Quarks at the modular S 4 cusp

Abstract We analyse the possibility of describing quark masses, mixing and CP violation in S 4 ′ $$ {S}_4^{\prime } $$ modular flavour models without flavons. We focus on the case where the closeness of the modulus to the point of residual ℤ 3 ST $$ {\mathbb{Z}}_3^{ST} $$ symmetry (the cusp) plays a role in generating quark mass hierarchies and discuss the role modular form normalisations play in such constructions. We find that fitting quark data requires explicit CP breaking, unless a second modulus is introduced.