the massive detectors Hyper-Kamiokande [4] and DUNE [5].These projects are going to establish CP violation due to the lepton Kobayashi-Maskawa (KM) phase [6], possible lepton counterpart of the quark CP violation [7].They will also determine the neutrino mass ordering at high confidence level by utilizing the Earth matter effect [8,9].Of course, the flagship projects will be challenged by the ongoing [10][11][12] and the other upcoming experiments, for example, ESSνSB [13], JUNO [14], T2KK [15], 1 INO [17], IceCube-Gen2/PINGU [18] and KM3NeT/ORCA [19], which compete for the same goals.
Toward establishing the three-flavor mixing scheme, in particular in the absence of a confirmed anomaly beyond the neutrino-mass embedded Standard Model (νSM), one of the most important topics in the future would be the high-precision paradigm test. 2 In this context, leptonic unitarity tests, either by closing the unitarity triangle [21], or by an alternative method of constraining the models of unitarity violation (UV) at high-energy [22,23] or low-energy scales [24][25][26], are extensively discussed. 3The references in Refs.[27][28][29][30][31][32][33][34][35], for example, include the past and the subsequent development of the subjects.A summary of the current constraints on UV is given e.g., in Refs.[26,36].
It was observed that in the 3 × 3 active neutrino subspace the evolution of the system can be formulated on the same footing in low-scale as well as high-scale UV scenarios [25,26].Nonetheless, dynamics of the three neutrino system with non-unitary mixing in matter has not been investigated in sufficient depth.Apart from numerically implemented calculation carried out in some of the aforementioned references, only a very limited effort was devoted to analytical understanding of the system so far.This is in sharp contrast to the fact that a great amount of effort was devoted to understanding the three-flavor neutrino oscillation. 4A general result known to us so far is the exact S matrix with non-unitarity in matter with constant density [25] calculated by using the Kimura-Takamura-Yokomakura-type construction [37].It allows us to obtain the exact expression of the oscillation probability with non-unitarity.
In a previous paper [38], we started a systematic investigation of the analytic structure of the three-neutrino evolution in matter with non-unitarity.We have used the so-called α parametrization [23] to implement non-unitarity in the three-neutrino system.Under the Particle Data Group (PDG) convention U PDG [39] of the flavor-mixing matrix U MNS , it parametrizes the 3 × 3 non-unitary mixing matrix N as Using a perturbative framework dubbed as the "helio-UV perturbation theory" (a UV-extended version of [40]) with two kinds of expansion parameters, the helio-to-terrestial ratio ≈ m 2 21 / m 2   31   and the α parameters, we computed the oscillation probability valid to the first-order in the expansion parameters.The region of validity of the perturbative framework spans the one around the atmospheric-scale enhanced oscillations which cover the relevant region for the ongoing and the next-generation long-baseline (LBL) accelerator neutrino oscillation experiments.The possibility of application to the data from the near-future facilities and the currently almost non-understood properties of the system may justify the examination even though it is to the first order in expansion.
In our view, the most significant observation in Ref. [38] is that the νSM CP phase δ and the complex α parameters defined above have an intriguing phase correlation of the form [e −iδ ᾱμe , e −iδ ᾱτe , ᾱτμ ].What is unique in the phase correlation is that it universally holds in all the oscillation channels as well as unitary and non-unitary parts of the oscillation probability.One should note that the definition PTEP 2020, 113B01 I. Martinez-Soler and H. Minakata of the α parameters, and consequently the precise form of the correlation between the CP phases, depends on the phase convention of the lepton flavor mixing Maki-Nakagawa-Sakata (MNS) matrix; see Sect.4.1. 5 puzzling feature of the δ-α parameter phase correlation in Ref. [38] is that it disappears in the SOL convention of U MNS in which e ±iδ is attached to s 12 .This triggered a skepticism of the nature of phase correlation, which may allow the following two alternative interpretations: (1) Existence of the SOL phase convention of U MNS in which δ and α phase correlation is absent implies that the CP phase correlation is not physical, but an artifact of an inadequate choice of U MNS phase convention.(2) Physics must be U MNS convention independent.In all the other conventions of U MNS except for the SOL, there exists δ-α parameter phase correlation.Therefore, the existence of phase correlation is generic and it must be physical.
If the interpretation (1) and the reasoning behind it are correct, δ and α phase correlation must be absent under the SOL convention of U MNS everywhere in the allowed kinematical regions.Conversely, if we see a non-vanishing phase correlation in the oscillation probability calculated with the SOL convention somewhere, it implies that the interpretation (1) cannot be true.We will show throughout this paper that the interpretation (2) holds by investigation of the system in the region of the solar-scale enhanced oscillation.[38] In this paper, we discuss the physics of neutrino flavor transformation in the region of solar-scale enhanced oscillation. 6We will try to achieve the following two goals:

The goal of this paper by itself, and in combining a companion work
• To examine the system of the three-flavor neutrinos in the SOL convention (e ±iδ attached to s 12 ) of U MNS in the region of the enhanced solar oscillation, which will testify to the physical reality of the correlation between the νSM-UV α parameter phases.
• To understand the νSM-UV parameter correlation in a more generic context and in the wider kinematical region by combining the results of this and previous works [38].
A few words on the examination of the "solar region" should be said.We feel that an immense need exists for the real understanding of parameter correlation in theories with non-unitarity, in particular outside the region investigated in Ref. [38].The natural "field of research" for this purpose is the region of solar-scale enhanced oscillation, the unique place for enhancement other than the atmospheric one in our world of the three generation leptons.The feature can be seen clearly in the "terrestrial-friendly" region of the E vs. L plot, e.g., in Refs.[41,42]; the latter of those works also provides a brief summary of recent activities on atmospheric neutrinos at low energies.We note that the solar region has been the target of investigation for a long time, see e.g., Refs.[43][44][45][46] and possibly others that we may have missed, mainly in the context of atmospheric neutrino observation at low energies.It should also be mentioned that this topic is now receiving renewed interest [41,42] given the new possibilities of gigantic detectors such as JUNO [47], DUNE [48], and Hyper-K [49].Thus, the second goal of this paper is to achieve a deeper understanding of parameter correlation by combining knowledge in the regions of atmospheric-scale and the solar-scale enhanced oscillations.
Very recently, we have formulated a perturbative framework in the νSM, dubbed as the "solar resonance perturbation theory" [41], the validity of which is around the very region of our interest.We extend this perturbative framework to include the effect of UV, by treating the α parameters as the additional expansion parameters.Using this framework, we investigate dynamics of the three neutrino evolution with a non-unitary mixing matrix under the constant matter density approximation, with particular attention to the parameter correlation.We will show that the system displays a rich, new phenomenon of clustering of the νSM and the UV variables.
Nonetheless, we find it insufficient to rely on analytic treatment based on perturbation theory to extract the characteristic feature of the system due to a new and intricate feature of the parameter correlation.For this reason we rely also on exact numerical analyses as well as the perturbative formula we derive in this paper to elucidate the physics of the parameter correlation in the region of enhanced solar-scale oscillation.It will be particularly illuminating when our analysis is done in a style of comparative study between the solar-and atmospheric-scale oscillation regions, as can be seen in Sect.7. We hope that such understanding will eventually help analyze data for leptonic unitarity tests.
In Sect.3, we introduce the concept of parameter correlations by describing a pedagogical example of the three-neutrino system with the non-standard interactions.In Sect.4, we give a step-by-step formulation of the perturbative framework which is to be utilized in analyzing features of the threeneutrino evolution with a non-unitary mixing matrix.The prescription for computing S matrix elements is given with the help of the tilde basis S matrix elements summarized in Appendix B. In Sect.5, a general formula for the oscillation probability is derived, and applied to the computation of the appearance probability in the ν μ → ν e channel.This section together with Appendix D.1 contains the explicit expression of the oscillation probability in the ν μ → ν e channel to the first order in expansion parameters.In Sect.6, we discuss the characteristic features of the correlation between the νSM CP phase and UV α parameters in the region of validity of our perturbative framework.In Sect.7, the physics of neutrino flavor transformation with UV is discussed paying a particular attention to parameter correlation, contrasting between the regions of the solar-and atmospheric-scale enhanced oscillations.In Sect.8, we give the concluding remarks.

