Breakings of the neutrino μ-τ reflection symmetry

The neutrino μ-τ reflection symmetry has been attracting a lot of attention as it predicts the interesting results θ23 = π/4 and δ = ±π/2. But it is reasonable to consider breakings of such a symmetry either from the theoretical considerations or on the basis of experimental results. We thus perform a systematic study for the possible symmetry-breaking patterns and their implications for the mixing parameters. The general treatment is applied to some specific symmetry breaking arising from the renormalization group effects for illustration.


Introduction
The discovery of neutrino oscillations indicates that neutrinos are massive and bear flavor mixing. The neutrino mixing arises from the mismatch between their mass and flavor eigenstates, and is described by a 3 × 3 unitary matrix U = U † l U ν (with U l and U ν being respectively the unitary matrix for diagonalizing the charged-lepton mass matrix M l M † l and neutrino mass matrix M ν ). In the standard parametrization, U reads where θ ij (for ij = 12, 13, 23) are the mixing angles (with c ij = cos θ ij and s ij = sin θ ij ) and δ is the Dirac CP phase. P ν = Diag(e iρ , e iσ , 1) contains two Majorana CP phases ρ and σ, while P φ = Diag(e iφ 1 , e iφ 2 , e iφ 3 ) consists of three unphysical phases φ 1,2,3 that can be removed via the charged-lepton field rephasing. In addition, neutrino oscillations are also controlled by two mass-squared differences ∆m 2 ij = m 2 i − m 2 j (for ij = 21, 31). Thanks to various neutrino-oscillation experiments [1], the neutrino mixing parameters have been measured to a good accuracy. A global-fit result [2] for them is given by sin 2 θ 12 = 0.308 ± 0.017 , ∆m 2 21 = (7.54 ± 0.24) × 10 −5 eV 2 , sin 2 θ 13 = 0.0234 ± 0.0020 , |∆m 2 31 | = (2.47 ± 0.06) × 10 −3 eV 2 . (1.2) Note that the sign of ∆m 2 31 remains undetermined, allowing for two possible neutrino mass orderings m 1 < m 2 < m 3 (referred to as the normal hierarchy and NH for short) or m 3 < m 1 < m 2 (the inverted hierarchy and IH for short). The absolute neutrino mass scale is not known either, but subject to the constraint m 1 + m 2 + m 3 < 0.23 eV from cosmological observations [3]. Particularly noteworthy, a recent result from the NOvA experiment (θ 23 = 39.5 • ± 1.7 • or 52.1 • ± 1.7 • in the NH case) disfavors the popular maximal mixing scenario θ 23 = 45 • with 2.6σ significance [4]. On the other hand, it is -1 -
How to understand the neutrino mixing pattern poses an interesting question. As symmetries (e.g., the SU(3) q quark flavor symmetry) have been serving as a guideline for understanding the particle physics, they may play a similar role in addressing the flavor issues. Along this line, many discrete groups have been proposed as the lepton flavor symmetry [6,7]. A simplest example is the µ-τ permutation symmetry [8][9][10][11][12]: in the basis of M l being diagonal, M ν should keep unchanged with respect to the transformation ν µ ↔ ν τ and thus feature M eµ = M eτ and M µµ = M τ τ (with M αβ for α, β = e, µ, τ being the matrix elements of M ν ). Such a symmetry (which results in θ 23 = π/4 and θ 13 = 0) was historically motivated by the experimental facts that θ 23 takes a value close to π/4 while θ 13 was only constrained by sin 2 2θ 13 < 0.18 [13] (and thus might be negligibly small) at the time. However, the relatively large θ 13 0.15 observed recently [14] requires a significant breaking of this symmetry unless neutrinos are quasi-degenerate in masses [15]. Hence we need to go beyond this simple possibility to accommodate the experimental results in a better way. In this connection, the µ-τ reflection symmetry [12,16] may serve as a unique alternative: when M l is diagonal, M ν should remain invariant under the transformation 1 and thus be characterized by In addition to allowing for an arbitrary θ 13 , this symmetry predicts θ 23 = π/4 and δ = ±π/2 [19] which are close to the present data, thereby having been attracting a lot of interest [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. Moreover, ρ and σ are required to take the trivial values 0 or π/2. Nevertheless, it is hard to believe that the µ-τ reflection symmetry can remain as an exact one. On the experimental side, the aforementioned results seem to hint towards θ 23 = π/4 (and possibly δ = ±π/2). On the theoretical side, flavor symmetries are generally implemented at a superhigh energy scale and so the renormalization group (RG) running effect may provide a source for the symmetry breaking as we will see. In view of these considerations, it is worthwhile to consider the breaking of this symmetry. In the next section, we perform a systematic study of the possible symmetry-breaking patterns and their implications for the mixing parameters. First of all, we establish an equation set relating the symmetry-breaking parameters in an M ν of approximate µ-τ reflection symmetry and the deviations of mixing parameters from their special values taken in the symmetry context. While the numerical results for these equations are analyzed in section 2.1, some analytical approximations will be derived in section 2.2 to explain the corresponding numerical results. In section 2.3 the general treatment is applied to some specific symmetry breaking arising from the RG running effect. Finally, we summarize our main results in section 3.
