(g-2)e,μ in an extended inverse type-III seesaw model

ðg−2Þe;μin an extended inverse type-III seesaw model Pablo Escribano ,1,* Jorge Terol-Calvo ,2,3,†and Avelino Vicente 1,4,‡ 1Instituto de Física Corpuscular, CSIC-Universitat de Val`encia, 46980 Paterna, Spain 2Instituto de Astrofísica de Canarias, C/Vía Láctea, s/n, 38205 La Laguna, Tenerife, Spain 3Universidad de La Laguna, Departamento de Astrofísica, La Laguna, Tenerife, Spain 4Departament de Física Teòrica, Universitat de Val`encia, 46100 Burjassot, Spain (Received 20 April 2021; accepted 26 May 2021; published 14 June 2021) There has been a longstanding discrepancy between the experimental measurements of the electron and muon anomalous magnetic moments and their predicted values in the Standard Model. This is particularly relevant in the case of the muon g−2, which has attracted a remarkable interest in the community after the long-awaited announcement of the first results by the Muon g−2collaboration at Fermilab, which confirms a previous measurement by the E821 experiment at Brookhaven and enlarges the statistical significance of the discrepancy, now at 4.2σ. In this paper we consider an extension of the inverse type-III seesaw with a pair of vectorlike leptons that induces masses for neutrinos at the electroweak scale and show that one can accommodate the electron and muon anomalous magnetic moments, while being compatible with all relevant experimental constraints. DOI: 10.1103/PhysRevD.103.115018 I. INTRODUCTION The charged leptons anomalous magnetic moments, al¼gl−2 2;ð1Þ with l¼e,μ,τ, are known to be powerful probes of new physics (NP) effects, potentially hidden in virtual loop contributions. Interestingly, there has been a longstanding discrepancy between the Standard Model (SM) prediction for the electron and muon anomalous magnetic moments and their experimentally determined values [1–7]. In the case of the electron g−2, the significance is slightly below ∼3σ, and hence not very significant at the moment. In contrast, the deviation has become particularly relevant in the case of the muon g−2, in particular after the Muon g−2experiment at Fermilab has published its longawaited first results [8]. Their measurement of aμperfectly agrees with the result obtained by the E821 experiment at Brookhaven [5] and, consequently, disagrees with the SM. Their combination leads to a 4.2σdiscrepancy with the SM prediction compiled by the theory community in [9]. In summary, the current status of the electron and muon g−2can be quantified as1 Δae¼aexp e−aSM e¼ð−87 36Þ×10−14; Δaμ¼aexp μ−aSM μ¼ð25.15.9Þ×10−10:ð2Þ New measurements and more refined theoretical calculations are definitely required to assess the relevance of these anomalies, and confirm whether these intriguing deviations are hints of NP [12], SM contributions not correctly taken into account or just statistical fluctuations.2 However, it is tempting to interpret them as a signal of the presence of new states beyond the SM (BSM). In this case, the g−2anomalies may hide valuable information about *[email protected].es †[email protected] ‡[email protected].es 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. 1The status of the electron g−2has recently changed by a new measurement of the fine-structure constant [10]. The new value differs by more than 5σto the previous one and affects the electron g−2anomaly, which gets reduced to just 1.6σand flips sign, see [11]. We will not include these results in our analysis, but note that it would be straightforward to accommodate a positive Δaein our model, as shown in Sec. IV. We also point out that this change in the fine-structure constant value has little impact on the muon g−2. 