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Leptonic Flavor from Modular $A_4$: UV Mediators and SMEFT Realizations

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Pith's one-line read Modular A4 flavor symmetry reduces the space of heavy lepton-flavor mediators to a finite, fully classifiable list, with TeV-to-hundred-TeV mass bounds.

desk verdict A useful but not fully checkable modular-A4/UV-mediator catalogue: the tree-level classification is solid and the phenomenology is careful, but the invariant counts that every bound inherits are asserted rather than proven. read the letter →

arxiv 2505.01535 v1 pith:EZBJIQXF submitted 2025-05-02 hep-ph

Adrián Moreno-Sánchez , Ajdin Palavrić This is my paper · ORCID
classification hep-ph
keywords modularA4SMEFTleptonflavorviolationUVmediatorsdiscretesymmetryformsmu-to-econversionWilsoncoefficients
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that imposing modular A4 flavor symmetry on the heavy fields that generate dimension-six lepton operators in the Standard Model Effective Field Theory is a sharp, predictive constraint. Working with 13 standard UV mediators (four scalars, six fermions, three vectors), it classifies, for each mediator, which A4 representations and modular weights are allowed when the UV interaction contains at most one insertion of the lowest-weight A4 modular form. From that classification it derives matching to lepton SMEFT operators and computes bounds on mediator masses from low-energy lepton-flavor-conserving data and from charged-lepton flavor-violating decays and conversions, including one-loop matching for scalars and fermions and leading-log RGE for vectors. A sympathetic reader would care because the result turns an otherwise arbitrary list of new-physics models into a finite, experimentally testable menu: for every allowed irrep there is a concrete lower bound on the mass scale, and some loop-induced bounds reach well beyond 100 TeV.

What carries the argument

The central object is the modular A4 flavor group, the rotation symmetry of a tetrahedron, together with its weight-2 triplet modular form $Y_3^{(2)}(\tau)$, whose components are explicit q-series, and the A4 tensor decomposition $3\otimes 3 = 1\oplus 1'\oplus 1''\oplus 3_S\oplus 3_A$. The field assignment in Eq. (17) fixes how SM leptons transform, and the requirement that UV interaction terms be A4 invariants with at most one $Y_3^{(2)}$ insertion selects the allowed mediator irreps. The modular weight then acts as an extra label distinguishing otherwise similar A4 representations, and the same machinery produces both the flavor tensors used in matching and the counting of independent parameters.

What would settle it

An independent, exhaustive enumeration of A4-invariant contractions involving the SM lepton fields, one mediator, and at most one insertion of $Y_3^{{(2)}}$ that turns up an invariant not listed in Tables X-XII would falsify the completeness claim; repeating the classification with the modulus set to τ = ω and finding a different set of allowed irreps would show the bounds are benchmark-dependent rather than intrinsic.

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Extended reading notes

Core claim

Within the modular A4 framework, with lepton doublets assigned as a weight-2 A4 triplet and right-handed charged leptons as three distinct weight-0 singlets, and with the modulus fixed to τ = i, the paper claims that the set of renormalizable couplings between SM leptons and the 13 mediators is completely classifiable at O(1) and O($Y_3^{{(2)}}$). The classification is organized by A4 tensor contractions and modular weight: each mediator irrep carries a label (A4 irrep, modular weight k), and the number of independent flavor parameters follows from the number of linearly independent invariants. After tree-level matching, one modular-form insertion in the UV becomes two in the Wilson coefficients; one-loop matching for scalar and fermion mediators and leading-log RGE for vectors modify the coefficients and, in many cases, provide the strongest constraints. Tables IV-IX list the resulting lower bounds on the mediator mass for every allowed irrep, with cLFV processes such as μ→eγ, μ→e conversion, and μ→eee often dominating. The paper presents this not as a single complete model but as a benchmark classification that maps the allowed parameter space of modular A4 UV completions.

Load-bearing premise

Everything depends on the benchmark choice that lepton doublets form a weight-2 A4 triplet, right-handed charged leptons form three distinct weight-0 singlets, and the symmetry-breaking complex modulus τ is fixed to i; change any of these and the list of allowed mediator irreps and every quoted mass bound shifts.

Editorial extensions

If this is right

  • Every allowed A4 irrep of a scalar, fermion, or vector mediator carries a concrete lower bound on the mass-to-coupling ratio; for the vector triplet at O(Y_3^{(2)}), RGE-enhanced μ→e conversion pushes the bound above 120 TeV.
  • Loop effects are not subdominant: for many irreps one-loop matching (scalars and fermions) or leading-log RGE (vectors) yields stronger constraints than tree level, and some observables such as μ→eγ arise only at one loop.
  • The classification gives a finite checklist for model building: a candidate UV completion based on modular A4 must place its mediators on Tables X-XII; irreps not listed are forbidden at the stated order.
  • Charged-lepton-flavor-violating transitions are generated at the same flavor power counting as flavor-conserving ones because the discrete symmetry does not suppress them, making μ→eee, μ→e conversion, and τ→ℓℓℓ prime probes.
  • Two-parameter flavor tensors admit controlled two-dimensional scans (Figs. 1-3), and multi-parameter cases can be profiled, giving a quantitative route to distinguish irreps experimentally.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The choice τ = i is a benchmark, not a prediction; redoing the classification at τ = ω or near i∞ would likely change which irreps are allowed and could move bounds by order-one factors, so the tables should be read as scenario-specific rather than universal.
  • Because one-loop matching breaks tree-level relations such as C_ϕℓ^{(1)} = -C_ϕℓ^{(3)} for the N mediator, radiative effects open flavor-violating channels that tree-level power counting would miss; the same mechanism could operate in other modular flavor setups, including those with more than one modular insertion.
  • The same A4 contraction machinery could be carried to dimension-8 operators or to quark-sector flavor structures, where modular weights and higher-weight modular forms would serve as additional spurions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This article extends the discrete-flavor SMEFT mediator classification of Ref. [109] to modular A4. It assigns the SM lepton fields as in Eq. (17), introduces the 13 scalar, fermionic, and vector mediators of Table I, and constructs A4 flavor invariants at O(1) and at one insertion of the lowest-weight modular form Y_3^(2). The output is a set of allowed irreps with parameter counts (Tabs. X-XII), tree-level SMEFT matching relations (Table I), one-loop matching for scalars and fermions, and leading-log RGE effects for vectors. The resulting Wilson coefficients are confronted with low-energy lepton-flavor-conserving fits and with |ΔL|=1 and |ΔL|=2 cLFV bounds, yielding mass-scale constraints in Tabs. IV-IX and two-parameter plots in Figs. 1-3. The central claim is that this classification is systematic and complete, and that the accompanying bounds form a useful benchmark for modular A4 model building.

