sign in sign up
<<
Home , Chirality (physics), Leptoquark

PHYSICAL REVIEW D 102, 075037 (2020)

Getting right: Single scalar leptoquark solutions − to the ðg 2Þe;μ puzzle † Innes Bigaran * and Raymond R. Volkas ARC Centre of Excellence for Physics at the Terascale, School of Physics, The University of Melbourne, Victoria 3010, Australia

(Received 10 March 2020; revised 5 August 2020; accepted 21 September 2020; published 28 October 2020)

We identify the two scalar leptoquarks capable of generating sign-dependent contributions to leptonic magnetic moments, R2 ∼ ð3; 2; 7=6Þ and S1 ∼ ð3; 1; −1=3Þ, as favored by current measurements. We consider the case in which the and sectors are decoupled, and real-valued Yukawa couplings are specified using an up-type mass-diagonal basis. Contributions to Δae arise from charm- containing loops and Δaμ from top-containing loops—hence avoiding dangerous LFV constraints, particularly from μ → eγ. The strongest constraints on these models arise from contributions to the Z leptonic decay widths, high-pT leptonic tails at the LHC, and from (semi)leptonic decays. To be a − 2 comprehensive solution to the ðg Þe=μ puzzle we find that the mass of either leptoquark must be ≲65 TeV. This analysis can be embedded within broader flavor studies, including those of hierarchical leptoquark coupling structures. It can also be straightforwardly adapted to accommodate future measurements of leptonic magnetic moments, such as those expected from the Muon g − 2 collaboration in the near future.

DOI: 10.1103/PhysRevD.102.075037

exp SM I. INTRODUCTION Δaμ ¼ aμ − aμ ; ð2Þ The remarkable agreement between measurements and 1 predictions of the muon and electron magnetic dipole corresponding to a 3.6σ anomaly : moments has long been testament to the success of −11 quantum field theory. Precise measurements of the Δaμ ¼ð286 63 43Þ × 10 : ð3Þ deviation of this observable from the classical, tree-level value, gl ¼ 2, give a sensitive probe of higher-order effects Similarly, recent experimental results have indicated a within the (SM) and beyond. SM correc- deviation for the electron magnetic moment, of 2.5σ tions are precisely known, and therefore these specify the significance [5]: SM quantity al , where −12 Δae ¼ −ð0.88 0.36Þ × 10 : ð4Þ 1 al ≡ ðg − 2Þl: ð1Þ 2 It is important to note that the sensitivity of al to NP at m n energy scale Λ scales generally as ml =Λ , for some integer n m − 2 , . This indicates that a heavier generally provides This makes anomalies in ðg Þl, particularly if these a more sensitive probe of NP. However, due to the short differ between lepton flavor, a very strong indication of new lifetime of the , a precise measurement of its magnetic physics (NP) effects at loop level [1,2]. moment (and, consequentially, any deviation from the SM) For the muon, there is persistent deviation between the is beyond the reach of current experiments. The important SM prediction and the measured value [3,4], issue to be addressed in this paper is that the discrepancies, Δaμ and Δae, are of opposite sign, which is a difficulty to *[email protected] 2 † be overcome when searching for a common explanation. [email protected]

Published by the American Physical Society under the terms of 1The uncertainty values refer to the experimental and theoretical the Creative Commons Attribution 4.0 International license. prediction uncertainties, respectively. See also the Appendix at the Further distribution of this work must maintain attribution to end of the conclusion. the author(s) and the published article’s title, journal citation, 2See, for example, Refs. [6–16] for alternative methods to and DOI. Funded by SCOAP3. explain these anomalies in a single model.

2470-0010=2020=102(7)=075037(13) 075037-1 Published by the American Physical Society INNES BIGARAN and RAYMOND R. VOLKAS PHYS. REV. D 102, 075037 (2020)

The leading candidates to explain these deviations ðg − 2Þl. The generic effective Lagrangian corresponding involve flavor-dependent, loop-level, NP effects. It has to contributions to al is given by: long been established that exotic scalar-only extensions to the SM are capable of generating sizeable corrections to L l¯ γ μ al σ μν l al ¼ e μA þ μνF ; ðg − 2Þl. Of particular interest are scalar leptoquark (LQ) 4ml models, which have proven to be useful for reconciling 1 ⊂ l¯γ μl l¯σ μν σl σl l other well-known flavor-dependent anomalies (e.g., see e μA þ 2 ie μνF ð LPL þ RPRÞ : ð6Þ Refs. [17–21]), provide a portal to generating radiative mass [22], and are embedded in a number of μν ∂ − ∂ σl where F ¼ μAν νAμ, and L=R parametrize the effec- theories of unification (e.g., see Ref. [23]). tive left- and right-chiral interactions. Equation (6) reveals, via coefficient matching, that A. Chirality of scalar LQ models Δ σl σl To begin characterizing scalar LQ models, we first need al ¼ imlð L þ RÞ: ð7Þ to introduce some terminology. Motivated by the intro- duction of direct lepton-quark couplings, rather than For the purpose of this discussion, for general scalar LQ separately considering lepton (L) or (B) number ðϕÞ models, the couplings to charged (l) and conservation, these are absorbed into the definition of a (q) can be expressed as follows: new conserved quantity [24] number, F; ðcÞ R L † Ll ¼ l ½y P þ y P q ϕ þ H:c: ð8Þ F ¼ 3B þ L: ð5Þ R L lðcÞ l 0 lðcÞ lc 2 F is well defined for each of the finite number of LQ where ¼ for jFj¼ and ¼ for jFj¼ . models, and characterizes the types of interactions medi- We consider the two leading-order topologies for these corrections illustrated in Fig. 1. Their contributions to σl ated. The jFj¼2 LQs couple to multiplets of the form lq, L=R are well established in the literature [17].ForF ¼ 0 and jFj¼0 couple to l¯q [25]. Table I gives an overview of scalar LQs: the scalar LQs and their gauge-group transformations, adopting the symbol notation from Ref. [17]. iN X σl c R 2 L 2 κ R L κ0 The important characteristic of models that can generate L ¼ 2 2 ½mlðjylqj þjylqj Þ þylqylqmq ; ð9Þ − 2 16π mϕ contributions to ðg Þðe=μÞ that are consistent with experi- q ment is the chirality of their Yukawa couplings. To generate σl − σl one-loop corrections whose sign can vary between lepton and R ¼ ½ L . In Eq. (9), flavors, the LQ must have mixed-chiral couplings; i.e., both left- and right-handed couplings to charged leptons are κðxqÞ¼QϕfSðxqÞ − fFðxqÞ; present. To see why this is, we begin with a brief overview 0 κ ðxqÞ¼QϕgSðxqÞ − gFðxqÞ; ð10Þ of the established calculations for al corrections from scalar LQ states. 2 2 where xq ¼ mq=mϕ. These contributions are proportional to the number of colors, N ¼ 3, and are summed over B. Scalar LQs for ðg − 2Þl c quark flavors q running in the loop. The of In this section, we follow the calculation procedure for l → l0γ from Ref. [17], but adapt it specifically for