Parameter correlation in neutrino oscillation with beyond-νSM extended settings
It may be useful to start the description of this paper by briefly recollecting some known features of parameter correlation in neutrino oscillation, in particular, in an extended setting that includes physics beyond the νSM.In this context, a general framework that is most frequently discussed is the one which includes the neutrinos' non-standard interactions (NSI) [8] in the flavor-basis Hamiltonian, where the ε parameters describe the flavor-dependent strengths of NSI and a denotes the matter potential (see Eq. ( 8)).We discuss only the so-called "propagation NSI".For a review of the physics of NSI in wider contexts, see e.g.Refs.[50][51][52].We note that the inclusion of the NSI Hamiltonian (2) brings an extra nine parameters into the νSM Hamiltonian with six degrees of freedom, the two m 2 , the three mixing angles, and the unique CP phase, under the influence of the matter potential background.

Emergence of collective variables involving νSM and NSI parameters
A large number of parameters with UV, which is more than doubled the νSM ones, makes it conceivable that the dynamics of neutrino oscillation naturally involves rich correlations among these variables. 7Here, we discuss only a particular type of correlation uncovered in Ref. [55] because, we believe, it illuminates the point.In that work, the authors formulated a perturbative framework of the system with NSI by using the three (the latter two assumed to be) small expansion parameters, ≡ m 2 21 / m 2 31 , s 13 ≡ sin θ 13 , and the ε parameters.They derived the formulas of the oscillation probability to the second order (the third order in the ν μ → ν e channel) in the expansion parameters, which is nothing but an extension of the Cervera et.al. formulas [56] to include NSI. 8 In this calculation the PDG convention of U MNS [39] is used.
An interesting and unexpected feature of the NSI-extended formulas is the emergence of the two sets of "collective variables": where an overall e −iδ is factored out from the matrix element S eμ to make the s 13 term δ free, through which e iδ dependences in 12 in Eq. ( 3) result.That is, if we replace s 13 ( m 2 31 /a) and c 12 s 12 ( m 2 21 /a) in the original formulas by 13 and 12 , respectively, the extended second-order formulas with full inclusion of NSI effects automatically appear [55].In fact, the procedure works for the third-order formula for P(ν μ → ν e ) as well.We note that the second-order computation of Ref. [55] includes the ν μ -ν τ sector, and the additional collective variables are identified.However, for simplicity, we do not discuss them here and refer the interested readers to Ref. [55].
The appearance of the cluster variables composed of the νSM and NSI parameters in Eq. (3) implies that there exists strong correlations between the νSM variables s 13 -δ and the NSI ε eμ -ε eτ parameters in such a way that they form the collective variable 13 to convert the Cervera et.al. formula to the NSI-extended version.A similar statement can be made for the other cluster variable PTEP 2020, 113B01 I. Martinez-Soler and H. Minakata Therefore, there is nothing strange in the parameter correlations among the νSM and new physics parameters.It appears that the phenomenon arises generically, at least under the environment that the matter effect is comparable to the vacuum effect.

Dynamical nature of the parameter correlation
We must point out, however, that the features of the parameter correlation depend on the values of the parameters involved, and also on the kinematical region of neutrino energy and baseline with background matter density.Therefore, depending upon the region of validity of the perturbative framework which is used to derive the correlation, the form of parameter correlation changes.We call all these features collectively the "dynamical nature" of the parameter correlation. 9e want to see explicitly whether a change in features of the correlation occurs when the values of the parameters involved are varied, or its effect is incorporated into the framework of perturbation theory.For this purpose let us go back to the collective variable correlation in Eg. (3).We know now the value of θ 13 is larger than that assumed at the time the Cervera et.al. formula was derived [39].The latest value from Daya Bay is s 13 = 0.148 [61], which is of the order of √ = 0.176.Then, we need higher-order corrections of s 13 , up to the fourth-order terms, to match to the second-order accuracy in [62,63].When this is carried out with the inclusion of NSI [63], it is seen that part of the additional terms generated do not fit to the form of collective variables given in Eq. ( 3).Therefore, when we make θ 13 larger, the parameter correlation which produced the collective variables (3) starts to dissolve.Thus, the analysis of this particular example reveals the dynamical nature of the parameter correlation in the neutrino propagation with NSI.We expect that overseeing the results of computations of the oscillation probabilities in this and the previous papers [38] will reveal the similar dynamical behavior of the parameter correlation in the three-flavor neutrino evolution in matter with non-unitary mixing.

Phase correlation through NSI-UV parameter correspondence?
Can we extract information about the δ-α parameter phase correlation from the collective variables (3)?The answer is Yes if we assume a "uniform chemical composition model" of the matter.As far as the propagation NSI is concerned, there is one-to-one mapping between NSI ε parameters and the UV α parameters, as noticed by Blennow et.al. [26] under the assumption N n = N e -equal neutron and proton number densities in a charge-neutral medium.Of course, an extension to the more generic case of N e = rN n [38] can be easily done without altering the conclusion.For the purpose of the present discussion, one also has to "approve" the procedure by which the e iδ dependence of the collective variables (3) is fixed.That is, removing an overall phase from the matrix element S eμ to make the s 13 term δ-free, as done in Ref. [55].
Assuming that the two conditions above are met, this leads to the collective variables in Eq.
where we have to use the α parameters defined in the PDG convention of U MNS .The emerged correlation between δ and the α parameters is consistent with the canonical phase combination obtained in Ref. [38] in the PDG convention.For the relationships between the α parameters with the various U MNS conventions, see Sect.4.1.This is not unreasonable because the regions of validity of the perturbative frameworks in Refs.[55] and [38] overlap.

