Flavored QCD axion and Modular invariance

A four-dimensional effective model with $G_{\mathrm{S}\mathrm{M}}\times SL(2,\mathbb{Z})\times U(1{)}_X$ is proposed in string-derived supergravity framework, where $G_{\mathrm{S}\mathrm{M}}$ 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{ä}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 ${\delta }_X^{\mathrm{G}\mathrm{S}}$ is generically non-zero, vanishing $U(1{)}_X$ anomalies cause gauge boson decoupling and ${\delta }_X^{\mathrm{G}\mathrm{S}}\to 0$, yielding a massless global $U(1{)}_X$ without a Nambu-Goldstone mode. We show that the modulus vacuum expectation value stabilizes near $⟨\tau ⟩\approx i$, 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 $m_a\approx 0.9\times {10}^{-2}$ eV and photon coupling $\mathrm{\mid }{g}_{a\gamma \gamma }\mathrm{\mid }\approx 1.7\times {10}^{-13}{\mathrm{G}\mathrm{e}\mathrm{V}}^{-1}$, 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.