Eur. Phys. J. C (2023) 83: 579
https://doi.org/10.1140/epjc/s10052-023-11727-0

Regular Article - Theoretical Physics

$A_4$ modular flavour model of quark mass hierarchies close to the fixed point $\tau =\omega$

¹ SISSA/INFN, Via Bonomea 265, 34136, Trieste, Italy
² Kavli IPMU (WPI), UTIAS, The University of Tokyo, 277-8583, Kashiwa, Chiba, Japan
³ Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784, Sofia, Bulgaria
⁴ Department of Physics, Niigata University, Ikarashi 2, 950-2181, Niigata, Japan

^b morimitsutanimoto@yahoo.co.jp

Received: 6 May 2023
Accepted: 19 June 2023
Published online: 7 July 2023

Abstract

We investigate the possibility to describe the quark mass hierarchies as well as the CKM quark mixing matrix without fine-tuning in a quark flavour model with modular $A_4$ symmetry. The quark mass hierarchies are considered in the vicinity of the fixed point $\tau =\omega \equiv exp(i2\pi /3)$ (the left cusp of the fundamental domain of the modular group), $\tau$ being the VEV of the modulus. The model involves modular forms of level 3 and weights 6, 4 and 2, and contains eight constants, only two of which, $g_u$ and $g_d$, can be a source of CP violation in addition to the VEV of the modulus, $\tau =\omega +ϵ$, ${(ϵ)}^{\ast }\ne ϵ$, $|ϵ|\ll 1$. We find that in the case of real (CP-conserving) $g_u$ and $g_d$ and common $\tau$ ($ϵ$) in the down-type and up-type quark sectors, the down-type quark mass hierarchies can be reproduced without fine tuning with $|ϵ|\cong 0.03$, all other constants being of the same order in magnitude, and correspond approximately to ${1:|ϵ|:|ϵ|}^2$. The up-type quark mass hierarchies can be achieved with the same $|ϵ|\cong 0.03$ but allowing $g_u\sim \mathcal{O}(10)$ and correspond to $1:|ϵ|/|g_u{|:|ϵ|}^2/{|g_u|}^2$. In this setting the reproduction of the value of the CKM element $|V_{\text{cb}}|$ is problematic. A much more severe problem is the correct description of the CP violation in the quark sector since it arises as a higher order correction in $ϵ$ with $|ϵ|\ll 1$. We show that this problem is somewhat alleviated in the case of complex $g_u$ and $g_d$ although the rephasing invariant $J_{\text{CP}}$ is larger by a factor of $\sim 1.8$ than the correct value. A correct no-fine-tuned description of the quark mass hierarchies, the quark mixing and CP violation is possible with all constants being of the same order in magnitude and complex $g_u$ and $g_d$, if one allows different values of $ϵ$ in the down-type and up-type quark sectors, or in a modification of the considered model which involves modular forms of level 3 and weights 8, 6 and 4.