Constraining low-scale flavor models with (g-2)μ and lepton flavor violation

Constraining low-scale flavor models with ðg−2Þμ and lepton flavor violation M. L. López-Ibáñez,1,2,* Aurora Melis,3,†M. Jay P´erez ,4,‡Moinul Hossain Rahat ,5,§ and Oscar Vives 6,∥ 1CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China 2Departmento de Física, Campus de Rabanales Edificio C2, Universidad de Córdoba, E-14071 Córdoba, Spain 3Laboratory of High Energy and Computational Physics, NICPB, Rävala 10, 10143 Tallinn, Estonia 4Valencia College, Osceola Science Department, Kissimmee, Florida 34744, USA 5Institute for Fundamental Theory, Department of Physics, University of Florida, Gainesville, Florida 32611, USA 6Departament de Fsica Torica, Universitat de Valncia, Dr. Moliner 50, E-46100 Burjassot and IFIC, Universitat de Valncia and CSIC, E-46071 Paterna, Spain (Received 10 January 2022; accepted 4 February 2022; published 23 February 2022) We present here two concrete examples of models where a sub-TeV scale breaking of their respective T13 and A5flavor symmetries is able to account for the recently observed discrepancy in the muon anomalous magnetic moment, ðg−2Þμ. Similarities in the flavor structures of the charged-lepton Yukawa matrix and dipole matrix yielding ðg−2Þμgive rise to strong constraints on low-scale flavor models when bounds from lepton flavor violation (LFV) are imposed. These constraints place stringent limits on the offdiagonal Yukawa structure, suggesting a mostly (quasi)diagonal texture for models with a low flavor breaking scale Λf. We argue that many of the popular flavor models in the literature designed to explain the fermion masses and mixings are not suitable for reproducing the observed discrepancy in ðg−2Þμ, which requires a delicate balance of maintaining a low flavor scale while simultaneously satisfying strong LFV constraints. DOI: 10.1103/PhysRevD.105.035021 I. INTRODUCTION At the dawn of the LHC era, the physics community awaited with bated breath for new electroweak physics whose discovery many were convinced was just around the corner. Defying expectations, more than ten years later and after the ends of Run1 and Run2, a clear sign of new physics has yet to emerge. That this is so especially puzzling, given the common conviction that the Standard Model (SM) cannot be the ultimate theory of everything due to the many questions it leaves unanswered. Among others, it cannot explain the presence of three families of fermions with different masses and mixings or accommodate the existence of dark matter in the Universe. New physics associated with these “open questions”is certainly required; the only remaining doubt is at what scale this new physics is hiding. In the case of dark matter, thermally obtaining the correct relic abundance seems to point to the electroweak scale as the scale associated with the dark matter particles, although other mass scales are still possible. In contrast, flavor physics, whose theoretical constraints in the SM come from dimensionless Yukawa couplings, does not provide any obvious hints as to the scale of the physics responsible for generating these couplings. Fortunately, there exist other flavorful couplings, such as dipole moments, with nonvanishing mass dimensions that could provide some information on the scale of the new physics responsible for the observed flavor structures in the SM. Any observation of new contributions to these operators with flavor structures beyond those of the SM Yukawa couplings would most certainly help shed light on the origins of flavor; when coupled with the constraints from other lepton flavor violating (LFV) processes, they may also provide bounds on the masses of new particles responsible for generating them. *[email protected] †[email protected] ‡[email protected] §[email protected] ∥[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 105, 035021 (2022) 2470-0010=2022=105(3)=035021(21) 035021-1 Published by the American Physical Society Admittedly, deviations in the dipole matrices from their SM predictions may give only indirect evidence for an underlying theory of flavor. New physics satisfying the conservative ansatz of minimal flavor violation (MFV), where the Yukawa couplings are the only source of flavorchanging interactions, can still generate new contributions to the dipole moments beyond those of the SM. However, even when the underlying mechanism responsible for generating the flavor structures of the Yukawa and dipole couplings are fundamentally the same, differences in their Oð1Þcoefficients stemming from the details of how they are generated may mean that these matrices are not exactly proportional. A framework, using effective operators, for how this could occur is given in Refs. [1,2]. There, the one-loop radiative corrections to the fermion masses, stemming from the breaking of a low-energy flavor symmetry, also generated the dipole matrices. Although the same underlying flavor symmetry controlled the flavor structure of both operators, differences in their Oð1Þcoefficients arose from the distinct ways of attaching the photon line to the diagrams generating the dipole transitions. In such models, the rotation from the flavor to mass basis which diagonalizes the fermion masses would not simultaneously diagonalize the dipole matrices, which may lead to strong constraints from LFVobservables. This sensitivity to any deviation between the flavor structures of these two