Significance. If the classification is correct, the paper provides a useful atlas of modular A4 mediator irreps and a transparent benchmark map of current experimental constraints; this is a worthwhile extension of Ref. [109]. Strengths include the detailed invariant tables, the use of modern experimental limits, the explicit cross-check of one-loop matching with Matchete and Ref. [149], and the absence of circularity: no model parameters are fitted to the observables used to constrain them. The main reservation is that the completeness and linear independence of the invariant enumerations, on which every bound depends, are asserted rather than demonstrated. This is a correctness risk, not an internal inconsistency.

major comments (3)
  1. [Sec. III, Tabs. X-XII] The parameter counts in Tabs. X-XII are load-bearing, since they feed every low-energy, cLFV, one-loop, and RGE bound in Tabs. IV-IX, but the paper asserts rather than demonstrates that the listed invariants are exhaustive and linearly independent. This matters especially for high-multiplicity entries such as the B triplet at O(Y_3^(2)) with 16 invariants, the W triplet with 7, and the φ, S2, and L3 triplets with 6; A4 tensor identities can reduce the rank of the invariant basis. The one-loop matching cross-check with Matchete and Ref. [149] does not cover this enumeration. Please include either an explicit linear-algebra verification of the ranks, or a reproducible enumeration artifact (e.g., a small code or supplementary table) that establishes both exhaustiveness and independence.
  2. [Sec. IV.B.1, IV.B.2, Tabs. V and VII] The one-loop matching contributions for scalar and fermionic mediators are not displayed anywhere; the text only reports that they were obtained with Matchete and cross-checked against Ref. [149]. Since several of the strongest bounds, such as φ (3,-2) at 61.9 TeV from μ→eγ and Ξ1 (3,2) at 38.9 TeV, rest on these contributions, and since the paper emphasizes that one-loop effects can dominate, the relevant formulas should be collected in an appendix or the supplemental material so the results can be checked and reproduced. Without them, the central claim that loop effects become leading in a significant number of cases cannot be verified from the manuscript as written.
  3. [Sec. IV.B, Tabs. IV-IX] The quoted bounds for multi-parameter irreps assume all independent couplings are set to ~1, as stated in Sec. IV.B. For tensors with up to 16 independent parameters, this is a single, non-generic point in a high-dimensional space; the paper does not quantify how the bounds shift under order-one variations of the couplings or when accidental cancellations are allowed. The conclusion does call the study a phenomenological benchmark, but the table captions and abstract should make this explicit so the numbers in Tabs. IV-IX are not read as robust exclusions. A short sensitivity statement for at least one multi-parameter irrep, e.g., the B (3,2) irrep, would greatly strengthen the presentation.
minor comments (5)
  1. [Table IV caption] The caption refers to the 'last two columns' for |ΔLα|=2, but the table has a single |ΔLα|=2 column; the column description should be corrected.
  2. [Eq. (16), Sec. II.B] The normalization of the modular forms in Eq. (16) should be stated explicitly, since the numerical values of Y_i(τ) at τ=i enter the Wilson coefficients and hence all quoted bounds; the q-expansions are shown, but the overall normalization convention is not defined.
  3. [Tabs. IV-IX] The mass bounds are quoted to two decimal places despite the unit-coupling benchmark and leading-log approximations; rounding to one or two significant digits would better reflect the precision of the assumptions.
  4. [Table IX caption] Please state explicitly in the caption of Tab. IX that the vector RGE treatment is a leading-log estimate and does not include full one-loop matching; the main text says this, but the table caption does not.
  5. [Sec. II.B, Eq. (17)] The choice of the conventional A4 field assignment and of τ=i is an assumption; a brief caveat in the table captions or in the introduction to Sec. IV would help prevent readers from treating the bounds as independent of these choices.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's mass bounds are direct translations of external experimental limits under stated modular-A4 assumptions, and no predicted quantity is fitted from the data it constrains.

full rationale

The derivation chain is self-contained against external benchmarks. The modular A4 field assignment in Eq. (17) and the benchmark tau = i are stated inputs, not outputs of the analysis; the paper does not fit these choices to the cLFV or low-energy observables used to derive the mass bounds. The flavor-invariant enumeration in Sec. III and Tables X-XII is constructed directly from the A4 tensor-product decompositions in Eqs. (6)-(10) and the modular forms in Eq. (15). No equation in the paper reduces a 'predicted' bound to a fitted parameter: the bounds in Tables IV-IX are obtained by matching the resulting Wilson coefficients to external experimental limits from Tables II and III and to published low-energy fits [118-120], with couplings set to unity as an explicit benchmark in Sec. IV.B. The one-loop matching was computed with Matchete and cross-checked against Ref. [149], providing an independent check of the Wilson-coefficient step. The same-author citations ([25,109]) supply context and the exact-A4 predecessor, but the modular extension and the invariant counts rest on the explicit group-theoretic rules and modular-form expansions reproduced in the paper, so no load-bearing claim reduces to a self-citation. The skeptical concern that completeness of the A4 invariant enumeration is asserted rather than machine-verified is a correctness and verification risk, not a circularity: an omission or miscount would change the input parameter counts, but those counts are not themselves derived from the observables they constrain. The paper's own concluding caveat that the bounds are a simplified phenomenological benchmark reinforces that the mass limits are conditional translations of experimental data, not predictions that presuppose the conclusions.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The classification rests on the standard modular A4 assignment of lepton fields, the restriction to one modular form insertion, the benchmark tau=i, and the simplification of unit couplings. These are all stated assumptions, not outputs derived from the low-energy data.

free parameters (2)
  • Modulus tau = i (fixed)
    Chosen at the self-dual point tau=i with residual Z2 symmetry (Sec. II.B). It is not fitted to data; the modular form values and all flavor tensors depend on this choice.
  • Mediator couplings = 1 (benchmark for all independent couplings)
    For irreps with more than one independent flavor invariant, all couplings are set to ~1 to extract mass bounds (Sec. IV.B). Bounds scale inversely with the actual coupling sizes.
assumptions (6)
  • domain assumption SM lepton doublets transform as an A4 triplet of modular weight 2; right-handed charged leptons are distinct A4 singlets of weight 0.
    Adopted in Eq. (17) and standard in modular A4 model building; fixes the allowed flavor invariants and all derived bounds.
  • ad hoc to paper UV interactions contain at most one insertion of the lowest-weight modular form Y3^(2), so tree-level SMEFT Wilson coefficients carry up to two modular form insertions.
    Restriction stated in Sec. I and III to make the classification finite; higher-order insertions are deferred.
  • domain assumption The analysis is performed below the SUSY-breaking scale, so the non-supersymmetric SMEFT is the correct low-energy description.
    Stated in Sec. I; the modular structure is assumed to originate from a supersymmetric UV theory.
  • domain assumption The modulus tau is fixed to the self-dual point i with residual Z2 symmetry.
    Chosen as an enhanced-symmetry benchmark in Sec. II.B; small deviations are discussed only qualitatively in footnote 10.
  • ad hoc to paper For flavor tensors with several independent invariants, all couplings are set to ~1, giving benchmark bounds rather than profiled fits.
    Stated in Sec. IV.B; this simplifies the mass-bound extraction and is acknowledged as a limitation in the Conclusion.
  • standard math The dimension-6 SMEFT operator basis (Ref. [2]) and the complete tree-level matching dictionary for general mediators (Ref. [24]) are correct.
    Background for the matching; these are standard, widely used results.

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Cite this review

Pith. "Pith review of Leptonic Flavor from Modular $A_4$: UV Mediators and SMEFT Realizations." pith.science (2026). https://pith.science/paper/EZBJIQXF

@misc{pith2026250501535,
  author       = {Pith},
  title        = {Pith review of: Leptonic Flavor from Modular $A_4$: UV Mediators and SMEFT Realizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EZBJIQXF}},
  note         = {Machine review of arXiv:2505.01535}
}
abstract

The absence of direct evidence for new physics at current collider energies motivates the study of indirect effects within the framework of the Standard Model Effective Field Theory (SMEFT). In this work, we investigate the role of the $A_4$ discrete flavor symmetry in constraining the UV dynamics, giving rise to dimension-6 SMEFT operators involving leptons. We consider renormalizable interactions between SM fields and heavy mediators, classified according to their $A_4$ transformation properties in the modular framework, where Yukawa structures are encoded in modular forms. Restricting our analysis to UV interactions with at most a single modular form insertion, we study the resulting classification of $A_4$ UV mediator irreps and explore their implications across a range of observables, including both lepton-flavor-conserving and lepton-flavor-violating processes. A detailed phenomenological analysis is carried out, incorporating both tree-level and one-loop matching contributions for scalar and fermionic mediators, as well as leading-log RGE effects in case of vectors.