Formulating perturbation theory around the solar-scale enhancement with non-unitarity
The discussion of physics in this paper necessitates a new analytical framework to illuminate the effect of a non-unitary mixing matrix in the region of the solar-scale enhanced oscillations; the UV extended version of the "solar-resonance perturbation theory" [41].

Neutrino evolution in the vacuum mass eigenstate basis
As is customary in our formulation of the three active neutrino evolution in matter with UV [38], we start from the evolution equation in the vacuum mass eigenstate basis, the justification of which is given in Refs.[25,26]. 10With use of the "check basis" for the vacuum mass eigenstate basis, it takes the form of a Schrödinger equation: with Hamiltonian where E is neutrino energy and m 2 ji ≡ m 2 j − m 2 i .A usual phase redefinition of neutrino wave function is done to leave only the mass squared differences.N denotes the non-unitary flavor mixing matrix which relates the flavor neutrino states to the vacuum mass eigenstates as where β (and the other Greek indices) runs over e, μ, τ , while the mass eigenstate index i (and the other Latin indices) runs over 1, 2, and 3.It must be noticed that the neutrino evolution described by Eq. ( 5) is unitary, as is obvious from the hermitian Hamiltonian (6).The apparent inconsistency between the unitary evolution and the non-unitarity of the flavor-basis S matrix, one of the points of emphasis in Ref. [38], will be resolved in Sect.4.7.Note that, due to a limited number of appropriate symbols, the notations for the various bases may not always be the same in our series of papers.The functions a(x) and b(x) in ( 17) denote the Wolfenstein matter potential [8] due to the charged current (CC) and the neutral current (NC) reactions, respectively.
Here, G F is the Fermi constant, and N e and N n are the electron and neutron number densities in matter.ρ and Y e denote, respectively, the matter density and the number of electrons per nucleon in matter.We define the following notations for simplicity to be used in the discussions hereafter in this paper: For simplicity and clarity we will work with the uniform matter density approximation in this paper.However, it is not difficult to extend our treatment to a varying matter density case if adiabaticity holds.
Throughout this paper, due to the reasoning mentioned in Sect. 1, we use the SOL convention of the U MNS matrix, the standard 3 × 3 unitary flavor mixing matrix where we have used the obvious notations s ij ≡ sin θ ij etc. and δ denotes the lepton KM phase [6], or the νSM CP violating phase.We use the term "SOL" because the phase factor e ±iδ is attached to the "solar angle" s 12 .It is physically equivalent to the commonly used PDG convention [39] in which the phase factor is attached to s 13 .We use the α parametrization of a non-unitary mixing matrix [23] defined in the U SOL convention: As seen in Eq. (11), and discussed in detail in Ref. [38], the definition of the α matrix depends on the phase convention of the flavor-mixing matrix U MNS .Consistent with the notation used in Ref. [38], we denote the α matrix elements in the SOL convention as α βγ .
The other convention of the MNS matrix which is heavily used in Ref. [38] is the "ATM" convention in which e ±iδ is attached to the "atmospheric angle" s 23 : The α parameters defined in the ATM and PDG conventions of U MNS are denoted as α βγ and ᾱβγ , respectively, in Ref. [38], and we follow that notation in this paper.We recapitulate here the relationships between the α parameters defined with the PDG ( ᾱ), ATM (α) and the SOL ( α) conventions of U MNS : where we note that the diagonal α parameters are equal among the three conventions.

Region of validity, expansion parameters, and the target sensitivity
In this section, we aim at constructing the perturbative framework which is valid at around the solar oscillation maximum, m 2 21 L/4E ∼ O (1).Given the formula Hence, our perturbative framework must fully take into account the MSW effect caused by the Earth matter effect.A more detailed discussion of the region of validity without the UV effect is given in Ref. [41].
As in the solar resonance perturbation theory, we will have the "effective" expansion parameter in the νSM sector, A exp = c 13 s 13 a/ m 2 31 ∼ 10 −3 , as discussed in Sect.4.8.The reason for having such a very small expansion parameter is due to the special structure of our perturbative Hamiltonian.
To formulate our perturbative framework with UV, we use α βγ , defined in Eq. ( 11), for the extra expansion parameters.That is, we assume that deviation from unitarity is small.Therefore, α βγ 1 holds for all β and γ .Though we follow basically the same procedure as in Ref. [41], we give a step-by-step presentation of the formulation because of the additional complexities associated with the inclusion of UV, and to make this paper self-contained.
What would be a reachable or a possible target sensitivity to α βγ in the context of unitarity test?For the sake of rough estimation, we assume momentarily a perfect knowledge of the νSM mixing parameters.Then, let us ask: Which level of sensitivity to UV α βγ parameters could one expect given that the accuracy of measurement of P βα ≡ P(ν β → ν α ) − P(ν β → ν α ) νSM (see Eq. ( 67)) is of the order of, for example, 10 −2 or 10 −4 ?Notice that P βα is the non-unitary contribution to the oscillation probability.The former number is more or less the situation at the current time or in the near future, while the latter is taken arbitrarily as an expectation in a foreseeable future.Since P βα ∼ α βγ , we would expect the constraints on α βγ parameters to be of the order of 10 −2 , or 10 −4 , respectively.
We note that once the accuracy of measurement reaches a "perturbative regime" the first-order UV correction is sufficient, as far as qualitative discussions are concerned.The second-order computation yields terms of the order of α 2 βγ ∼ 10 −4 , or ∼ 10 −8 , respectively, far beyond the accuracy of the P βα measurement in each era.This is the reason why we restrict ourselves to the first-order formulas in this and companion papers [38].
In low-scale UV scenarios, the probability leaking term as well as the flux "mis-normalization" term in the appearance channels are of the order of ∼ |W | 4 , where W denotes collectively the activesterile mixing matrix elements [25].Due to unitarity in the whole 3 + N sterile space, α βγ must be of the order of W 2 .Then, the leaking and the mis-normalization terms are of the order of α 2 βγ ∼ 10 −4 9/39 PTEP 2020, 113B01 I. Martinez-Soler and H. Minakata or 10 −8 in the above two regimes, respectively, which are far too small compared to the accuracy of P βα measurement in each era.This constitutes one of the serious problems in their determination.

Transformation to the tilde basis
We transform to a different basis to formulate our perturbation theory for solar-scale enhancement.It is the tilde basis with Hamiltonian Notice that the term "tilde basis" has no connection to our notation of α parameters in the SOL convention.The Hamiltonian in the tilde basis is given by where each term of the right-hand side of ( 18) is given by  19) In this paper, we restrict ourselves to the perturbative correction to the first order in the expansion parameters.There is a number of reasons for this limitation.It certainly simplifies our discussion of the δ-α parameter phase correlation, though we will make a brief comment on the effect of H (2) UV on the correlation in Sect.6.2.Unfortunately, the expression of the first-order UV correction to the oscillation probability is sufficiently complex at this order, as we will see in Sects.5.1 and 5.2 and Appendix D.1.We do not consider our restriction to the first order in α βγ a serious limitation because the framework anticipates a precision era of neutrino experiment for the unitarity test in which the condition α βγ 1 should be justified.