operators make low-energy flavor models that can simultaneously explain both operators successfully rich targets for study. Excitingly, recent experimental results are beginning to hint at just such deviations. The new results from the Muon g−2experiment at Fermilab [3] have confirmed the longstanding discrepancy between the SM prediction for the anomalous magnetic moment of the muon, aμ¼ðgμ−2Þ=2 [4],1and the previous Brookhaven National Laboratory measurements [35,36]. This outcome makes a strong case for new physics interacting with SM muons at scales not far above the electroweak scale. On the other hand, a similar 2σ tension between theory and experiments for the electron g−2is now under dispute. In the presence of an independent and sufficiently precise measurement of α, one can employ ðg−2Þeas a test for new physics [37]. This has become possible in recent years, and the most precise result, obtained by employing matter-wave interferometry with cesium-133 atoms [38], highlighted the discrepancy in ðg−2Þe. Other hints may come from B physics, where similar 3–4σdeviations are providing hints for violations of lepton universality [39]. As stated above, a precise determination of the flavor structure of dipole moments would be crucial in determining the mechanism responsible for flavor in the SM. Of course, one possibility is that we find the flavor structure of the dipole matrix proportional to the SM Yukawa couplings, as in a MFV scenario, providing only a lower bound on the scale of flavor-dependent interactions. Nevertheless, given that Yukawa couplings do not restrict the scale of flavor breaking, it is still possible to have a flavor symmetry broken at a low scale, such that the contributions to the dipole moments are sizable. In this work, we follow up on the ideas explored by some of the authors in Refs. [1,2] by providing explicit constructions of flavor models capable of realizing these ideas. As a proof of concept, we build two explicit models with low-energy flavor symmetries based on the groups T13 and A5, capable of explaining the muon g−2discrepancy while satisfying all constraints from LFV dipole transitions. We show that the required absence of LFV transitions while maintaining a sizable contribution to the muon anomalous magnetic moment restricts the structure of the flavor symmetry or, alternatively, the scale of flavor symmetry breaking. The paper is organized as follows. In Sec. II, we review the salient features of the framework proposed in Refs. [1,2] for constraining low-scale flavor models using the anomalous magnetic moment of the muon and LFV observables in the language of effective field theories. Sections III and IV contain the main results of this work: explicit constructions of low-scale flavor models based on the flavor groups T13 and A5capable of producing sizable contributions to the muons anomalous magnetic moment while evading LFV constraints. We provide some clarifying remarks and comments on the general applicability of such a framework to other flavor models and possible obstacles to extending such an analysis in Sec. V. Finally, we summarize and conclude in Sec. IV. II. CONNECTING THE CHARGED LEPTON YUKAWA STRUCTURE TO g−2 We begin by briefly reviewing the idea introduced in Refs. [1,2] that the observed discrepancy in the anomalous magnetic moment of a muon can be entirely explained by accounting for contributions from the breaking of a lowenergy flavor symmetry. Such a framework works by exploiting the similar flavor structures present in the dipole matrix describing the anomalous magnetic moment, lepton flavor violating transitions and electric dipole moment, and the Yukawa matrix. When both operators are generated from the same underlying theory of flavor, stringent constraints from LFV observables then restrict the offdiagonal entries of the charged lepton Yukawas. As is well known, the charged-lepton Yukawa matrix is not completely determined in the Standard Model; the only constraint is that its diagonalization should yield the 1The community consensus value of muon g−2in the SM is based on latest evaluations of the contributions from quantum electrodynamics (QED) to the tenth order [5,6], hadronic vacuum polarization [7–14], hadronic light-by-light [15–29], and electroweak processes [30–34]. LÓPEZ-IBÁÑEZ, MELIS, P´ EREZ, RAHAT, and VIVES PHYS. REV. D 105, 035021 (2022) 035021-2 hierarchical charged lepton mass ratios ye=yμ¼me=mμ≃ 0.005 and yμ=yτ¼mμ=mτ≃0.059. A simple approach for generating such hierarchies is through the spontaneous breaking of an underlying flavor symmetry, transmitted to the SM fermions by heavy messengers `alaFroggattNielsen (FN) [40–42]. Integrating out these heavy messengers yields effective Yukawa interactions of the form ¯ LlHðφ1φ2…φn=Λn fÞ; where ¯ Land lare the SM SUð2Þdoublets and singlets, respectively. The φkare flavon fields, gauge-singlet scalars transforming nontrivially under the chosen flavor symmetry which encodes its breaking, and Λfis a scale associated with the underlying flavor dynamics. In the electroweak vacuum, such operators yield the mass matrix vYij ¯ Lilj, where v≃246 GeV is the Higgs vacuum expectation value (VEV), and the Yukawa matrix Yij contains the relevant suppression factors hφki=Λf. Since we are only concerned with mass ratios, the flavor symmetry breaking scale Λfremains unresolved at this stage. This scale can be determined if the physics responsible for new contributions