TABLE I. Scalar LQs and their transformation properties, under the hypercharge convention Q ¼ I3 þ Y. The penultimate column indicates whether the model is able to generate one-loop (1L) corrections to the muon and electron magnetic moments with opposite sign—i.e., the LQ has mixed-chiral couplings. 3 ⊗ 2 ⊗ 1 − 2 Symbol SUð ÞC SUð ÞL Uð ÞY ðg Þl at 1L jFj ˜ S1 ð3; 1; −4=3Þ ✗ 2 S1 ð3; 1; −1=3Þ ✓ 2 S3 ð3; 3; −1=3Þ ✗ 2 ¯ S1 ð3; 1; 2=3Þ ✗ 2 FIG. 1. Dominant contributions to the lepton magnetic moment R2 ð3; 2; 7=6Þ ✓ 0 from scalar LQs. Arrows indicating fermion flow are omitted as ˜ there are multiple valid assignments possible for these topologies, R2 ð3; 2; 1=6Þ ✗ 0 each of which will be considered in calculations.

075037-2 GETTING CHIRALITY RIGHT: SINGLE SCALAR LEPTOQUARK … PHYS. REV. D 102, 075037 (2020) the field ϕ is given by Qϕ, and the loop functions in (10) are C. Is there a no-go theorem for single scalar LQs − [17,26]: to generate ðg 2Þe=μ? As the mixed-chiral contributions in Eq. (13) are propor- tional to the mass of the quark in the loop, we may initially, x þ 1 x logðxÞ f ðxÞ¼ − ; naively, restrict LQ couplings to represent a top-philic S 4ðx − 1Þ2 2ðx − 1Þ3 coupling texture, which maximizes this mq enhancement. x2 − 5x − 2 x logðxÞ However, we will begin here by discussing why such a f ðxÞ¼ þ ; F 12ðx − 1Þ3 2ðx − 1Þ4 model is severely disfavored. In Ref. [7], the authors show that, for a single-field 1 logðxÞ g ðxÞ¼ − ; extension of the SM, with the muon and electron sector not S x − 1 ðx − 1Þ2 decoupled, the anomalies in Δaμ=e are incompatible with x − 3 logðxÞ the rare decay μ → eγ. For the scalar leptoquarks discussed gFðxÞ¼ 2 þ 3 : ð11Þ 2ðx − 1Þ ðx − 1Þ in Sec. IB, from Fig. 1 if the quark (qi) coupling to the leptoquark (ϕ) is identical for both charged-leptons, then the μ → eγ transition can be obtained by combining a Therefore, we conclude via Eq. (7) that subset of the leptoquark vertices involved in generating Δaμ=e. As a result, the following expression holds: 3 X ml R 2 L 2 2 2 Δal ¼ − ½mlðjy j þjy j Þκðx Þ mμ 8π2 2 l l q μ → γ e Δ Δ ∼ 9 10−5 mϕ q Br½ e ¼ j aμ aej × ; ð14Þ 64πΓμ me L R 0 þ mqReðyl yl Þκ ðxqÞ: ð12Þ where Γμ is the full decay width of the muon. This algebraic expression is consistent with that quoted in [7], within the For a scalar LQ with maximally chiral Yukawa couplings, justifiable limit that me ≪ mμ. The numerical value is a the second term will not be present and contributions from result of inputting the results from (3) and (4), which is each propagator will be of definite relative sign. However, eight orders of magnitude above the current experimental for mixed-chiral scalar LQs, we can access terms propor- bound on this process from the MEG collaboration [28]: tional to κ0, allowing us to vary the sign of the NP contribution. As these contributions scale proportional to Br½μ → eγMEG < 4.2 × 10−13: ð15Þ mq, we expect the dominant contributions to be those with the third-generation quarks entering the loop. For this reason, the single-leptoquark solution to this Taking ml ≪ mq, the like-handed terms are subdomi- — process seems to be heavily disfavored. However, the nant to the mixed-handed contributions exactly the terms validity of Eq. (14) relies on the assumption that the quark required for generating contributions with relative sign. in the loop of Fig. 1 is the same for both the muon and This leaves the following: electron Δal contributions. This no-go result could be avoided by relaxing this assumption. 3 X Without a like-quark coupling between the leptoquark and ml L R 0 Δal ∼− m Re y y κ x : 13 8π2 2 q ð l;q l;qÞ ð qÞ ð Þ both charged-lepton generations, we can avoid the constraint mϕ ∶ ≪ fq ml mqg from Br½μ → eγ by restricting any contributions to be a higher-order process. The same philosophy applies to other lepton-flavor violating processes such as Z → eμ, KL → eμ For jFj¼2 LQs, the above applies but with Qϕ ↦ −Qϕ, and muon-electron conversion in nuclei. The relative success R ↦ L L ↦ R yl yl and yl yl . of such a restriction is dependent on the flavor ansatz for For mixed-chiral scalar LQ models, this provides a portal Yukawa couplings, as will be further discussed in Section II to flavor-dependent sign allocation for the correction. We A. Althoughthis decouplingmay reduce the dominance of the have the clear prospect of meeting current experimental Δ 3 mixed-chiral term to al, it remains to be seen whether such a measurements with simple, single-scalar field extensions, model could still be successful. such as those identified in Table I. Please note that we simply require the mixed-chiral term to dominate, but that II. MODELS OF INTEREST the like-chiral terms will also be included in later calcu- lations (see Sec. III). As summarized in Table I, the S1 and R2 leptoquarks are − 2 able to induce opposite-sign ðg Þe=μ contributions via 3We consider only a single-scalar solution, whereas with the mixed-chiral contribution in Eq. (13). These LQs have multiple scalar LQ models the idea of using LQ mixing to also garnered recent attention in other flavor anomaly generate (at least) ðg − 2Þμ was explored in Ref. [27]. studies [29–34]. The relevant LQ couplings for each