2The theoretical calculation of the electron and muon anomalous magnetic moments is a challenging task and has led to some controversies along the years. For instance, a recent calculation of the hadronic vacuum polarization contribution by the BudapestMarseilles-Wuppertal collaboration [13] brings the SM prediction for the muon g−2into agreement with the experimental value, hence ruling out any discrepancy. However, it has been pointed out that this result in turn leads to some tension with electroweak data [14–17]. PHYSICAL REVIEW D 103, 115018 (2021) 2470-0010=2021=103(11)=115018(17) 115018-1 Published by the American Physical Society the shape of the underlying model. In particular, the sign difference between Δaeand Δaμand the sizable value of jΔaejwould indicate that the NP contributions do not scale with the square of the charged lepton masses [18]. This calls for a nontrivial extension of the SM. The inverse type-III seesaw model (ISS3) is obtained by replacing the fermionic SUð2ÞLsinglets in the original inverse type-I seesaw [19] by SUð2ÞLtriplets. This variant has already been studied [20–26], although not extensively, and many of its phenomenological features are still to be investigated.3In fact, there are several phenomenological directions of interest in the ISS3. The fact that triplets couple to the SM gauge bosons allow for new production mechanisms at the LHC, where one can also look for lepton number violating signatures. Lepton flavor violation might also be an interesting subject to explore in this model, which may offer some differences with respect to the more common inverse type-I seesaw [34]. Finally, the potentially sizable mixing between the charged components of the type-III triplet and the SM charged leptons may also lead to observable Z→lþ il− jdecays with i≠j. We refer to [35] for a recent analysis of these and other relevant observables in the presence of light fermion triplets. While many models have been put forward to address the discrepancy between theory and experiment in the electron and muon g−2, the main motivation for the ISS3 is not to accommodate the existing deviations, but to induce nonzero masses for neutrinos. It is therefore natural to investigate whether the model can account for the experimental values for the electron and muon anomalous magnetic moments in the region of parameter space that can reproduce the observed neutrino masses and leptonic mixing angles, measured in oscillation experiments, while being compatible with the bounds obtained at colliders and low-energy experiments. In this paper we show that these constraints preclude the ISS3 from inducing large contributions to ðg−2Þe;μ. More importantly, the ISS3 contributions are negative, making it impossible to address the existing discrepancy in the muon g−2. This motivates a minimal extension of the model that keeps its most relevant features but provides additional ingredients to generate the required contributions to the electron and muon g−2.We find that the introduction of a pair of vectorlike (VL) lepton doublets with sizable couplings to electrons and muons can explain both anomalies and simultaneously satisfy all the experimental constraints. It is the aim of this paper to study the electron and muon g−2in this model, which we denote as the ISS3VL. The muon g−2has been considered in a wide variety of contexts, in many cases in connection to neutrino mass generation. This includes models based on the inverse seesaw mechanism [36–44] and/or with VL leptons [45–65].See also [66] for a recent work in the context of a radiative neutrino mass model including triplet fermions. Finally, we note that the muon g−2has also been considered as a motivation for a muon collider [67–69]. The rest of the manuscript is organized as follows. In Sec. II the basic features of the ISS3VL model are introduced, including the generation of charged and neutral lepton masses. In Sec. III we compute the charged lepton g−2values and provide simplified approximate expressions. A numerical analysis is performed in Sec. IV. After arguing that the pure ISS3 model cannot address the anomalies, we explore the parameter space of the ISS3VL model and obtain results for the electron and muon g−2compatible with the relevant experimental constraints. Finally, we discuss our results and conclude in Sec. V. Appendixes Aand Bcontain additional details, such as analytical expressions for the couplings of interest to our calculation and full expressions for the charged lepton anomalous magnetic moments. II. THE MODEL The ISS3VL is an extension of the leptonic sector of the SM with the addition of six right-handed Weyl fermion SUð2ÞLtriplets with vanishing hypercharge, ΣAand Σ0 A (A¼1, 2, 3), and a VL copy of the SM lepton doublet, LL and LR. The ΣAand