Figures

Figures reproduced from arXiv: 2505.01535 by the authors.

Figure 1
Figure 1. FIG. 1. Complementary overview of the constraints on three scalar irreps leading to two independent parameters in the [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Overview of the complementary constraints on four [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Overview of the combined constraints on the vector irreps whose flavor tensors contain two independent parameters. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Two-dimensional profiled likelihood contours for the [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Profiled two-dimensional likelihoods for all [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Two-dimensional profiled likelihood contours for the [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Two-dimensional profiled likelihood contours in case of [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Two-dimensional profiled likelihood contours in case of [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Two-dimensional profiled likelihood contours in case of [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]

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Reference graph

Works this paper leans on

157 extracted references · 20 canonical work pages · cited by 1 Pith paper

  1. [149]

    Davidson, Completeness and complementarity for µ → eγµ → e¯ee and µA → eA, JHEP 02, 172, arXiv:2010.00317 [hep-ph]

    S. Davidson, Completeness and complementarity for µ → eγµ → e¯ee and µA → eA, JHEP 02, 172, arXiv:2010.00317 [hep-ph]

    arXiv 2010

  2. [109]

    G.-J. Ding, S. F. King, X.-G. Liu, and J.-N. Lu, Mod- ular S4 and A4 symmetries and their fixed points: new predictive examples of lepton mixing, JHEP 12, 030, arXiv:1910.03460 [hep-ph]

    arXiv 1910

  3. [1]

    I, three of them, more specifically S1, S2 and Ξ 1, cou- ple to the SM fermion bilinears of the form ¯fcf, where f∈{ ℓ,e} represents the lepton fields

    Scalars Among the four scalar mediators included in Tab. I, three of them, more specifically S1, S2 and Ξ 1, cou- ple to the SM fermion bilinears of the form ¯fcf, where f∈{ ℓ,e} represents the lepton fields. For these media- tors, the coupling structure of this type imposes signifi- cant constraints on the allowed flavor tensor structures, which can be d...

  4. [2]

    The classification of the A4 irreps along with the extraction of the flavor tensors proceeds analogously to the scalar cases

    Fermions We now proceed with the analysis of the fermionic me- diators. The classification of the A4 irreps along with the extraction of the flavor tensors proceeds analogously to the scalar cases. In all cases presented in Tab. I, the NP fermion couples to a single SM lepton, through the bilin- ear of the form ¯F Φ(†)f, where f∈{ℓ,e} and Φ∈{ϕ, ˜ϕ}, which...

  5. [3]

    Vectors Finally, we examine the three vector mediators enu- merated in Tab. I. Two of them ( B andW) couple to the vector bilinears of the form ¯fγµf, where f∈{ ℓ,e}, resulting in a rich structure once A4 symmetry is im- posed, with numerous independent A4 invariants per- mitted. In contrast, L3 mediator couples to the ¯ecγµℓ bilinear, mirroring the coupl...

    2016

  6. [4]

    IV and V, respectively

    Scalars Our analysis starts with the UV scalar mediators, whose corresponding tree-level and one-loop constraints are summarized in Tabs. IV and V, respectively. In the following discussion, we analyze each scalar mediator in- dividually. As previously noted, for those A4 irreps per- mitting multiple independent flavor invariants, for the corresponding co...

  7. [5]

    Fermions The next part of our phenomenological analysis is ded- icated to fermionic UV mediators. Tab. VI summarizes the tree-level constraints for the various fermionic A4 ir- reps. As indicated in Tab. VI, all fermionic irreps are subject to bounds from the low-energy combined fit. The low-energy constraints lead to bounds ranging from about 3 TeV to 10...

  8. [6]

    Flavor Physics at the High Energy Frontier

    Vectors In the final stage of our analysis, we focus on the phe- nomenological implications of vector mediators. As out- lined at the beginning of Sec. IV, in line with the bottom- up approach, our EFT framework remains agnostic to the details of UV completions and the mechanisms of spontaneous symmetry breaking associated with vector mediators. Therefore...

Show all 157 references
  1. [7]

    Buchmuller and D

    W. Buchmuller and D. Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation, Nucl. Phys. B 268, 621 (1986)

    1986

  2. [8]

    Grzadkowski, M

    B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 10, 085, arXiv:1008.4884 [hep-ph]

    arXiv

  3. [9]

    Brivio and M

    I. Brivio and M. Trott, The Standard Model as an Effective Field Theory, Phys. Rept. 793, 1 (2019), arXiv:1706.08945 [hep-ph]

    2019 arXiv

  4. [10]

    Isidori, F

    G. Isidori, F. Wilsch, and D. Wyler, The Stan- dard Model effective field theory at work (2023), arXiv:2303.16922 [hep-ph]

    2023 arXiv

  5. [11]

    G. F. Giudice, C. Grojean, A. Pomarol, and R. Rattazzi, The Strongly-Interacting Light Higgs, JHEP 06, 045, arXiv:hep-ph/0703164

    arXiv

  6. [12]

    Henning, X

    B. Henning, X. Lu, and H. Murayama, How to use the Standard Model effective field theory, JHEP 01, 023, arXiv:1412.1837 [hep-ph]

    arXiv

  7. [13]

    Henning, X

    B. Henning, X. Lu, T. Melia, and H. Murayama, 2, 84, 30, 993, 560, 15456, 11962, 261485, ...: Higher dimen- sion operators in the SM EFT, JHEP08, 016, [Erratum: JHEP 09, 019 (2019)], arXiv:1512.03433 [hep-ph]

    2019 arXiv

  8. [14]

    R. M. Fonseca, Enumerating the operators of an ef- fective field theory, Phys. Rev. D 101, 035040 (2020), arXiv:1907.12584 [hep-ph]

    2020 arXiv

  9. [15]

    Lehman, Extending the Standard Model Effective Field Theory with the Complete Set of Dimension- 7 Operators, Phys

    L. Lehman, Extending the Standard Model Effective Field Theory with the Complete Set of Dimension- 7 Operators, Phys. Rev. D 90, 125023 (2014), arXiv:1410.4193 [hep-ph]

    2014 arXiv

  10. [16]

    X.-X. Li, Z. Ren, and J.-H. Yub, Complete tree-level dictionary between simplified BSM models and SMEFT d≤7 operators, Phys. Rev. D 109, 095041 (2024), arXiv:2307.10380 [hep-ph]

    2024 arXiv

  11. [17]

    Ren and J.-H

    Z. Ren and J.-H. Yu, A complete set of the dimension-8 Green’s basis operators in the Standard Model effective field theory, JHEP 02, 134, arXiv:2211.01420 [hep-ph]

    arXiv

  12. [18]

    H.-L. Li, Z. Ren, M.-L. Xiao, J.-H. Yu, and Y.-H. Zheng, Complete set of dimension-nine operators in the stan- 15 UV Field (A4,k) Order Invariants # of parameters S1∼ (1,1)1 (3,4) O(1) y(1) S1 [S† 1]3[¯ℓℓc]3A 1 (1,2) O(¯Y (2) 3 ) y(1) S1 [S† 1]1[[¯ℓℓc]3A ¯Y (2) 3 (τ)]1 1 (1′,2...

