Flavored QCD axion and Modular invariance

TL;DR

The paper develops a 4D effective theory from string-derived supergravity with symmetry $G_{ m SM}\times SL(2,\mathbb{Z})\times U(1)_X$, using modular invariance to fix Yukawa textures as modular forms in the modulus $\tau$ and employing anomaly-cancellation (including a Green-Schwarz mechanism) to constrain the chiral spectrum. A modulus stabilization mechanism drives $\langle\tau\rangle$ near $i$, spontaneously breaking $SL(2,\mathbb{Z})$ and yielding a predictive, flavored $U(1)_X$ that acts as a PQ-like symmetry; after $U(1)_X$ breaking, a flavored QCD axion emerges with calculable mass and photon coupling, while flavor-violating axion couplings are suppressed to $O(\lambda^4)$. The quark/lepton mass matrices and mixings arise from modular-invariant Yukawas with unit-magnitude complex coefficients, yielding CKM/PMNS structures and a seesaw-generated neutrino spectrum consistent with data, as well as a predicted axion with $m_a\approx 9.12\times10^{-3}$ eV and $|g_{a\gamma\gamma}|\approx 1.69\times10^{-13}$ GeV$^{-1}$ for $f_A=4\times10^{10}$ GeV. The framework links the PQ scale to the flavor sector and neutrino seesaw scale, providing a cohesive picture of flavor, CP, and axion phenomenology within a modular-invariant string-derived setting.

Abstract

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{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 $δ^{\rm GS}_X$ is generically non-zero, vanishing $U(1)_X$ anomalies cause gauge boson decoupling and $δ^{\rm GS}_X\rightarrow 0$, yielding a massless global $U(1)_X$ without a Nambu-Goldstone mode. We show that the modulus vacuum expectation value stabilizes near $\langleτ\rangle \approx i$, 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 $|g_{aγγ}|\approx1.7\times10^{-13}\,{\rm GeV}^{-1}$, and unlike the ordinary case, suppresses flavor-violating axion couplings to $s,d$ quarks and $μ,e$ leptons to $\mathcal{O}(λ^4)$ (with $λ$ the Cabibbo angle). It also yields normal neutrino mass hierarchy consistent with oscillation data, $0νββ$-decay rate, and cosmological and astrophysical measurements.

Figures (4)

• Figure 1: Contour plot of $\partial V^{(2)}_F/\partial\tau|_{\tau_0,s_0,\sigma_0}=0$, where $\tau=z_1+i z_2$. Blue curve is for $\partial V^{(2)}_F/\partial z_2=0$ and red line for $\partial V^{(2)}_F/\partial z_1=0$, where $s_0=1$ is used for simplicity.
• Figure 2: Plot of $|g_{a\gamma\gamma}|$ versus $m_a$ for Kim-Shifman-Vainshtein-Zakharov (KSVZ) KSVZ (black dashed line), Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) DFSZ (blue dotted line), and the model (red solid line) in terms of $E/\delta^G_X=0,8/3$, and $386/201$, respectively. With $f_A=4\times10^{10}$ GeV of Eq.(\ref{['sss']}) and $z=0.47$ our model predicts $m_a=9.12\times10^{-3}$ eV and $|g_{a\gamma\gamma}|=1.69\times10^{-13}\,{\rm GeV}^{-1}$ (black point).
• Figure 3: Plots for $0\nu\beta\beta$-decay rate as a function of the neutrino masses $m_{\nu_i}$ (left) and leptonic Dirac CP phase $\delta_{CP}$ as a function of the atmospheric mixing angle $\theta_{23}$ (right). Vertical and horizontal dashed lines represent the $1\sigma$ bounds for $\theta_{23}$ and $\delta_{CP}$, respectively, in Table-\ref{['exp_nu']}.
• Figure 4: Plots for $0\nu\beta\beta$-decay rate as a function of leptonic Dirac CP phase $\delta_{CP}$ (left) and $\sum m_{\nu}=m_{\nu_1}+m_{\nu_2}+m_{\nu_3}$ as a function of $m_{\rm lightest}$ (right). Vertical dashed lines indicate the $1\sigma$ bound for $\delta_{CP}$ listed in Table-\ref{['exp_nu']}.