075037-3 INNES BIGARAN and RAYMOND R. VOLKAS PHYS. REV. D 102, 075037 (2020) extension, represented here as 3 × 3 Yukawa coupling correspondent directly with the L of the alternative quark matrices, are given by4: type. In the “up-type’ formalism, the relevant down-type quark LS1 c λ c λ † int ¼ðLL LQQL þ eR euuRÞS1 þ H:c:; ð16Þ couplings are related to those of the up-type quarks via the CKM matrix. This flavor ansatz allows us to explicitly LR2 λ λ † forbid one-loop contributions to LFV processes by decou- int ¼ðLL LuuR þ eR eQQLÞR2 þ H:c: ð17Þ pling the muon and electron sectors. The doublet, R2, can be expressed in terms of its electric The alternative basis choice would be to select the down- charge-definite components: type quark couplings to fix, and allow associated up-type quark couplings to be generated via CKM mixing, thus 5=3 “ ” R2 defining the down-type mass diagonal basis. This for- R2 ∼ ; ð18Þ − 2 2=3 malism for tackling ðg Þe=μ, for which models with R2 single scalar leptoquarks have been shown to be ruled out by a significant contribution to μ → eγ [16]. with charges as indicated by the superscripts. We assume Of course, the validity of a particular field content in NP negligible mass splitting between the components of the models is basis independent; however, we typically model- multiplet, i.e, build around choices of specific nonzero coupling textures. This may lead to missing allowed parameter space if one ≈ 5 3 ≈ 2 3 mR2 m = m = ; R2 R2 were to consider only one of these two Yukawa cou- pling bases. so as to avoid constraints from electroweak oblique corrections [17]. 2. Model framework under the “up-type” flavor ansatz A. COUPLING FRAMEWORK There are two independent coupling matrices for each model, and the interaction Lagrangians may be “ ” “ ” 1. Up-type vs down-type mass-diagonal basis reexpressed as for Yukawa couplings

S1 SLQ c c † When rotated into the flavor eigenbasis, we have two L ⊃ y ½e uL;j − Vjkv dL;kS1 “ ” ij L;i L;i choices to redefine the couplings: either an up-type or Seu c † “down-type” mass-diagonal basis. Practically, this involves þ yij eR;iuR;jS1 þ H:c:; ð20Þ a choice of which couplings we “fix” to a particular texture, 2 3 † 5 3 † – LR2 ⊃ RLu ν = ; − = ; and which we allow to be generated by CKM (Cabibbo yij ½ L;iuR;jR2 eL;iuR;jR2 – Kobayashi Maskawa) mixing. ReQ 5=3;† 2=3;† The “up-type” mass-diagonal Yukawas are defined in þ yij eR;i½uL;jR2 þ VjkdL;kR2 accordance with the mappings þ H:c: ð21Þ

R λ R ↦ Seu L λ L ↦ SLQ 5=3 e eu u y ; e LQ u y ; Recalling the discussion in Sec. I, we note that R2 , and S1 † RLu † ReQ both have left- and right-handed couplings to charged LeλLuRu ↦ y ; LeλeQLu ↦ y : ð19Þ leptons and SM up-type quarks. It is these couplings that − 2 Here, L and R represent the basis mapping between the we have deemed most important for the ðg Þl anoma- † lies, and so we will fix the up-type couplings to ϕ, and gauge and flavor eigenstates, and V ¼ LuLd is the standard CKM matrix.5 allow down-type interactions to be generated only by virtue By “mass-diagonal” we refer to the mathematical out- of their model-dependent relationships to these. We will come of this choice. The nonphysical, left-handed rotation discuss these couplings further in Sec. II A. κ0 matrix (L) for either the “up-type” or “down-type” quarks The parameters for each model, as per (10), are given 2 ≪ 2 is set to the identity, and we allow the CKM to be by the following, assuming mq mϕ:

2 4 7 2 m We implement the interaction basis with in their κ0 ðm Þ¼ þ log q ; S1 q 6 3 2 flavor eigenstates (i.e., no PMNS rotation) and consider only mS1 terms in which these fields couple as leptoquarks. We impose 2 conservation of Uð1Þ to forbid cou- 1 2 mq B κ0 ðm Þ¼− − log : ð22Þ plings. R2 q 6 3 m2 5As neutrino masses are negligible for the phenomenology of R2 this paper, we keep neutrinos in the flavor eigenbasis and set the PMNS (Pontecorvo–Maki–Nakagawa–Sakata) matrix to the Their mixed-chiral Δal contributions are therefore, via identity throughout all subsequent calculations. Eq. (13), given by

075037-4 GETTING CHIRALITY RIGHT: SINGLE SCALAR LEPTOQUARK … PHYS. REV. D 102, 075037 (2020) 0 1 0 1 mlm 7 m 00 0 0 S1 q S1 L R × × Δal ∼− − 2 log Reðyl yl Þ; ð23Þ B C B C 4π2 2 4 q q 1 ∼ 0 0 2 ∼ 00 mS1 mq Texture @ × A; Texture @ × A: 000 000 mlm 1 m ð27Þ ΔaR2 ∼ q − 2 log R2 Re yLyR ; 24 l 4π2 2 4 ð lq lqÞ ð Þ mR2 mq For the mixed-chiral interaction to be present, and thus enhanced, both the left- and right-handed couplings need to where, for S1, have the same nonzero texture.

yR ySeu yL ySLQ ¼ and ¼ ; ð25Þ 1. Texture 1: the charmphilic solution for ðg − 2Þμ

A “charmphilic” model for Δaμ was explored for both and, for R2, R2 and S1 by Kowalska et al. [35]. Notably, for generating Δaμ via charm-muon coupling, the R2 model is found to be yR ¼ −yRLu and yL ¼ yReQ: ð26Þ entirely ruled out by a combination of flavor constraints and searches in the LHC dimuon channel [35]. As a solution for ðg − 2Þμ, the muon-charm coupling for S1 was explored For consistency between F 0 and F 2 models, and j j¼ j j¼ and found to have highly constrained but nonzero allowed with Eq. (6), we have labeled the chirality to be that of the parameter space. quark field in the associated interaction term. Furthermore, For S1 under texture 1, the most constraining process on the strongest contributions in the R2 model arise only from Kþ → πþνν 5=3 the muon NP couplings is from the decay . interactions of the R component of the LQ doublet. The Couplings between the strange-quark and neutrinos 2=3 component R does not have couplings of both are unavoidably generated via CKM mixing, where to charged leptons. Vcs ∼ Oð1Þ. For nonzero left-handed couplings to the Input quark masses are the MS masses at the scale where charm, there will be non-negligible contributions to this “ ” each contribution is integrated out, i.e, mcðmcÞ and decay width. Following the derivation from Ref. [35], the mtðmtÞ. The Yukawa couplings here are at the high scale 2σ bound on Br(Kþ → πþνν)(Table I) generates the fol- (mϕ), where we have neglected running of these NP lowing constraint: contributions between this and lower energy scales. We ∼O 10% SLQ −2 mϕ estimate via numerical trial that this introduces ð Þ jy22 j < 5.26 × 10 : ð28Þ effect. For larger LQ masses, a more careful EFT (effective TeV field theory) analysis may be preferable, but within the Note that larger LQ masses allow a weakening of this scope of this work we have neglected such effects for bound when applied to the coupling ySLQ. However, such Δ 22 calculation of al. mass and coupling combinations are found to be heavily constrained by LHC dimuon channel searches, via B. Coupling textures pp → μμðjÞ. The S1 model is found to viable only within two sigma of the Δaμ central value, by a combination of The Yukawa coupling values themselves are a priori completely arbitrary. The mixed chiral terms shown in (23) flavor constraints and searches in the LHC constraints [35]. and (24) are directly proportional to the mass of the quark The results of Ref. [35] motivate consideration of the in the loop in Fig. 1—so an enhancement of this term is best alternative coupling structure, texture 2, in this work. For achieved by LQ couplings to higher-generation quarks. both R2 and S1, we will explore a novel, minimal parameter Ideally, contributions to the muon and electron magnetic space, and extend the single-leptoquark model to both electron and muon magnetic moments. moments would both be enhanced proportional to mt; however, Section II C has shown that this is incompatible with constraints from Brðμ → eγÞ. The next-best thing is to 2. Texture 2: establishing the focus of this work couple these charged leptons separately to either the top or As argued above, the most sensible approach hereon will the charm; in both models, it is only the up-type quarks that be to implement texture 2 in (27), i.e., for the remainder of have both chiralities of the required coupling. this work we will consider the following nonzero couplings In this framework, there are two options for textures for both models: to generate Δal ≠ 0, l ∈ fe; μg, labeled texture 1 and 0 1 0 1 texture 2. We reiterate that the convention adopted here is 0 × 0 0 × 0 B C B C such that the rows are labeled as charged-lepton, and yL ∼ @ 00× A; yR ∼ @ 00× A: ð29Þ columns as up-type quark, generations. Here, nonzero 000 000 couplings are indicated by a cross:

075037-5 INNES BIGARAN and RAYMOND R. VOLKAS PHYS. REV. D 102, 075037 (2020)

