⚛️ phenomenology

Deviation from $\mu-\tau$ reflection symmetry under radiative corrections in the minimal seesaw framework

This paper investigates how radiative corrections within the minimal seesaw framework break the high-energy $\mu$ - $\tau$ reflection symmetry, demonstrating that the resulting low-energy neutrino parameters remain consistent with current experimental data while showing deviations that increase with $\tan\beta$ .

Original authors: Prokash Pegu, Chandan Duarah

Published 2026-09-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Prokash Pegu, Chandan Duarah

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Neutrinos are ghostly particles that zip through the universe in staggering numbers, passing through planets and people without ever leaving a trace. For decades, scientists believed these particles were massless, but experiments in the late twentieth century proved otherwise. We now know they have a tiny mass and that they can change their identity as they travel, shifting from one type to another in a process called oscillation. This behavior is governed by a mixing pattern, a set of rules that determines how the different types of neutrinos blend together. Among the many patterns physicists have proposed to explain this mixing, one stands out for its elegant simplicity: a symmetry that treats two specific types of neutrinos, the muon and the tau, as mirror images of each other. This idea, known as mu-tau reflection symmetry, predicts that the mixing between these two types should be perfectly balanced, resulting in a specific, maximal angle of mixing, and that the particles should violate a fundamental symmetry of nature called CP symmetry in a very specific way.

However, the universe rarely obeys perfect rules. Recent measurements suggest that while this symmetry is a good approximation, it is not exact. The mixing angle is close to, but not exactly, the perfect value, and the CP violation is near, but not precisely at, the predicted maximum. This small discrepancy raises a compelling question: if the symmetry exists at the highest energy levels of the early universe, why does it appear broken in the world we observe today? A team of researchers from Dibrugarh University in India set out to answer this by looking at how the laws of physics change as energy scales drop. They investigated whether the tiny differences we see today could be the result of natural, gradual shifts that occur as the universe cools, rather than a fundamental flaw in the symmetry itself.

The researchers focused on a specific theoretical framework called the minimal seesaw, a model that explains why neutrinos are so light by introducing two heavy, unseen particles into the mix. In this model, one of the three known neutrinos is predicted to be completely massless. The team assumed that at the incredibly high energy scale of the early universe, roughly one hundred trillion times the energy of a proton, the mu-tau reflection symmetry was perfect. They then used mathematical tools to trace how the properties of these particles would evolve as the energy dropped from that primordial scale down to the energy levels we can test in modern laboratories. They calculated how the mass of the neutrinos and the angles of their mixing would shift due to quantum effects, which are tiny fluctuations that happen constantly at the subatomic level.

To make their calculations concrete, the team worked within a popular extension of the Standard Model of particle physics that includes supersymmetry, a theory suggesting that every known particle has a heavier partner. They tested their ideas under three different scenarios for how the forces between these particles behave, represented by a value called tan beta, which can be thought of as a dial controlling the strength of certain interactions. They ran their simulations for three settings of this dial: ten, thirty, and fifty. For each setting, they started with the perfect, symmetric values at the high-energy scale and watched how the numbers changed as they ran them down to the energy of the top quark, a heavy particle discovered in the late 1990s.

The results showed that the symmetry does indeed break, but in a very controlled way. As the energy scale decreased, the perfect mixing angle and the specific CP phase shifted slightly, moving away from their ideal values. This shift was not random; it depended directly on the strength of the interactions. When the researchers increased the value of the interaction dial, the amount of deviation from the perfect symmetry grew larger. For the scenario where the neutrino masses are arranged in a normal order, with the lightest being the first, the calculations showed that the mixing angle for the atmospheric neutrinos shifted just enough to move slightly away from the perfect halfway point, suggesting a preference for one side of the spectrum over the other. This small shift was consistent with current experimental data, which shows the angle is close to, but not exactly, the perfect value.