Definitions of F and K matrices
To make expressions of the S matrix and the oscillation probability as compact as possible, it is important to introduce the new matrix notations F and K: The explicit expressions of the elements F ij and K ij defined in Eqs. ( 22) and ( 23), respectively, are given in Appendix A. By using these notations, the first-order Hamiltonian in the tilde basis ( 20) can be written as

Formulating perturbation theory with the hat basis
We use the "renormalized basis" such that the zeroth-order and the perturbed Hamiltonian takes the form H = H 0 + H 1 .H 0 (we discuss H 1 later) is given by To formulate the solar-resonance perturbation theory with UV, we transform to the "hat basis", which diagonalizes H 0 : with Hamiltonian where U ϕ is parametrized as where The three eigenvalues of the zeroth-order Hamiltonian H 0 in ( 25) is given by 11 Then, the Hamiltonian in the hat basis is given by Ĥ = Ĥ0 + ĤνSM1 + ĤUV1 , where where the K matrix is defined in Eq. ( 23), and the simplified notations are hereafter used: c ϕ = cos ϕ and s ϕ = sin ϕ.Notice that we have omitted the second-order ĤUV , though one can easily compute it from (21) if necessary.

Calculation of Ŝ and S matrices
To calculate Ŝ(x) we define (x) as Then, (x) obeys the evolution equation where Notice that Ĥ1 = Ĥ νSM 1 + Ĥ UV 1 as in Eq. ( 32).Then, (x) can be computed perturbatively as 11 Notice that one can show that and the Ŝ matrix is given by Using Ĥ1 = Ĥ νSM 1 + Ĥ UV 1 in Eq. ( 32), the Ŝ matrix of the νSM part is given to the zeroth and first orders in the effective expansion parameter s 13 a h 3 −h 1 by where we have used the fact that b is spatially constant as a consequence of the uniform matter density approximation.Then, the νSM part of the tilde basis S matrix is given by where

S
(1 νSM can be written in the form where Notice that S νSM respects the generalized T invariance.Now, we must compute the UV parameter related part of Ŝ and S matrices.By remembering 35) is given by Due to frequent usage of the factors in the parenthesis above we give the formula for them here: PTEP 2020, 113B01 I. Martinez-Soler and H. Minakata Notice that S (0) νSM is nothing but S (−) ϕ .Then, H UV 1 takes a simple form: where we have introduced another simplifying matrix notation ≡ S ϕ and its elements ij .The explicit expressions of ij are given in Appendix A.
Since U ϕ rotation back to the tilde basis removes U † ϕ and U ϕ in Eq. ( 45), it is simpler to go directly to the calculation of the tilde basis S matrix:

S(x)
(1) Hereafter, the subscript "EV" is used for the S and Ŝ matrices to indicate that they describe unitary evolution.We note that the subscript "UV" placed on (x) (1) UV implies that it is the first-order contribution from the UV part of H 1 , not to be confused with the "UV" subscript showing the genuine non-unitary nature of the part of the probability which will be defined in Eq. (56).The computed results of the elements of S(x) (1) EV are given in Appendix B. Notice that again S(x) (1) EV respects the generalized T invariance.
Thus, we have computed all the tilde-basis S matrix elements to the first order as The first and the second terms are given, respectively, in Eqs. ( 40) and ( 41) with (42), and the third in Appendix B.

The relations between various bases and computation of the flavor-basis S matrix
We first 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: Then, the flavor-basis S matrix is related to Ŝ and S matrices as Using the formula (51), it is straightforward to compute the flavor-basis S matrix elements.Notice, however, that U 13 is free from CP phase δ due to our choice of the SOL convention of the U MNS matrix in Eq. (10).
The flavor-basis S matrix has the structure S flavor = (1 − α)S prop (1 − α) † where S prop ≡ U 23 U 13 SU † 13 U † 23 describes the unitary evolution despite the presence of non-unitary mixing [38].The factors (1 − α) and (1 − α) † , parts of the N matrix which project the mass eigenstates to the flavor states and vice versa, may be interpreted as playing the analogous roles as the "production NSI" and "detection NSI" which induce non-unitarity [64].Notice, however, that the production and detection NSI in this case are not independent from the "propagation NSI", but are solely determined by the latter.

Effective expansion parameter with and without the UV effect
As announced in Sect.4.2, the expression of S (1) νSM in Eq. ( 41) with (42) tells us that we have another expansion parameter [41] A exp ≡ c which is very small.The reason for such a "generated by the framework" expansion parameter is the special feature of the perturbed Hamiltonian in Eq. (32).
In fact, our perturbative framework is peculiar from the beginning, in the sense that the key nonperturbed part of the Hamiltonian (25), its top-left 2 × 2 sub-matrix, is smaller in size than the 33-element by a factor of ∼ 30, and is comparable with Ĥ νSM 1 in Eq. (32).The secret for the emergence of the very small effective expansion parameter (52) is that the 33-element decouples in the leading order and appears in the perturbative corrections only in the energy denominator, making them smaller for the larger ratio of m 2  31 / m 2 21 .The latter property holds because of the special structure of perturbative Hamiltonian Ĥ νSM 1 with non-vanishing elements only in the third row and third column.
With the inclusion of the UV Hamiltonian (24), however, the size of the first-order correction is controlled not only by A exp in Eq. ( 52) but also by the magnitudes of α β,γ .In computing the higherorder corrections the energy denominator suppression does not work for all the terms because the last property, "non-vanishing elements in the third row and third column only", ceases to hold in the first-order Hamiltonian.This is confirmed by looking into the formulas of the oscillation probabilities in Sect. 5

