A large number of parameters with UV, which is more than doubled the SM ones, makes it conceivable that the dynamics of neutrino oscillation naturally involves rich correlations among these variables.⁷ Here, 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, , , and the parameters. They derived the formulas of the oscillation probability to the second order (the third order in the 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 [39] is used.

An interesting and unexpected feature of the NSI-extended formulas is the emergence of the two sets of “collective variables”: (3) where an overall is factored out from the matrix element to make the term free, through which dependences in in Eq. (3) result. That is, if we replace and in the original formulas by and , 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 as well. We note that the second-order computation of Ref. [55] includes the – 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 SM and NSI parameters in Eq. (3) implies that there exists strong correlations between the SM variables – and the NSI – parameters in such a way that they form the collective variable to convert the Cervera et. al. formula to the NSI-extended version. A similar statement can be made for the other cluster variable as well. The NSI-extended second-order formula derived in this way serves for understanding the – confusion uncovered in Ref. [60] in a more complete manner, in such a way that the effects of and the CP phase are also included. It also predicts the occurrence of the similar correlation among the variables to produce the collective variable , the feature of which could be confirmed by experiments at low energies, ; this possibility was revisited recently [41,42].

Therefore, there is nothing strange in the parameter correlations among the SM 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.

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.⁹

We 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 is larger than that assumed at the time the Cervera et. al. formula was derived [39]. The latest value from Daya Bay is [61], which is of the order of . Then, we need higher-order corrections of , 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 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.

Can we extract information about the – 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 parameters and the UV parameters, as noticed by Blennow et. al. [26] under the assumption —equal neutron and proton number densities in a charge-neutral medium. Of course, an extension to the more generic case of [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 dependence of the collective variables (3) is fixed. That is, removing an overall phase from the matrix element to make the term -free, as done in Ref. [55].

Assuming that the two conditions above are met, this leads to the collective variables in Eq. (3) written by the UV parameters, (4) where we have to use the parameters defined in the PDG convention of . The emerged correlation between and the parameters is consistent with the canonical phase combination obtained in Ref. [38] in the PDG convention. For the relationships between the parameters with the various conventions, see Sect. 4.1. This is not unreasonable because the regions of validity of the perturbative frameworks in Refs. [55] and [38] overlap.

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].¹⁰ With use of the “check basis” for the vacuum mass eigenstate basis, it takes the form of a Schrödinger equation: (5) with Hamiltonian (6) where is neutrino energy and . A usual phase redefinition of neutrino wave function is done to leave only the mass squared differences. denotes the non-unitary flavor mixing matrix which relates the flavor neutrino states to the vacuum mass eigenstates as (7) where (and the other Greek indices) runs over , while the mass eigenstate index (and the other Latin indices) runs over and . 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 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 and in (17) denote the Wolfenstein matter potential [8] due to the charged current (CC) and the neutral current (NC) reactions, respectively.

(8)

Here, is the Fermi constant, and and are the electron and neutron number densities in matter. and 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: (9)

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 matrix, the standard unitary flavor mixing matrix (10) where we have used the obvious notations etc. and denotes the lepton KM phase [6], or the SM CP violating phase. We use the term “SOL” because the phase factor is attached to the “solar angle” . It is physically equivalent to the commonly used PDG convention [39] in which the phase factor is attached to .

We use the parametrization of a non-unitary mixing matrix [23] defined in the convention: (11)

As seen in Eq. (11), and discussed in detail in Ref. [38], the definition of the matrix depends on the phase convention of the flavor-mixing matrix . Consistent with the notation used in Ref. [38], we denote the matrix elements in the SOL convention as .

The other convention of the MNS matrix which is heavily used in Ref. [38] is the “ATM” convention in which is attached to the “atmospheric angle” : (12)

The parameters defined in the ATM and PDG conventions of are denoted as and , respectively, in Ref. [38], and we follow that notation in this paper. We recapitulate here the relationships between the parameters defined with the PDG (), ATM () and the SOL () conventions of : (13) where we note that the diagonal parameters are equal among the three conventions.

In this section, we aim at constructing the perturbative framework which is valid at around the solar oscillation maximum, . Given the formula (14) it implies neutrino energy MeV and baseline km. In this region, the matter potential is comparable in size to the vacuum effect represented by , (15)

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 SM sector, , 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 , defined in Eq. (11), for the extra expansion parameters. That is, we assume that deviation from unitarity is small. Therefore, holds for all and . 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 in the context of unitarity test? For the sake of rough estimation, we assume momentarily a perfect knowledge of the SM mixing parameters. Then, let us ask: Which level of sensitivity to UV parameters could one expect given that the accuracy of measurement of (see Eq. (67)) is of the order of, for example, or ? Notice that 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 , we would expect the constraints on parameters to be of the order of , or , 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 , or , respectively, far beyond the accuracy of the 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 , where denotes collectively the active-sterile mixing matrix elements [25]. Due to unitarity in the whole space, must be of the order of . Then, the leaking and the mis-normalization terms are of the order of or in the above two regimes, respectively, which are far too small compared to the accuracy of measurement in each era. This constitutes one of the serious problems in their determination.