Also, for the remainder of this work we will restrict our effects in leptonic charged-current charmed decays, input couplings to real values in order to circumvent and related constraints from the kaon sector [41]. constraints from CP-violating observables.6 Contributions to mesonic decays are parametrized using Assuming that the mixed-chiral terms of (23) and (24) a low-energy EFT, and as such the Wilson coefficients dominate, we begin by deriving rough analytic bounds on the (WCs) corresponding to SMEFT (standard model effective products of relevant couplings. The current experimental field theory) operators outlined in Table III are run and values for Δae (4) and Δaμ (3) yield the following for S1: matched to low-energy prior to the calculation of observ- ables. This is achieved using the combination of WILSON 2 mS1 [42] and FLAVIO [43] packages. yL yR ¼ −ð0.35 0.09Þ × ; ð30Þ 23 23 10 TeV 2 1. Effective electroweak couplings L R mS1 y12y12 ¼ð0.46 0.19Þ × ; ð31Þ We follow the calculation of effective Z couplings to 10 TeV charged-leptons l, and associated observables, from Ref. [40]. The couplings g are the effective left- and and for R2: fLðRÞ right-handed couplings of the Z to , f. The m 2 effective Lagrangian for describing these interactions is yL yR ¼ð0.20 0.05Þ × R2 ; ð32Þ 23 23 10 TeV given by g X 2 LZ γμ ij ij L R mR2 eff ¼ fi ½gf PL þ gf PRfjZμ; ð34Þ y12y12 ¼ −ð0.41 0.17Þ × : ð33Þ cosðθ Þ L R 10 TeV W i;j θ Note that the loop functions in (11) depend explicitly on where W is the weak-mixing angle, and, for i ¼ j,we ii fi the leptoquark mass, and so the mϕ dependence cannot relabel the coupling g ≡ g for clarity. fLðRÞ LðRÞ truly be completely factored out. The above bounds rely on The strongest contributions to these corrections come the assumption that evaluating (11) for mϕ ¼ 1 TeV gives a from top-containing loops that have the same topology as reasonable numerical approximation to their behavior in a those in Fig. 1, but with a Z rather than line viable mass range. attached. The same couplings are relevant for these con- – Although relations (30) (33) guide us to the relative tributions as to Δaμ—which motivates careful consider- sizes of these couplings, we will proceed with a more ation of Z → μμ in our numerical study. For this study, we careful phenomenological analysis to grasp the full scope enforce 2σ agreement for the effective left- and right- of these models. μ handed couplings about the central values (gL=R) quoted in Table II. We do not directly consider the correlation C. Consideration of relevant constraints between these two parameters. Regardless of which of the two models we consider, the We also follow the calculation of Ref. [40] to consider crossing- between the topologies in Fig. 1 and any sizeable contribution to the invisible width of the Z contributions of the form l → l0ðγ=ZÞ, we expect the boson—particularly for R2, where neutrino LQ couplings strong constraints to be from Z → ll0 processes [19,37– are not purely CKM generated. We enforce 2σ agreement 40]. As we have decoupled the leptonic sectors, the for the observable effective number of neutrinos, Nν,in strongest constraints of this form will be from LFU which contributions to this width are manifest (Table II). processes, particularly from muon-top couplings i.e., Although electroweak constraints provide significant l ¼ l0 ¼ μ. By virtue of the structure of Eqs. (20) and bounds on the top-muon coupling in these models, they (21), couplings to neutrinos generate contributions to the are not a sensitive probe for the charm-electron coupling invisible width of the Z. These effective electroweak necessary to generate Δae. Loop corrections to Z → ll couplings provide the most stringent constraints on muon from charm-containing loops scale proportional to the ∝ 2 2 couplings to the . Z-boson threshold, mZ=mϕ, notably without a top-mass The couplings between the electron and the charm-quark enhancement. For constraining these couplings, we turn to give a more rich phenomenology. They are most strongly complementary constraints from both high- and low-energy constrained by dimension six contact interactions, which physics. manifest in high-pT lepton tails at the LHC, short-distance

2. Contact interactions and high pT leptonic tails 6The parameter space could, of course, be extended to complex couplings, but to do so we would need to carefully consider Our model generates NP effects in dimension-6 SMEFT (among other things) the contribution of the diagrams in Fig. 1 to operators, contributing to pp → ll (dilepton production) electric dipole moments [36]. and pp → lν (monolepton production), via tree-level

075037-6 GETTING CHIRALITY RIGHT: SINGLE SCALAR LEPTOQUARK … PHYS. REV. D 102, 075037 (2020)

