Complete Operator Basis for the modular invariant SMEFT

TL;DR

The work constructs a modular-flavor SMEFT based on the two-sector group $A_4^{(q)}\times A_4^{(e)}$, treating the moduli $\tau_q$ and $\tau_e$ as spurions under a MFV-like organizing principle. Using Hilbert-series techniques, the authors count and explicitly construct all independent modular-invariant operators up to dimension 7 in the holomorphic case, obtaining 6 dim-5, 2961 dim-6, and 360 dim-7 operators, with explicit dim-5 and dim-6 examples. They show two equivalent Hilbert-series bases and validate the operator structures via explicit tensor product constructions, confirming a finite and complete basis under holomorphic modular forms generated from the weight-2 triplet $Y^{(2)}_{\mathbf 3}$. They also extend the discussion to non-holomorphic modular forms (polyharmonic Maaß forms), arguing that without a minimal organizing principle the operator set would proliferate, and propose a constrained construction that preserves finiteness and reproduces the Weinberg operator and related dim-6 lepton-sector structures. The results provide a structured framework for systematic flavor-invariant SMEFT analyses with modular symmetry, linking modular form algebra to explicit effective operators and highlighting compatibility with a modular SUSY UV viewpoint.

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 $τ_q$ 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.

Figures (1)

• Figure 1: $U(3)^5$ is broken down to $U(1)^{11}$ by the $A_4$ modular symmetry.