Leptonic unitarity triangles: RGE running effects and $\mu$-$\tau$ reflection symmetry breaking

There are six leptonic unitarity triangles (LUTs) defined by six orthogonality conditions of the three-family lepton flavor mixing matrix in the complex plane. In the framework of the standard model or the minimal supersymmetric standard model, the evolutions of sides and inner angles of the six LUTs from a superhigh energy scale $\Lambda_{\rm H}^{}$ to the electroweak scale $\Lambda_{\rm EW}^{}$ due to the renormalization-group equation (RGE) running are derived in the integral form for both Dirac and Majorana neutrinos. Furthermore, the LUTs as an intuitively geometrical language are applied to the description of the RGE-induced $\mu$-$\tau$ reflection symmetry breaking analytically and numerically.


Introduction
In the recent twenty years, a series of neutrino oscillation experiments have definitely proved that neutrinos have masses and lepton flavors mix with one another [1]. The latter can be described by the well-known Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U [2,3], which connects three neutrino mass eigenstates (ν 1 , ν 2 , ν 3 ) and flavor eigenstates (ν e , ν µ , ν τ ) by in the basis where the mass eigenstates of three charged leptons are identical with their flavor eigenstates. According to the orthogonality of the rows and columns of U, one may define six leptonic unitarity triangles (LUTs) in the complex plane as a geometrical language to intuitively describe lepton flavor mixing and CP violation [4]. The six triangles are which are insensitive to the Majorana phases; and whose orientations are fixed by the Majorana phases 1 [5]. The areas of these LUTs are all equal to |J |/2, where J means the Jarlskog invariant of U describing leptonic CP violation in neutrino oscillations and can be defined through The subscripts (α, β, γ) and (i, j, k) in this paper always run over (e, µ, τ ) and (1,2,3), respectively, if not otherwise specified. The six LUTs consist of eighteen vector sides in the complex plane shown in Eqs. (2) and (3) and nine inner angles which can be expressed as 1 In the definitions of six LUTs in Eqs. (2) and (3) In Eq. (5), α, β and γ run cyclically over e, µ and τ ; i, j and k run cyclically over 1, 2 and 3; η φ = 1 for J < 0 and η φ = −1 for J > 0 2 . The language of LUTs has been discussed in a number of papers [6,7,8,9,10,11,12] since it was introduced into the lepton sector. These papers mainly focus on the following aspects: • The reconstruction of LUTs through future precision neutrino oscillation and nonoscillation experiments will be a useful and intuitive geometric way to demonstrate CP violation in the lepton sector, and this will be complementary to the direct measurements of CP asymmetries [6,7]. Furthermore, testing whether the LUTs are close will provide tests of the unitarity of the PMNS matrix, which might be violated due to the existence of sterile neutrinos [4,5,8].
• One can directly use the sides and inner angles of the LUTs to describe neutrinoneutrino oscillations, neutrino-antineutrino oscillations and neutrino decays, where the inner angles of the LUTs have definite physical meaning [9,10]. The shapes of the LUTs can be reformed either by terrestrial matter effects, or by renormalization-groupequation (RGE) running effects, or by some other new physics effects, implying the corrections of such effects to lepton flavor mixing and CP violation [11,12]. There are also discussions about the underlying phenomenological meaning of special shapes of the LUTs [10].