In the alternative scenario, where the masses are arranged in an inverted order, the behavior was slightly different. Here, the radiative corrections pushed the mixing angle in the opposite direction, favoring the other side of the spectrum. This distinction is important because it offers a way to test which mass arrangement is correct in nature. The researchers found that for the inverted order, the predicted value for the CP phase, a measure of how the particles violate symmetry, landed remarkably close to the value currently favored by global experiments. Specifically, one of their cases predicted a value of approximately 270.7 degrees, which is very near the experimental best fit of 274 degrees. This suggests that the slight imperfections we see today could naturally arise from the evolution of a perfect symmetry in the early universe.

The study also looked at the total mass of the neutrinos and the effective mass that would be measured in experiments searching for a rare type of radioactive decay. In all their scenarios, the predicted total mass remained well below the strict upper limits set by cosmological observations, and the effective mass for the decay experiment stayed within the current experimental bounds. This consistency across multiple different measurements strengthens the case that the minimal seesaw framework, combined with the idea of a high-energy symmetry that breaks gradually, is a viable description of reality. The researchers concluded that the deviations from perfect symmetry are not a sign that the symmetry is wrong, but rather a signature of how the universe evolves. The magnitude of these deviations grows as the interaction strength increases, providing a clear, testable prediction for future experiments. By showing that a perfect symmetry at the highest energies can naturally evolve into the slightly imperfect pattern we observe today, the work offers a compelling bridge between the elegant theories of the early universe and the messy, precise data of the present day.

Technical Summary: Deviation from $\mu-\tau$ Reflection Symmetry under Radiative Corrections in the Minimal Seesaw Framework

Problem Statement and Motivation
Neutrino oscillation experiments have established that neutrinos possess mass and that flavor eigenstates are mixtures of mass eigenstates. While parameters such as the solar mixing angle ( $\theta_{12}$ ), reactor mixing angle ( $\theta_{13}$ ), and mass-squared differences ( $\Delta m^2_{21}, |\Delta m^2_{31}|$ ) are measured with high precision, the atmospheric mixing angle ( $\theta_{23}$ ) and the Dirac CP phase ( $\delta$ ) remain subjects of investigation. Recent global analyses suggest that $\theta_{23}$ is close to maximal ( $45^\circ$ ) and $\delta$ is near $270^\circ$ (specifically $274^\circ$ for Inverted Ordering), hinting at an underlying symmetry. The $\mu-\tau$ reflection symmetry is a theoretical framework that naturally predicts maximal atmospheric mixing ( $\theta_{23} = \pi/4$ ) and maximal CP violation ( $\delta = \pi/2$ or $3\pi/2$ ). However, since experimental data indicates small but significant deviations from these exact values, the symmetry is likely approximate. This work investigates the breaking of $\mu-\tau$ reflection symmetry induced by radiative corrections within the minimal seesaw framework, assuming the symmetry is exact at a high-energy seesaw scale ( $\Lambda_{\mu\tau} = 10^{14}$ GeV) and evolves to the electroweak scale ( $\Lambda_{EW} = 172.76$ GeV).

Methodology
The authors employ the minimal seesaw framework, which introduces only two right-handed neutrino fields, resulting in one massless light neutrino and a single physical Majorana CP phase. The analysis is conducted within the Minimal Supersymmetric Standard Model (MSSM) with a SUSY-breaking scale ( $\Lambda_S$ ) of 1 TeV.

Analytical Derivation: Starting from the one-loop Renormalization Group Equation (RGE) for the effective Majorana neutrino mass matrix $M_\nu$ , the authors utilize the integral solution method. They derive analytical expressions relating the low-energy parameters (masses, mixing angles, and CP phases) to their high-energy counterparts. The symmetry breaking is parameterized by the factor $\epsilon$ , arising from the running of the $\tau$ -lepton Yukawa coupling, and the overall scaling factor $I_\alpha$ .
Symmetry Implementation: The $\mu-\tau$ reflection symmetry is imposed at the high scale $\Lambda_{\mu\tau}$ . This fixes the high-energy values of $\theta_{23}$ to $\pi/4$ and $\delta$ to $\pi/2$ (Case-I) or $3\pi/2$ (Case-II). The Majorana and unphysical phases are assigned specific values to maintain the Harrison-Scott form of the mixing matrix.
Numerical Analysis: The derived analytical expressions are used to numerically estimate low-energy parameters at $\Lambda_{EW}$ . The analysis is performed for both Normal Ordering (NO, $m_1=0$ ) and Inverted Ordering (IO, $m_3=0$ ) scenarios. The study considers three representative values of $\tan\beta$ (10, 30, and 50) to investigate the dependence of radiative corrections on this parameter. High-energy input parameters (masses and mixing angles not fixed by symmetry) are constrained by correlation plots to ensure low-energy predictions fall within current experimental $3\sigma$ ranges.