We transform to a different basis to formulate our perturbation theory for solar-scale enhancement. It is the tilde basis (16) with Hamiltonian (17)

Notice that the term “tilde basis” has no connection to our notation of parameters in the SOL convention. The Hamiltonian in the tilde basis is given by (18) where each term of the right-hand side of (18) is given by (19) (20) (21)

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 – parameter phase correlation, though we will make a brief comment on the effect of 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 a serious limitation because the framework anticipates a precision era of neutrino experiment for the unitarity test in which the condition should be justified.

To make expressions of the matrix and the oscillation probability as compact as possible, it is important to introduce the new matrix notations and : (22) (23)

The explicit expressions of the elements and 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 (24)

We use the “renormalized basis” such that the zeroth-order and the perturbed Hamiltonian takes the form . (we discuss later) is given by (25)

To formulate the solar-resonance perturbation theory with UV, we transform to the “hat basis”, which diagonalizes : (26) with Hamiltonian (27) where is parametrized as (28) is determined such that is diagonal, which leads to (29) where (30)

The three eigenvalues of the zeroth-order Hamiltonian in (25) is given by¹¹ (31)

Then, the Hamiltonian in the hat basis is given by , where (32) where the matrix is defined in Eq. (23), and the simplified notations are hereafter used: and . Notice that we have omitted the second-order , though one can easily compute it from (21) if necessary.

To calculate we define as (33)

Then, obeys the evolution equation (34) where (35)

Notice that as in Eq. (32). Then, can be computed perturbatively as (36) and the matrix is given by (37)

Using in Eq. (32), the matrix of the SM part is given to the zeroth and first orders in the effective expansion parameter by (38) where we have used the fact that is spatially constant as a consequence of the uniform matter density approximation.

Then, the SM part of the tilde basis matrix is given by (39) where (40) can be written in the form (41) where (42)

Notice that respects the generalized T invariance.

Now, we must compute the UV parameter related part of and matrices. By remembering , the UV part of in Eq. (35) is given by (43)

Due to frequent usage of the factors in the parenthesis above we give the formula for them here: (44)

Notice that is nothing but . Then, takes a simple form: (45) where we have introduced another simplifying matrix notation and its elements . The explicit expressions of are given in Appendix A.

Since rotation back to the tilde basis removes and in Eq. (45), it is simpler to go directly to the calculation of the tilde basis matrix: (46)

Hereafter, the subscript “EV” is used for the and matrices to indicate that they describe unitary evolution. We note that the subscript “UV” placed on implies that it is the first-order contribution from the UV part of , 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 are given in Appendix B. Notice that again respects the generalized T invariance.

Thus, we have computed all the tilde-basis matrix elements to the first order as (47)

The first and the second terms are given, respectively, in Eqs. (40) and (41) with (42), and the third in Appendix B.

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: (48)

The non-unitary transformation is involved from the check basis to the flavor basis: (49)

The relationship between the flavor basis Hamiltonian and the hat basis one is (50)

Then, the flavor-basis matrix is related to and matrices as (51)

Using the formula (51), it is straightforward to compute the flavor-basis matrix elements. Notice, however, that is free from CP phase due to our choice of the SOL convention of the matrix in Eq. (10).

The flavor-basis matrix has the structure where describes the unitary evolution despite the presence of non-unitary mixing [38]. The factors and , parts of the 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.

As announced in Sect. 4.2, the expression of in Eq. (41) with (42) tells us that we have another expansion parameter [41] (52) 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 non-perturbed part of the Hamiltonian (25), its top-left sub-matrix, is smaller in size than the 33-element by a factor of , and is comparable with 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 . The latter property holds because of the special structure of perturbative Hamiltonian 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 in Eq. (52) but also by the magnitudes of . In computing the higher-order 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. 5.1, Appendix D.1, and Sect. 5.2.

We first introduce the decomposition of . After computation of all the terms, we assemble them according to the types of variables involved. See Eq. (23) and (A2) in Appendix A for the definitions and the explicit expressions of the , respectively. For bookkeeping purposes we decompose into the following four terms: (58) where the subscripts “D” and “OD” refer to the diagonal and the off-diagonal variables. The organization inside each term is largely determined such that the symmetry under the transformation is manifest. See Sect. 5.3 for the symmetry.