In Ref. [12], the RGE running behaviors of inner angles of the LUTs have been discussed in the differential form. In this paper, we aim to study how the sides and inner angles of the LUTs evolve in the integral form due to the RGE running from an arbitrary superhigh energy scale Λ H to the electroweak scale Λ EW in the framework of the standard model (SM) or the minimal supersymmetric standard model (MSSM). Both the cases of Dirac and Majorana neutrinos will be considered. We get the RGE-induced corrections to the LUTs by performing perturbative expansions. The final analytical results are independent of the parametrization of U. Assuming the µ-τ reflection symmetry [13,14] to be satisfied at a superhigh energy Λ µτ , the corresponding △ i should be isosceles triangles; △ µ and △ τ are congruent with each other. When running down to Λ EW , the µ-τ reflection symmetry will be broken due the RGE running effects, leading to the deviations of the LUTs from their special shapes at Λ µτ . So the corrections to the LUTs from Λ µτ to Λ EW can be used to intuitively describe the corresponding RGE-induced µ-τ reflection symmetry breaking, and thus it is meaningful to explore how the LUTs can be reformed analytically and numerically in this case. The rest of this paper is organized as follows. In section 2, we derive the RGE-induced connections of the sides and inner angles of the LUTs between Λ EW and Λ H in the integral form in the framework of the SM or the MSSM, where both Dirac and Majorana neutrinos are considered. Section 3 is devoted to simplifying the analytically approximate results in section 2 by assuming the µ-τ reflection symmetry at Λ µτ . In section 4, the RGE-induced deviations of the LUTs from the µ-τ reflection symmetry limits will be studied numerically by scanning the complete parameter space, where the smallest neutrino mass and the MSSM parameter tan β at Λ EW run in the reasonable ranges [0, 0.1] eV and [10,50], respectively, just as the way taken in Ref. [15]. The normal mass ordering (NMO) and inverted mass ordering (IMO) of Dirac or Majorana neutrinos will be considered. Finally, section 5 is a summary of our main results.

The case of Dirac neutrinos
Before a decisive measurement of the neutrinoless double-beta decay [16] verifies the Majorana nature of massive neutrinos, it is meaningful to consider the cases of both Dirac and Majorana neutrinos theoretically [17]. The evolution of the Dirac neutrino mass matrix from Λ H to Λ EW in the integral form can be written as [18] where M ν and M ′ ν are the Dirac neutrino mass matrix at Λ H and Λ EW , respectively. Note that the notations with a prime superscript in the following text denote the parameters at Λ EW and those without such a superscript stand for the corresponding parameters at Λ H if not otherwise specified. Here we define T l = Diag{I e , I µ , I τ } and where (M ν ) αβ = i m i U αi U βi . By solving the equation By inserting the tau-dominance approximation of T l and Eq. (26) into Eq. (23), and expanding it in ǫ up to the first order, we can get the analytical approximations of |U ′ αi | 2 at Λ EW : and and with R ij αβ denoting the real parts of U αi U * αj U βi U * βj . The vector sides of △ ′ α at Λ EW turn out to be: for △ ′ e ; and for △ ′ µ ; and for △ ′ τ . The above analytical approximations of |U ′ can not be derived in this way, implying that it is impossible to get any information on the Majorana phases at Λ EW . However, we can calculate |U ′ αi U ′ * αj | 2 from Eqs. (27)- (29) and fix the shapes of △ ′ i without their orientations. In appendix B, we list the nine inner angles of △ ′ α or △ ′ i at Λ EW running from Λ H for Majorana neutrinos. With the help of Eq. (4) and the vector sides in Eqs. (30)-(32), the Jarlskog invariant J ′ at Λ EW for Majorana neutrinos can be given by where I ij αβ denote the imaginary parts of U αi U * αj U βi U * βj . Some discussions about the analytical results above for both Dirac and Majorana neutrinos are as follows: • The approximate expressions of |U ′ ei | 2 and U ′ µi U ′ * τ i are similar to those of |U ′ µi | 2 and U ′ τ i U ′ * ei , respectively. The analytical results for Majorana neutrinos are not equivalent to those for Dirac neutrinos even if one turns off the Majorana phases by setting their values to be zeros. In both cases, the corrections to the LUTs depend a lot on the magnitudes of the lightest neutrino mass and the small quantity ǫ. The evolutions of the sides U ′ α3 U ′ * β3 and the inner angles φ α3 are more stable against the RGE running.
• Different from the Dirac case, J ′ of Majorana neutrinos is in general nonzero even assuming J at Λ H to be zero, and vice versa. One can conclude from Eq. (33) that there may exist leptonic CP violation at Λ EW unless all the Dirac and Majorana phases at the superhigh energy vanish. This observation is consistent with the analysis in Refs. [12,24].