Neutrino oscillation probability to the first order in expansion
The oscillation probability can be calculated using the formula We denote the flavor-basis S matrices corresponding to νSM and S (1) νSM and S (1) EV , respectively, as they are related through Eq. (51).To the first order we have Then, we are ready to calculate the expressions of the oscillation probabilities using the formula (53) to the first order in the expansion parameters.Following Ref. [38], we categorize P(ν β → ν α ) into the three types of terms: UV , where The subscripts "EV" and "UV" refer the unitary evolution part and the genuine non-unitary contribution, terminology defined in Ref. [38].The first term in Eq. (56), νSM , is already computed in Ref. [41].Hence, we do not repeat the calculation, but urge the readers to go to this reference.Notice that use of the SOL convention of U MNS does not alter the expression of the oscillation probabilities.The rest of the terms in Eq. ( 56) can be computed straightforwardly using the expressions of the tilde-basis S matrix which are given explicitly in Appendix B, and the α matrix defined in Eq. (11).
Unfortunately, the resulting expression of the oscillation probability even at first order is far from simple.Therefore, to give a feeling to the readers, we will show in the next two subsections a part of the first-order probability in the ν μ → ν e channel.In Sect.5.1, one of the five terms in P(ν μ → ν e ) (1) EV is given, and the whole expression of P(ν μ → ν e ) (1) UV is in Sect.5.2.We leave the rest of the terms of P(ν μ → ν e ) (1) EV to Appendix D.1.In this appendix, we also give a practical suggestion to the readers on how to compute the oscillation probabilities in the ν μ -ν τ sector.
For notational simplicity, we define, following Ref.[41], the reduced Jarlskog factor in matter as which is proportional to the reduced Jarlskog factor in vacuum, J r ≡ c 23 s 23 c 2 13 s 13 c 12 s 12 [65].We have used Eq. ( 29) in the second equality in (57).
In this paper, we do not discuss numerical accuracy of the first-order oscillation probability because (1) the νSM part, which is controlled by A exp ∼ 10 −3 , is known to be very accurate already in the 16/39 PTEP 2020, 113B01 I. Martinez-Soler and H. Minakata first order [41], and (2) the accuracy of the UV-related part is trivial; the smaller the α βγ , the better the accuracy. 12.1.Unitary evolution part of the first-order probability P(ν μ → ν e ) (1)

EV
We first introduce the decomposition of P(ν μ → ν e ) (1) EV .After computation of all the terms, we assemble them according to the types of K ij variables involved.See Eq. ( 23) and (A2) in Appendix A for the definitions and the explicit expressions of the K ij , respectively.For bookkeeping purposes we decompose P(ν μ → ν e ) (1) EV into the following four terms: where the subscripts "D" and "OD" refer to the diagonal and the off-diagonal K ij variables.The organization inside each term is largely determined such that the symmetry under the transformation ϕ → ϕ + (π/2) is manifest.See Sect.5.3 for the ϕ symmetry.
Here, we only present the first term in Eq. ( 58), P(ν μ → ν e ) (1) EV | D-OD , leaving the others to Appendix D.1: We first note that the α parameter dependence is expressed through the K jj elements; see its definition and the expression Eq. ( 23) and (A2), respectively.It is noticeable that the diagonal K jj elements organize themselves into the form of difference, K 22 − K 11 type combinations, as it should be, because it comes from the rephasing invariance. 13This leads to the similar structure expressed by the diagonal α parameters; see Sect.6.1.

UV
To calculate P(ν μ → ν e ) (1) UV defined in the last line in Eq. ( 56), we need the expressions of zerothorder elements of νSM matrix S (0) νSM , which are given in Appendix C. They can be easily obtained from the tilde-basis S matrix in Eq. (40).Using the S (0) matrix elements P(ν μ → ν e ) (1) UV can be readily calculated as Here, the dependence on the α βγ is manifest.The feature of the diagonal α parameter correlation is vastly different from that of the unitary evolution part P(ν μ → ν e ) (1) EV , as will be discussed in Sect.6.1.

Symmetry of the oscillation probability
It is observed in Ref. [41] that for each matter-dressed mixing angle φ there is an invariance under the transformation φ → φ + (π/2).φ can be θ 13 or θ 12 in matter. 14In our system in this paper, the oscillation probability is invariant under the transformation which induces the following transformations simultaneously: Hence, J mr → −J mr under the transformation.
It is interesting to observe explicitly that the symmetry is respected by P(ν μ → ν e ) (1) EV and P(ν μ → ν e ) (1) UV , the former of which is given in Sect.5.1 and Appendix D.1, and the latter in Sect.5.2.The nature of the symmetry is identified as the "dynamical symmetry", not a symmetry in the Hamiltonian [41].Yet, it serves for a powerful consistency check of the calculation.

Dynamical correlation between νSM and the UV α parameters
In this section, we discuss correlations between νSM and the UV α parameters, including the clustering of the latter, which are manifested in the oscillation probabilities calculated in Sects.5.1, 5.2 and Appendix D.1. 15

Diagonal α parameter correlation
As discussed in Ref. [38], the diagonal α parameters have particular types of correlations in the evolution part of the probability a b − 1 α ee + α μμ , and which arises due to the rephasing invariance.It becomes manifest in the would-be flavor basis Of course, it must hold in regions of the solar-scale enhanced oscillation.In our expressions of P(ν μ → ν e ) (1) EV given in Sect.5.1 and in Appendix D.1, it is hidden in the diagonal K jj parameters in the form of K jj − K ii :16 We note that K 33 − K 11 is not independent of the above two expressions as it is obtained by adding them.See (23) for definition of K ij , andAppendixA for their explicit expressions.Though the diagonal α parameter correlation is written in terms of the SOL convention α jj variables, it is independent of the convention of U MNS because the variables do not depend on the convention.

Correlations between νSM phase δ and the α parameters
In view of the expressions of the first-order probability and its UV-related but unitary part in Sect.Therefore, the δ-complex-α parameter correlation does exist in the SOL convention of U MNS , which is in marked contrast to the feature of no δ-α parameter phase correlation in the region of the atmospheric scale enhanced oscillation [38].Notice that K 21 e iδ = K 12 e −iδ * and K 32 e −iδ = K 23 e iδ * , and therefore they do not introduce correlations independent of those in Eq. ( 65).In fact, the feature of the e ±iδ -K blob correlation can be traced back to the form of ij given in Appendix A. One can also conclude from the features of α μe vs e ±iδ correlation seen in Eq. ( 64) and ( 65) there is no definite "chiral" combination α μe e iδ and/or α τ e e iδ , nor α τ μ e ±iδ .Consideration of the non-unitary part of the probability (60) does not change the conclusion.
To summarize, the feature of the δ-α parameter correlation at around the solar scale enhanced oscillation is different from the one in the region of the atmospheric scale oscillation discussed in Ref. [38], most notably, in the following two aspects: • The correlation between the νSM phase δ and the α parameters does exist in the SOL convention of U MNS in the region of the solar scale enhanced oscillation.
• However, the correlation does not have the "chiral" form, α βγ e ±iδ .Rather it takes the form of correlation between e ±iδ and the blobs composed of the α parameters.
Since the correlation between δ and the K 12 -K 23 cluster variables lives in matrix elements, which are the building block of the perturbation series, it is obvious that the correlation prevails to higher orders in perturbation theory in the unitary evolution part.