Σ0 Atriplets are introduced in order to generate neutrino masses via the inverse type-III seesaw mechanism.4They can be distinguished by their different lepton numbers, with LðΣÞ¼þ1and LðΣ0Þ¼−1. Nevertheless, lepton number will be explicitly broken in the ISS3VL. Therefore, this lepton number assignment is arbitrary. The new fermionic fields LLand LRhave the same representations under the SUð3Þc×SUð2ÞL×Uð1ÞY gauge group, and both are doublets under SUð2ÞL. The full particle content of the ISS3VL model and the representations of all fields under the SUð3Þc×SUð2ÞL×Uð1ÞY gauge group are shown in Table I. As usual, the SM SUð2ÞLdoublets can be decomposed as qL¼u dL ;lL¼ν eL ;H¼Hþ H0:ð3Þ The Σand Σ0triplets can also be decomposed into SUð2ÞL components. With ΣA¼ðΣ1;Σ2;Σ3ÞA, they can be conveniently written in the usual 2×2matrix notation according to (the same holds for the primed states) 3See also [27,28] for discussions of generalized inverse seesaw models, including versions with Dirac neutrinos, [29–32] for four references studying the phenomenology of light fermion triplets and [33] for a recent work on the inverse seesaw with spontaneous violation of lepton number. 4In order to simplify the notation, we will not denote the chirality of the ΣA≡ΣRAand Σ0 A≡Σ0 RAfermions explicitly. ESCRIBANO, TEROL-CALVO, and VICENTE PHYS. REV. D 103, 115018 (2021) 115018-2 ΣA¼1 ffiffiffi 2 pτ·  ΣA¼Σ0 A=ffiffiffi 2 pΣþ A Σ− A−Σ0 A=ffiffiffi 2 p;ð4Þ where τAare the usual Pauli matrices and the states with well-defined electric charge are given by Σ0 A¼Σ3 A;Σ A¼Σ1 A∓iΣ2 A ffiffiffi 2 p:ð5Þ Finally, the VL leptons LL;R can decomposed as LL;R ¼N EL;R :ð6Þ Under the above working assumptions, the most general Yukawa Lagrangian allowed by all symmetries can be written as LY¼LSM YþLISS3 YþLVL Y;ð7Þ where −LSM Y¼¯ qLYuuR ˜ Hþ¯ qLYddRHþ¯ lLYeeRHþH:c:ð8Þ is the usual SM Lagrangian, with ˜ H¼iτ2H, and Yu;d;e are 3×3Yukawa matrices in flavor space. Here and in the following we omit SUð2ÞLcontractions and flavor indices to simplify the notation. The terms −LISS3 Y¼ffiffiffi 2 p¯ ΣYΣlL ˜ H†þ¯ ΣMΣΣ0cþ1 2 ¯ Σ0μΣ0cþH:c:; ð9Þ correspond to the usual ISS3 extension. Here YΣ,MΣand μ are 3×3matrices, the latter two with dimensions of mass. Finally, the VL leptons allows for additional Lagrangian terms, given by −LVL Y¼ffiffiffi 2 p¯ ΣλLLL ˜ H†þ¯ eRλRLL ˜ H†þ¯ LLMLLR þ¯ lLϵLRþH:c:ð10Þ Here λLand λRare dimensionless 3×1vectors and MLa parameter with dimensions of mass. The 1×3vector ϵhas dimensions of mass, and will be assumed to vanish for simplicity.5The guiding principle when writing the Yukawa Lagragian in Eq. (7), in particular the piece in Eq. (9), is the conservation of lepton number, only allowed to be broken by the ¯ Σ0μΣ0cterm. In fact, in the absence of the Majorana mass μ, the Lagrangian would have an additional Uð1ÞLglobal symmetry. In the following we will consider μ≪MΣ, corresponding to a slightly broken lepton number, in the spirit of the original inverse seesaw mechanism.6 The scalar potential of the model is the same as in the SM, V¼m2jHj2þλjHj4;ð11Þ with m2a parameter with dimensions of ½mass2. Therefore, electroweak symmetry breaking takes place in the usual way, with hHi¼ 1 ffiffiffi 2 p0 v;ð12Þ with v≃246 GeV the SM Higgs vacuum expectation value. After electroweak symmetry breaking, several terms in the Yukawa Lagrangian in Eq. (7) induce mixings in the neutral and charged lepton sectors. In the bases n≡nL¼ðνL;ðΣ0Þc;ðΣ00Þc;NL;Nc RÞ,fL¼ðeL;ðΣþÞc; ðΣ0þÞc;E LÞand fR¼ðeR;Σ−;Σ0−;E RÞ, the neutral and charged fermion mass terms read −Lm¼1 2 ¯ ncMNnþfLMCfRþH:c:; ð13Þ with the mass matrices given by MN¼ 0 B B B B B B @ 0mT D000 mD0MΣmL0 0MT Σμ00 0mT L00ML 000ML0 1 C C C C C C A ;ð14Þ and TABLE I. Particle content of the ISS3VL. qL,lL,uR,dR,eR, and Hare the usual SM fields. qLuRdRlLeRΣΣ 0LLLRH SUð3ÞC3¯ 3¯ 31111111 SUð2ÞL211 2 1332 22 Uð1ÞY 1 6 2 3−1 3−1 2−100−1 2−1 2 1 2 GENERATIONS 333 3 3331 11 5The ϵterm contributes to the electron, muon and tau masses and is therefore constrained to be small. 