Key Contributions

Analytical Expressions: The paper derives explicit analytical formulas for all low-energy neutrino parameters (masses, mixing angles, Dirac and Majorana CP phases, Jarlskog invariant, and effective Majorana mass) in terms of high-energy parameters for both NO and IO scenarios within the minimal seesaw framework.
Radiative Breaking Mechanism: It quantifies the deviation of low-energy parameters from their high-energy symmetric values specifically due to one-loop radiative corrections in the MSSM, distinguishing this from scenarios where the SUSY-breaking scale is taken at the electroweak scale.
Dependence on $\tan\beta$ : The study systematically investigates how the magnitude of symmetry breaking scales with $\tan\beta$ , a crucial parameter in the MSSM.

Results

Normal Ordering (NO): For NO, appropriate high-energy input parameters yield low-energy predictions consistent with current experimental data for all three values of $\tan\beta$ (10, 30, 50). The predicted $\theta_{23}$ shifts slightly from $45^\circ$ into the second octant ( $\theta_{23} > 45^\circ$ ) due to radiative corrections. The sum of neutrino masses remains below the cosmological bound ( $\sum m_i < 0.12$ eV), and the effective Majorana mass $|\langle m \rangle_{ee}|$ satisfies current limits.
Inverted Ordering (IO): For IO, the same high-energy inputs that work for $\tan\beta = 10$ fail to satisfy the mass-squared difference constraints for $\tan\beta = 30$ and $50$. However, by adjusting the high-energy mass inputs, the authors recover consistency with experimental data for all $\tan\beta$ values. A notable result is that radiative corrections in the IO scenario shift $\theta_{23}$ toward the first octant ( $\theta_{23} < 45^\circ$ ), opposite to the NO trend. Furthermore, for Case-II ( $\delta = 3\pi/2$ ), the radiatively corrected low-energy $\delta$ is found to be approximately $270.7^\circ$ (for $\tan\beta=50$ ), which is remarkably close to the global best-fit value of $274^\circ$ .
Deviation Trends: In both NO and IO scenarios, the magnitude of deviations ( $\Delta m_i$ , $\Delta \theta_{ij}$ , $\Delta \delta$ ) from high-energy values increases with increasing $\tan\beta$ . This is attributed to the increase in the radiative correction parameter $|\epsilon|$ as $\tan\beta$ grows.

Significance and Claims
The paper claims that the minimal seesaw framework, combined with $\mu-\tau$ reflection symmetry at a high scale and radiative corrections, provides a viable mechanism to explain the observed neutrino mixing pattern. Specifically, it demonstrates that:

The symmetry can be exact at the high scale and broken naturally by renormalization group running to match low-energy data.
The framework successfully accommodates current experimental constraints for both mass orderings across a range of $\tan\beta$ values, provided appropriate high-energy inputs are selected.
The specific prediction of $\delta \approx 270.7^\circ$ in the IO scenario (Case-II) offers a compelling theoretical explanation for the experimentally observed value of $\delta \approx 274^\circ$ .
The study highlights the distinct behavior of $\theta_{23}$ octants under radiative corrections for NO (shift to second octant) versus IO (shift to first octant), providing a potential discriminant between the two mass orderings.

The authors conclude that their analytical approach establishes a direct connection between high-energy symmetry parameters and low-energy observables, offering a robust testable framework for understanding lepton flavor mixing.

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