Renormalization group evolution induced breaking of $μ-τ$ reflection symmetry in MSSM with effects of variation of $tanβ$

Technical Summary: Renormalization Group Evolution Induced Breaking of $\mu-\tau$ Reflection Symmetry in MSSM with Effects of Variation of $\tan\beta$

Problem Statement
Neutrino oscillation data indicates that the atmospheric mixing angle $\theta_{23}$ and the Dirac CP-violating phase $\delta$ are close to maximal values, a feature suggestive of an underlying $\mu-\tau$ reflection symmetry. This symmetry predicts exact maximal mixing ($\theta_{23} = \pi/4$) and specific CP phases ($\delta = \pi/2$ or $3\pi/2$). However, current global analyses reveal small but non-zero deviations from these exact values. The paper investigates whether these deviations can be naturally explained by the Renormalization Group (RG) evolution of neutrino parameters. Specifically, the study focuses on the Minimal Supersymmetric Standard Model (MSSM) and examines how the variation of the ratio of Higgs vacuum expectation values, $\tan\beta \equiv v_u/v_d$, influences the breaking of $\mu-\tau$ reflection symmetry as energy scales run from a high flavor symmetry scale ($\Lambda_{FS}$) down to the electroweak scale.

Methodology
The authors employ a numerical approach to solve a system of nineteen coupled one-loop Renormalization Group Equations (RGEs). The methodology involves the following steps:

1. Theoretical Framework: The analysis assumes exact $\mu-\tau$ reflection symmetry at a high flavor symmetry scale $\Lambda_{FS}$. The evolution proceeds in two stages:

   • MSSM Regime: From $\Lambda_{FS}$ down to the SUSY breaking scale $\Lambda_s = 7$ TeV.
   • Standard Model (SM) Regime: From $\Lambda_s$ down to the electroweak scale (identified with the top quark mass, $m_t$).
   • Appropriate matching conditions are applied at $\Lambda_s$ to ensure continuity of gauge couplings and to rescale Yukawa couplings ($y_t, y_b, y_\tau$) based on $\tan\beta$.

2. Parameter Space:

   • Fixed Inputs: The symmetry imposes maximal values for $\theta_{23}$, $\delta$, and the Majorana phases at $\Lambda_{FS}$. The gauge and Yukawa couplings are evolved upward from experimental low-energy values to determine their high-scale inputs.
   • Free Parameters: The three neutrino mass eigenvalues ($m_1, m_2, m_3$) and the mixing angles $\theta_{12}$ and $\theta_{13}$ are treated as free parameters at $\Lambda_{FS}$.
   • Scenarios: The study covers both Normal Ordering (NO, $m_1 < m_2 < m_3$) and Inverted Ordering (IO, $m_3 < m_1 < m_2$). For each ordering, two cases are analyzed corresponding to the two maximal predictions for $\delta$ (Case I: $\delta = \pi/2$; Case II: $\delta = 3\pi/2$).
   • $\tan\beta$ Variation: Three benchmark values for $\tan\beta$ are tested: 10, 30, and 58.

3. Numerical Implementation: The coupled differential equations are solved using a Python code. The free parameters at the high scale are tuned via a trial-and-error approach to ensure that the evolved low-energy values fall within the $3\sigma$ allowed ranges of global neutrino oscillation data and satisfy the cosmological bound on the sum of neutrino masses ($\sum m_i < 0.12$ eV).

Key Results
The numerical analysis yields the following observations regarding the RG-induced breaking of symmetry:

• Dependence on $\tan\beta$: The magnitude of RG running effects is strongly dependent on $\tan\beta$.
  • For small $\tan\beta$ (e.g., 10), the deviations of mixing angles and CP phases from their high-scale symmetric values are negligible.
  • For large $\tan\beta$ (e.g., 58), the deviations become significantly more pronounced. This is attributed to the enhancement of the tau Yukawa coupling ($y_\tau \propto 1/\cos\beta$) in the MSSM, which dominates the RG evolution of neutrino parameters.
• Mixing Angles:
  • $\theta_{12}$ and $\theta_{13}$ show mild decreases as the energy scale lowers.
  • $\theta_{23}$ exhibits a slight increase in the Normal Ordering (NO) scenario, moving into the second octant (values $> 45^\circ$), while in the Inverted Ordering (IO) scenario, it slightly decreases, remaining in the first octant.
  • The deviations in $\theta_{23}$ are more distinct for $\tan\beta = 58$ compared to lower values.
• CP-Violating Phases:
  • The Dirac phase $\delta$ and Majorana phases generally show very small deviations for $\tan\beta = 10$ and $30$.
  • For $\tan\beta = 58$, the evolution curves for the CP phases (particularly $\delta$, $\alpha$, and $\beta$) diverge noticeably from the lower $\tan\beta$ cases, indicating stronger radiative corrections.
• Mass Eigenvalues: The running of the mass eigenvalues is found to be largely insensitive to variations in $\tan\beta$ across all scenarios. The mass-squared differences and the total neutrino mass sum remain consistent with experimental constraints for all tested $\tan\beta$ values, provided the high-scale inputs are appropriately tuned.

Significance and Claims
The paper claims to provide a detailed investigation into the radiative breaking of $\mu-\tau$ reflection symmetry within the MSSM, specifically isolating the impact of $\tan\beta$. Unlike previous works that may have fixed $\tan\beta$ or varied the SUSY scale, this study fixes the SUSY breaking scale ($\Lambda_s = 7$ TeV) and systematically varies $\tan\beta$.

The authors conclude that while $\mu-\tau$ reflection symmetry remains largely stable under RG evolution, the parameter $\tan\beta$ acts as a crucial control knob. Larger values of $\tan\beta$ amplify the radiative corrections, potentially generating the small deviations from maximality observed in current experimental data. The study demonstrates that it is possible to reproduce the low-energy experimental constraints (including the $3\sigma$ ranges for mixing angles and mass differences) starting from exact symmetry at the high scale, provided the high-scale free parameters are carefully chosen and the $\tan\beta$ dependence is accounted for. The work reinforces the established understanding that RG effects are more significant in the MSSM at large $\tan\beta$ compared to the SM or MSSM at small $\tan\beta$.