6In principle, a term of the form ¯ Σμ0Σcis also allowed by all symmetries. However, it is well known that such a term would contribute to neutrino masses in a subdominant way if μand μ0 are of the same order, see for instance [28]. Therefore, we neglect this term in the following. ðg−2Þe;μIN AN EXTENDED INVERSE TYPE-III …PHYS. REV. D 103, 115018 (2021) 115018-3 MC¼ 0 B B B B B @ meffiffiffi 2 pmT D00 00MΣ0 0MT Σμ0 mT Rffiffiffi 2 pmT L0−ML 1 C C C C C A :ð15Þ Here we have defined mD¼v ffiffiffi 2 pYΣ;m e¼v ffiffiffi 2 pYe;m L¼v ffiffiffi 2 pλL and mR¼v ffiffiffi 2 pλR:ð16Þ We note that the neutral lepton mass matrix MNis 11 ×11, whereas the charged lepton mass matrix MCis 10 ×10. They can be brought to diagonal form by means of the unitary transformations U,VLand VR, defined by UMNU†¼diagðmNiÞ;ð17Þ VLMCVR†¼diagðmχiÞ;ð18Þ resulting in the 11 neutral (Majorana) fermion masses mNi and the 10 charged (Dirac) fermion masses mχj, with i¼ 1;…;11 and j¼1;…;10. In the following we will assume the hierarchy of energy scales μ≪mD;m L;m R≪MΣ;ML;ð19Þ which allows one to obtain approximate expressions for the physical lepton masses. We note that a small μparameter is justified through ’t Hooft naturalness criterion [70]. In the case of the neutral leptons, one finds 3 light states, to be identified with the standard light neutrinos. Their mass matrix is approximately given by mν≈mT DðMT ΣÞ−1μM−1 ΣmD;ð20Þ with corrections of the order of the small ratios ðmD=MΣÞ2 and ðmL=MLÞ2. This result is proportional to the μ parameter. It is then clear that sizable YΣYukawa couplings and triplets at the TeV scale are consistent with light neutrino masses due to the suppression by the μterm. This is the inverse seesaw mechanism. III. CHARGED LEPTON ANOMALOUS MAGNETIC MOMENTS The charged lepton magnetic moments can be described by the effective Hamiltonian [71] H¼cij ¯ ljσμνPRliFμν þH:c:; ð21Þ where PR¼ð1þγ5Þ=2is the usual right-handed chiral projector, Fμν the electromagnetic field strength tensor and lidenote the light charged lepton mass eigenstates, equivalent in our scenario to χ1;2;3. The anomalous magnetic moment is given in terms of the real components of the diagonal ccoefficients as ai¼−2mi eðcii þc iiÞ¼−4mi eRe cii;ð22Þ whereas the imaginary components would in turn induce electric dipole moments. The ISS3VL has the ingredients to induce large charged lepton anomalous magnetic moments, namely, light new particles with sizable couplings to the charged leptons. In the ISS3VL, new contributions to the charged lepton anomalous magnetic moments are induced at the 1-loop level, as shown in Fig. 1. While these diagrams also exist in the SM, in the ISS3VL the mass eigenstates Niand χi include new heavy states beyond the SM leptons. Moreover, the couplings of the SM states get modified due to mixings with the new BSM states. The amplitudes of these Feynman diagrams are given by7 iMW l¼Zd4q ð2πÞ4¯ ulðp0Þiγβ½ðgL χNWÞljPLþðgR χNWÞljPRi = qþmNj q2−m2 Nj ×iγα½ðgL χNWÞjlPLþðgR χNWÞjlPR −iðgα˜α−ðp−qÞαðp−qÞ˜α m2 WÞ ðp−qÞ2−m2 W iΓμ˜α ˜ β × −iðgβ ˜ β−ðp0−qÞβðp0−qÞ˜ β m2 WÞ ðp0−qÞ2−m2 W ulðpÞεμ;ð23Þ 7In order to obtain the correct sign for the ISS3VL contributions to the electron and muon g−2one must use a consistent set of sign conventions for the Feynman rules of the model. We used the useful Ref. [72] to guarantee the consistency of the amplitudes in Eqs. (23)–(25). ESCRIBANO, TEROL-CALVO, and VICENTE PHYS. REV. D 103, 115018 (2021) 115018-4 iMZ l¼Zd4q ð2πÞ4¯ ulðp0Þiγβ½ðgL χZÞilPLþðgR χZÞilPRið = p0− = qÞþmχi ðp0−qÞ2−m2 χi ×iγμ½ðgL χγÞiiPLþðgR χγÞiiPRið = p− = qÞþmχi ðp−qÞ2−m2 χi iγα½ðgL χZÞliPLþðgR χZÞ2iPR × −iðgαβ −qαqβ m2 ZÞ q2−m2 Z ulðpÞεμ;ð24Þ iMh l¼Zd4q ð2πÞ4¯ ulðp0Þi½ðgL χhÞilPLþðgR χhÞilPRið = p0− = qÞþmχi ðp0−qÞ2−m2 χi ×iγμ½ðgL χγÞiiPLþðgR χγÞiiPRið = p− = qÞþmχi ðp−qÞ2−m2 χi i½ðgL χhÞliPLþðgR χhÞliPR ×i q2−m2 h ulðpÞεμ:ð25Þ Here PL¼ð1−γ5Þ=2is the left-handed chiral projector, εμ is the photon polarization 4-vector and the couplings gL;R χNW, gL;R χZ,gL;R χγ ,gL;R χhand Γare defined in Appendix A. A sum over the indices i,jis implicit in these expressions, while l is the index of the external charged lepton. We have computed the amplitudes in Eqs. (23)–(25) with the help of Package-X [73]. After projecting onto the operator in Eq. (21), one obtains analytical expressions for the contributions to the ccoefficient, which can then be translated into contributions to the charged leptons g−2 thanks to the relation in Eq. (22). The