    2021 arXiv

  13. [19]

    H.-L. Li, Z. Ren, J. Shu, M.-L. Xiao, J.-H. Yu, and Y.- H. Zheng, Complete set of dimension-eight operators in the standard model effective field theory, Phys. Rev. D 104, 015026 (2021), arXiv:2005.00008 [hep-ph]

    2021 arXiv

  14. [20]

    C. W. Murphy, Dimension-8 operators in the Stan- dard Model Effective Field Theory, JHEP 10, 174, arXiv:2005.00059 [hep-ph]

    2005 arXiv

  15. [21]

    Liao and X.-D

    Y. Liao and X.-D. Ma, Renormalization Group Evolu- tion of Dimension-seven Baryon- and Lepton-number- violating Operators, JHEP 11, 043, arXiv:1607.07309 [hep-ph]

    arXiv

  16. [22]

    Liao and X.-D

    Y. Liao and X.-D. Ma, An explicit construction of the dimension-9 operator basis in the standard model effec- tive field theory, JHEP 11, 152, arXiv:2007.08125 [hep- ph]

    2007 arXiv

  17. [23]

    R. V. Harlander, T. Kempkens, and M. C. Schaaf, Stan- dard model effective field theory up to mass dimension 12, Phys. Rev. D 108, 055020 (2023), arXiv:2305.06832 [hep-ph]

    2023 arXiv

  18. [24]

    R. V. Harlander and M. C. Schaaf, AutoEFT: Au- tomated operator construction for effective field the- ories, Comput. Phys. Commun. 300, 109198 (2024), arXiv:2309.15783 [hep-ph]

    2024 arXiv

  19. [25]

    R. K. Ellis et al., Physics Briefing Book: Input for the European Strategy for Particle Physics Update 2020 (2019), arXiv:1910.11775 [hep-ex]

    2019 arXiv

  20. [26]

    Greljo, A

    A. Greljo, A. Palavri´ c, and A. E. Thomsen, Adding Fla- vor to the SMEFT, JHEP 10, 010, arXiv:2203.09561 [hep-ph]

    arXiv

  21. [27]

    D. A. Faroughy, G. Isidori, F. Wilsch, and K. Ya- mamoto, Flavour symmetries in the SMEFT, JHEP 08, 166, arXiv:2005.05366 [hep-ph]

    2005 arXiv

  22. [28]

    Bruggisser, D

    S. Bruggisser, D. van Dyk, and S. Westhoff, Resolving the flavor structure in the MFV-SMEFT, JHEP02, 225, arXiv:2212.02532 [hep-ph]

    arXiv

  23. [29]

    C. S. Machado, S. Renner, and D. Sutherland, Building blocks of the flavourful SMEFT RG, JHEP 03, 226, arXiv:2210.09316 [hep-ph]. 16 UV Field (A4,k) Order Invariants # of parameters N∼(1,1)0 (3,2) O(1) λ(1) N [ ¯NRℓ]1 1 (1,4) O(¯Y(2) 3 ) λ(1) N [ ¯NR]1[ℓ¯Y(2) 3 ]1 1 (1′,4) O(¯...

    arXiv

  24. [30]

    de Blas, J

    J. de Blas, J. C. Criado, M. Perez-Victoria, and J. San- tiago, Effective description of general extensions of the Standard Model: the complete tree-level dictionary, JHEP 03, 109, arXiv:1711.10391 [hep-ph]

    arXiv

  25. [31]

    Greljo and A

    A. Greljo and A. Palavri´ c, Leading directions in the SMEFT, JHEP 09, 009, arXiv:2305.08898 [hep-ph]

    arXiv

  26. [32]

    Greljo, A

    A. Greljo, A. Palavri´ c, and A. Smolkoviˇ c, Leading di- rections in the SMEFT: Renormalization effects, Phys. Rev. D 109, 075033 (2024), arXiv:2312.09179 [hep-ph]

    2024 arXiv

  27. [33]

    Morisi and E

    S. Morisi and E. Peinado, An A(4) model for lepton masses and mixings, Phys. Rev. D 80, 113011 (2009), arXiv:0910.4389 [hep-ph]

    2009 arXiv

  28. [34]

    P. M. Ferreira, L. Lavoura, and P. O. Ludl, A new A4 model for lepton mixing, Phys. Lett. B 726, 767 (2013), arXiv:1306.1500 [hep-ph]

    2013 arXiv

  29. [35]

    G.-J. Ding, S. F. King, and X.-G. Liu, Modular A4 sym- metry models of neutrinos and charged leptons, JHEP 09, 074, arXiv:1907.11714 [hep-ph]

    1907 arXiv

  30. [36]

    Feruglio, C

    F. Feruglio, C. Hagedorn, Y. Lin, and L. Merlo, Lepton Flavour Violation in Models with A(4) Flavour Sym- metry, Nucl. Phys. B 809, 218 (2009), arXiv:0807.3160 [hep-ph]

    2009 arXiv

  31. [37]

    Holthausen, M

    M. Holthausen, M. Lindner, and M. A. Schmidt, Lepton flavor at the electroweak scale: A complete A4 model, Phys. Rev. D 87, 033006 (2013), arXiv:1211.5143 [hep- ph]

    2013 arXiv

  32. [38]

    Abbas, Fermion masses and mixing in modular A4 Symmetry, Phys

    M. Abbas, Fermion masses and mixing in modular A4 Symmetry, Phys. Rev. D 103, 056016 (2021), arXiv:2002.01929 [hep-ph]

    2021 arXiv

  33. [39]

    Morisi, M

    S. Morisi, M. Nebot, K. M. Patel, E. Peinado, and J. W. F. Valle, Quark-Lepton Mass Relation and CKM mixing in an A4 Extension of the Minimal Super- symmetric Standard Model, Phys. Rev. D 88, 036001 (2013), arXiv:1303.4394 [hep-ph]

    2013 arXiv

  34. [40]

    L. M. G. De La Vega, R. Ferro-Hernandez, and E. Peinado, Simple A4 models for dark matter stabil- ity with texture zeros, Phys. Rev. D 99, 055044 (2019), arXiv:1811.10619 [hep-ph]

    2019 arXiv

  35. [41]

    Peinado, An A4 model for neutrinos, J

    E. Peinado, An A4 model for neutrinos, J. Phys. Conf. Ser. 259, 012093 (2010), arXiv:1010.2614 [hep-ph]. 17 UV Field (A4,k) Order Invariants # of parameters B∼ (1,1)0 (1,0) O(1) gℓ(1) B [Bµ]1[¯ℓℓ]1⊕ge(1) B [Bµ]1[¯e]1[e]1⊕ge(2) B [Bµ]1[¯e]1′[e]1′′⊕ge(3) B [Bµ]1[¯e]1′′[e]1′ 4 (1...