• The direct connections of the LUTs between two energy scales, which have been established above, are independent of the parametrization of U and complementary to the differential forms in Ref. [12]. They can also reproduce the analytical approximations of neutrino masses, flavor mixing angles and the Dirac CP phase in other references [15,18,25] by taking a specific parametrization. Note that the accuracy of the approximate results above and in section 3 will be very poor if the neutrino masses are strongly degenerate, i.e., the smallest neutrino mass is big enough. Considering the fact that the combination of Planck and baryon acoustic oscillation (BAO) measurements gives the limit of the sum of three light neutrino masses as i m i < 0.12 eV at 95% confidence level [26], one can use the analytical approximations to understand most part of the parameter space. We plan to explicitly study the case of nearly degenerate neutrino masses elsewhere.

LUTs and RGE-induced µ-τ reflection symmetry breaking
The µ-τ reflection symmetry of the neutrino sector serving as the minimal discrete flavor symmetry to explain the lepton flavor mixing and CP violation has been extensively studied for both Dirac and Majorana neutrinos [14]. One of the usual ways is that by assuming the µ-τ reflection symmetry at a superhigh energy scale Λ µτ , we confront its RGE-induced breaking effects at Λ EW with current experiment data [14,18,27,25]. This can be connected with the corresponding reformations of the LUTs below.

The case of Dirac neutrinos
If massive neutrinos are the Dirac particles, the µ-τ reflection symmetry means that the effective Dirac neutrino mass term is invariant under the flavor and charge-conjugation transformations below: where ν αL and N αR for α = e, µ, τ denote the left-handed and right-handed neutrino fields, respectively. This results in the constraint conditions of ( There are eight choices of (η 1 , η 2 , η 3 ) while all of them are identical with one another through redefining the relevant phases of charged lepton and Dirac neutrino fields. Given the µ-τ reflection symmetry of Dirac neutrinos at a superhigh energy scale Λ µτ , we have |U µi | = |U τ i |. Hence the corresponding △ i are isosceles triangles, each with two equal sides |U µj U * µk | = |U τ j U * τ k |; and the two LUTs △ µ and △ τ are congruent with each other with three pairs of equal sides |U τ i U * ei | = |U ei U * µi |. The deviations of the LUTs at Λ EW from these special shapes at Λ µτ due to the RGE running can demonstrate the RGE-induced µ-τ reflection symmetry breaking intuitively. Let us define to describe the deviations of ∆ ′ i from their µ-τ reflection symmetry limits, and to show how the LUTs ∆ ′ µ and ∆ ′ τ can be reformed as compared with their µ-τ reflection symmetry limits. With the help of |U µi | = |U τ i | together with Eqs. (14)- (16), the analytical approximations of the six asymmetries in Eqs. (35) and (36) can be expressed as: and where |U µi | 2 = |U τ i | 2 has been be replaced by (1 − |U ei | 2 )/2. We can see that S 1 △ µτ and S 2 △ µτ are most sensitive to the neutrino mass ordering. The absolute values of S µτ △ 3 and S 3 △ µτ should be smaller because of the smallness of ∆ 21 and |U e3 | 2 . The Jarlskog invariant J ′ at Λ EW running from Λ µτ can be written as whose magnitude is proportional to the area of the LUTs at Λ EW . Taking account of . The asymmetries of these three pairs of inner angles satisfy i (φ ′ µi − φ ′ τ i ) = 0.