Nature of the δ-α parameter correlation: Is it real?
The result in Ref. [38] shows that the SOL convention of U MNS is the unique case in the atmosphericscale enhanced oscillation in which the δ-α parameter correlation is absent.Then, the first itemized statement above indicates that there is no U MNS convention in which the phase correlation is absent both at around the atmospheric-and the solar-scale enhanced oscillations.Thus we can now conclude that the δ-α parameter correlations seen in this and the previous paper [38] are all physical.That is, it cannot be wiped away by a U MNS convention choice.
In fact, it is very likely that, in the solar-scale enhanced oscillation region, the phase correlation exists with all three conventions of U MNS .The oscillation probability in the other U MNS conventions can be obtained simply by using the translation rule, Eq. ( 13). 17Then, we observe in the ATM and PDG conventions even more complicated correlations between e ±iδ and the blobs composed of the α parameters inside which some of the α parameters are attached with e ±iδ .
One may wonder why the features of the correlation between δ and the α parameters are so different between the regions of the atmospheric-and the solar-scale enhanced oscillations, but it is entirely normal.As we have learned in Sect.3, the nature of the parameter correlation in neutrino evolution with the inclusion of outside-νSM ingredients is dynamical.The features depend on the values of the relevant parameters as well as the kinematical regions where different degrees of freedom play the dominant role.The dynamical nature of the phase correlation will be demonstrated in a visible way in Sect.7.

Clustering of the α parameters
In addition to the δ−(blob of the α parameters) correlation, we observe a feature which may be called the "clustering of the α parameters" in the unitary evolution part of the first-order oscillation probability P(ν μ → ν e ) calculated in Sects.5.1 and Appendix D.1.We can identify the following "clustering variables" at the level of the S (1) EV matrix elements: where we have not listed (s 2 ϕ K 13 + c ϕ s ϕ K 23 e iδ ) and (c ϕ s ϕ K 13 − s 2 ϕ K 23 e iδ ).They are not dynamically independent from the ones in Eq. ( 66) because they can be generated by the symmetry transformation (61) from the second and the third in Eq. (66).Also there exists the exceptional, isolated one K 12 e −iδ in Eq. (D2).
In Eq. ( 66), we did not quote the diagonal variables which come as a form of the difference, for example (K 22 − K 11 ), because these combinations are enforced by rephasing invariance.However, these diagonal α parameter differences often come in a particular combination with the other cluster variables, e.g., as cos 2ϕ (K 22 − K 11 ) + sin 2ϕ K 12 e −iδ + K 21 e iδ , 17 To close a possible loophole in this statement, we performed an explicit construction of the solar resonance perturbation theory extended with the UV effect using the ATM convention of U MNS .A preliminary investigation reveals that the same δ-(cluster of the α parameters) correlation as in Eq. ( 65) survives, but inside K ij α βγ must be transformed to α βγ (α matrix elements in the ATM convention) by the transformation rule (13).This is the expected result and apparently there is no loophole in our prescription.59) and (D2) -(D4) for all the above examples of blobs.It seems that the appearance of such cluster variables as well as the correlation between δ and the α parameter blobs are worth some attention though we do not quite understand the cause of this phenomenon.

Physics of neutrino flavor transformation with non-unitary mixing matrix
Up to this section, we have aimed at analytical understanding of the system around the region of solar-scale enhanced oscillation; the "solar region", for short.Likewise, we use below the simplified terminology "atmospheric region" for a region of enhanced atmospheric-scale oscillation.Now, we discuss the physics of neutrino flavor transformation in the solar region.However, we do it in comparison with that of the atmospheric region as it proves to be more revealing.We try to illuminate some new aspects of the system of the three-flavor active neutrinos with non-unitary mixing matrix by using the numerical method together with our first-order formula.
We use the PDG convention of U MNS in all the computations in this section, because it is used in most of the analyses of neutrino flavor transformations.We also depart from our "official" notations ᾱβγ of the α parameters in the PDG convention defined in Sect.4.1, and simply denote them as α βγ in this section beyond the next subsection.

Use of the exact and perturbative oscillation probabilities: General convention of U MNS
Towards the goal, we utilize the perturbative oscillation probability derived in Sect.5, as well as the exact formula for the probability based on numerical integration of the evolution equation, the latter of which is valid even for a varied matter density. 18t was pointed out in Ref. [38] that the α matrix depends on the convention of U MNS .By using this property, one can derive a probability formula in the PDG or ATM conventions using the substitution rule from the α parameter in the SOL convention to the ᾱ parameter in the PDG, or the α parameter in the ATM conventions.See Eq. ( 13) in Sect.4.1.One can also transform to a general U MNS convention by using the phase redefinition U (β, γ ) defined in Ref. [38].Notice that the translation rule applies not only in the perturbative formulas but also in the exact formulas.

Overview of the effect of UV
The first step to understand the effect of non-unitarity which is brought into the νSM three neutrino system by introducing the α matrix is to know where and how strongly the UV α parameters affect the neutrino flavor transformation.For this purpose, we turn on each α βγ parameter one by one and calculate the non-unitary contribution to the appearance probability P μe defined by where P(ν μ → ν e ) in Eq. ( 67) denotes the appearance probability in the ν μ → ν e channel with the UV effect fully implemented.Both P(ν μ → ν e ) and P(ν μ → ν e ) νSM are computed numerically.In all the calculations in this section, the matter density is taken to be ρ = 3.2 g cm −3 over the entire baseline.
In Fig. 1 we show P μe by using color grading guided by the contour lines.In each panel we turn on one of α βγ , from the top left-hand to the bottom right-hand panels, in order: α ee , α μμ , α τ τ , α μe , α τ e , and α τ μ . 19In this section, we turn on only one of the α βγ parameters in each panel, except for the top right-hand and bottom two panels in Fig. 2. To have an insight into the required accuracy of the P(ν μ → ν e ) measurement to improve the current bounds by a factor of 2, we take the value of each α βγ as half of the bound obtained by Blennow et al. [26] with the positive sign.Figure 1 as a whole displays how large the UV effect is depending upon the energy E and the baseline L.
The "mountain ridges" roughly follow the line of L/E = constant.The atmospheric and the solar MSW enhancements are visible, respectively, at around E ∼ 10 GeV and near the upper end of L = 10 4 km, and E several ×100 MeV and E several ×1000 km.
We observe two salient features: • P μe is at most ± 1% in all the panels in Fig. 1, which means a 1% level measurement of the probability is necessary for a factor of 2 improvement of the bounds.• P μe changes sign depending upon which α βγ is turned on, and on the region of kinematical phase space, e.g., in the atmospheric region, or the solar region.
The 1% accuracy measurement of the probability is mentioned at the end of Sect.4.2 in relation to the possible target accuracy of constraining UV α parameters.
For the second point above, we notice in Fig. 1 that with turning on α τ τ (middle left-hand panel) P μe is positive in the solar region and negative in the atmospheric region.On the other hand, this tendency is reversed completely with α τ e (bottom left-hand panel), and less completely with α μe (middle right-hand panel).In the other cases, P μe is negative in both the regions of the atmosphericscale and solar-scale enhancement.This means that if we turn on all α βγ at once, the effect of each element may cancel each other at least partly.One must also take into account the fact that, since we do not know a priori the sign of the α parameters, the pattern of the cancellation can be more complicated when all the parameters are turned on with arbitrary signs, or if phases are attached to the off-diagonal α parameters.This implies that (1) determination of the UV α βγ parameters (assuming their existence) could have additional difficulties due to confusion and degeneracy caused by the cancellation between the effect of different α parameters, and (2) the bound on UV obtained by using the "one α βγ turned on at one time" procedure could have made the bound artificially stronger than the one obtained with the proper procedure of "all α βγ turned on but the rest of them marginalized".

