Complete operator basis for the modular invariant SMEFT

A bstract We implement modular flavor symmetries within the Standard Model Effective Field Theory (SMEFT) framework, using the flavor group A₄(q) A 4 q × A₄⁽ᵉ⁾ A 4 e with distinct moduli τ q and τ e , and assigning different modular weights to right-handed quarks using simplest weight assignment. By treating the moduli as non-dynamical spurions, adopting the MFV-like assumption, and neglecting effects associated with Im τ , we systematically construct a finite set of independent modular-invariant higher-dimensional operators via the Hilbert-series techniques. In the holomorphic A 4 scenario, where all modular forms derive from the weight-2 triplet Y₃⁽²⁾ Y 3 2 , we present two equivalent Hilbert-series bases. This establishes that higher-dimensional operators can be formally organized as [Yr^(kY),Y_r^^(kY^),O]₁ Y r k Y Y r ′ k Y ′ ∗ O 1 singlets. We subsequently enumerate all independent operators up to dimension 7 under this assumption and provide explicit constructions for all dimension-5 operators as well as baryon- and lepton-number conserving dimension-6 operators. Relaxing holomorphicity to the non-holomorphic case of polyharmonic Maaß forms, considering that non-holomorphic modular forms are not closed under multiplication, adopting the holomorphic organizing idea would generically lead to an infinite proliferation of modular-invariant structures. To retain a finite and complete operator basis, we therefore impose the same minimal formal organizing principle, which reproduces the benchmark Weinberg operator and the corresponding dimension-6 operators.