The case of Majorana neutrinos
When it comes to the Majorana neutrinos, the µ-τ reflection symmetry implies the effective Majorana mass term should stay unchanged under the flavor and charge-conjugation transformations of neutrino fields: ν eL ↔ ν c eR , ν µL ↔ ν c τ R and ν τ L ↔ ν c µR . This results in the limits to the elements of neutrino mass matrix M ν : Four of the eight choices of (η 1 , η 2 , η 3 ) are independent because we can not redefine the Majorana neutrino fields to change the sign of arbitrary column of U just like the Dirac case. Given the µ-τ reflection symmetry at Λ µτ , one gets |U µi | = |U τ i |, which results in three isosceles LUTs △ i with |U µj U * µk | = |U τ j U * τ k | and a pair of congruent triangles △ µ and △ τ ) with |U τ i U * ei | = |U ei U * µi | just as the Dirac case. So the asymmetries defined in Eqs. (35) and (36) can be used to denote the deviations of LUTs of the Majorana neutrinos at Λ EW from their special shapes at Λ µτ . The analytical approximations of these asymmetries in this case can be obtained with the help of U µi = η i U * τ i and Eqs. (27)-(39). The results are demonstrating the deviations of △ ′ i at Λ EW from their isosceles shapes at Λ µτ ; and showing the deviations of △ ′ µ and △ ′ τ at Λ EW from their congruent shapes at Λ µτ . From Eqs. (41) and (42), we find that S 1 △ µτ and S 2 △ µτ are most sensitive to the neutrino mass ordering; S 3 △ µτ and S µτ △ 3 are smaller due to the suppression of ∆ 21 and |U e3 | 2 . This conclusion is the same as the Dirac case. The connection of the Jarlskog invariants of Majorana neutrinos between Λ EW and Λ µτ can be written as It is clear to see that the analytical approximations of S µτ △ i , S i △ µτ and J ′ for the Majorana neutrinos include more odd terms of η i (i.e., η i η j for i = j) compared with their counterparts for the Dirac neutrinos. These terms can be directly connected with the Majorana phases and have complicated influence on the LUT reformations at Λ EW .

Numerical analysis
Before we start the numerical analysis, let us first parametrize U as for the Dirac neutrinos with c ij ≡ cos θ ij and s ij ≡ sin θ ij . For the Majorana neutrinos, one has to add the Majorana phase matrix P ν ≡ Diag {e iρ , e iσ , 1} on the right side of Eq. (44). U ′ at Λ EW has the same form as U with the corresponding set of flavor mixing angles and CP phases (θ ′ 12 , θ ′ 13 , θ ′ 23 , δ ′ , ρ ′ , σ ′ ). According to the specific parametrization of U in Eq. (44), we interpret the constraints of the µ-τ reflection symmetry as two conditions for the Dirac neutrinos: θ 23 = π/4 and δ = ±π/2, and four conditions for the Majorana neutrinos: θ 23 = π/4, δ = ±π/2, ρ = 0 or π/2 and σ = 0 or π/2. The correspondences between the eight choices of (δ, ρ, σ) and the four independent cases of (η 1 , η 2 , η 3 ) have been listed in Table 1. Given the fact that the global-fit analysis of current neutrino oscillation data has implied a preference of δ around −π/2 [28,29], we only focus on the case δ = −π/2 at Λ µτ for both Dirac and Majorana neutrinos. The framework of the MSSM is typically chosen because the RGE-induced µ-τ reflection symmetry breaking is always very small in the SM. Table 1: The correspondences between (δ, ρ, σ) and (η 1 , η 2 , η 3 ) in the µ-τ reflection symmetry limit for the Majorana neutrinos.
To show the deviations of the six LUTs at Λ EW from their special shapes at Λ µτ , which can be described by the asymmetries defined in section 3, the numerical analysis similar to that in Ref. [15] has been done. Both the NMO (m 2 ) cases of the Dirac or Majorana neutrinos will be taken into account. Note that there are four choices of the two Majorana phases at Λ µτ , which need to be considered separately, too. In each case, we first run the relevant RGEs from Λ µτ ∼ 10 14 down to Λ EW by inputting the corresponding µ-τ reflection symmetry constraint conditions of flavor mixing angles and CP phases at Λ µτ and allowing the smallest neutrino mass (m ′ 1 for the NMO case and m ′ 3 for the IMO case) at Λ EW and the MSSM parameter tan β to vary in the reasonable ranges [0, 0.1] eV and [10,50], respectively. For each given values of m ′ 1 or m ′ 3 and tan β, the other parameters (sin 2 θ 12 , sin 