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Downloaded from https://academic.oup.com/ptep/article/2020/11/113B01/5877439 by guest on 27 January 2021 PTEP 2020, 113B01 I. Martinez-Soler and H. Minakata cision required to measure the probability.Therefore, further discussion of the question of how to disentangle the effects of different alpha parameters is called for.

Unitary vs. non-unitary pieces of the UV related oscillation probability
The UV α parameter related part of the probability P μe decomposes into two parts, the unitary evolution part P(ν μ → ν e ) EV and the genuine non-unitary part P(ν μ → ν e ) UV [38]; see Eq. ( 55).Then, a natural question is which part is larger or dominating, and whether they mutually tend to add up or cancel each other out.
These questions are answered by Fig. 2. In the top two panels the whole UV effects, P μe = P(ν μ → ν e ) UV + P(ν μ → ν e ) EV are presented, with α μe = 0.014 only in the left-hand panel, and with α ee = 0.012 and α μμ = 0.011 in the right-hand panel.The values of the α parameters are the same as used in Fig. 1, and hence the left-hand panel overlaps with a part of the middle-right-hand panel of Fig. 1.
The decomposition of P μe into P(ν μ → ν e ) UV and P(ν μ → ν e ) EV is displayed in the middle (α μe = 0.014 case) and bottom (α ee = 0.012 and α μμ = 0.011 case) panels of Fig. 2, respectively.We restrict ourselves into the two choices of the α parameters because P(ν μ → ν e ) UV in the first order depends only on the two combinations α μe and α ee + α μμ .P(ν μ → ν e ) EV is computed by using the formula UV with the first-order expression of P(ν μ → ν e ) UV , and hence P(ν μ → ν e ) EV is accurate only to the first order.
An overall feature is that in wide areas in Fig. 2 P(ν μ → ν e ) UV and P(ν μ → ν e ) EV tend to cancel each other out.In looking into the figure more closely, however, we observe a little more intricate features.In the α μe = 0.014 case (middle panels), above the L/E = 10 4 km/230 MeV line, P(ν μ → ν e ) UV contributes to lift up the probability, enhancing the yellow regions of P(ν μ → ν e ) EV into the thicker ones in P μe .Below the line, P(ν μ → ν e ) UV is more dominating in the blue solar resonance region, but is partially cancelled by P(ν μ → ν e ) EV .The cancellation is even more prominent in the bottom panels, the case with α ee + α μμ turned on.The overall feature of the colorgraded contour of P μe is similar to that of P(ν μ → ν e ) UV , but P(ν μ → ν e ) EV over-cancels the peaks of P(ν μ → ν e ) UV above the L/E = 10 4 km/230 MeV line.
This feature of cancellation is akin to, but is much more prominent compared to, that observed in the "atmospheric region" in Ref. [38].Unfortunately, we cannot offer a physical explanation as to why the cancellation between P(ν μ → ν e ) UV and P(ν μ → ν e ) EV takes place, or why the feature is common to both the atmospheric and the solar regions.In most of the regions it acts as a partial "hiding mechanism" of non-unitarity since a less prominent effect is left in the observable, the appearance probability P(ν μ → ν e ).To obtain the information of the genuine non-unitary part P(ν μ → ν e ) UV , it must be complemented by measurement of departure from unitarity, P(ν μ → ν e ) + P(ν μ → ν μ ) + P(ν μ → ν τ ) = 1.

νSM-α parameter phase correlation: The atmospheric vs. solar regions
We have learned in the previous section that the features of the parameter correlation between the νSM and the UV new physics parameters in the solar region is different from the ones in the atmospheric region.A new δ-(blobs of the α parameters) correlation is observed.Then, it is natural to ask the question: What is the feature of νSM-UV parameter CP phase correlation in the solar region, and which characteristic difference does it have from those in the atmospheric region?To discuss correlation between δ and phases of the off-diagonal α parameters, we parametrize the latter as 26/39 where βγ = μe, τ e, τ μ.To make the phase correlation clearly visible, we use P μe ≡ P(ν μ → ν e ) − P(ν μ → ν e ) νSM defined in Eq. ( 67), not the probability itself.In Fig. 3, the non-unitary contribution to the appearance probability P μe computed by turning on α μe only is presented on the φ μe -δ plane by showing the equi-contours of P μe with color grading.In Figs. 4 and 5, the results of the similar exercises are presented, the case with α τ e turned on (Fig. 4), and the one with α τ μ (Fig. 5).In Figs. 3, 4, and 5, we use a large value α βγ = 0.1 to enhance effects of the phase correlation, which merits higher visibility.The global features of the δ-α parameter phase correlation shown in Figs. 3, 4 and 5 are: • The linear, oblique correlation seen in the case of both α μe = 0 (Fig. 3) and α τ e = 0 (Fig. 4) in the atmospheric region shown in the upper panels, but no clearly visible correlation in any of the other panels.
• The absolute value of | P μe | is larger in the panels with baseline L = 12000 km than those with L = 3000 km by a factor of ∼ 5.This statement applies to all the panels including both the atmospheric and solar regions.Let us start from a discussion of the phase correlation seen in the atmospheric region-the top two panels in Figs. 3, 4 and 5.The linear, oblique correlations seen in Figs. 3 and 4, φ μe -δ and φ τ e -δ correlations, respectively, and the lack of visible correlation between φ τ μ and δ shown in Fig. 5 (all in the upper two panels) is perfectly consistent with the "canonical phase combination" [38]   On the other hand, the features of the phase correlation in the solar region shown in the lower panels in Figs. 3, 4 and 5 are more subtle and not easy to understand.In some panels, the equal-P μe contours are vertical, which may imply that there is no significant correlation between δ and α parameter phases.In the other, there exists "circular-shaped correlation" with positive and negative signs of P μe in the two-dimensional phase space.Notice that in the panels with vertical correlation and with "circular correlation", the δ (in-)dependence cannot be understood as a remnant of insufficient subtraction of the νSM part.This is because the values of P μe and its variation in φ or δ directions can be as large as ∼ 0.1, of the order of the α parameter that is turned on.The feature of the φ-δ phase correlation, in particular, the coexistence of the vertical and circular shaped correlations is not understood, regrettably, by our analytic framework. 21 With regard to the baseline dependence of the strength of the correlation, it might be that | P μe | itself is larger at the longer baseline of L = 12000 km among the two baselines we have chosen to display in Figs. 3, 4 and 5.
The features of the α parameter phase-νSM δ correlation in the atmospheric and the solar regions presented in Figs. 3, 4 and 5 testify that the nature of the correlation is quite dynamical, confirming our view stated in Sect.3. Unfortunately, physical understanding of the features of the phase correlation in the solar region are not yet achieved, which calls for further studies.