    2010 arXiv

  36. [42]

    Pramanick and A

    S. Pramanick and A. Raychaudhuri, A4-based seesaw model for realistic neutrino masses and mixing, Phys. Rev. D 93, 033007 (2016), arXiv:1508.02330 [hep-ph]

    2016 arXiv

  37. [43]

    Altarelli and D

    G. Altarelli and D. Meloni, A Simplest A4 Model for Tri-Bimaximal Neutrino Mixing, J. Phys. G 36, 085005 (2009), arXiv:0905.0620 [hep-ph]

    2009 arXiv

  38. [44]

    Pramanick, Radiative generation of realistic neutrino mixing with A4, Nucl

    S. Pramanick, Radiative generation of realistic neutrino mixing with A4, Nucl. Phys. B 963, 115282 (2021), arXiv:1903.04208 [hep-ph]

    2021 arXiv

  39. [45]

    Carrolo, J

    S. Carrolo, J. C. Romao, and J. P. Silva, Conditions for global minimum in the A4 symmetric 3HDM, Eur. Phys. J. C 82, 749 (2022), arXiv:2207.02928 [hep-ph]

    2022 arXiv

  40. [46]

    G.-J. Ding, S. F. King, and J.-N. Lu, SO(10) mod- els with A 4 modular symmetry, JHEP 11, 007, arXiv:2108.09655 [hep-ph]

    arXiv

  41. [47]

    S. F. King and C. Luhn, A4 models of tri-bimaximal- reactor mixing, JHEP 03, 036, arXiv:1112.1959 [hep- ph]

    1959 arXiv

  42. [48]

    Ding, S.-Y

    G.-J. Ding, S.-Y. Jiang, S. F. King, J.-N. Lu, and B.- Y. Qu, Pati-Salam models with A4 modular symmetry, (2024), arXiv:2404.06520 [hep-ph]

    2024 arXiv

  43. [49]

    Kadosh and E

    A. Kadosh and E. Pallante, An A(4) flavor model for quarks and leptons in warped geometry, JHEP 08, 115, arXiv:1004.0321 [hep-ph]

    arXiv

  44. [50]

    Barman, N

    A. Barman, N. K. Francis, and H. Bora, Neutrino mixing phenomenology: A4 discrete flavor symmetry with type-I seesaw mechanism, Mod. Phys. Lett. A 39, 2350200 (2024), arXiv:2306.11461 [hep-ph]

    2024 arXiv

  45. [51]

    Borah and B

    D. Borah and B. Karmakar, Linear seesaw for Dirac neutrinos withA4 flavour symmetry, Phys. Lett. B789, 59 (2019), arXiv:1806.10685 [hep-ph]

    2019 arXiv

  46. [52]

    M. S. Boucenna, M. Hirsch, S. Morisi, E. Peinado, M. Taoso, and J. W. F. Valle, Phenomenology of Dark Matter from A4 Flavor Symmetry, JHEP 05, 037, arXiv:1101.2874 [hep-ph]

    arXiv

  47. [53]

    Memenga, W

    N. Memenga, W. Rodejohann, and H. Zhang, A4 flavor symmetry model for Dirac neutrinos and sizable Ue3, Phys. Rev. D 87, 053021 (2013), arXiv:1301.2963 [hep- ph]

    2013 arXiv

  48. [54]

    Morisi, M

    S. Morisi, M. Picariello, and E. Torrente-Lujan, Model for fermion masses and lepton mixing in SO(10) x A(4), Phys. Rev. D75, 075015 (2007), arXiv:hep-ph/0702034

    2007 arXiv

  49. [55]

    Nomura and H

    T. Nomura and H. Okada, A modular A4 symmetric model of dark matter and neutrino, Phys. Lett. B 797, 134799 (2019), arXiv:1904.03937 [hep-ph]

    2019 arXiv

  50. [56]

    Zhang, A modular A4 symmetry realization of two- zero textures of the Majorana neutrino mass matrix, Nucl

    D. Zhang, A modular A4 symmetry realization of two- zero textures of the Majorana neutrino mass matrix, Nucl. Phys. B 952, 114935 (2020), arXiv:1910.07869 18 [hep-ph]

    2020 arXiv

  51. [57]

    Asaka, Y

    T. Asaka, Y. Heo, T. H. Tatsuishi, and T. Yoshida, ModularA4 invariance and leptogenesis, JHEP 01, 144, arXiv:1909.06520 [hep-ph]

    1909 arXiv

  52. [58]

    Kumar, P

    R. Kumar, P. Mishra, M. K. Behera, R. Mohanta, and R. Srivastava, Predictions from scoto-seesaw with A4 modular symmetry, Phys. Lett. B 853, 138635 (2024), arXiv:2310.02363 [hep-ph]

    2024 arXiv

  53. [59]

    Okada and Y

    H. Okada and Y. Orikasa, A radiative seesaw model in modular A4 symmetry, (2019), arXiv:1907.13520 [hep- ph]

    2019 arXiv

  54. [60]

    Gogoi, N

    J. Gogoi, N. Gautam, and M. K. Das, Neutrino masses and mixing in minimal inverse seesaw using A4 modular symmetry, Int. J. Mod. Phys. A 38, 2350022 (2023), arXiv:2207.10546 [hep-ph]

    2023 arXiv

  55. [61]

    M. K. Singh, S. R. Singh, and N. N. Singh, Modular A4 symmetry in 3+1 active-sterile neutrino masses and mixings, (2023), arXiv:2303.10922 [hep-ph]

    2023 arXiv

  56. [62]

    Kobayashi, N

    T. Kobayashi, N. Omoto, Y. Shimizu, K. Tak- agi, M. Tanimoto, and T. H. Tatsuishi, Modular A4 invariance and neutrino mixing, JHEP 11, 196, arXiv:1808.03012 [hep-ph]

    arXiv

  57. [63]

    J. C. Criado and F. Feruglio, Modular Invariance Faces Precision Neutrino Data, SciPost Phys. 5, 042 (2018), arXiv:1807.01125 [hep-ph]

    2018 arXiv

  58. [64]

    Feruglio, Are neutrino masses modular forms?, in From My Vast Repertoire ...: Guido Altarelli’s Legacy , edited by A

    F. Feruglio, Are neutrino masses modular forms?, in From My Vast Repertoire ...: Guido Altarelli’s Legacy , edited by A. Levy, S. Forte, and G. Ridolfi (2019) pp. 227–266, arXiv:1706.08749 [hep-ph]

    2019 arXiv

  59. [65]

    Altarelli and F

    G. Altarelli and F. Feruglio, Discrete Flavor Symmetries and Models of Neutrino Mixing, Rev. Mod. Phys. 82, 2701 (2010), arXiv:1002.0211 [hep-ph]

    2010 arXiv

  60. [66]

    Centelles Chuli´ a, R

    S. Centelles Chuli´ a, R. Kumar, O. Popov, and R. Srivas- tava, Neutrino mass sum rules from modular A4 symme- try, Phys. Rev. D109, 035016 (2024), arXiv:2308.08981 [hep-ph]

    2024 arXiv

  61. [67]

    Kumar, N

    R. Kumar, N. Nath, and R. Srivastava, Cutting the Sco- togenic loop: Adding flavor to Dark Matter, (2024), arXiv:2406.00188 [hep-ph]

    2024 arXiv

  62. [68]

    Chauhan, P

    G. Chauhan, P. S. B. Dev, I. Dubovyk, B. Dziewit, W. Flieger, K. Grzanka, J. Gluza, B. Karmakar, and S. Zieba, Phenomenology of lepton masses and mixing with discrete flavor symmetries, Prog. Part. Nucl. Phys. 138, 104126 (2024), arXiv:2310.20681 [hep-ph]

    2024 arXiv

  63. [69]

    Borah and B

    D. Borah and B. Karmakar, A4 flavour model for Dirac neutrinos: Type I and inverse seesaw, Phys. Lett. B 780, 461 (2018), arXiv:1712.06407 [hep-ph]

    2018 arXiv

  64. [70]

    Karmakar and A

    B. Karmakar and A. Sil, Nonzero θ13 and leptogenesis in a type-I seesaw model withA4 symmetry, Phys. Rev. D 91, 013004 (2015), arXiv:1407.5826 [hep-ph]

    2015 arXiv

  65. [71]

    Feruglio and A

    F. Feruglio and A. Romanino, Lepton flavor symmetries, Rev. Mod. Phys. 93, 015007 (2021), arXiv:1912.06028 [hep-ph]

    2021 arXiv

  66. [72]