to the dipole operator, which mitigates the observed discrepancy in the muon anomalous magnetic moment, is the very same flavor symmetry which determines the Yukawa couplings; in this case, the dipole operator ¯ LσμνPRlFμν will inherit the same basic flavor structure as the Yukawa interactions. After spontaneous symmetry breaking, it can be expressed as L⊃ev 8π2Cijð¯ LiσμνPRljÞFμν þH:c:; i;j ¼1;2;3;ð2:1Þ where the dipole matrix Cij in the flavor basis, expressed in units of GeV−2, contains similar factors of hφki=Λf. Such tree-level Yukawa couplings will receive radiative correction from loops involving the flavons appearing in tree-level diagrams. Because of their similar flavor structures, it is expected that the same loops contributing to the Yij would also contribute to the Cij. Crucially however, each entry in the dipole matrix will, in general, be multiplied by a different Oð1Þfactor related to the different ways of inserting the external photon line in the dipole operator. This implies that the dipole matrix is not exactly proportional to the Yukawa matrix, and the transformation from the flavor basis to the mass basis will not diagonalize the dipole matrix.2 In terms of the dipole matrix, and after rotating to the mass basis, the new physics contributions to the leptonic anomalous magnetic moment Δalare given by Δal¼mlv 2π2ReðCllÞ;l¼e; μ;τ;ð2:2Þ while the imaginary parts of Cll are related to the leptonic electric dipole moments (EDMs),3 dl¼ev 4π2ImðCllÞ:ð2:3Þ Similarly, off-diagonal couplings in the dipole matrix, in the basis of a diagonal Yl, contribute to LFV processes, in particular, to the radiative decays, BRðl→l0γÞ BRðl→l0ν¯ ν0Þ¼3α ffiffiffi 2 pπG3 Fm2 lðjCll0j2þjCl0lj2Þ:ð2:4Þ The constraints on the dipole matrix from these processes are summarized in Table I. Following Ref. [2], we can assume that the entries of the dipole matrix are of the order of the corresponding entries of the Yukawa matrix in the flavor basis, Cll0≳κYll0 Λ2 f , where we have omitted the different Oð1Þcoefficients that, in general, can appear in each entry, and κis a global factor that takes care of the relative size of the tree-level Yukawa to the loop contribution to the mass. For example, a typical value of κ≃1=8is expected when the dipole is loop suppressed with respect to the mass, while κ≃2π2≃20 if the mass has a radiative origin [2]. Using the observed discrepancy in the measured muon anomalous magnetic moment to fix the scale Λf, we can then employ the limits on various entries of the dipole operators to obtain the following constraints on the flavor structure of the leptonic Yukawa matrix: Yl≈yτ0 B @ λ5≲λ8.6≲λ2.8 ≲λ8.6λ2≲λ2.7 ≲λ2.8≲λ2.71 1 C A;ð2:5Þ where λ≃0.225 is the Wolfenstein parameter.4 In view of the above constraints, our strategy for model building is as follows: (i) We propose a charged-lepton Yukawa matrix in the flavor basis, arising from the breaking of an underlying flavor symmetry, so that the ratios of its eigenvalues yield the observed charged-lepton mass ratios. (ii) The dipole matrix is then generated with the same flavor structure as the loop corrections to the 2A similar effect can occur in the soft terms of supersymmetric flavor models generated by a FN mechanism mediated by supergravity, see [43–46]. 3See Refs. [47–50], for example, for a combined explanation of ðg−2Þe;μand the relation to EDMs. 4The (11) element is determined by requiring the observed mass ratio me=mτ≈λ5, which could also be accommodated if Y13 lY31 l∼λ5, while Y11 l≪λ5. CONSTRAINING LOW-SCALE FLAVOR MODELS WITH …PHYS. REV. D 105, 035021 (2022) 035021-3 Yukawas, except for possible differences in the Oð1Þ factors of each entry, determined by the ways of attaching the external photon line in the generating one-loop diagram. (iii) Rotating to the mass basis, we match the contribution of the (22) element of the dipole matrix to the anomalous magnetic moment of the muon, so that the experimentally observed value of ðg−2Þμis reproduced. From this, we determine the scale Λf. (iv) We impose upper bounds from LFV observables to the off-diagonal entries of the dipole matrix in the mass basis, which, in turn, restricts the structure of the Yukawa matrix. In general, simultaneously satisfying all of the above constraints in a realistic model is nontrivial and may not be possible for a low flavor scale Λf. As a proof of concept that it can be done, in principle, in the following two sections we build two explicit models based on the flavor groups T13 and A5, respectively. III. A T13 MODEL T13 ≡Z13⋊Z3is an Oð39Þfinite subgroup of SUð3Þ [60–64]. It contains two generators aand b, related to its Z13 and Z3subgroups. These generators are nontrivially related to each other, yielding the presentation ha; bja13 ¼b3¼I; bab−1¼a3i: The group has two distinct complex 3-dimensional representations, a trivial singlet, and a complex singlet. It is the smallest discrete subgroup of SUð3Þwith two inequivalent complex triplet representations, 31and 32. Each element of the triplets has a unique Z13 charge, 31∶ðρ1;ρ3;ρ9Þ;¯ 31∶ðρ12;ρ10;ρ4Þ; 32∶ðρ2;ρ6;ρ5Þ;¯ 32∶ðρ11;ρ7;ρ8Þ;ð3:1Þ where ρ13 ≡1. The Kronecker products and ClebschGordan coefficients of the group are summarized in Appendix A. An interesting feature of T13 is that the Kronecker product of two triplets places the diagonal and off-diagonal terms in different representations. This can be useful in model building, specifically when there are different