total ISS3VL contribution to Δalcan be written as8 (a) (b) (c) FIG. 1. Feynman diagrams that contribute to the charged lepton anomalous magnetic moment at the 1-loop level in the ISS3VL. Here χiand Njdenote any of the charged and neutral lepton mass eigenstates, respectively. Momenta are shown in blue. (a) W. (b) Z. (c) h. 8Higher-order contributions, such as those induced by 2-loop Barr-Zee diagrams [74], will be neglected in the following. ðg−2Þe;μIN AN EXTENDED INVERSE TYPE-III …PHYS. REV. D 103, 115018 (2021) 115018-5 Δal¼ΔalðWÞþΔalðZÞþΔalðhÞ:ð26Þ Assuming that the ISS3VL states Niand χiare much heavier than the SM states (this is, assuming the mass hierarchy mNi;m χi≫mh;m W;m Z≫ml), one can find approximate expressions for the three contributions: ΔalðWÞ ≃ml 32π2m2 W4 3ml1−3m2 W 4m2 Ni11 þ6log m2 W m2 NiðC2 χNWÞil −mNi1−3m2 W m2 Ni3þ2log m2 W m2 NiðD2 χNWÞil;ð27Þ ΔalðZÞ≃ml 32π2m2 Z−5 3mlðC2 χZÞilþmχiðD2 χZÞil;ð28Þ ΔalðhÞ≃ml 32π2m2 χi1 3mlðC2 χhÞilþmχiðD2 χhÞil:ð29Þ Here we have defined the coupling combinations ðC2 YÞil≡jðgL YÞilj2þjðgR YÞilj2and ðD2 YÞil≡ðgL YÞilðgR YÞ ilþðgL YÞ ilðgR YÞil;ð30Þ with Y¼χNW,χZ,χh. Again, the indices iand l denote the BSM particle running in the loop and the charged lepton, respectively. We have checked that Eqs. (27),(28), and (29) reproduce the ISS3VL contributions to the charged leptons anomalous magnetic moments in very good approximation. Nevertheless, full expressions are given in Appendix Band used in the numerical analysis presented in the next section. We note that ΔalðWÞ,ΔalðZÞ, and ΔalðhÞcontain contributions proportional to mNgL χNWgR χNW,mχgL χZgR χZ, and mχgL χhgR χh, respectively, proportional to the mass of the fermion in the loop. These terms are usually called chirally enhanced contributions and they typically dominate due to the large masses of the heavy fermions running in the loop. IV. PHENOMENOLOGICAL DISCUSSION We proceed now to present our phenomenological exploration of the parameter space of the ISS3VL. A. Experimental constraints Let us first discuss how we fix the parameters of the model in order to reproduce the measured lepton masses and mixings. The YeYukawa matrix will be fixed to the same values as in the SM, hence neglecting corrections from the mixing between the SM charged lepton states and the charged components of the Σand Σ0triplets. These corrections are multiplicative and enter at order ∼ðmD=MΣÞ2and can thus be safely neglected. The same argument applies to the mixing with the charged components of the VL leptons, which enter at order ∼ðmR=MLÞ2. Without loss of generality, we will work in the basis in which MΣis diagonal and μis a general complex symmetric matrix. In this case, YΣand μmust be properly fixed in order to reproduce neutrino oscillation data [75]. In principle, one can fix the entries of μ to some input values and express the YΣYukawa matrix by means of the master parametrization [76,77], which in this case reduces to a modified Casas-Ibarra parametrization [78]. While this is perfectly valid, one generically obtains YΣmatrices with sizable off-diagonal entries unless some input parameters are tuned very finely. Due to the strong constraints from the nonobservation of lepton flavor violating processes, this excludes most of the parameter points. Therefore, we take the alternative choice of fixing YΣto specific input values, diagonal for simplicity, and computing μby inverting Eq. (20) as μ¼MT ΣðmT DÞ−1mνm−1 DMΣ;ð31Þ where mν¼U ν ˆ mνU† ν. Here Uνis the leptonic mixing matrix measured in oscillation experiments, given in terms of 3 mixing angles and 3 CP-violating phases, while ˆ mνis a diagonal matrix containing the physical neutrino mass eigenvalues. Equation (31) guarantees that all the parameter points considered in our numerical analysis are compatible with neutrino oscillation data. In our analysis we use the results of the global fit in [75] and we consider both normal and inverted neutrino mass orderings. In order to ensure compatibility with constraints from flavor and electroweak precision data we use the bounds derived in [35], where a global analysis is performed in the context of general type-III seesaw models. The limits provided in this reference are given for the 3×3matrix