    S. T. Petcov and A. V. Titov, Assessing the Viability of A4, S4 and A5 Flavour Symmetries for Description of Neutrino Mixing, Phys. Rev. D 97, 115045 (2018), arXiv:1804.00182 [hep-ph]

    2018 arXiv

  67. [73]

    S. F. King and C. Luhn, Neutrino Mass and Mixing with Discrete Symmetry, Rept. Prog. Phys. 76, 056201 (2013), arXiv:1301.1340 [hep-ph]

    2013 arXiv

  68. [74]

    Feruglio, M

    F. Feruglio, M. Parriciatu, A. Strumia, and A. Titov, Solving the strong CP problem without axions, JHEP 08, 214, arXiv:2406.01689 [hep-ph]

    arXiv

  69. [75]

    Feruglio, A

    F. Feruglio, A. Strumia, and A. Titov, Modular invariance and the QCD angle, JHEP 07, 027, arXiv:2305.08908 [hep-ph]

    arXiv

  70. [76]

    G.-J. Ding, L. L. Everett, and A. J. Stuart, Golden Ra- tio Neutrino Mixing and A5 Flavor Symmetry, Nucl. Phys. B 857, 219 (2012), arXiv:1110.1688 [hep-ph]

    2012 arXiv

  71. [77]

    P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, Modular A 5 symmetry for flavour model build- ing, JHEP 04, 174, arXiv:1812.02158 [hep-ph]

    arXiv

  72. [78]

    Di Iura, Lepton mixing and neutrino masses from A5 and CP, in Prospects in Neutrino Physics (2016) arXiv:1604.08391 [hep-ph]

    A. Di Iura, Lepton mixing and neutrino masses from A5 and CP, in Prospects in Neutrino Physics (2016) arXiv:1604.08391 [hep-ph]

    2016 arXiv

  73. [79]

    Di Iura, C

    A. Di Iura, C. Hagedorn, and D. Meloni, Lepton mixing from the interplay of the alternating group A 5 and CP, JHEP 08, 037, arXiv:1503.04140 [hep-ph]

    arXiv

  74. [80]

    I. K. Cooper, S. F. King, and A. J. Stuart, A Golden A5 Model of Leptons with a Minimal NLO Correction, Nucl. Phys. B 875, 650 (2013), arXiv:1212.1066 [hep- ph]

    2013 arXiv

  75. [81]

    G.-J. Ding, S. F. King, and X.-G. Liu, Neutrino mass and mixing with A5 modular symmetry, Phys. Rev. D 100, 115005 (2019), arXiv:1903.12588 [hep-ph]

    2019 arXiv

  76. [82]

    Li and G.-J

    C.-C. Li and G.-J. Ding, Lepton Mixing in A5 Fam- ily Symmetry and Generalized CP, JHEP 05, 100, arXiv:1503.03711 [hep-ph]

    arXiv

  77. [83]

    Turner, Mixing angle and phase predictions from A5 with generalised CP, in Topical Research Meeting on Prospects in Neutrino Physics (2015) arXiv:1506.06898 [hep-ph]

    J. Turner, Mixing angle and phase predictions from A5 with generalised CP, in Topical Research Meeting on Prospects in Neutrino Physics (2015) arXiv:1506.06898 [hep-ph]

    2015 arXiv

  78. [84]

    Puyam and N

    V. Puyam and N. N. Singh, A5 symmetry and deviation from Golden Ratio mixing with charged lepton flavor violation, (2023), arXiv:2308.05944 [hep-ph]

    2023 arXiv

  79. [85]

    Gehrlein, J

    J. Gehrlein, J. P. Oppermann, D. Sch¨ afer, and M. Spin- rath, An SU(5)× A5 golden ratio flavour model, Nucl. Phys. B 890, 539 (2014), arXiv:1410.2057 [hep-ph]

    2014 arXiv

  80. [86]

    Hernandez and A

    D. Hernandez and A. Y. Smirnov, Lepton mixing and discrete symmetries, Phys. Rev. D 86, 053014 (2012), arXiv:1204.0445 [hep-ph]

    2012 arXiv

  81. [87]

    de Medeiros Varzielas and J

    I. de Medeiros Varzielas and J. a. Louren¸ co, Two A5 modular symmetries for Golden Ratio 2 mixing, Nucl. Phys. B 984, 115974 (2022), arXiv:2206.14869 [hep-ph]

    2022 arXiv

  82. [88]

    Ballett, S

    P. Ballett, S. Pascoli, and J. Turner, Mixing angle and phase correlations from A5 with generalized CP and their prospects for discovery, Phys. Rev. D 92, 093008 (2015), arXiv:1503.07543 [hep-ph]

    2015 arXiv

  83. [89]

    Feruglio and A

    F. Feruglio and A. Paris, The Golden Ratio Predic- tion for the Solar Angle from a Natural Model with A5 Flavour Symmetry, JHEP 03, 101, arXiv:1101.0393 [hep-ph]

    arXiv

  84. [90]

    L. L. Everett and A. J. Stuart, Icosahedral (A(5)) Fam- ily Symmetry and the Golden Ratio Prediction for So- lar Neutrino Mixing, Phys. Rev. D 79, 085005 (2009), arXiv:0812.1057 [hep-ph]

    2009 arXiv

  85. [91]

    Gehrlein, S

    J. Gehrlein, S. T. Petcov, M. Spinrath, and X. Zhang, Leptogenesis in an SU(5) x A5 Golden Ratio Flavour Model: Addendum 10.1016/j.nuclphysb.2015.08.019 (2015), [Addendum: Nucl.Phys.B 899, 617–630 (2015)], arXiv:1508.07930 [hep-ph]

    2015 arXiv

  86. [92]

    Gehrlein, S

    J. Gehrlein, S. T. Petcov, M. Spinrath, and X. Zhang, Leptogenesis in an SU(5) × A5 Golden Ratio Flavour Model, Nucl. Phys. B896, 311 (2015), arXiv:1502.00110 [hep-ph]

    2015 arXiv

  87. [93]

    C.-S. Chen, T. W. Kephart, and T.-C. Yuan, An A5 Model of Four Lepton Generations, JHEP 04, 015, arXiv:1011.3199 [hep-ph]. 19

    arXiv

  88. [94]

    Kobayashi, Y

    T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, New A4 lepton flavor model from S4 modular symmetry, JHEP 02, 097, arXiv:1907.09141 [hep-ph]

    1907 arXiv

  89. [95]

    Ishimori, Y

    H. Ishimori, Y. Shimizu, and M. Tanimoto, S(4) Fla- vor Model of Quarks and Leptons, Prog. Theor. Phys. Suppl. 180, 61 (2010), arXiv:0904.2450 [hep-ph]

    2010 arXiv

  90. [96]

    Thapa, S

    B. Thapa, S. Barman, S. Bora, and N. K. Francis, A minimal inverse seesaw model with S 4 flavour symme- try, JHEP 11, 154, arXiv:2305.09306 [hep-ph]

    arXiv

  91. [97]

    Ishimori and T

    H. Ishimori and T. Kobayashi, Towards Minimal S4 Lepton Flavor Model, JHEP 10, 082, arXiv:1106.3604 [hep-ph]

    arXiv

  92. [98]

    Hagedorn, M

    C. Hagedorn, M. Lindner, and R. N. Mohapatra, S(4) flavor symmetry and fermion masses: Towards a grand unified theory of flavor, JHEP 06, 042, arXiv:hep- ph/0602244

  93. [99]

    Bazzocchi and L

    F. Bazzocchi and L. Merlo, Neutrino Mixings and the S4 Discrete Flavour Symmetry, Fortsch. Phys. 61, 571 (2013), arXiv:1205.5135 [hep-ph]