constraints on the diagonal and off-diagonal elements of the Yukawa matrices. Assuming the SM SUð2Þdoublets and singlets transform as triplets, while the Higgs transforms trivially under T13, the Yukawa matrix elements are then generated from the Kronecker product of two lepton triplets. If the two triplet representations are the same, each diagonal and off-diagonal pair can be tagged with a distinct Z13 charge, as given by Eq. (3.1). Although T13 has most often found use in describing popular mixing patterns (such as tribimaximal mixing) in the neutrino sector,5our aim here is to employ itin building a new model capable of producing the flavor structure for the charged-lepton Yukawa matrix given in Eq. (2.5).Weleave an extension of such a model to the neutrino sector for a future work; however, as an example of a model which describes both the charged lepton and neutrino sectors, we presenta modifiedA5modelfoundin theliteratureinSec.IV. A. Tree-level model As a concrete example, we take theSUð2Þlepton doublets andsingletstobothtransformasthe31representationofT13: ¯ L≡ð¯ L1;¯ L2;¯ L3Þ∼31and l≡ðl1;l2;l3Þ∼31. In this basis, the entries ðYlÞij are given by the coefficients of ¯ Ljli. Our objective is to build a minimal Yukawa matrix whose diagonalization yields the hierarchical chargedlepton mass ratios. An intuitive understanding of how to generate the desired Yukawa structure can be gained from the fermion bilinears. From the T13 Clebsch-Gordan coefficients, we have TABLE I. Current experimental limits on the sizes of the dipole matrix entries. Observable Limit on coefficient (GeV−2) C.L. (%) Δae¼ð4.83.0Þ×10−13 [51] ReðCeeÞ≈½−0.2;1.7×10−10 95 Δaμ¼ð251 59Þ×10−11 [3,4,35] ReðCμμÞ≈½1.0;2.8×10−995 −0.007 <Δaτ<0.005 [52,53] ReðCττÞ≈½−5.9;5.0×10−495 de<1.1×10−29 ecm [54] jImðCeeÞj≲9.0×10−17 90 dμ<1.9×10−19 ecm [55] jImðCμμÞj≲1.5×10−695% dτ<4.5×10−17 ecm [56] jImðCττÞj≲3.7×10−495 BRðμ→eγÞ≤4.2×10−13 [57] jCeμj;jCμej≲3.9×10−14 90 BRðτ→eγÞ≤3.3×10−8[58] jCeτj;jCτej≲4.3×10−10 90 BRðτ→μγÞ<4.2×10−8[59] jCμτj;jCτμj≲5.0×10−10 90 5See Refs. [65–72] for the application of T13 in beyond the Standard Model building. LÓPEZ-IBÁÑEZ, MELIS, P´ EREZ, RAHAT, and VIVES PHYS. REV. D 105, 035021 (2022) 035021-4 0 B @ ¯ L1 ¯ L2 ¯ L3 1 C A31 ⊗0 B @ l1 l2 l3 1 C A31 ¼0 B @ ¯ L1l1 ¯ L2l2 ¯ L3l3 1 C A32 ⊕0 B @ ¯ L2l3 ¯ L3l1 ¯ L1l2 1 C A¯ 31 ⊕0 B @ ¯ L3l2 ¯ L1l3 ¯ L2l1 1 C A¯ 31 :ð3:2Þ As promised, the diagonal and off-diagonal elements of the Yukawa matrix are separated into the 32and ¯ 31representations, respectively. Each element has a unique Z13 charge according to Eq. (3.1) and can therefore be paired with a flavon of conjugate charge. For simplicity we take the SM Higgs to be a T13 singlet. The Yukawa matrix is then generated by dimension-5 and higher operators of the form ¯ LlHφ, where φis a (combination of) flavon(s) transforming as a triplet/antitriplet under T13. The observed hierarchy in the charged-lepton masses suggests a diagonal Yukawa matrix Yl∼diagðλ5;λ2;1Þ. Naively, one could generate this structure by employing three flavons, all transforming as a ¯ 32, cf. Eq. (3.2), and vacuum values λ5ð1;0;0Þ,λ2ð0;1;0Þ, and (0,0,1), respectively. In this case however, the hierarchy between the scale of the first and third flavon VEVs would be large, Oðλ5Þ. Another option is a quasidiagonal Yukawa structure, where Y11 lis zero, and the electron mass is generated from Oðλ5=2Þentries in the Y13–31 lelements. This can be achieved with a single flavon VEV λ5=2ð0;1;0Þtransforming as a 31, cf. Eq. (3.2), thanks to the separation of diagonals and offdiagonals terms. We also require a sizable loop correction to Y22 l,asa similar diagram will generate the contribution to ðg−2Þμin the dipole matrix. If Y22 lis generated by the product of two identical flavons φ22, one could expect a loop correction from the quartic coupling ðφ22φ 22Þ2. Here, we make use of the unique Z13 charges of the triplets to introduce a useful subscript notation for the flavons, distinguishing them by the elements of Ylto which they couple in vacuum. However, as is shown, the contribution to the dipole matrix is a product of a negative loop factor and the quartic flavon coupling. Since the required contribution to muon g−2is positive, the flavon quartic coupling must be negative. In this case, one should worry about the stability of the potential; a simple remedy is to include a negative mixed quartic coupling β2aðφaφ 22Þ2while keeping the couplings β2ðφ22φ 22Þ2and βaðφaφ aÞ2such that β2aþ ffiffiffiffiffiffiffiffiffi β2βa pis positive [73–77].T13 constrains this flavon φa to be of the same representation as φ22, so that the mixed quartic loop contributes to Y22 l. This implies that φamust be φ33, the flavon which couples to ¯ L3l3, and therefore that Y33 lbe generated by the product φ2 33 at tree level. Given the above reasoning, a minimal Yukawa structure emerges where the matrix elements Y13–31 l,Y33 l, and Y22 lare generated from the following Lagrangian: Le Y¼¯ LlH1 Λf φ13 þ1 Λ2 f φ22φ22 þ1 Λ2 f φ33φ33:ð3:3Þ The flavons transform under T13 as φ33 ∼¯ 31,φ22 ∼¯ 31, φ13 ∼31, aligning in the flavor vacuum along hφ33i¼ϵð0;0;1ÞΛf;hφ22i¼ϵ2ð0;1;0ÞΛf; hφ13i¼ϵ9=2ð0;1;0ÞΛf;ð3:4Þ where ϵ∼OðλÞ, fixed by the observed lepton mass ratios. Notice that we have to add a factor ϵin the VEVof φ33,so that the mass of τis reproduced with v= ffiffiffi 2 p¼174 GeV with an Oð1Þcoefficient. Furthermore, in this way, higher order operators like ¯ LlHφaðφ 33φ33Þnare