η, defined in our case in terms of the matrices MD¼mD 0and M¼0MΣ MT Σμ;ð32Þ as η¼1 2M† DðM†Þ−1M−1MD ¼1 2m† DðM† ΣÞ−1½I3þμðM ΣÞ−1ðMT ΣÞ−1μM−1 ΣmD ≈1 2m† DðM† ΣÞ−1M−1 ΣmD:ð33Þ In our analysis, we make sure that the bounds are respected by computing the ηmatrix in all the parameter points considered. As we will explain below, these limits imply very small BSM contributions in the ISS3, thus ESCRIBANO, TEROL-CALVO, and VICENTE PHYS. REV. D 103, 115018 (2021) 115018-6 motivating our ISS3VL extension. Furthermore, we also consider the decay widths for the processes Z→lþl− and h→lþl−, with l¼e,μ, which are also affected due to the mixing of the light charged leptons with the heavy states in our model. They are computed as ΓðZ→lþl−Þ¼ m3 Z 12πv2½jðgV χZÞllj2þjðgA χZÞllj2;ð34Þ Γðh→lþl−Þ¼mh 8π½jðgL χhÞllj2þjðgR χhÞllj2;ð35Þ with gV χZ¼gL χZþgR χZand gA χZ¼gR χZ−gL χZ. The Z→ lþl−decay turns out to provide an important constraint in our setup. In fact, it has been recently pointed out that this process potentially correlates with the charged leptons g−2[79]. We define the ratios RZll ¼ΓðZ→lþl−Þ ΓSMðZ→lþl−Þ;ð36Þ with ΓSMðZ→lþl−Þthe SM predicted decay width, and impose that RZll lies within the 95% CL range, which we estimate to be 0.995 <R Zee <1.003 and 0.993 < RZμμ <1.006 [80]. Regarding the Higgs boson decays, no constraints are actually obtained from them, since at present there is no hint for h→eþe−and evidence for h→μþμ−was only obtained recently [81]. Therefore, they will be considered as predicted observables, potentially correlated with Δal[79,82]. Finally, we also impose bounds from collider searches. The type-III seesaw triplets have been searched for at the LHC in multilepton final states, both by ATLAS [83] and CMS [84,85]. No excess above the expected SM backgrounds has been found, hence allowing the experimental collaborations to set limits on the triplet mass and couplings. Using a data sample obtained with proton collisions at ffiffiffi s p¼13 TeV and an integrated luminosity of 35.9fb−1, CMS reports a lower bound on the triplet mass of 840 GeV at 95% confidence level, if the triplet couplings are assumed to be lepton flavor universal [85]. While the flavor structure of the triplet couplings to leptons does not affect the heavy triplet pair production cross sections, driven by gauge interactions, it has an impact on the flavor composition of the multilepton signature. The limit changes if the assumption of lepton flavor universal couplings is dropped, resulting in a more stringent bound when the triplet couples mainly to electrons and a more relaxed bound when the triplet couples mainly to taus, with values ranging between 390 and 930 GeV. The bounds from the CMS collaboration in [85] are applied in our analysis. However, since the CMS analysis focuses on the standard type-III seesaw scenario, and does not consider the particular features of the ISS3VL model, several simplifying assumptions must be made. We define BAα¼ΓðΣ0 A→lαþbosonÞþΓðΣþ A→lαþbosonÞ Pα½ΓðΣ0 A→lαþbosonÞþΓðΣþ A→lαþbosonÞ; ð37Þ where Σ0 Aand Σþ Aare the quasi-Dirac pairs approximately formed by the mass eigenstates NiþNjand χiþχj, respectively. An implicit sum over the bosons in the final states is also assumed, including decays to W,Zand h. For instance, in a parameter point in which the lightest BSM states are mainly composed by the components of the Σ1triplet, we have i¼4,j¼5and ΓðΣ0 1→lαþ bosonÞ≡ΓðN4→Wl∓ÞþΓðN4→ZναÞþΓðN4→hναÞþ ΓðN5→Wl∓ÞþΓðN5→ZναÞþΓðN5→hναÞ. We note that BAe þBAμþBAτ¼1. This is the quantity that we use to confront each quasi-Dirac pair with the limits given on Figure 3 of [85]. Our approach approximates the total heavy triplet pair production to pp →Σ0 AΣþ A, which is known to give the dominant contribution at the LHC [86]. Furthermore, we apply two additional simplifications. First, since CMS assumes the neutral and charged components of the triplet to be mass degenerate, we adopt a conservative approach and take the lowest of them as the triplet mass to be used in the analysis. And second, we do not apply the CMS bounds to quasi-Dirac triplet pairs that are largely mixed with the VL leptons, since their production cross section is clearly reduced with respect to the pure triplet case.9We believe