Here, we only present the first term in Eq. (58), , leaving the others to Appendix D.1: (59)

We first note that the parameter dependence is expressed through the elements; see its definition and the expression Eq. (23) and (A2), respectively. It is noticeable that the diagonal elements organize themselves into the form of difference, type combinations, as it should be, because it comes from the rephasing invariance.¹³ This leads to the similar structure expressed by the diagonal parameters; see Sect. 6.1.

To calculate defined in the last line in Eq. (56), we need the expressions of zeroth-order elements of SM matrix , which are given in Appendix C. They can be easily obtained from the tilde-basis matrix in Eq. (40). Using the matrix elements can be readily calculated as (60)

Here, the dependence on the is manifest. The feature of the diagonal parameter correlation is vastly different from that of the unitary evolution part , as will be discussed in Sect. 6.1.

It is observed in Ref. [41] that for each matter-dressed mixing angle there is an invariance under the transformation . can be or in matter.¹⁴ In our system in this paper, the oscillation probability is invariant under the transformation (61) which induces the following transformations simultaneously: (62)

Hence, under the transformation.

It is interesting to observe explicitly that the symmetry is respected by and , 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.

As discussed in Ref. [38], the diagonal parameters have particular types of correlations in the evolution part of the probability (63) 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 given in Sect. 5.1 and in Appendix D.1, it is hidden in the diagonal parameters in the form of :¹⁶ (64)

We note that is not independent of the above two expressions as it is obtained by adding them. See (23) for definition of , and Appendix A for their explicit expressions. Though the diagonal parameter correlation is written in terms of the SOL convention variables, it is independent of the convention of because the variables do not depend on the convention.

In view of the expressions of the first-order probability and its UV-related but unitary part in Sect. 5.1 and Appendix D.1, we identify the following correlated pairs consisting of the – parameters, and , where the blobs of the parameters and can be written as (65)

Therefore, the –complex- parameter correlation does exist in the SOL convention of , which is in marked contrast to the feature of no – parameter phase correlation in the region of the atmospheric scale enhanced oscillation [38]. Notice that and , and therefore they do not introduce correlations independent of those in Eq. (65). In fact, the feature of the – blob correlation can be traced back to the form of given in Appendix A.

One can also conclude from the features of vs correlation seen in Eq. (64) and (65) there is no definite “chiral” combination and/or , nor . Consideration of the non-unitary part of the probability (60) does not change the conclusion.

To summarize, the feature of the – 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:

Since the correlation between and the – 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.

The result in Ref. [38] shows that the SOL convention of is the unique case in the atmospheric-scale enhanced oscillation in which the – parameter correlation is absent. Then, the first itemized statement above indicates that there is no 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 – parameter correlations seen in this and the previous paper [38] are all physical. That is, it cannot be wiped away by a 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 . The oscillation probability in the other conventions can be obtained simply by using the translation rule, Eq. (13).¹⁷ Then, we observe in the ATM and PDG conventions even more complicated correlations between and the blobs composed of the parameters inside which some of the parameters are attached with .

One may wonder why the features of the correlation between and the 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-SM 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.

In addition to the (blob of the parameters) correlation, we observe a feature which may be called the “clustering of the parameters” in the unitary evolution part of the first-order oscillation probability calculated in Sects. 5.1 and Appendix D.1. We can identify the following “clustering variables” at the level of the matrix elements: (66) where we have not listed and . 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 in Eq. (D2).

In Eq. (66), we did not quote the diagonal variables which come as a form of the difference, for example , because these combinations are enforced by rephasing invariance. However, these diagonal parameter differences often come in a particular combination with the other cluster variables, e.g., as , or . Moreover, the other blobs of variables , and , which are not visible at the level of the matrix, shows up in the oscillation probability. See Eqs. (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 and the parameter blobs are worth some attention though we do not quite understand the cause of this phenomenon.

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.¹⁸

It was pointed out in Ref. [38] that the matrix depends on the convention of . By using this property, one can derive a probability formula in the PDG or ATM conventions using the substitution rule from the parameter in the SOL convention to the parameter in the PDG, or the parameter in the ATM conventions. See Eq. (13) in Sect. 4.1. One can also transform to a general convention by using the phase redefinition defined in Ref. [38]. Notice that the translation rule applies not only in the perturbative formulas but also in the exact formulas.

