Modulus stabilization of modular flavor models in Jordan frame supergravity

TL;DR

The paper investigates modulus stabilization within modular flavor models by introducing a non-minimal scalar-curvature coupling ${\Phi(\tau,\bar{\tau})R}$ in Jordan-frame supergravity. Modular invariance tightly constrains the frame function, linking it to the Kähler potential through ${\Phi = -3 e^{-K/3}}$, and the Einstein-frame potential ${V_E}$ is reshaped by the Jordan-to-Einstein scale transformation. The analysis shows that ${V_E}$ can possess a stationary point at ${\tau = i\infty}$, yielding a runaway-type vacuum for certain parameter choices, while finite fixed points ${\tau = i, \omega}$ remain stationary due to CP and modular symmetries. Numerical studies in simplified setups reveal that the infinite boundary can be the global minimum (or a metastable minimum) depending on parameters like ${m,n}$ and ${\xi}$, with CP-breaking vacua and shifted minima arising from nontrivial holomorphic data in the modular forms. Overall, curvature–modulus interactions provide a natural mechanism to reshape the modulus potential, offering new avenues for flavor phenomenology and potential cosmological implications.

Abstract

We propose to discuss the modular flavor model and the stabilization of single modulus field in the Jordan frame supergravity with non-minimal scalar-curvature coupling of the form $Φ(τ,\barτ)R$. Modular invariance and positivity of the scale factor constrain stringently the form of the frame function, consequently the Kahler potential by the relation $Φ(τ,\barτ)=-3\exp[-K(τ,\barτ)/3]$. We discuss some general properties of scalar potentials after the scale transformation from the Jordan frame to the Einstein frame. We find that the shape of the resulting scalar potential in the Einstein frame is quite different from that of ordinary single modulus stabilization mechanism. The scalar potential could be stationary at the $i\infty$ fixed point, leading to a runaway type vacuum. We also discuss numerically the modulus stabilization for some simplified scenarios.