-3 -JHEP09(2017)023 It should be noted that Im( 1,2 ) and 3,4 will transform in a way as under the neutrino-field rephasing with ϕ 1,2,3 being some small parameters comparable to i . Taking advantage of such a freedom, one can always achieve 3,4 = 0 from the general case given by eq. (2.6). In the following discussions, we therefore concentrate on this particular case without loss of generality.
Starting from an M ν of the form in eq. (2.6) but with 3,4 = 0, we study the dependence of mixing parameters on 1,2 . To this end, we diagonalize such an M ν with one unitary matrix in a straightforward way (2.11) The mixing parameters in U are supposed to lie around those special values in eq. (2.3) and the corresponding deviations are some small quantities. By making perturbation expansions for these small quantities in eq. (2.11), one reaches the following relations connecting the mixing-parameter deviations with 1,2 have been taken.
As typical examples, the mixing-parameter deviations respectively arising from R 1 ≡ Re( 1 ) = 0.1 (in figure 1), I 1 ≡ Im( 1 ) = 0.1 (in figure 2), R 2 ≡ Re( 2 ) = 0.1 (in figure 3) and I 2 ≡ Im( 2 ) = 0.1 (in figure 4) are presented in such a way. Here and in the following, the black, red, green and blue colors are assigned to ∆θ, ∆δ, ∆ρ and ∆σ, respectively. To save space, the absolute value of a mixing-parameter deviation will be represented by a dashed line when it is negative. By contrast, the full line will be used when the mixing-parameter deviations are positive. In consideration of the experimental sensitivity, the region where the mixing-parameter deviations have magnitudes smaller than 0.01 has not been shown. In doing the calculations we have specified δ (0) = −π/2. If it takes the opposite value π/2, the mixing-parameter deviations produced by R 1 or R 2 will change in the way while those produced by I 1 or I 2 will change in the way The point is that eq. (2.13) stays invariant with respect to the transformations There are two immediate remarks for the above results. For the convenience of discussions, we use the contribution of R 1 to ∆δ as an illustration. But the following discussions apply to the contribution of each symmetry-breaking parameter to each mixing-parameter deviation. (1) The resulting ∆δ still can be inferred from the results in figure 1 when   ∆δ arising from a given value R 1 of R 1 (e.g., 0.1) has already been known (e.g., the red lines in figure 1), the ∆δ generated by another value R 1 of R 1 can be directly obtained as For example, the ∆δ produced by R 1 = 0.01 will be a tenth of the result given by figure 1.
