Flavored QCD axion and Modular invariance

A four-dimensional effective model with $G_{\rm SM}\times SL(2,\mathbb{Z}) \times U(1)_X$ is proposed in string-derived supergravity framework, where $G_{\rm SM}$ is the Standard Model (SM) gauge group and $U(1)_X$ is gauged. We show $SL(2,\mathbb{Z})$- and $U(1)_X$-mixed anomalies should vanish. Anomalies induced by K{\"a}hler transformations match those from gaugino chiral rotations. When SM fermions transform nontrivially under $SL(2,\mathbb{Z})$, and with vanishing gaugino contributions, the anomaly-free conditions are powerful enough to determine the quark and lepton flavor structures, set scales for $U(1)_X$ breaking, and ensure the strong CP phase remains unmodified. While the Green-Schwarz coefficient $\delta^{\rm GS}_X$ is generically non-zero, vanishing $U(1)_X$ anomalies cause gauge boson decoupling and $\delta^{\rm GS}_X\rightarrow 0$, yielding a massless global $U(1)_X$ without a Nambu-Goldstone mode. We show that the modulus vacuum expectation value stabilizes near $\langle\tau\rangle \approx i$, where exact $SL(2,\mathbb{Z})$ ($T$-duality) is spontaneously broken, removing residual modular symmetry. The framework predicts seesaw-generated neutrino masses and flavored axion properties, with all Yukawa coefficients constrained to unit-magnitude complex numbers. Our model reproduces current quark and lepton data, predicts an axion mass $m_a\approx0.9\times10^{-2}$ eV and photon coupling $|g_{a\gamma\gamma}|\approx1.7\times10^{-13}\,{\rm GeV}^{-1}$, and unlike the ordinary case, suppresses flavor-violating axion couplings to $s,d$ quarks and $\mu,e$ leptons to $\mathcal{O}(\lambda^4)$ (with $\lambda$ the Cabibbo angle). It also yields normal neutrino mass hierarchy consistent with oscillation data, $0\nu\beta\beta$-decay rate, and cosmological and astrophysical measurements.