that our assumptions conservatively adapt the CMS limits in [85] to our scenario. We note that ATLAS finds a similar bound on the triplet mass in the flavor universal scenario, ruling out (at 95% confidence level) values below 790 GeV [83]. Finally, LHC limits on VL leptons strongly depend on their decay modes, namely the flavor of the charged leptons produced in the final states [87]. In our analysis we will consider ML≥500 GeV, a conservative value that guarantees compatibility with current LHC searches. These limits are expected to be improved by the end of the LHC Run-III [88]. B. ðg−2Þe;μin the ISS3 Before studying the electron and muon g−2in the ISS3VL, let us discuss these observables in the context of the pure ISS3 and show that this model is unable to address the existing discrepancies. One can easily reach this conclusion by estimating the size of the 9In practice, we do not apply the CMS bounds in cases with large mixings. For instance, they are not applied to χiDirac states that combine a left-handed fermion that is mostly a type-III triplet with a right-handed fermion that is mostly a VL lepton, or vice versa. ðg−2Þe;μIN AN EXTENDED INVERSE TYPE-III …PHYS. REV. D 103, 115018 (2021) 115018-7 dominant contributions to the charged lepton g−2. Figure 2shows the dominant Wcontribution. Assuming that the chirally enhanced term in Eq. (27) dominates, one finds the estimate ΔalðWÞ∼− 1 32π2 mNml m2 W g2mD MΣ2 ll ml v∼−10−3mNm2 l m3 W ηll: ð38Þ First of all, we notice that this contribution is always negative since ηll >0. Therefore, it cannot accommodate the muon g−2anomaly, which requires Δaμ>0. This result was already found in early studies of the charged leptons anomalous magnetic moments in seesaw scenarios [89–91], as well as in [92]. Furthermore, the absolute value of ΔalðWÞis also too small to account for the anomalies. This implies that the electron g−2cannot be explained either in the ISS3. In this regard, we highlight the relevance of the ml=v factor in Eq. (38). This factor is not apparent when inspecting the analytical expressions for the couplings in Appendix A. In fact, the individual contributions to ΔalðWÞby the neutral fermions in the loop are larger than their sum, ΔalðWÞ, by a factor ∼v=ml. Therefore, a strong cancellation among them takes place. This cancellation can be easily understood due to the chirality-flipping nature of the dipole moment operator in Eq. (21). The factor ml=v is required to flip the chirality of the fermion line and induce a contribution to a dipole moment. One can now consider mN¼1TeV to obtain ΔaeðWÞ∼−5×10−13ηee;ð39Þ ΔaμðWÞ∼−2×10−8ημμ:ð40Þ Since ηee and ημμ are constrained to be smaller than ∼10−4 [35], these contributions fail to address the electron and muon g−2anomalies by several orders of magnitude. The same argument can be applied to the Zand hcontributions to find that they are actually even more suppressed. In summary, the suppression by the ml=mWchirality flip and the stringent bounds on ηll imply that the ISS3 cannot induce sizable contributions. This, added to the fact that the contributions to the muon g−2have the wrong sign, implies that the ISS3 cannot explain the deviations in the electron and muon anomalous magnetic moments. We now proceed to show that the additional ingredients in our extended model can alter this conclusion. C. ðg−2Þe;μin the ISS3VL As already discussed, the ISS3 cannot explain the experimental anomalies in the electron and muon anomalous magnetic moments. Therefore, we now consider its ISS3VL extension. In this case one has Wcontributions such as the one shown in Fig. 3. We can now derive an analogous estimate, along the same lines as in the case of the ISS3. One finds jΔalðWÞj∼1 32π2 mNml m2 W g2mD MΣll mL ML mR ML ∼10−3mNmlmLmR m2 WM2 Lffiffiffiffiffiffiffi ηll p:ð41Þ One can now choose mN¼1TeV, ML¼500 GeV, mL¼200 GeV, and mR¼10 GeV to obtain jΔaeðWÞj∼6×10−10 ffiffiffiffiffiffi ηee p;ð42Þ jΔaμðWÞj∼10−7ffiffiffiffiffiffi ημμ p:ð43Þ Therefore, even after the suppression given by ffiffiffiffiffiffiffi ηll p≲ 10−2these Wcontributions can address the current discrepancies with the electron and muon g−2measurements. Furthermore, the signs of these contributions are not fixed and can be properly adjusted