TABLE II. Processes most constraining on this model. Values Amplitude contributions to these high- and low-energy quoted without citation are from PDG [44]. Constraints from processes are related by a crossing-symmetry, and there- pp → ee are derived from Table 1 of Ref. [45], in conjunction fore provide complementary constraints.8 → ¯ ν with the expressions from Table III. For c dkei j, 95% con- Note that with increased LHC luminosity, the allowed fidence intervals for parameters (36) are quoted where accessible, regions for the effective interactions manifest in high-pT otherwise upper bounds on their magnitudes are given [41]. leptonic tails are forecast to shrink significantly [41,45], Process Observable Limits further restricting parameter space for both models. Here we limit our discussion to the present constraints. → l l μ μ −0 2689 11 0 2323 13 Z i j fgL;gRg f . ð Þ; þ . ð Þg Z → νν Nν 2.9840(82) Neutral currents and dilepton searches.—These processes D → eν Br <8.8 × 10−6 −5 provide the strongest constraints on the charged-current Ds → eν Br <8.3 × 10 −10 cc → ee transition generated at tree level, constraining the Kþ → πþνν Br ð1.7 1.1Þ × 10 0 → þ − 9þ6 10−12 magnitudes of Yukawa couplings responsible for correcting KL e e Br ð −4 Þ × Δae. The WCs and associated operators constrained by → ll Seu 0 648 pp jy12 j < . mϕ=TeV these processes are indicated in Table III. For these SLQ 0 537 [45] jy12 j < . mϕ=TeV processes, constraints on chirality-flipping scalar and ReQ 0 393 jy12 j < . mϕ=TeV tensor operators are irrelevant at this order, due to sup- RLu jy12 j <0.524 mϕ=TeV pression from a light-fermion mass insertion. In Table II we ReQ 0 904 jy23 j < . mϕ=TeV quote the numerical upper bounds on explicit Yukawa 111 −2 couplings, derived from the two-sigma limits on the c → dke¯iνj ϵ ∈½−0.52; 0.86 × 10 −1 VL corresponding SMEFT WCs, using ATLAS 36.1 fb [41] ϵ112 ∈½−0.28; 0.59 × 10−2 VL dataset [47]—Table 1 of Ref. [45]. For correlated WCs, fjϵ121j; jϵ122jg f<0.67;<0.42g × 10−2 VL VL e.g., Clqð1Þ and Clqð3Þ, we take the most constraining of the fjϵ111j; jϵ112jg f<0.72;<0.43g × 10−2 SL SL two as a conservative bound. fjϵ211j; jϵ212jg f<1.1;<0.68g × 10−2 SL SL Furthermore, in the R2 model, tree-level contributions to ϵ111 ϵ112 4 3 2 8 10−3 fj T j; j T jg f< . ;< . g × the process bb → μμ are generated via CKM mixing. ϵ211 ϵ212 6 6 4 0 10−3 fj T j; j T jg f< . ;< . g × Taking Vtb ∼ 1, this process constrains the magnitude of ReQ the Yukawa coupling y23 , resulting in the upper bound t-channel processes. These can be probed directly in the quoted in Table II. high-pT tails of Drell-Yan processes at the LHC [47].For these processes, numerical analyses find minimal interfer- Charged currents and monolepton searches.—Broadly ence between NP effective operator contributions and speaking, high-pT monolepton searches give strong con- a priori several NP operators can be simultaneously con- straints on the relevant scalar and tensor operators. strained [45,48]. Constraints on the relevant WCs are given for rescaled Under the assumption of no new degrees of freedom at or versions of those quoted in Table III, defined via the tree- below the electroweak scale, NP contributions can be level matching conditions between the SMEFT and the parametrized using the SMEFT: low-energy effective theory [41]: X L ⊃ O αβi 2 Vji αβ2j SMEFT Cα α: ð35Þ ϵ ¼ −v ½C ; VL lqð3Þ α V2i αβ V βα 2 ϵ i ¼ −v2 ji ½C j ; The analytic expressions for the relevant operators and SL lequð1Þ 2V2i corresponding WCs at a high scale (mϕ) can be found in αβ V βα 2 Table III. We emphasize that here we restrict our discussion ϵ i −v2 ji C j ; 36 T ¼ 2 ½ lequð3Þ ð Þ to constraints on tree-level contributions to canonically V2i 7 dimension-6 SMEFT operators. As noted in Section II C 1, ∼ 246 these processes will be most relevant for constraining input where v GeV is the electroweak vev, and Vij are CKM matrix elements. Note that ϵ is relevant for the S1 “charm-electron” Yukawa couplings. VL For second-generation quark processes generated in these model but not for R2. models, effective interactions are also constrained by low- In Table II, we quote constraints on the relations in (36), 139 −1 energy (semi)leptonic meson decays (Section II C 3). derived via Ref. [41] from the ATLAS fb [49] and CMS 35.9 fb−1[50] datasets. These limits will be 7As our models do not induce corrections to the W vertex at tree level, those constraints from Ref. [41] are omitted from this 8For a review of these limits in the context of charm decays and discussion. high-pT limits, see Ref. [41].

075037-7 INNES BIGARAN and RAYMOND R. VOLKAS PHYS. REV. D 102, 075037 (2020)

TABLE III. Effective contact interactions in both LQ models. Labeling of WCs ðijklÞ indicates fermion flavor, and is ordered by appearance of the fermion in the effective operator. Following the SMEFT basis as outlined in Ref. [46], σa are the Pauli matrices (a ¼ 1, T 2 2, 3) and is understood to indicate transposition of the SUð ÞL indices, exclusively. Note that we have adopted the SMEFT Warsaw-up basis, a variant of the standard Warsaw basis where the up-type quark mass matrix (rather than the down type) is diagonal, and each of the nonphysical rotation matrices appearing in (19) are set to the identity matrices.

LQ, ϕ SMEFT Operator Wilson Coefficient LHC high − pT constraints 1 1 þ − S1 ð Þ μ ijkl SLQ; SLQ pp → l l O ¼½L γμL ½Q γ Q C 1 ¼ 4 2 y y lq L L L L lqð Þ mϕ ik jl ð3Þ μ ijkl ijkl ¯ þ − O γ σ γ σ − c → dkeiνj, pp → l l lq ¼½LL μ aLL½QL aQL Clqð3Þ ¼ Clqð1Þ μ 1 þ − O e γμe u γ u ijkl Seu; Seu pp → l l eu ¼½ R R½ R R Ceu ¼ 2 2 y y mϕ ik jl ð1Þ T ijkl 1 SLQ Seu; c → d e¯ ν O ¼½L e iσ2½Q u C 1 ¼ 2 2 y y k i j lequ L R L R lequð Þ mϕ ik jl ð3Þ μν ijkl 1 ijkl O σ σ σ T − c → dke¯iνj lequ ¼½LL μνeRi 2½QL uR Clequð3Þ ¼ 4 Clequð1Þ

