Radiative correction on moduli stabilization in modular flavor symmetric models

We study the radiative corrections to the stabilization of the complex structure modulus τ in modular flavor symmetric models. We discuss the possibility of obtaining the vacuum expectation value of τ in the vicinity of the fixed point where residual symmetries remain unbroken. As concrete examples, we analyze the one-loop Coleman-Weinberg potential in the A4 modular flavor models. We show that the one-loop correction may lead to the slight deviation from the tree-level result, which may realize a phenomenologically preferred value of the complex structure modulus τ, particularly when the number of species contributing to the one-loop correction is large enough.