2 θ 13 , ∆ sol , ∆ atm ) at Λ µτ are scanned over wide enough ranges by means of the MultiNest program [30] and their counterparts at Λ EW ∆ ′ sol = m ′2 2 − m ′2 1 and ∆ ′ atm = m ′2 3 − (m ′2 1 + m ′2 2 )/2 have been defined to keep consistent with the notations in Ref. [28]. From each scan, we can get a set of parameters at Λ EW which will be confronted with the latest global-fit results of current neutrino oscillation data by where ξ i ∈ {sin 2 θ ′ 12 , sin 2 θ ′ 13 , sin 2 θ ′ 23 , δ ′ , ∆ ′ sol , ∆ ′ atm } stand for the oscillation parameters yielded from the scan; ξ i and σ i denote the best-fit values and averaged 1σ errors of ξ i from the globalfit analysis in Ref. [28], respectively. The best-fit values and 3σ ranges of S i △ µτ , φ ′ µi − φ ′ τ i and J ′ are listed in Tables 2-6, corresponding to the minimal values χ 2 min of χ 2 and χ 2 ≤ 9 for one degree of freedom, respectively. Considering that the two asymmetries S µτ imply consistent deviations of the LUTs, we only demonstrate the numerical results of the latter. Some discussions about the numerical results are as follows: • Complementary to the analytical approximations in section 3, the numerical results generally reveal how the six LUTs can be reformed at Λ EW by assuming the µ-τ reflection symmetry at Λ µτ . The reformations depend a lot on the lightest neutrino mass, the neutrino mass ordering, the Majorana phases and tan β. From Tables 2-6, we find that the parameters running from Λ µτ and their corresponding best-fit values from the global analysis in Ref. [28] can not fit very well in the IMO case, leading to big values of χ 2 min . This is mainly because the running direction of θ 23 from Λ µτ to Λ EW is opposite to its best-fit value in this case [15,18]. The lightest neutrino mass m ′ 3 and tan β are limited to smaller ranges by χ 2 ≤ 9 in Tables 3-5.
• The deviations of the six LUTs are small for the case (ρ, σ) = (π/2, π/2) in Table 6 but their values can be very big in some other cases. For example, the two asymmetries φ ′ µ1 − φ ′ τ 1 and φ ′ µ2 − φ ′ τ 2 may reach about 180 • in magnitude because of the smallness of the corresponding J ′ . We also notice that J ′ running from Λ µτ can not be zero due to the nonzero value of J constrained by the µ-τ reflection symmetry conditions. It is easy to understand this point from Eqs. (39) and (43).
• The smallest χ 2 min for the Dirac and Majorana neutrinos come from the best-fit results of the NMO case in Table 2 and Table 5, respectively. The corresponding LUTs together with their counterparts at Λ µτ have been specifically shown in Fig. 1 and Fig. 2. The blue triangles with χ 2 min ≃ 0.01 stand for the LUTs at Λ EW and almost overlap the LUTs implied by the best-fit values of the global analysis in Ref. [28], while the red ones denote the corresponding LUTs at Λ µτ . When comparing the two figures, we find that the blue LUTs at Λ EW differ with each other only in the orientations of △ i caused by the Majorana phases, while the red ones are very different. The numerical analysis of deviations of the six LUTs at Λ EW from their µ-τ reflection symmetry limits at Λ µτ for the Dirac neutrinos in the framework of the MSSM, by inputting (θ 23 , δ) = (π/4, −π/2) at Λ µτ and allowing the smallest neutrino mass (m ′ 1 for the NMO case and m ′ 3 for the IMO case) and the MSSM parameter tan β to vary in the ranges [0, 0.1] eV and [10,50]    : The numerical analysis of deviations of the six LUTs at Λ EW from their µ-τ reflection symmetry limits at Λ µτ for the Majorana neutrinos in the framework of the MSSM, by inputting (θ 23 , δ, ρ, σ) = (π/4, −π/2, 0, π/2) at Λ µτ and allowing the smallest neutrino mass at Λ EW (m ′ 1 for the NMO case and m ′ 3 for the IMO case) and the MSSM parameter tan β to vary in the ranges [0, 0.1] eV and [10,50]     : The numerical analysis of deviations of the six LUTs at Λ EW from their µ-τ reflection symmetry limits at Λ µτ for the Majorana neutrinos in the framework of the MSSM, by inputting (θ 23 , δ, ρ, σ) = (π/4, −π/2, π/2, π/2) at Λ µτ and allowing the smallest neutrino mass at Λ EW (m ′ 1 for the NMO case and m ′ 3 for the IMO case) and the MSSM parameter tan β to vary in the ranges [0, 0.1] eV and [10,50] tan β 50 (10, 50) 10 (10, 50)