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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.23060 |
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| _version_ | 1866918315847319552 |
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| author | Kang, Luo-Jia Sun, Hao Yu, Jiang-Hao |
| author_facet | Kang, Luo-Jia Sun, Hao Yu, Jiang-Hao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_23060 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complete Operator Basis for the modular invariant SMEFT Kang, Luo-Jia Sun, Hao Yu, Jiang-Hao High Energy Physics - Phenomenology 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. |
| title | Complete Operator Basis for the modular invariant SMEFT |
| topic | High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2601.23060 |