Quarks at the modular S4 cusp

A bstract We analyse the possibility of describing quark masses, mixing and CP violation in S₄^ S 4 ′ modular flavour models without flavons. We focus on the case where the closeness of the modulus to the point of residual Z₃^ST 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.