Complete Operator Basis for the modular invariant SMEFT

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 $τ_q$ 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 $\mathrm{Im},τ$, 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 $Y^{(2)}{\mathbf{3}}$, we present two equivalent Hilbert-series bases. This establishes that higher-dimensional operators can be formally organized as $[Y{\mathbf{r}}^{(k_Y)},{Y_{\mathbf{r}’}^{(k_Y’)}}^{*},\mathcal{O}]_{\mathbf{1}}$ 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.

💡 Research Summary

The authors present a systematic construction of a complete operator basis for the Standard Model Effective Field Theory (SMEFT) when the theory is endowed with modular flavor symmetries. They focus on the finite modular group Γ₃≃A₄ and consider two independent copies of the flavor group, A₄^{(q)} for quarks and A₄^{(e)} for leptons, each associated with its own modulus τ_q and τ_e. By treating the moduli as non‑dynamical spurions (i.e. background parameters) and neglecting any explicit dependence on Im τ, the analysis mimics a Minimal Flavor Violation (MFV) scenario where the only sources of flavor breaking are the modular forms themselves.

The key building block is the weight‑2 triplet Y^{(2)}_{\mathbf{3}}(τ), from which all higher‑weight modular forms can be generated through symmetric tensor products. Consequently, any higher‑dimensional SMEFT operator can be written schematically as a singlet under A₄: \