Concluding remarks
In this paper, we have attempted to achieve an understanding of the physics of the three-flavor neutrino system with non-unitary mixing matrix.We have focused our discussion on elucidating the nature of parameter correlations in such a system, in particular the correlation between the νSM and the UV new physics parameters.We do this in the region of the solar-scale oscillations, the "solar 20 If the probability calculated by first-order helio-UV perturbation theory [38] is sufficiently accurate, there should be no δ dependence in the upper two panels in Fig. 5, because the δ dependence would have been eliminated by the subtraction of P(ν μ → ν e ) νSM .Obviously, this is not the case.Notice that the results presented in Figs. 3, 4 and 5 are accurate as they do not rely on perturbative treatment.This means that the perturbative treatment fails to provide accurate description of the probability, which is natural due to the large value of 0.1 taken for α τ μ .In fact, the remaining δ dependence is up to a few ×10 −3 level for L = 3000 km, and is of the order ∼ ±0.02 for L = 12000 km, so that our interpretation may be valid. 21It appears that there are some regularities which may be relevant for understanding the phase correlation in the solar region.That is, we often observe "red" ( P μe > 0) and "blue" ( P μe < 0) vertical contours in central region, φ ∼ π.It is likely that the central "red" vertical correlation corresponds to the region of negative P μe in Fig. 1, whereas the central "blue" vertical correlation corresponds to the region of positive P μe in Fig. 1.For the latter, we refer to the lower right-hand panel of Fig. 4, and the case of α τ e = 0.1, E = 200 MeV and L = 5000 km.

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region" for short, in this paper.It nicely complements the one given in our previous paper [38] which dealt with the region of atmospheric-scale oscillations, the "atmospheric region".
Towards this goal, we have formulated a new perturbative framework to discuss effect of a non-unitary mixing matrix in the solar region, the UV extended version of the "solar-resonance perturbation theory" [41].It was necessary to resolve the question raised in Ref. [38] which casts doubt on the physical reality of the correlation between the νSM δ and the phases of UV α parameters.However, in turn, the framework serves as a powerful analytic machinery for analyzing the features of parameter correlation in the solar region.The skepticism about the reality of the phase correlation, which is described in detail in Sect. 1, is cleared up by showing that the phase correlation does exist in the solar region with the SOL (e ±iδ attached to s 12 ) convention of U MNS .See Sect.6.
In fact, we have uncovered that the features of the νSM-UV parameter correlations are much more profound than we thought.This point can be illuminated most clearly by contrasting the atmospheric region to the solar one.In the atmospheric region, the most notable feature is the νSM δ-UV α parameter phase correlation of the "chiral type", [e −iδ α μe , e −iδ α τ e , α τ μ ] in the PDG convention of U MNS [38].This picture no longer holds in the solar region, and the correlation takes the form of δ-(blobs of the α parameters) correlation as we saw in Sect.6.2.Another interesting observation in this context is that when we move the kinematical region from E/L = 200 MeV/3000 km to E/L = 300 MeV/5000 km, the δ-φ μe correlation takes vastly different forms, as shown in Fig. 3, where φ μe denotes the phase of α μe .
We have utilized the analytic framework developed in this paper as well as the numerical method to reveal more generic features of the effects of the UV α parameters.In addition to the ones mentioned above, we have observed that the effect of non-unitarity tends to cancel between the unitary evolution part (denoted as "EV") and the non-unitary part (denoted as "UV") of the probability, and between the different α βγ parameters.see Sect. 7.
One of the most intriguing features of the parameter correlation is that the form of the correlation depends also on the values of the mixing parameters.The phenomenon is briefly mentioned in Sect.3.2; as θ 13 becomes larger, the correlation seen at smaller θ 13 starts to dissolve.Since we cannot control the values of the mixing angles or m 2 by ourselves, the discussion might look appealing only to an academic interest.However, we believe that it merits deepening our understanding on the mechanism and the cause of parameter correlation.We are not able to explore this point further in this paper, and a focused investigation on this issue is called for.
All these features of the parameter correlation may be summarized by the term "dynamical nature of the parameter correlation".
Finally, we remark that the occurrence of dynamical correlations between the parameters in systems with many degrees of freedom is very common, as discussed in Sect.3. The rich variety of correlations we encountered in our system with non-unitarity adds another example to this list.If one chooses the way of testing leptonic unitarity by setting up a class of models with UV and confrontation of them with experimental data, understanding the system with UV would be an indispensable step in carrying out this task.Yet we must emphasize that our understanding of the system, e.g., of the parameter correlation, is far from sufficient, generically in the system with new physics beyond the νSM.
On the experimental side, if we want to utilize the low-energy region with the solar-scale enhanced oscillation, in the context of the precision unitarity test, a possible advantage of the Kamioka-Korea identical two-detector setup [16,67] may be worth renewed attention.Fortunately, the construction of Hyper-K has been started, which may act as the Kamioka site detector in an extended plan of the two-detector complex [15,49].
The explicit expressions of elements F ij , K ij and ij defined, respectively, in Eqs. ( 22), ( 23) and ( 45) are given as follows:

Appendix B. The first-order tilde-basis unitary evolution S matrix elements
Here, we present the result of unitary S matrix elements which come from the first-order UV parameter related part of the Hamiltonian.Appendix D.2.The neutrino oscillation probability in the ν μ -ν τ sector

S(x) EV
We refrain from explicit computation of the oscillation probabilities in the ν μ -ν τ sector.The reason is that the expression is too lengthy and not particularly structure-revealing beyond that which we 36/39

it implies neutrino energy E = ( 1 − 5 )
× 100 MeV and baseline L = (1 − 10) × 1000 km.In this region, the matter potential is comparable in size to the vacuum effect represented by m

Fig. 3 .
Fig. 3. P μe ≡ P(ν μ → ν e ) − P(ν μ → ν e ) νSM is presented in the φ μe -δ plane by color graduation, which is calculated by turning on α μe = 0.1 only.The top two panels are in the atmospheric region with energy E = 10 GeV, and the middle two panels are in the solar region with energy E = 200 MeV.The baseline is taken as L = 3000 km (left-hand panel) and L = 12000 km (right-hand panel), in both the top and middle panels.The bottom panel is in the solar region with E = 300 MeV and L = 5000 km.

Fig. 4 .
Fig. 4. P μe ≡ P(ν μ → ν e ) − P(ν μ → ν e ) νSM is presented in the φ μe -δ plane by color graduation, which is calculated by turning on α τ e = 0.1 only.The upper two panels are in the atmospheric region with energy E = 10 GeV, and the lower two panels are in the solar region with energy E = 200 MeV.The baseline is taken as L = 3000 km (left-hand panel) and L = 12000 km (right-hand panel), in both the upper and lower panels.