(2) But there is one thing to take care of: in some cases (in particular the case of quasidegenerate neutrino mass spectrum combined with ρ (0) = σ (0) ) ∆δ seems to have gained a contribution of ≥ O(1) from R 1 = 0.1. For instance, as shown by the red line of the JHEP09(2017)023    ]. This would never be a realistic result but some signal for the breakdown of our approximation method used to derive eq. (2.13). Nevertheless, the ratio ∆δ/R 1 which takes a value of 100 in this specific case (implying that even a small R 1 = 0.001 can lead to some sizable ∆δ 0.1 = 6 • ) is still useful. To keep ∆δ within an acceptably small range in such a case (remember that it is right the possible closeness of θ 23 and δ to π/4 and −π/2 that motivates us to study the µ-τ reflection symmetry), R 1 should be of ≤ O(0.001). If R 1 were unfortunately much greater than 0.001, the resulting ∆δ would be of O(1). And one has to invoke an exact method (instead of the approximation one adopted here) to obtain the precise value of ∆δ. But this is beyond the scope of our interest, because it would be insignificant to discuss the µ-τ reflection symmetry any more if δ turned out to be far from ±π/2.
Given small values of symmetry-breaking parameters (i.e., R 1 = 0.1, I 1 = 0.1, R 2 = 0.1 and I 2 = 0.1), the resulting ∆θ appears as:    On the other hand, the resulting ∆δ turns out to be:     most sensitive to R 2 while ∆δ to all the symmetry-breaking parameters. In magnitude, ∆δ is generally much greater than ∆θ.

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For illustration, we give a toy example to show how to make use of the above results. In this connection, we discuss how the global-fit results θ 23 / • = 41.6 +1.5 −1.2 and δ/ • = 261 +51 −59 in the NH case [5] may arise from an approximate µ-τ reflection symmetry. (For simplicity, only the best-fit results will be used.) An interesting possibility is that a single R Finally, we discuss the consequences of breaking of µ-τ reflection symmetry on the allowed range of effective Majorana neutrino mass |M ee | which directly controls the rates of neutrinoless double-beta decays [36][37][38][39]. (2) In the IH case, the value of |Re(M ee )| is mainly determined by the first two components as the third one is highly suppressed. Because of m 1 m 2 in the IH case, |Re(M ee )| approximates to m 1 (or m 1 (c 2 12 − s 2 12 )) for ρ (0) = σ (0) (or ρ (0) = σ (0) ).

RG induced symmetry breaking
This section is devoted to the RG-induced breaking of µ-τ reflection symmetry. A flavor symmetry [6,7] together with the associated new fields is usually introduced at an energy scale Λ FS much higher than the electroweak (EW) one Λ EW . In this case one must consider the RG running effect when confronting the flavor-symmetry model with the low-energy data [40]. During the RG evolution process the significant difference between m µ and m τ can serve as a unique source for the breaking of µ-τ reflection symmetry. The general symmetry breaking studied in the above thus finds an interesting application in such a specific situation [41][42][43][44]. The energy dependence of neutrino mass matrix is described by its RG equation, which at the one-loop level appears as [45][46][47][48][49]

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Here t is defined as ln(µ/µ 0 ) with µ denoting the renormalization scale, whereas C and α read In eq. (2.29) the α-term is flavor universal and therefore just contributes an overall rescaling factor (which will be referred to as I α ), while the other two terms may modify the structure of M ν . In the basis under study, the Yukawa coupling matrix of three charged leptons is given by Y l = Diag(y e , y µ , y τ ). In light of y e y µ y τ , it is reasonable to neglect the contributions of y e and y µ . Integration of the RG equation enables us to connect the neutrino mass matrix M ν (Λ FS ) at Λ FS with the corresponding one at Λ EW in a manner as [50,51] M where I τ Diag{1, 1, 1 − ∆ τ } and In the SM case, the RG running effect is negligible due to the smallness of y τ 0.01 (which renders ∆ τ O(10 −5 ) ). By contrast, y 2 τ = (1 + tan 2 β)m 2 τ /v 2 can