by fixing the signs of the relevant Yukawa couplings. We note that the loop in Fig. 3 is proportional to the product YΣλLλRwhich, as shown below, will be crucial for the resulting values for Δalin the ISS3VL model. Similar hcontributions are also found, again proportional to the YΣλLλRproduct. Therefore, the model is in principle capable of producing sizable contributions to the electron and muon g−2. We now proceed to confirm this by performing a detailed numerical analysis of the parameter space of the model. Since we are interested FIG. 3. Dominant Wcontribution in the ISS3VL. Mass insertions are represented by white blobs. FIG. 2. Dominant Wcontribution in the ISS3. Mass insertions are represented by white blobs. ESCRIBANO, TEROL-CALVO, and VICENTE PHYS. REV. D 103, 115018 (2021) 115018-8 in Δaeand Δaμ, we fix ðλLÞ3¼ðλRÞ3¼0and randomly scan within the following parameter ranges: Parameter Min Max ðMΣÞii 850 GeV 1.5 TeV ML500 GeV 1.5 TeV ðYΣÞii 0.05 0.2 ðλLÞ1−ffiffiffiffiffi 4π p−0.1 ðλLÞ20.1 ffiffiffiffiffi 4π p ðλRÞ10.05 0.5 ðλRÞ20.05 0.5 Some comments about the chosen ranges are in order. First, the ranges for the mass parameters ðMΣÞii and ML have been selected following the discussion on LHC bounds of Sec. IVA. Many of the parameter points in our scan were ruled out due to LHC searches for triplets, but we also find that a substantial fraction pass the test. The ranges for the Yukawa couplings have been chosen in order to maximize the resulting Δal. The usual ISS3 Yukawas ðYΣÞii have been scanned around their maximal values compatible with the ηii bounds. A relative sign between ðλLÞ1and ðλLÞ2has been introduced in order to obtain Δae<0and Δaμ>0, as required by the experimental hints. Finally, the corrections to ΓðZ→lþl−Þtend to be too large unless ðλRÞ1;2≲0.5. Our results are based on a random scan with 50.000 parameter points, out of which about 12%–13% pass all the experimental tests. We have selected normal neutrino mass ordering. However, we have also run a second scan with inverted ordering and found the same qualitative results. As already explained, we consider a scenario with diagonal YΣ and MΣmatrices. In this case, the lepton mixing angles encoded in the matrix Uνare generated by the off-diagonal entries of the μmatrix and all lepton flavor violating processes are strongly suppressed. For this reason, the bounds on the ηij entries, with i≠j, are easily satisfied. In contrast, the bounds on the diagonal elements of the η matrix turn out to be very important, removing a significant amount of the parameter points considered and implying the approximate bounds ðYΣÞ11 ≲0.2and ðYΣÞ22 ≲0.15 for triplet masses of the order of the TeV. Another very important constraint in our setup is provided by the decay Z→lþl−. The mixing between the SM charged leptons and the new charged BSM states from the Σand Σ0triplets and LL;R VL doublets reduces ΓðZ→lþl−Þwith respect to its SM value. This has a strong impact on the mD=MΣ and mR=MLratios. Since these ratios must be sizable in order to induce large contributions to Δal, see Fig. 3, this limit is crucial for the correct evaluation of our scenario. Finally, the CMS limits discussed in Sec. IVA also have an impact, discarding some of the parameter points in our scan. Our choice of a diagonal YΣ(in the basis in which MΣis diagonal too) implies that the mixing among different triplets is typically very small. In this case, and unless there is a large mixing with the VL neutral leptons, two of the heavy neutral mass eigenstates are given in good approximation by the Σ1−Σ0 1quasi-Dirac pair and couple FIG. 4. Δae(left) and Δaμ(right) as a function of the product ðYΣÞiiðλLÞiðλRÞi. Blue dots correspond to parameter points that pass all the experimental constraints, whereas gray points are experimentally excluded. The horizontal dashed lines represent the central values for Δaeand Δaμ, whereas 1σ(3σ) regions are displayed as yellow (green) bands. FIG. 5. Δaμas a function of the combination ðYΣÞ22ðλLÞ2ðλRÞ2=½ðMΣÞ2M2 L. Dashed line, horizontal bands and color code as in Fig. 4. ðg−2Þe;μIN AN EXTENDED INVERSE TYPE-III …PHYS. REV. D 103, 115018 (2021) 115018-9 [52] J. Kawamura, S. Raby, and A. 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