Predictions of effective Majorana neutrino mass under radiative corrections to $μ-τ$ reflection symmetry

Technical Summary: Predictions of Effective Majorana Neutrino Mass under Radiative Corrections to $\mu-\tau$ Reflection Symmetry

Problem Statement
The search for neutrinoless double beta decay ($0\nu\beta\beta$) is a primary objective in neutrino physics, as its observation would confirm lepton number violation and the Majorana nature of neutrinos. The decay rate is governed by the effective Majorana neutrino mass, $|\langle m \rangle_{ee}|$. While current experiments, particularly KamLAND-Zen, have established an upper bound of $|\langle m \rangle_{ee}| < (0.028-0.122)$ eV, theoretical models must predict values consistent with this limit. This work addresses the prediction of $|\langle m \rangle_{ee}|$ within the framework of $\mu-\tau$ reflection symmetry, specifically investigating how radiative corrections (Renormalization Group running) induce deviations from this symmetry as the energy scale flows from the seesaw scale ($\Lambda_{\mu\tau} \approx 10^{14}$ GeV) to the electroweak scale ($\Lambda_{EW} \approx 172.76$ GeV).

Methodology
The authors extend their previous work by numerically estimating $|\langle m \rangle_{ee}|$ at the electroweak scale within the Minimal Supersymmetric Standard Model (MSSM). The methodology involves the following steps:

1. Symmetry Assumption: The study assumes exact $\mu-\tau$ reflection symmetry at the high-energy flavor symmetry scale ($\Lambda_{\mu\tau}$). This symmetry imposes specific constraints on the lepton mixing matrix and the neutrino mass matrix, fixing the atmospheric mixing angle $\theta_{23}$ to $\pi/4$ and the Dirac CP phase $\delta$ to $\pi/2$ or $3\pi/2$. It also constrains the Majorana phases ($\alpha, \beta$) and unphysical phases.
2. Radiative Corrections: The breaking of this symmetry is modeled via one-loop Renormalization Group Equations (RGEs). The evolution of the effective Majorana neutrino mass matrix $M_\nu$ is described by the integral solution method, accounting for the running of parameters from $\Lambda_{\mu\tau}$ to $\Lambda_{EW}$.
3. Analytical Derivation: The authors derive analytical expressions relating the low-energy parameters (mass eigenvalues $m_i$, mixing angles $\theta_{ij}$, and CP phases) to their high-energy counterparts. These expressions include correction terms dependent on the parameter $\epsilon$, which arises from the dominance of the tau-lepton Yukawa coupling in the RGEs.
4. Numerical Analysis: The study performs numerical simulations for two neutrino mass ordering scenarios: Normal Ordering (NO, $m_1 < m_2 < m_3$) and Inverted Ordering (IO, $m_3 < m_1 < m_2$).
   • Input Parameters: High-energy mass eigenvalues and mixing angles ($\theta_{12}, \theta_{13}$) are treated as free parameters, chosen to ensure low-energy predictions align with global 3$\sigma$ oscillation data and cosmological bounds ($\sum |m_i| < 0.12$ eV).
   • SUSY Parameters: The analysis considers three SUSY breaking scales ($\Lambda_s = 1, 7, 14$ TeV) and two values of $\tan\beta$ (30 and 58).
   • Calculation: Using the derived analytical relations, the authors compute the low-energy values of all mixing parameters and subsequently calculate $|\langle m \rangle_{ee}|$ using the standard formula involving the PMNS matrix elements and mass eigenvalues.

Key Contributions and Results
The paper provides a comprehensive numerical prediction of $|\langle m \rangle_{ee}|$ under the specific scenario of radiative breaking of $\mu-\tau$ reflection symmetry. The key findings are:

• Consistency with Experimental Bounds: All predicted values of $|\langle m \rangle_{ee}|$ across the various scenarios (NO/IO, different $\Lambda_s$, and $\tan\beta$) are consistent with the current KamLAND-Zen upper bound of $|\langle m \rangle_{ee}| < (0.028-0.122)$ eV.
• Normal Ordering (NO) Results:
  • For $\tan\beta = 30$, predictions range from $0.024189$ eV to $0.026337$ eV as $\Lambda_s$ increases. These values lie strictly below the experimental interval $(0.028-0.122)$ eV.
  • For $\tan\beta = 58$, predictions range from $0.013733$ eV to $0.031013$ eV. The value for $\Lambda_s = 1$ TeV is below the interval, while values for higher $\Lambda_s$ fall within the interval.
  • In the NO scenario, $|\langle m \rangle_{ee}|$ generally increases with increasing $\Lambda_s$.
• Inverted Ordering (IO) Results:
  • For both $\tan\beta = 30$ and $58$, the predicted values of $|\langle m \rangle_{ee}|$ are larger than in the NO scenario, ranging approximately between $0.049$ eV and $0.055$ eV.
  • All IO predictions lie within the experimental interval $(0.028-0.122)$ eV.
  • Unlike the NO scenario, in the IO scenario, $|\langle m \rangle_{ee}|$ decreases as the SUSY breaking scale $\Lambda_s$ increases.
• Case Independence: The study finds that the predicted values of $|\langle m \rangle_{ee}|$ are identical for Case-I ($\delta_{\mu\tau} = \pi/2$) and Case-II ($\delta_{\mu\tau} = 3\pi/2$) for any given set of parameters.

Significance
The paper claims that its primary significance lies in demonstrating that the effective Majorana neutrino mass $|\langle m \rangle_{ee}|$, when calculated under the assumption of $\mu-\tau$ reflection symmetry at the high scale and its subsequent radiative breaking, yields predictions that are fully compatible with the most stringent current experimental limits. The work extends previous analyses by explicitly computing these low-energy values for a range of SUSY breaking scales and $\tan\beta$ values within the MSSM framework. The authors conclude that the model remains a viable theoretical framework for describing neutrino properties, as the radiative corrections do not drive the effective mass beyond the exclusion limits set by the KamLAND-Zen collaboration.