be enhanced by a large tan β in the MSSM case. Given Λ FS 10 13 GeV, for example, the value of ∆ τ depends on tan β in a way as ∆ τ 0.042 tan β 50 2 . (2.33) With the help of eq. (2.31), one will get the RG-corrected neutrino mass matrix at Λ EW from a neutrino mass matrix respecting the µ-τ reflection symmetry at Λ FS . By means of the above-mentioned treatment, one may arrange M ν (Λ EW ) in a form as given by eq. (2.6) with 2 = 2 1 = ∆ τ and 3,4 = 0. The relations between the mixing-parameter deviations and ∆ τ can therefore be obtained by simply taking R 2 = 2R 1 = ∆ τ and I 1,2 = 0 in eq. (2.13). As before, the mixing-parameter deviations have a linear dependence on ∆ τ and thus a square dependence on tan β. In figures 6-7 we display the mixing-parameter deviations respectively arising from tan β = 50 and tan β = 30 against the absolute neutrino mass scale for various combinations of ρ (0) and σ (0) . By analogy with eq. (2.18) (with a square dependence in place of the linear dependence), the mixing-parameter deviations generated by other values of tan β can also be inferred from these results. Qualitatively, the mixing-parameter deviations produced by ∆ τ closely resemble those resulting from R 1,2 in a few aspects: (1) Their magnitudes tend to grow with the absolute neutrino mass   scale (except in the case of [ρ (0) , σ (0) ] = [π/2, π/2]). (2) ∆θ generally receives a modest contribution from ∆ τ in most cases, while ∆δ, ∆ρ and ∆σ may get remarkably magnified for ρ (0) = σ (0) . (3) |∆δ| and |∆ρ| are comparable to each other, while |∆σ| is somewhat smaller. (4) ∆θ is always positive (negative) in the NH (IH) case. Quantitatively, the allowed ∆θ and ∆δ from ∆ τ ≤ 0.04 (for tan β ≤ 50) are as follows: (1) In the case of m 1 m 2 m 3 or m 3 m 1 m 2 , |∆θ| is smaller than 0.02. When the neutrino masses are quasi-degenerate (except in the case of [ρ (0) , σ (0) ] = [π/2, π/2]), |∆θ| may receive a contribution up to 0. 3 To summarize, the µ-τ reflection symmetry deserves particular attention as it leads to the interesting results θ 23 = π/4 and δ = ±π/2 (which are close to the current experimental data) as well as trivial Majorana phases. Nevertheless, it is reasonable for us to consider the breaking of such a symmetry either from the theoretical considerations (e.g., the RG running effect may provide a source for the symmetry breaking) or on the basis of experimental results (e.g., the newly-reported NOvA result disfavors the maximal mixing scenario at a 2.6σ level). Consequently, we have performed a systematic study for the possible symmetry-breaking patterns and their implications for the mixing parameters. We first define some parameters measuring the symmetry-breaking strengths and then derive an equation set relating them with the deviations of mixing parameters from the special values taken in the symmetry context. By solving these equations in both a numerical and analytical way, the sensitivity of mixing-parameter deviations to the symmetrybreaking parameters for various neutrino mass schemes and the Majorana-phase combinations are investigated in some detail. It turns out that ∆θ is most sensitive to R 2 while ∆δ, ∆ρ and ∆σ to all the symmetry-breaking parameters. And the magnitudes of ∆δ, ∆ρ and ∆σ are generally much greater than that of ∆θ. This means that the symmetry-breaking pattern with a sizable R 2 will be favored if ∆θ is notable, but any symmetry-breaking pattern may cause significant ∆δ, ∆ρ and ∆σ. The mixing-parameter deviations tend to get remarkably magnified (a symmetry-breaking parameter of ≤ O(0.01) may give rise to some mixing-parameter deviations of ≥ O(0.1)) in the case of m 1 m 2 m 3 combined with ρ (0) = σ (0) . But they will be highly suppressed and inconsiderable in the case of m 1 m 2 m 3 combined with [ρ (0) , σ (0) ] = [π/2, π/2]. With these general results as guide, one may easily find an appropriate specific way to break the µ-τ reflection symmetry so as to generate the required mixing-parameter deviations when necessary. Finally, as a unique illustration, the general treatment is applied to the specific symmetry breaking induced by the RG running effect.