The first step to understand the effect of non-unitarity which is brought into the SM three neutrino system by introducing the matrix is to know where and how strongly the UV parameters affect the neutrino flavor transformation. For this purpose, we turn on each parameter one by one and calculate the non-unitary contribution to the appearance probability defined by (67) where in Eq. (67) denotes the appearance probability in the channel with the UV effect fully implemented. Both and are computed numerically. In all the calculations in this section, the matter density is taken to be g cm over the entire baseline.

In Fig. 1 we show by using color grading guided by the contour lines. In each panel we turn on one of , from the top left-hand to the bottom right-hand panels, in order: , , , , , and .¹⁹ In this section, we turn on only one of the 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 measurement to improve the current bounds by a factor of 2, we take the value of each 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 and the baseline . The “mountain ridges” roughly follow the line of constant. The atmospheric and the solar MSW enhancements are visible, respectively, at around GeV and near the upper end of km, and several MeV and several km.

We observe two salient features:

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 parameters.

For the second point above, we notice in Fig. 1 that with turning on (middle left-hand panel) is positive in the solar region and negative in the atmospheric region. On the other hand, this tendency is reversed completely with (bottom left-hand panel), and less completely with (middle right-hand panel). In the other cases, is negative in both the regions of the atmospheric-scale and solar-scale enhancement. This means that if we turn on all 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 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 parameters. This implies that (1) determination of the UV parameters (assuming their existence) could have additional difficulties due to confusion and degeneracy caused by the cancellation between the effect of different parameters, and (2) the bound on UV obtained by using the “one turned on at one time” procedure could have made the bound artificially stronger than the one obtained with the proper procedure of “all turned on but the rest of them marginalized”.

The features of possible cancellation between the effect of parameters may add another difficulty to the task of identifying their effects, an already highly non-trivial one due to high precision required to measure the probability. Therefore, further discussion of the question of how to disentangle the effects of different alpha parameters is called for.

The UV parameter related part of the probability decomposes into two parts, the unitary evolution part and the genuine non-unitary part [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, are presented, with only in the left-hand panel, and with and in the right-hand panel. The values of the 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 into and is displayed in the middle ( case) and bottom ( and case) panels of Fig. 2, respectively. We restrict ourselves into the two choices of the parameters because in the first order depends only on the two combinations and . is computed by using the formula with the first-order expression of , and hence is accurate only to the first order.

An overall feature is that in wide areas in Fig. 2 and tend to cancel each other out. In looking into the figure more closely, however, we observe a little more intricate features. In the case (middle panels), above the line, contributes to lift up the probability, enhancing the yellow regions of into the thicker ones in . Below the line, is more dominating in the blue solar resonance region, but is partially cancelled by . The cancellation is even more prominent in the bottom panels, the case with turned on. The overall feature of the color-graded contour of is similar to that of , but over-cancels the peaks of above the 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 and 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 . To obtain the information of the genuine non-unitary part , it must be complemented by measurement of departure from unitarity, .

We have learned in the previous section that the features of the parameter correlation between the SM and the UV new physics parameters in the solar region is different from the ones in the atmospheric region. A new –(blobs of the parameters) correlation is observed. Then, it is natural to ask the question: What is the feature of SM–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 and phases of the off-diagonal parameters, we parametrize the latter as (68) where . To make the phase correlation clearly visible, we use defined in Eq. (67), not the probability itself.

In Fig. 3, the non-unitary contribution to the appearance probability computed by turning on only is presented on the – plane by showing the equi-contours of with color grading. In Figs. 4 and 5, the results of the similar exercises are presented, the case with turned on (Fig. 4), and the one with (Fig. 5). In Figs. 3, 4, and 5, we use a large value to enhance effects of the phase correlation, which merits higher visibility. The global features of the – parameter phase correlation shown in Figs. 3, 4 and 5 are:

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, – and – correlations, respectively, and the lack of visible correlation between and shown in Fig. 5 (all in the upper two panels) is perfectly consistent with the “canonical phase combination” [38]²⁰ (69) which holds under the PDG convention of . One should note the non-trivial convention dependence: In the ATM phase convention of (in which is attached to ), the phase correlation takes the form [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- contours are vertical, which may imply that there is no significant correlation between and parameter phases. In the other, there exists “circular-shaped correlation” with positive and negative signs of in the two-dimensional phase space. Notice that in the panels with vertical correlation and with “circular correlation”, the (in-)dependence cannot be understood as a remnant of insufficient subtraction of the SM part. This is because the values of and its variation in or directions can be as large as , of the order of the parameter that is turned on. The feature of the – 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 itself is larger at the longer baseline of km among the two baselines we have chosen to display in Figs. 3, 4 and 5.

The features of the parameter phase–SM 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.