    2013 arXiv

  94. [100]

    Zhang and S

    X. Zhang and S. Zhou, Inverse seesaw model with a modular S 4 symmetry: lepton flavor mixing and warm dark matter, JCAP 09, 043, arXiv:2106.03433 [hep-ph]

    arXiv

  95. [101]

    Ishimori, Y

    H. Ishimori, Y. Shimizu, M. Tanimoto, and A. Watan- abe, Neutrino masses and mixing from S4 flavor twist- ing, Phys. Rev. D 83, 033004 (2011), arXiv:1010.3805 [hep-ph]

    2011 arXiv

  96. [102]

    Izawa, S4 Lepton Flavor Model with 3HDM, (2023), arXiv:2312.16545 [hep-ph]

    Y. Izawa, S4 Lepton Flavor Model with 3HDM, (2023), arXiv:2312.16545 [hep-ph]

    2023 arXiv

  97. [103]

    Bazzocchi and S

    F. Bazzocchi and S. Morisi, S(4) as a natural flavor symmetry for lepton mixing, Phys. Rev. D 80, 096005 (2009), arXiv:0811.0345 [hep-ph]

    2009 arXiv

  98. [104]

    Wang and S

    X. Wang and S. Zhou, The minimal seesaw model with a modular S4 symmetry, JHEP 05, 017, arXiv:1910.09473 [hep-ph]

    1910 arXiv

  99. [105]

    Nomura and H

    T. Nomura and H. Okada, Linear seesaw model with a modular S 4 flavor symmetry *, Chin. Phys. C 46, 053101 (2022), arXiv:2109.04157 [hep-ph]

    2022 arXiv

  100. [106]

    J. T. Penedo and S. T. Petcov, Lepton Masses and Mix- ing from ModularS4 Symmetry, Nucl. Phys. B939, 292 (2019), arXiv:1806.11040 [hep-ph]

    2019 arXiv

  101. [107]

    P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, Modular S4 models of lepton masses and mixing, JHEP 04, 005, arXiv:1811.04933 [hep-ph]

    arXiv

  102. [108]

    S. F. King and Y.-L. Zhou, Twin modular S4 with SU(5) GUT, JHEP 04, 291, arXiv:2103.02633 [hep-ph]

    arXiv

  103. [110]

    Okada and Y

    H. Okada and Y. Orikasa, Neutrino mass model with a modular S4 symmetry, (2019), arXiv:1908.08409 [hep- ph]

    2019 arXiv

  104. [111]

    M. K. Behera, P. Ittisamai, C. Pongkitivanichkul, and P. Uttayarat, Phenomenology of Inverse Seesaw Using S3 Modular Symmetry, (2025), arXiv:2504.12954 [hep- ph]

    2025 arXiv

  105. [112]

    J. C. Criado, F. Feruglio, and S. J. D. King, Modular Invariant Models of Lepton Masses at Levels 4 and 5, JHEP 02, 001, arXiv:1908.11867 [hep-ph]

    1908 arXiv

  106. [113]

    Belfkir, M

    M. Belfkir, M. A. Loualidi, and S. Nasri, Fermion Masses and Mixing in Pati–Salam Unification with S3 Modular Symmetry, PTEP 2025, 033B05 (2025), arXiv:2501.00302 [hep-ph]

    2025 arXiv

  107. [114]

    Ishimori, T

    H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada, and M. Tanimoto, Non-Abelian Discrete Symmetries in Particle Physics, Prog. Theor. Phys. Suppl. 183, 1 (2010), arXiv:1003.3552 [hep-th]

    2010 arXiv

  108. [115]

    Palavri´ c, Discrete leptonic flavor symmetries: UV mediators and phenomenology, Phys

    A. Palavri´ c, Discrete leptonic flavor symmetries: UV mediators and phenomenology, Phys. Rev. D 110, 115025 (2024), arXiv:2408.16044 [hep-ph]

    2024 arXiv

  109. [116]

    Kobayashi, H

    T. Kobayashi, H. Otsuka, M. Tanimoto, and K. Ya- mamoto, Modular symmetry in the SMEFT, Phys. Rev. D 105, 055022 (2022), arXiv:2112.00493 [hep-ph]

    2022 arXiv

  110. [117]

    Kobayashi and M

    T. Kobayashi and M. Tanimoto, Modular flavor sym- metric models (2023) arXiv:2307.03384 [hep-ph]

    2023 arXiv

  111. [118]

    M. K. Behera, S. Singirala, S. Mishra, and R. Mohanta, A modular A 4 symmetric scotogenic model for neutrino mass and dark matter, J. Phys. G 49, 035002 (2022), arXiv:2009.01806 [hep-ph]

    2022 arXiv

  112. [119]

    S. T. Petcov and M. Tanimoto, A4 modular flavour model of quark mass hierarchies close to the fixed point τ =ω, Eur. Phys. J. C83, 579 (2023), arXiv:2212.13336 [hep-ph]

    2023 arXiv

  113. [120]

    S. T. Petcov and M. Tanimoto, A 4 modular flavour model of quark mass hierarchies close to the fixed point τ = i∞, JHEP 08, 086, arXiv:2306.05730 [hep-ph]

    arXiv

  114. [121]

    Yao, J.-N

    C.-Y. Yao, J.-N. Lu, and G.-J. Ding, Modular Invariant A4 Models for Quarks and Leptons with Generalized CP Symmetry, JHEP 05, 102, arXiv:2012.13390 [hep-ph]

    2012 arXiv

  115. [122]

    Nomura, H

    T. Nomura, H. Okada, and Y. Orikasa, Quark and lep- ton flavor model with leptoquarks in a modularA4 sym- metry, Eur. Phys. J. C81, 947 (2021), arXiv:2106.12375 [hep-ph]

    2021 arXiv

  116. [123]

    Okada and M

    H. Okada and M. Tanimoto, Towards unification of quark and lepton flavors inA4 modular invariance, Eur. Phys. J. C 81, 52 (2021), arXiv:1905.13421 [hep-ph]

    2021 arXiv

  117. [124]

    Falkowski, M

    A. Falkowski, M. Gonz´ alez-Alonso, and K. Mimouni, Compilation of low-energy constraints on 4-fermion op- erators in the SMEFT, JHEP08, 123, arXiv:1706.03783 [hep-ph]

    arXiv

  118. [125]

    Bres´ o-Pla, A

    V. Bres´ o-Pla, A. Falkowski, M. Gonz´ alez-Alonso, and K. Mons´ alvez-Pozo, EFT analysis of New Physics at COHERENT (2023), arXiv:2301.07036 [hep-ph]

    2023 arXiv

  119. [126]

    Falkowski and K

    A. Falkowski and K. Mimouni, Model independent constraints on four-lepton operators, JHEP 02, 086, arXiv:1511.07434 [hep-ph]

    arXiv

  120. [127]

    Schael et al

    S. Schael et al. (ALEPH, DELPHI, L3, OPAL, LEP Electroweak), Electroweak Measurements in Electron- Positron Collisions at W-Boson-Pair Energies at LEP, Phys. Rept. 532, 119 (2013), arXiv:1302.3415 [hep-ex]

    2013 arXiv

  121. [128]

    Hanai et al

    H. Hanai et al. (VENUS), Measurement of tau polariza- tion in e+ e- annihilation at s**(1/2) = 58-GeV, Phys. Lett. B 403, 155 (1997), arXiv:hep-ex/9703003

    1997 arXiv

  122. [129]

    Falkowski and D

    A. Falkowski and D. Straub, Flavourful SMEFT like- lihood for Higgs and electroweak data, JHEP 04, 066, arXiv:1911.07866 [hep-ph]

