Complete Operator Basis for the modular invariant SMEFT

Abstract

We implement modular flavor symmetries within the Standard Model Effective Field Theory (SMEFT) framework, using the flavor group $A_4^{(q)}\times A_4^{(e)}$ with distinct moduli $\ddot{I}{„}_q$ and $\ddot{I}{„}_e$, 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 $\mathrm{Im}\ddot{I}„$, 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 $Y_3^{(2)}$, we present two equivalent Hilbert-series bases. This establishes that higher-dimensional operators can be formally organized as $[Y_{\mathbf{r}}^{(k_Y)},{Y_{{\mathbf{r}}^{\prime }}^{(k_Y^{\prime })}}^{\ast },\mathcal{O}{]}_1$ 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 Maas 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.

82 pages, 10 tables