Abstract
A
bstract
We implement modular flavor symmetries within the Standard Model Effective Field Theory (SMEFT) framework, using the flavor group
$$ {A}_4^{(q)} $$
A
4
q
×
$$ {A}_4^{(e)} $$
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
$$ {Y}_{\mathbf{3}}^{(2)} $$
Y
3
2
, we present two equivalent Hilbert-series bases. This establishes that higher-dimensional operators can be formally organized as
$$ {\left[Y{\mathbf{r}}^{\left({k}_Y\right)},{Y}_{{\mathbf{r}}^{\prime}}^{{\left({k}_Y^{\prime}\right)}^{\ast }},\mathcal{O}\right]}_{\mathbf{1}} $$
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.