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Perturbations to $μ-τ$ reflection symmetry due to renormalization group running effects

Chandan Kumar Borah, Chandan Duarah

TL;DR

This work investigates whether μ-τ reflection symmetry, which fixes $θ_{23}$ to be maximal and $δ$ to be near maximal, remains compatible with low-energy data after radiative corrections. By deriving and numerically solving the one-loop RGEs for the full set of neutrino parameters via a two-step MSSM→SM running from $Λ_{FS}=10^{14}\mathrm{GeV}$ to $Λ_{EW}$, and by exploring Normal and Inverted ordering with two δ-cases, the authors show that RG effects induce only small deviations from the high-scale symmetry. The low-energy predictions for masses, mixing angles, and CP phases stay within the 3σ ranges of global oscillation data and satisfy the cosmological bound $\sum m_i<0.12$ eV, supporting μ-τ reflection symmetry as a robust high-scale flavor structure. The findings also reveal distinct yet mild sensitivities to the SUSY-breaking scale in the MSSM regime, informing how high-scale symmetries translate into observable lepton mixing patterns.

Abstract

$μ-τ$ reflection symmetry is an attractive flavour symmetry in lepton mixing, which accommodates maximal values of atmospheric mixing angle ($θ_{23}=π/4$) and Dirac CP phase ($δ=π/2/3π/2$). Another significance of this symmetry is that it does not constrain $θ_{13}$ to be zero. As the recent results from $T2K$ and $NOνA$ experiments indicate a near-maximal value of the Dirac CP phase, the role of $μ-τ$ reflection symmetry becomes more prominent. In this work, we study RG running effects as a perturbation to the $μ-τ$ reflection symmetry. Assuming the symmetry to be preserved at the seesaw scale, we study the deviations of mass eigenvalues and lepton mixing parameters at the electroweak scale due to RG running. We derive the one-loop RGEs of the mass eigenvalues and mixing parameters and solve them numerically. Numerical analysis shows that the deviations from $μ-τ$ reflection symmetry are consistent with $3σ$ range of global oscillation data.

Perturbations to $μ-τ$ reflection symmetry due to renormalization group running effects

TL;DR

This work investigates whether μ-τ reflection symmetry, which fixes to be maximal and to be near maximal, remains compatible with low-energy data after radiative corrections. By deriving and numerically solving the one-loop RGEs for the full set of neutrino parameters via a two-step MSSM→SM running from to , and by exploring Normal and Inverted ordering with two δ-cases, the authors show that RG effects induce only small deviations from the high-scale symmetry. The low-energy predictions for masses, mixing angles, and CP phases stay within the 3σ ranges of global oscillation data and satisfy the cosmological bound eV, supporting μ-τ reflection symmetry as a robust high-scale flavor structure. The findings also reveal distinct yet mild sensitivities to the SUSY-breaking scale in the MSSM regime, informing how high-scale symmetries translate into observable lepton mixing patterns.

Abstract

reflection symmetry is an attractive flavour symmetry in lepton mixing, which accommodates maximal values of atmospheric mixing angle () and Dirac CP phase (). Another significance of this symmetry is that it does not constrain to be zero. As the recent results from and experiments indicate a near-maximal value of the Dirac CP phase, the role of reflection symmetry becomes more prominent. In this work, we study RG running effects as a perturbation to the reflection symmetry. Assuming the symmetry to be preserved at the seesaw scale, we study the deviations of mass eigenvalues and lepton mixing parameters at the electroweak scale due to RG running. We derive the one-loop RGEs of the mass eigenvalues and mixing parameters and solve them numerically. Numerical analysis shows that the deviations from reflection symmetry are consistent with range of global oscillation data.
Paper Structure (7 sections, 72 equations, 8 figures, 5 tables)

This paper contains 7 sections, 72 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Basics of $\mu-\tau$ reflection symmetry and corresponding lepton mixing matrix
  3. RG running of neutrino parameters
  4. Numerical analysis and results
  5. Normal Order ($m_1<m_2<m_3$)
  6. Inverted Order ($m_3<m_1<m_2$)
  7. Summary and discussion

Figures (8)

  • Figure 1: Evolution of mass eigenvalues and mixing angles with energy scale for NO and case-I with three different values of $\Lambda_s$. Solid and dashed portions of each curve represent the evolution in the MSSM and SM regions respectively. Magenta , green and red colors respectively stands for $\Lambda_s=1\ TeV,\ 7\ TeV\ \text{and}\ 14\ TeV$.
  • Figure 2: Evolution of CP phases with energy scale for NO and case-I with three different values of $\Lambda_s$. Solid and dashed portions of each curve represent the evolution in the MSSM and SM regions respectively. Magenta , green and red colors respectively stands for $\Lambda_s=1\ TeV,\ 7\ TeV\ \text{and}\ 14\ TeV$
  • Figure 3: Evolution of mass eigenvalues and mixing angles with energy scale for NO and case-II with three different values of $\Lambda_s$. Solid and dashed portions of each curve represent the evolution in the MSSM and SM regions respectively. Magenta , green and red colors respectively stands for $\Lambda_s=1\ TeV,\ 7\ TeV\ \text{and}\ 14\ TeV$.
  • Figure 4: Evolution of CP phases with energy scale for NO and case-II with three different values of $\Lambda_s$. Solid and dashed portions of each curve represent the evolution in the MSSM and SM regions respectively. Magenta , green and red colors respectively stands for $\Lambda_s=1\ TeV,\ 7\ TeV\ \text{and}\ 14\ TeV$.
  • Figure 5: Evolution of mass eigenvalues and mixing angles with energy scale for IO and case-I with three different values of $\Lambda_s$. Solid and dashed portions of each curve represent the evolution in the MSSM and SM regions respectively. Magenta , green and red colors respectively stands for $\Lambda_s=1\ TeV,\ 7\ TeV\ \text{and}\ 14\ TeV$.
  • ...and 3 more figures