suppressed by a factor ϵ2nwith respect to the leading order contribution. At tree level, the operators in Eq. (3.3) can be generated using heavy vectorlike messengers as mediators, as shown in the Feynman diagrams of Figs. 1(a)–1(c). In principle, without specifying the full UV theory, one would naively expect mediators of all possible representations, generating additional vertices. The simplest way to forbid such dangerous operators is then to restrict the messenger spectrum, which we do here by including only the three mediators Δ,χ, and χ0. This is the minimal set of mediators required to generate the operators in Eq. (3.3). The mediators are SUð2Þsinglets, and their T13 charges, along with the charges of the other fields in the model, are listed in Table II. We note that one could also generate the same effective operators by coupling the Higgs to linstead of ¯ Lin the relevant diagrams, in which case some of the mediators would be SUð2Þdoublets. The choice in Figs. 1(a)–1(c) is motivated by the relatively relaxed bounds on singlet mediator masses compared to doublets [78]. As we see later, successfully reproducing the correct value for ðg−2Þμrequires a low flavor scale Λfin this model, which might be in tension with LHC bounds on SUð2Þ doublet mediators [78]. While limiting the allowed mediators restricts the possible vertices to a great extent, dangerous terms allowed by T13 still remain. To further protect the desired flavor structure of Ylagainst unwanted contributions from these vertices, we introduce an Abelian Zn“shaping symmetry”; the origin of this shaping symmetry and the restricted mediator spectrum must then be addressed in a UV completion of the model. CONSTRAINING LOW-SCALE FLAVOR MODELS WITH …PHYS. REV. D 105, 035021 (2022) 035021-5 B. “Shaping”symmetry There are six “expected”vertices involving two fermions and a scalar in this model, as seen from Figs. 1(a)–1(c): ¯ LΔH,l¯ Δφ13,l¯ χφ22,l¯ χ0φ33,χ¯ Δφ22, and χ0¯ Δφ33. On the other hand, ten “unwanted”vertices are still allowed by the T13 and SM charges, listed below in two categories: ðiÞl¯ Δφ 33;l¯ Δφ 22;l¯ χφ 13;lχ0φ 13;χ¯ Δφ 13;χ0¯ Δφ 13;ð3:5Þ ðiiÞl¯ χφ33;lχ0φ22;χ¯ Δφ33;χ0¯ Δφ22:ð3:6Þ These vertices can be prevented by introducing a Zn “shaping”symmetry. As there are nine fields and six vertices in the model, at least three of the fields will have independent Zncharges. Denoting these charges with a ½·notation, suppose ½¯ L¼x,½l¼yand ½H¼z. Then, from the desired vertices, one has that ½Δ¼−x−z; ½φ13¼−x−y−z; ½φ22¼−xþyþz 2; ½φ33¼n 2−xþyþz 2;½χ¼−xþy−z 2; ½χ0¼−xþy−z 2−n 2;ð3:7Þ modulo n. Note that the quartic scalar vertex ðφ33φ 22Þ2 required by the model is always allowed. The charges of φ22 and φ33 (and the corresponding mediators χand χ0) have been separated by n=2so that the otherwise allowed operator ¯ LlHφ22φ33, which contributes to Y23−32 lat Oðϵ3Þ, is no longer permitted. The remaining dangerous operators in category (ii) are prevented by any Znsymmetry for the charge assignment given by Eq. (3.7). This follows from the fact that operators in the first category are not allowed when FIG. 1. Feynman diagrams for effective operators generating the charged lepton Yukawa matrix. Here, × denotes a mass insertion. Panels (a)–(c) show the tree-level contributions, while panels (d) and (e) show loop corrections. For panel (e), the mediator χk≡χðχ0Þ when φb≡φ22ðφ33Þ. TABLE II. Transformation properties of matter, scalar, and messenger fields. Here, η4¼1. The Z4“shaping”symmetry prevents unwanted tree-level operators. Fields ¯ LlHφ33 φ22 φ13 Δχχ 0 SUð2ÞL212 1 1 1 111 T13 31311¯ 31¯ 3131¯ 3111 Z4η11η1η1η3η2η2η3η1 LÓPEZ-IBÁÑEZ, MELIS, P´ EREZ, RAHAT, and VIVES PHYS. REV. D 105, 035021 (2022) 035021-6 3ðxþyþzÞ≠0mod n: ð3:8Þ On the other hand, from Eq. (3.7), requiring all the field charges to be integers, we have xþyþz¼0mod 2;ð3:9Þ y−z−x¼0mod 2;ð3:10Þ and n¼0mod 2:ð3:11Þ The case for n¼2is ruled out as then Eqs. (3.8) and (3.9) would be in conflict. The next case n¼4is viable, for example, with the charges shown in Table II using y¼0,x¼z¼1. C. Loop corrections With these Zncharges, the flavon potential is restricted to have the following form: Vφ¼−μ2 1φ 13φ13 −μ2 2φ 22φ22 −μ2 3φ 33φ33 þðAφ 13φ2 22 þH:c:Þþβ1 2ðφ 13φ13Þ2 þβ2 2ðφ 22φ22Þ2þβ3 2ðφ 33φ33Þ2 þβ4 2ðφ 22φ33Þ2þH:c:þ…;ð3:12Þ where the dots stand for additional operators related to all possible contractions of quartic couplings of the form φ aφaφ bφb. The μiare the masses of the flavons, presumably OðΛfÞ, and A∼OðΛfÞis a cubic coupling. The βiare quartic couplings. The cubic and quartic couplings present in the potential induce loop corrections to the Yukawas. In Fig. 1(d), the VEVof φ 13 contributes to Y13−31 l, similar to the diagram in Fig. 1(a). The conjugate of the cubic term yields another diagram like Fig. 1(a), with ðφ 22φ 22Þcoupling to the φ13. Fortunately however, the bilinear ðφ 22φ 22Þin the representation of φ 13 is zero in the vacuum and does not contribute to Y22 l. The quartic interactions yield loop corrections through Feynman diagrams of the form of Fig. 1(e). The pure quartic terms ðφaφ aÞ2contribute to Ya l. Since there is no tree-level diagram with two φ13’s coupling to fermions, Y13–31 ldoes not receive any loop correction from the pure quartic term. On the other hand, the mixed quartic terms ðφ 22φ33Þ2and ðφ 33φ22Þ2contribute to Y22 land Y33 l, respectively. D. Predictions The Yukawa matrix generated by the operators of Eq. (3.3) can be