    1911 arXiv

  123. [130]

    Efrati, A

    A. Efrati, A. Falkowski, and Y. Soreq, Electroweak con- straints on flavorful effective theories, JHEP 07, 018, arXiv:1503.07872 [hep-ph]

    arXiv

  124. [131]

    Patrignani et al

    C. Patrignani et al. (Particle Data Group), Review of Particle Physics, Chin. Phys. C 40, 100001 (2016)

    2016

  125. [132]

    Geiregat et al

    D. Geiregat et al. (CHARM-II), First observation of neutrino trident production, Phys. Lett. B 245, 271 (1990)

    1990

  126. [133]

    S. R. Mishra et al. (CCFR), Neutrino Tridents and W Z Interference, Phys. Rev. Lett. 66, 3117 (1991)

    1991

  127. [134]

    Altmannshofer, S

    W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, 20 Neutrino Trident Production: A Powerful Probe of New Physics with Neutrino Beams, Phys. Rev. Lett. 113, 091801 (2014), arXiv:1406.2332 [hep-ph]

    2014 arXiv

  128. [135]

    Heeck and M

    J. Heeck and M. Sokhashvili, Lepton flavor viola- tion by two units, Phys. Lett. B 852, 138621 (2024), arXiv:2401.09580 [hep-ph]

    2024 arXiv

  129. [136]

    Heeck, Interpretation of Lepton Flavor Violation, Phys

    J. Heeck, Interpretation of Lepton Flavor Violation, Phys. Rev. D 95, 015022 (2017), arXiv:1610.07623 [hep- ph]

    2017 arXiv

  130. [137]

    Conlin and A

    R. Conlin and A. A. Petrov, Muonium-antimuonium os- cillations in effective field theory, Phys. Rev. D 102, 095001 (2020), arXiv:2005.10276 [hep-ph]

    2020 arXiv

  131. [138]

    Banerjee, Searches for Lepton Flavor Violation in Tau Decays at Belle II, Universe 8, 480 (2022), arXiv:2209.11639 [hep-ex]

    S. Banerjee, Searches for Lepton Flavor Violation in Tau Decays at Belle II, Universe 8, 480 (2022), arXiv:2209.11639 [hep-ex]

    2022 arXiv

  132. [139]

    Bigaran, X.-G

    I. Bigaran, X.-G. He, M. A. Schmidt, G. Valencia, and R. Volkas, Lepton-flavor-violating tau decays from trial- ity, Phys. Rev. D 107, 055001 (2023), arXiv:2212.09760 [hep-ph]

    2023 arXiv

  133. [140]

    Crivellin, S

    A. Crivellin, S. Najjari, and J. Rosiek, Lepton Flavor Vi- olation in the Standard Model with general Dimension- Six Operators, JHEP 04, 167, arXiv:1312.0634 [hep-ph]

    arXiv

  134. [141]

    Hayasaka et al., Search for Lepton Flavor Violating Tau Decays into Three Leptons with 719 Million Pro- duced Tau+Tau- Pairs, Phys

    K. Hayasaka et al., Search for Lepton Flavor Violating Tau Decays into Three Leptons with 719 Million Pro- duced Tau+Tau- Pairs, Phys. Lett. B 687, 139 (2010), arXiv:1001.3221 [hep-ex]

    2010 arXiv

  135. [142]

    Calibbi and G

    L. Calibbi and G. Signorelli, Charged Lepton Flavour Violation: An Experimental and Theoretical Introduc- tion, Riv. Nuovo Cim. 41, 71 (2018), arXiv:1709.00294 [hep-ph]

    2018 arXiv

  136. [143]

    A. M. Baldini et al. (MEG), Search for the lepton flavour violating decay µ+ → e+γ with the full dataset of the MEG experiment, Eur. Phys. J. C 76, 434 (2016), arXiv:1605.05081 [hep-ex]

    2016 arXiv

  137. [144]

    Aubert et al

    B. Aubert et al. (BaBar), Searches for Lepton Flavor Vi- olation in the Decays tau+- —> e+- gamma and tau+- —> mu+- gamma, Phys. Rev. Lett.104, 021802 (2010), arXiv:0908.2381 [hep-ex]

    2010 arXiv

  138. [145]

    Abdesselam et al

    A. Abdesselam et al. (Belle), Search for lepton-flavor- violating tau-lepton decays to ℓγ at Belle, JHEP 10, 19, arXiv:2103.12994 [hep-ex]

    arXiv

  139. [146]

    Bellgardt et al

    U. Bellgardt et al. (SINDRUM), Search for the Decay µ+→e+e+e−, Nucl. Phys. B 299, 1 (1988)

    1988

  140. [147]

    W. H. Bertl et al. (SINDRUM II), A Search for muon to electron conversion in muonic gold, Eur. Phys. J. C 47, 337 (2006)

    2006

  141. [148]

    Calibbi, X

    L. Calibbi, X. Marcano, and J. Roy, Z lepton flavour vi- olation as a probe for new physics at futuree+e− collid- ers, Eur. Phys. J. C 81, 1054 (2021), arXiv:2107.10273 [hep-ph]

    2021 arXiv

  142. [150]

    M. Ardu, S. Davidson, and S. Lavignac, Constrain- ing new physics models from µ → e observables in bottom-up EFT, Eur. Phys. J. C 84, 458 (2024), arXiv:2401.06214 [hep-ph]

    2024 arXiv

  143. [151]

    G. M. Pruna and A. Signer, The µ→eγ decay in a sys- tematic effective field theory approach with dimension 6 operators, JHEP 10, 014, arXiv:1408.3565 [hep-ph]

    arXiv

  144. [152]

    Uesaka, M

    Y. Uesaka, M. Yamanaka, and Y. Kuno, µ-→e-γ in a muonic atom as a probe for effective lepton-flavor- violating operators involving photon fields, Phys. Rev. D 111, 035017 (2025), arXiv:2411.10304 [hep-ph]

    2025 arXiv

  145. [153]

    Willmann et al., New bounds from searching for muo- nium to anti-muonium conversion, Phys

    L. Willmann et al., New bounds from searching for muo- nium to anti-muonium conversion, Phys. Rev. Lett. 82, 49 (1999), arXiv:hep-ex/9807011

    1999 arXiv

  146. [154]

    Fuentes-Mart´ ın, M

    J. Fuentes-Mart´ ın, M. K¨ onig, J. Pag` es, A. E. Thomsen, and F. Wilsch, A proof of concept for matchete: an au- tomated tool for matching effective theories, Eur. Phys. J. C 83, 662 (2023), arXiv:2212.04510 [hep-ph]

    2023 arXiv

  147. [155]

    Gargalionis, J

    J. Gargalionis, J. Quevillon, P. N. H. Vuong, and T. You, Linear Standard Model extensions in the SMEFT at one loop and Tera-Z, (2024), arXiv:2412.01759 [hep-ph]

    2024

  148. [156]

    E. E. Jenkins, A. V. Manohar, and P. Stoffer, Low- Energy Effective Field Theory below the Electroweak Scale: Operators and Matching, JHEP 03, 016, arXiv:1709.04486 [hep-ph]

    arXiv

  149. [157]

    Alonso, E

    R. Alonso, E. E. Jenkins, A. V. Manohar, and M. Trott, Renormalization Group Evolution of the Stan- dard Model Dimension Six Operators III: Gauge Cou- pling Dependence and Phenomenology, JHEP 04, 159, arXiv:1312.2014 [hep-ph]. 21 SUPPLEMENTAL MATERIAL Analysis of Mediators wit...

    2014 arXiv

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