expressed as Yl¼ϵ20 B @ 00ϵ5=2 0ϵ20 ϵ5=20y1 1 C A;ð3:13Þ where we have introduced an Oð1Þcoefficient y1to account for a slightly different magnitude in the VEV of φ33 with respect to the others. From Eq. (3.13), the following mass ratios are obtained: me mτ¼1þy2 1 2ϵ51−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ4ϵ5 y2 1 s≃ϵ5 y2 1 ∼0.0003;ð3:14Þ mμ mτ¼2ϵ2 y1 1 1þffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ4ϵ5 y2 1 q≃ϵ2 y1 ∼0.045;ð3:15Þ which, to a first approximation, are consistent with observations for ϵ∼0.15 and y1∼0.5. Radiative corrections to the Yukawa matrix of Eq. (3.13) are generated by the diagrams in Fig. 1(d) and 1(e), where the flavon lines can be closed through a cubic coupling Aor quartic coupling βi. They yield δYl¼ϵ2f1ðx2 φÞ 32π2 ×0 B @ 002y2y4ϵ5=2 0ðβ2y2þβ4y3Þϵ20 2y2y4ϵ5=20y1ðβ3y3þβ4y2Þ 1 C A; ð3:16Þ where f1ðxÞis the loop function given by f1ðxÞ¼1þlog x−x ð1−xÞ2<0;ð3:17Þ and xφ≡mφ=Mχ, with mφand Mχgeneric masses for the flavons and heavy mediators. In principle, the radiative correction for each element has a different loop factor f1ðm2 φa=M2 χÞ, where mφacorresponds to the mass of the flavon contributing to the particular element. If mφa∼hφai, the masses of these flavons are expected to differ by some orders of magnitude. However, within the loop function, this variation can effectively be approximated as a change in the Oð1Þcoefficient. We therefore take the loop factor as common to all elements, introducing new Oð1Þcoefficients y2and y3to take into account the possible differences in mass of φ22 and φ33. Similarly, the cubic coupling Acan be absorbed in the Oð1Þ factor y4≡A=Mχ. The dipole operator is generated through diagrams similar to those that give radiative correction to the Yukawa couplings, albeit with a photon radiated from the charged particle inside the loop, see Fig. 2. CONSTRAINING LOW-SCALE FLAVOR MODELS WITH …PHYS. REV. D 105, 035021 (2022) 035021-7 The resulting matrix therefore has the same basic structure as the radiative correction to the Yukawas in Eq. (3.16). However, as it is a dimension-6 operator, an explicit dependence on the flavor scale Λfis present,6 Cl¼f2ðx2 φÞ 16 ϵ2 Λ2 f ×0 B @ 002y2y4ϵ5=2 0ðβ2y2þβ4y3Þϵ20 2y2y4ϵ5=20y1ðβ3y3þβ4y2Þ 1 C A; ð3:18Þ where again we have taken the loop function f2ðxÞ, given by f2ðxÞ¼−1þ4xð1þlogxÞ−x2ð5−2logxÞ 2ð1−xÞ4<0;ð3:19Þ as common to all elements while capturing the differences in the flavon masses through the factors y2and y3. In total, nine free parameters determine the chargedlepton masses and anomalous magnetic moments in this model. To find their best-fit values, we scan and minimize over the χ2function given in Eq. (C1). The fine tuning of the model is also computed for each fit, as indicated in Eq. (C2). In the left panel of Fig. 3, we display a set of points with fine tuning less than 10 [79] which provide good fits to both the masses and ðg−2Þμ. Mediator masses of up to 430 GeV (705 GeV) can accommodate the measured discrepancy in the anomalous magnetic of the muon at the 1σ(3σ) level. We select as a benchmark point the one with the heaviest mediator mass which exactly reproduces the central value of ðg−2Þμ, see Table III. As detailed in Table IV, our representative point in Table III successfully reproduces both the charged lepton masses and the anomalous magnetic moment of the muon. Our model contribution to the electron g−2remains small, so that the total prediction (SM þflavor model) and the experimental result in Ref. [51] remain compatible at 1.6σ. FIG. 2. Diagrams contributing to the dipole operators. FIG. 3. Branching ratio of the LFV decay τ→eγvs mass of the heavy mediator (∼Λf) for several points that correctly reproduce the mass of the charged leptons and Δaμ. Our benchmark point in Table III corresponds to the red diamond. 6We assume here that this scale, where the flavor dynamics act, is roughly the same as the mediator masses. LÓPEZ-IBÁÑEZ, MELIS, P´ EREZ, RAHAT, and VIVES PHYS. REV. D 105, 035021 (2022) 035021-8 Regarding LFV, while the transitions μ→eand τ→μare absent in this model, the flavor-changing decay τ→eγhas a sizable branching ratio which is predicted to be below present limits but still testable with future sensitivity [80] (see Fig. 3, left). Finally, the right panel of Fig. 3displays how the model prediction for the anomalous magnetic moment of the muon varies when the general mass for the heavy mediators is modified for our chosen benchmark point. In this case, we scan around the parameter values in Table III and select sets which correctly reproduce the charged-lepton masses. We note that for this specific point in the parameter space a maximum mass of 710 GeV can be reached if the prediction for the muon g−2is allowed to fall within the 3σrange. The relevant collider bounds on the masses of the mediators in our model come from ATLAS searches for vectorlike leptons (VLLs) [81], which exclude SUð2Þ singlets with masses in the range of 114–176 GeV and CMS searches for SUð2Þdoublets coupling only to taus [82], which rule out masses between 120–790 GeV. We note however that these limits have been obtained assuming simplified models. A more detailed discussion can be found in Ref. [78], where the authors interpret the experimental findings in the context of flavorful VLLs. They conclude that the masses of these particles have to be above 300 GeV (800 GeV) if they transform as singlets (doublets) of SUð2Þ. These limits are satisfied in our model, where the heavy mediators are defined as SUð2Þsinglets. If they had instead transformed as doublets, there would be tension with these bounds, as the largest possible mass capable of reproducing ðg−2Þμin our model is around 700 GeV. IV. AN A5MODEL A5is the non-Abelian discrete group composed of the even permutations on five objects. It has 60 elements and five irreducible representations (irrep): one singlet 1,two triplets 3and 30, one tetraplet 4, and one pentaplet 5 (12þ2×32þ42þ52¼60). It can be generated by two elements, sand t, and is given by the presentation [83] hs; tjs2¼ðstÞ3¼t5¼Ii:ð4:1Þ The Kronecker products and Clebsch-Gordan coefficients of A5are given in Appendix B. A5has been used extensively as a flavor symmetry for explaining the observed lepton masses and mixings.7 Typically, the flavor symmetry is broken down into different residual symmetries in the charged lepton and in the neutrino sector. These residual symmetries constrain the form of the flavon VEVs in each sector. Following the analyses of Refs. [88–90], we consider Ge¼Z5as the residual symmetry in the charged-lepton sector. This implies that, in vacuum, the flavons φe imust be symmetric under the transformations of the generators Qi corresponding to the representation iof A5[87,91], Qihφe ii¼hφe ii:ð4:2Þ Under this condition, nonzero VEVs are possible only for the triplet and pentaplet representations, and their vacuum alignments are of the form hφe 3i¼ðϵ3;0;0ÞΛf;hφe 30i¼ðϵ30;0;0ÞΛf; hφe 5i¼ðϵ5;0;0;0;0ÞΛf;ð4:3Þ where ϵ3;ϵ30, and ϵ5are dimensionless real parameters. On the other hand, following Ref. [91], the neutrino sector respects a different residual symmetry, Gν¼Z2×CP, and the VEVs invariant under this symmetry are hφν 4i¼0 B B @ wr−iwi ð1þ2φÞwr−iwi ð1þ2φÞwrþiwi wrþiwi 1 C C A ;hφν 5i¼ 0 B B B B B B B B @ ffiffi2 3 qðzr1þzr2Þ −zr1þiφzi zr2−izi zr2þizi zr1þiφzi 1 C C C C C C C C A ; ð4:4Þ where wr;w i;z r1;z r2, and ziare real parameters of mass dimension one. An appropriate Zn“shaping”symmetry then keeps the flavons φeand φνrestricted to the charged lepton and the neutrino sectors, respectively. A. Tree-level model Following Ref. [87], we assign the leptons to triplets under A5:¯ L≡ð¯ L1;¯ L2;¯ L3Þ∼3and l≡ðl1;l3;l2Þ∼3, where the ordering of the components of the fields has been TABLE III. Benchmark point for the T13 model. y1y2y3y4β2β3β4ϵxφ 0.50 0.50 3.50 0.70 0.25 1.56 −6.24 0.15 0.08 TABLE IV. Predictions at the benchmark point of the T13 model. me(MeV) mμ(GeV) mτ(GeV) Δaμ 0.507 0.103 1.806 2.51 ×10−9 ΔaeΔaτBRðτ→eγÞMχ(GeV) 3.64 ×10−15 −1.07 ×10−76.52 ×10−9388 7Some early examples include Refs. [84–87]. CONSTRAINING LOW-SCALE FLAVOR MODELS WITH …PHYS. REV. D 105, 035021 (2022) 035021-9 0 B @ j1i j2i j3i 1 C A30 ⊗0 B B B @ j10i j20i j30i j40i 1 C C C A4 ¼0 B B B @ −ffiffiffi 2 pj2ij30i−ffiffiffi 2 pj3ij20i ffiffiffi 2 pj1ij10iþj2ij40i−j3ij30i ffiffiffi 2 pj1ij40i−j2ij20iþj3ij10i 1 C A3 ⊕0 B B B B B @ j1ij10iþ ffiffiffi 2 pj3ij30i j1ij20i−ffiffiffi 2 pj3ij40i −j1ij30iþ ffiffiffi 2 pj2ij10i −j1ij40i−ffiffiffi 2 pj2ij20i 1 C C C C C A4 ⊕ 0 B B B B B B @ ffiffiffi 6 pj2ij30i−ffiffiffi 6 pj3ij20i ffiffiffi 2 pj1ij10i−3j2ij40i−j3ij30i 2ffiffiffi 2 pj1ij20iþ2j3ij40i −2ffiffiffi 2 pj1ij30i−2j2ij10i −ffiffiffi 2 pj1ij40iþj2ij20iþ3j3ij10i 1 C C C C C C A5 0 B @ j1i j2i j3i 1 C A3 ⊗ 0 B B B B B B @ j10i j20i j30i j40i j50i 1 C C C C C C A5 ¼0 B @ −2j1ij10iþ ffiffiffi 3 pj2ij50iþ ffiffiffi 3 pj3ij20i ffiffiffi 3 pj1ij20iþj2ij10i−ffiffiffi 6 pj3ij30i ffiffiffi 3 pj1ij50i−ffiffiffi 6 pj2ij40iþj3ij10i 1 C A3 ⊕0 B @ffiffiffi 3 pj1ij10iþj2ij50iþj3ij20i j1ij30i−ffiffiffi 2 pj2ij20i−ffiffiffi 2 pj3ij40i j1ij40i−ffiffiffi 2 pj2ij30i−ffiffiffi 2 pj3ij50i 1 C A30 ⊕0 B B B @ 2ffiffiffi 2 pj1ij20i−ffiffiffi 6 pj2ij10iþj3ij30i −ffiffiffi 2 pj1ij30iþ2j2ij20i−3j3ij40i ffiffiffi 2 pj1ij40iþ3j2ij30i−2j3ij50i −2ffiffiffi 2 pj1ij50i−j2ij40iþ ffiffiffi 6 pj3ij10i 1 C C C A4 ⊕ 0 B B B B B B B B @ ffiffiffi 3 pj2ij50i−ffiffiffi 3 pj3ij20i −j1ij20i−ffiffiffi 3 pj2ij10i−ffiffiffi 2 pj3ij30i −2j1ij30i−ffiffiffi 2 pj2ij20i 2j1ij40iþ ffiffiffi 2 pj3ij50i j1ij50iþ ffiffiffi 2 pj2ij40iþ ffiffiffi 3 pj3ij10i 1 C C C C C C C C A5 0 B @ j1i j2i j3i 1 C A30 ⊗0 B B B B B @ j10i j20i j30i j40i j50i 1 C C C C C A5 ¼0 B @ffiffiffi 3 pj1ij10iþj2ij40iþj3ij30i j1ij20i−ffiffiffi 2 pj2ij50i−ffiffiffi 2 pj3ij40i j1ij50i−ffiffiffi 2 pj2ij30i−ffiffiffi 2 pj3ij20i 1 C A3 ⊕0 B @ −2j1ij10iþ ffiffiffi 3 pj2ij40iþ ffiffiffi 3 pj3ij30i ffiffiffi 3 pj1ij30iþj2ij10i−ffiffiffi 6 pj3ij50i ffiffiffi 3 pj1ij40i−ffiffiffi 6 pj2ij20iþj3ij10i 1 C A30 ⊕0 B B B B B @ ffiffiffi 2 pj1ij20iþ3j2ij50i−2j3ij40i 2ffiffiffi 2 pj1ij30i−ffiffiffi 6 pj2ij10iþj3ij50i −2ffiffiffi 2 pj1ij40i−j2ij20iþ ffiffiffi 6 pj3ij10i −ffiffiffi 2 pj1ij50iþ2j2ij30i−3j3ij20i 1 C C C C C A4 ⊕ 0 B B B B B B B B @ ffiffiffi 3 pj2ij40i−ffiffiffi 3 pj3ij30i 2j1ij20iþ ffiffiffi 2 pj3ij40i −j1ij30i−ffiffiffi 3 pj2ij10i−ffiffiffi 2 pj3ij50i j1ij40iþ ffiffiffi 2 pj2ij20iþ ffiffiffi 3 pj3ij10i −2j1ij50i−ffiffiffi 2 pj2ij30i 1 C C C C C C C C A5 LÓPEZ-IBÁÑEZ, MELIS, P´ EREZ, RAHAT, and VIVES PHYS. REV. D 105, 035021 (2022) 035021-16 0 B B B B B @ j1i j2i j3i j4i 1 C C C C C A4 ⊗0 B B B B B @ j10i j20i j30i j40i 1 C C C C C A4 ¼ðj1ij40iþj2ij30iþj3ij20iþj4ij10iÞ1S⊕0 B @ −j1ij40iþj2ij30i−j3ij20iþj4ij10i ffiffiffi 2 pj2ij40i−ffiffiffi 2 pj4ij20i ffiffiffi 2 pj1ij30i−ffiffiffi 2 pj3ij10i 1 C A3A ⊕0 B @ j1ij40iþj2ij30i−j3ij20i−j4ij10i ffiffiffi 2 pj3ij40i−ffiffiffi 2 pj4ij30i ffiffiffi 2 pj1ij20i−ffiffiffi 2 pj2ij10i 1 C A30A ⊕0 B B B B B @ j2ij40iþj3ij30iþj4ij20i j1ij10iþj3ij40iþj4ij30i j1ij20iþj2ij10iþj4ij40i j1ij30iþj2ij20iþj3ij10i 1 C C C C C A4S ⊕ 0 B B B B B B B B @ ffiffiffi 3 pj1ij40i−ffiffiffi 3 pj2ij30i−ffiffiffi 3 pj3ij20iþ ffiffiffi 3 pj4ij10i −ffiffiffi 2 pj2ij40iþ2ffiffiffi 2 pj3ij30i−ffiffiffi 2 pj4ij20i −2ffiffiffi 2 pj1ij10iþ ffiffiffi 2 pj3ij40iþ ffiffiffi 2 pj4ij30i ffiffiffi 2 pj1ij20iþ ffiffiffi 2 pj2ij10i−2ffiffiffi 2 pj4ij40i −ffiffiffi 2 pj1ij30iþ2ffiffiffi 2 pj2ij20i−ffiffiffi 2 pj3ij10i 1 C C C C C C C C A5S 0 B B B B B @ j1i j2i j3i j4i 1 C C C C C A4 ⊗ 0 B B B B B B @ j10i j20i j30i j40i j50i 1 C C C C C C A5 ¼0 B @ 2ffiffiffi 2 pj1ij50i−ffiffiffi 2 pj2ij40iþ ffiffiffi 2 pj3ij30i−2ffiffiffi 2 pj4ij20i −ffiffiffi 6 pj1ij10iþ2j2ij50iþ3j3ij40i−j4ij30i j1ij40i−3j2ij30i−2j3ij20iþ ffiffiffi 6 pj4ij10i 1 C A3 ⊕0 B @ffiffiffi 2 pj1ij50iþ2ffiffiffi 2 pj2ij40i−2ffiffiffi 2 pj3ij30i−ffiffiffi 2 pj4ij20i 3j1ij20i−ffiffiffi 6 pj2ij10i−j3ij50iþ2j4ij40i −2j1ij30iþj2ij20iþ ffiffiffi 6 pj3ij10i−3j4ij50i 1 C A30 ⊕0 B B B B B @ ffiffiffi 3 pj1ij10i−ffiffiffi 2 pj2ij50iþ ffiffiffi 2 pj3ij40i−2ffiffiffi 2 pj4ij30i −ffiffiffi 2 pj1ij20i−ffiffiffi 3 pj2ij10iþ2ffiffiffi 2 pj3ij50iþ ffiffiffi 2 pj4ij40i ffiffiffi 2 pj1ij30iþ2ffiffiffi 2 pj2ij20i−ffiffiffi 3 pj3ij10i−ffiffiffi 2 pj4ij50i −2ffiffiffi 2 pj1ij40iþ ffiffiffi 2 pj2ij30i−ffiffiffi 2 pj3ij20iþ ffiffiffi 3 pj4ij10i 1 C C C C C A4 ⊕ 0 B B B B B B @ ffiffiffi 2 pj1ij50i−ffiffiffi 2 pj2ij40i−ffiffiffi 2 pj3ij30iþ ffiffiffi 2 pj4ij20i −ffiffiffi 2 pj1ij10i−ffiffiffi 3 pj3ij40i−ffiffiffi 3 pj4ij30i ffiffiffi 3 pj1ij20iþ ffiffiffi 2 pj2ij10iþ ffiffiffi 3 pj3ij50i ffiffiffi 3 pj2ij20iþ ffiffiffi 2 pj3ij10iþ ffiffiffi 3 pj4ij50i −ffiffiffi 3 pj1ij40i−ffiffiffi 3 pj2ij30i−ffiffiffi 2 pj4ij10i 1 C C C C C C A51 ⊕ 0 B B B B B B @ 2j1ij50iþ4j2ij40iþ4j3ij30iþ2j4ij20i 4j1ij10iþ2ffiffiffi 6 pj2ij50i −ffiffiffi 6 pj1ij20iþ2j2ij10i−ffiffiffi 6 pj3ij50iþ2ffiffiffi 6 pj4ij40i 2ffiffiffi 6 pj1ij30i−ffiffiffi 6 pj2ij20iþ2j3ij10i−ffiffiffi 6 pj4ij50i 2ffiffiffi 6 pj3ij20iþ4j4ij10i 1 C C C C C C A52 CONSTRAINING LOW-SCALE FLAVOR MODELS WITH …PHYS. REV. D 105, 035021 (2022) 035021-17 0 B B B B B B @ j1i j2i j3i j4i j5i 1 C C C C C C A5 ⊗ 0 B B B B B B @ j10i j20i j30i j40i j50i 1 C C C C C C A5 ¼ðj1ij10iþj2ij50iþj3ij40iþj4ij30iþj5ij20iÞ1S ⊕0 B @ j2ij50iþ2j3ij40i−2j4ij30i−j5ij20i −ffiffiffi 3 pj1ij20iþ ffiffiffi 3 pj2ij10iþ ffiffiffi 2 pj3ij50i−ffiffiffi 2 pj5ij30i ffiffiffi 3 pj1ij50iþ ffiffiffi 2 pj2ij40i−ffiffiffi 2 pj4ij20i−ffiffiffi 3 pj5ij10i 1 C A3A ⊕0 B @ 2j2ij50i−j3ij40iþj4ij30i−2j5ij20i ffiffiffi 3 pj1ij30i−ffiffiffi 3 pj3ij10iþ ffiffiffi 2 pj4ij50i−ffiffiffi 2 pj5ij40i −ffiffiffi 3 pj1ij40iþ ffiffiffi 2 pj2ij30i−ffiffiffi 2 pj3ij20iþ ffiffiffi 3 pj4ij10i 1 C A30A ⊕0 B B B B B @ 3ffiffiffi 2 pj1ij20iþ3ffiffiffi 2 pj2ij10i−ffiffiffi 3 pj3ij50iþ4ffiffiffi 3 pj4ij40i−ffiffiffi 3 pj5ij30i 3ffiffiffi 2 pj1ij30iþ4ffiffiffi 3 pj2ij20iþ3ffiffiffi 2 pj3ij10i−ffiffiffi 3 pj4ij50i−ffiffiffi 3 pj5ij40i 3ffiffiffi 2 pj1ij40i−ffiffiffi 3 pj2ij30i−ffiffiffi 3 pj3ij20iþ3ffiffiffi 2 pj4ij10iþ4ffiffiffi 3 pj5ij50i 3ffiffiffi 2 pj1ij50i−ffiffiffi 3 pj2ij40iþ4ffiffiffi 3 pj3ij30i−ffiffiffi 3 pj4ij20iþ3ffiffiffi 2 pj5ij10i 1 C C C C C A4S ⊕0 B B B B B @ ffiffiffi 2 pj1ij20i−ffiffiffi 2 pj2ij10iþ ffiffiffi 3 pj3ij50i−ffiffiffi 3 pj5ij30i −ffiffiffi 2 pj1ij30iþ ffiffiffi 2 pj3ij10iþ ffiffiffi 3 pj4ij50i−ffiffiffi 3 pj5ij40i −ffiffiffi 2 pj1ij40i−ffiffiffi 3 pj2ij30iþ ffiffiffi 3 pj3ij20iþ ffiffiffi 2 pj4ij10i ffiffiffi 2 pj1ij50i−ffiffiffi 3 pj2ij40iþ ffiffiffi 3 pj4ij20i−ffiffiffi 2 pj5ij10i 1 C C C C C A4A ⊕ 0 B B B B B B B B @ 2j1ij10iþj2ij50i−2j3ij40i−2j4ij30iþj5ij20i j1ij20iþj2ij10iþ ffiffiffi 6 pj3ij50iþ ffiffiffi 6 pj5ij30i −2j1ij30iþ ffiffiffi 6 pj2ij20i−2j3ij10i −2j1ij40i−2j4ij10iþ ffiffiffi 6 pj5ij50i j1ij50iþ ffiffiffi 6 pj2ij40iþ ffiffiffi 6 pj4ij20iþj5ij10i 1 C C C C C C C C A5S;1 ⊕ 0 B B B B B B B B @ 2j1ij10i−2j2ij50iþj3ij40iþj4ij30i−2j5ij20i −2j1ij20i−2j2ij10iþ ffiffiffi 6 pj4ij40i j1ij30iþj3ij10iþ ffiffiffi 6 pj4ij50iþ ffiffiffi 6 pj5ij40i j1ij40iþ ffiffiffi 6 pj2ij30iþ ffiffiffi 6 pj3ij20iþj4ij10i −2j1ij50iþ ffiffiffi 6 pj3ij30i−2j5ij10i 1 C C C C C C C C A5S;2 APPENDIX C: DETAILS OF THE FIT The optimization problem of finding a good benchmark point for the model is based on minimizing the following cost function: χ2¼χ2 Oþχ2 c¼X khOki− ˆ Ok σOk2 þX ijcij−1 σc2 ;ðC1Þ LÓPEZ-IBÁÑEZ, MELIS, P´ EREZ, RAHAT, and VIVES PHYS. REV. D 105, 035021 (2022) 035021-18 where Okstands for the observable with measured value hOkiσOkand model prediction ˆ Ok. In our case, k∈½1;4 runs over the three lepton masses and the anomalous magnetic moment of the muon. The χ2 cimplies a normal distribution for the absolute value of the coefficients ciwith standard deviation σci. It is a prior that expresses the requirement that a successful flavor model should feature nonhierarchical coefficients and alleviates overfitting. The fine-tuning of the model is determined as [79] ΔFN ≡Maxk;ijδK;ij;δK;i ≡ci ˆ Ok δ ˆ Ok δci ;ðC2Þ where K∈½1;4and i∈½1;7run over the number of observables and parameters, respectively. [1] L. Calibbi, M. L. López-Ibáñez, A. Melis, and O. Vives, Muon and electron g−2and lepton masses in flavor models, J. High Energy Phys. 06 (2020) 087. [2] L. Calibbi, M. L. López-Ibáñez, A. Melis, and O. Vives, Implications of the Muon g-2 result on the flavour structure of the lepton mass matrix, Eur. Phys. J. 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