μ 1 þ − R2 O γ γ ijkl − RLu; RLu pp → l l lu ¼½LL μLL½uR uR C ¼ 2 2 y y lu mϕ jk il ð1Þ T ijkl 1 ReQ; RLu c → d e¯ ν O ¼½L e iσ2½Q u C 1 ¼ 2 2 y y k i j lequ L R L R lequð Þ mϕ jk il ð3Þ μν ijkl 1 ijkl ¯ O σ σ σ T c → dkeiνj lequ ¼½LL μνeRi 2½QL uR Clequð3Þ ¼ 4 Clequð1Þ O γ γμ ijkl − 1 ReQ; ReQ pp → lþl− qe ¼½QL μQL½eR eR Cqe ¼ 2 2 y y mϕ li kj

→ ν numerically accounted for in our phenomenological study BrðDðsÞ e Þ constraints within two-sigma of their central (Section III). values, for a sample of indicative allowed regions—includ- ing, but not exclusive to, that shown in Figs. 2 and 3.We 3. Mesonic decays: D-meson decays found that this subset of mesonic decays did not to rule out any additional parameter space, beyond what was already As the lepton Yukawa sector is explicitly flavor decoupled in these models, this leads to mesonic con- excluded. The experimental values used for these cross straints governed primarily by charged-current decays checks can be found in Table II. → ν DðsÞ e . Contributions to these processes are generated by both models at tree level. 4. Mesonic decays: K-meson decays For both models in the considered parameter space, Interactions involving are generated by these LQ 2 D-meson decay constraints are significantly weaker than the couplings via SUð ÞL invariance of the associated inter- correlated high-pT leptonic constraints. We cross-checked action, explicitly manifest in Eqs. (20) and (21).

FIG. 2. S1 model for fixed benchmark LQ mass of 2 TeV: hashed regions indicate that they are ruled out by the labeled constraint, with L − L R conditions for each constraint outlined in Section III. The leftmost plot indicates the allowed region in the y12 y23 plane, with yij fixed by the Δal central values. Here, the shaded perturbativity constraint refers to that region being ruled out by nonperturbative generated right-handed couplings. The center and rightmost plots show the relevant constraints for a scan over a decoupled muon and electron sector, and here the shaded region shows the one- and two-sigma allowed regions about the associated Δal central values.

075037-8 GETTING CHIRALITY RIGHT: SINGLE SCALAR LEPTOQUARK … PHYS. REV. D 102, 075037 (2020)

FIG. 3. R2 model for fixed benchmark LQ mass of 2 TeV: hashed regions indicate that they are ruled out by the labeled constraint, with L − L R conditions for each constraint outlined in Section III. The leftmost plot illustrates a scan over the y12 y23 plane, with yij fixed by the Δal central values. Here, the shaded perturbativity constraint refers to that region being ruled out by nonperturbative generated right- handed couplings. The center and rightmost plots show the relevant constraints for a scan over a decoupled muon and electron sector, and here the shaded region shows the one- and two-sigma allowed regions about the associated Δal central values.

Following the derivations in Ref. [51], we rederive Conversely, for the R2 model the strongest constraint the relevant constraints in terms of the Yukawas in the from the kaon sector come from tree-level leptoquark “ ” 0 → þ − up-type coupling regime. For this, we assume the exchange to the helicity suppressed KL e e transition. Wolfenstein parametrization for the CKM matrix. Noting the vector nature of the effective contribution to this Namely the relevant combination of parameters for the process, and following again from Ref. [51], we derive an decays considered here is this: upper bound at 90% confidence on the associated coupling: 1 ∼−λ 1 − λ2 ∼ 0 22 mϕ VcdVcs 2 . ; ð37Þ jyReQj ≲ 9.5 × 10−2 : ð39Þ 12 TeV where λ ∼ 0.226 is the standard Wolfenstein param- eter [44]. We emphasize that for the purpose of these derivations Contributions to the theoretically clean rare process we have assumed real-valued Yukawa couplings for each BrðKþ → πþνν¯Þ are found to give competitive constraints model, and recognize that complementary constraints may for the S1 model. In Section II B 1 we highlighted the exist on any imaginary component if this assumption were importance of this constraint for the case of a charmphilic to be relaxed. The S1 leptoquark cannot contribute to this ðg − 2Þμ model, and it is similarly constraining for this process at tree level, and higher order contributions were coupling texture. As the effective interaction here is purely found to be weaker than those from the high-pT LHC vector in nature, it does not run in QCD and has negligible observables discussed in earlier subsections. electroweak running effects within the scope of this analysis. Following Ref. [51], considering the dominant III. PHENOMENOLOGY contribution at the 90% confidence level gives the follow- ing upper bound: In both models, the free parameters are summarized by

SLQ −2 mϕ L L R R jy j ≲ 4.09 × 10 : ð38Þ fy12;y23;y12;y23;mϕg; ð40Þ 12 TeV For real-valued Yukawa couplings, this process generates a where we stress that the right- and left-handed couplings ¯ more constraining bound for S1 than those from K − K are explicitly defined for each model as per (25) and (26). 0 ¯ mixing or from the process KL → π νν [52]. For the R2 Presently the scalar leptoquark mass, mϕ, is most strongly model, contributions to this process are loop-order and the directly constrained using LHC searches for the decay of constraints are negligible relative to those otherwise pairs of scalar LQ with couplings predominantly to first- considered. generation leptons [53]:

075037-9 INNES BIGARAN and RAYMOND R. VOLKAS PHYS. REV. D 102, 075037 (2020)

mϕ > 1.8 TeV at 95% C:L:; ð41Þ 1. Fixed about the Δal central values The leftmost plots illustrate the results of a scan fixing providing a conservative lower bound.9 the right-handed couplings via the method outlined at the For the majority of this section, we will aim to fix the beginning of Sec. III. Starting with S1, note particularly that values of Δal to a point within one sigma of the central contributions to effective Nν, via Z → νν, are suppressed values, reducing the parameter degrees of freedom to significantly enough (i.e., by CKM) to render these 5 − 2 ¼ 3. To achieve this, we scan logarithmically over negligible for the considered parameter space. This is positive perturbative left-handed couplings, and fix the not true for R2, and an effective Nν bound is found to right-handed couplings according to the full calculation of be competitive with that from Z → μμ. For the both LQs, Δal (12); the relative sign is absorbed into the allocation of this indicates a preferred allocation of couplings such that right-handed couplings. Since this equation is a polynomial R L ∼ O 10−2 L ≳ O 10−2 in yl , and we restrict the couplings to be real valued, we y12 ð Þ; and y23 ð Þ: ð42Þ R R solve for the value of yl for which (i) Reðyl Þ is perturba- R All points shown pass the requirement that Δa μ fall tive, and (ii) Imðyl Þ is minimal. We truncate the input value eð Þ R within the one-sigma region. These results act to demon- for yl to be the real component of this root. This is done algorithmically such that the input value is the closest fit to strate explicit existence of viable parameter space. these requirements, listed in order of preference. We then μ check the calculated value for Δal and identify points that 2. Decoupled and e sectors remain within one sigma of the central values. The center and leftmost plots in Figs. 2 and 3 illustrate The conditions for this parameter study are as follows, the planes of electron and muon couplings for each model. L R where experimental values implemented are as given in In each of these plots, the relevant ðy ;y Þl couplings are Table II. The below represent the requirements for values to scanned over perturbative values, without enforcing a fit pass constraints for each observable. with Δal. For these scans, the effectively decoupled Δ Δ 1σ L R l0 ≠ l (1) ae and aμ within ; pffiffiffiffiffiffi parameters ðy ;y Þl0 , where , are fixed to zero. (2) Yukawa coupling perturbativity bound jyj ≤ 4π; Shaded regions illustrate the one- and two-sigma agreement 2σ (3) Effective Nν within ; with the Δal values in (3) and (4). Hashed regions illustrate (4) Effective left- and right-handed Z-muon couplings the strength of the important constraints on these parameter within 2σ of their central values; subspace projections. These results demonstrate the impact (5) LHC high-pT leptonic constraints, as listed in of uncertainties on the Δal measurements on these models’ Table II, parameter space regions. (6) and mϕ > 1.8 TeV. Furthermore, we impose the following model-specific B. Parameter scan for S and R models constraints to the their respective parameter space: 1 2 (1) Kþ → πþνν¯ derived coupling bound in Eq. (38) Here we extend the benchmark region to include variation in the LQ mass for each model. We logarithmi- for S1, 0 → þ − cally vary the parameters within the following ranges: (2) and KL e e bound in Eq. (39) for R2. We will begin by identifying a benchmark region of pffiffiffiffiffiffi ∈ 1 8 100 12 23 ∈ 0 4π ⊂ Rþ parameter space for each model, then proceed to a full mϕ ½ . ; TeV;yL ;yL ½ ; : ð43Þ parameter grid scan. We aim to achieve an upper bound on the LQ masses capable of tackling the ðg − 2Þ puzzle, The right-handed couplings are fixed, as per the algorithm e=μ Δ and explore the emergent coupling structures of these for al detailed in earlier discussion, and we require that solutions. points satisfy the constraints outlined at the beginning of Sec. III. We emphasize that the conclusions here are based on the current central values and error ranges for the TeV A. S1 and R2 models with mϕ =2 anomalous magnetic moments, and should be revisited We first begin to explore the available parameter space when new experimental results are obtained. The results of by performing grid scans over coupling space projections, these scans are shown in Fig. 4. We have labeled the with a fixed mϕ ¼ 2 TeV in both LQ models. The results of couplings here explicitly in accordance with Eqs. (20) these scans are shown in Figs. 2 and 3. Constraints that do and (21). not explicitly appear in these figures were not competitive The left two plots in Figs. 4(a) and 4(b) can be thought of with those shown, within the illustrated region of interest. as illustrating the shrinkage and translation with mϕ of the constrained regions in the leftmost plots of Figs. 2 and 3. 9 Here, the colorbar for these indicates the maximum mϕ A novel study by the CMS collaboration considered searches for LQ with off-diagonal couplings to t − μ [54], although this value that is allowed for that parameter combination; these constraint is slightly weaker than that quoted above. masses are not unique for a particular coupling allocation

075037-10 GETTING CHIRALITY RIGHT: SINGLE SCALAR LEPTOQUARK … PHYS. REV. D 102, 075037 (2020)

FIG. 4. Preferred coupling substructures for S1 (a) and R2 (b) Δae=μ model: these figures illustrate the results a grid scan of the coupling values for parameter points, performed as outlined at the beginning of Sec. III. Colored points pass all constraints. In each subfigure, the left two plots demonstrate the preferred right- and left-handed coupling assignments for varying LQ mass, and share an associated color bar. The right two plots show the ratio of nonzero couplings for both coupling matrices, as they vary with the LQ mass. because of the allowed region about the central values of Eqs. (30)–(33), where we allow for the one-sigma regions Δal. As expected, Yukawa couplings show a trend towards about the central values of Δae=μ. higher magnitudes for larger LQ masses—reflecting The rapid convergence of S1 to this ratio would indicate Eqs. (23) and (24). The upper limit on LQ mass for the that this LQ model structure is more sensitive to variations S1 and R2 models under the specified constraints was found about the central values of Δal, and therefore improve- to be such that ments of these measurements (i.e., those expected in the near future from the Muon g − 2 collaboration [4])may ≲ 65 ≲ 68 mS1 TeV; and mR2 TeV: ð44Þ necessitate a reassessment of the viability of such a model. This information is not immediately evident from contrast In the right two plots in Figs. 4(a) and 4(b), the colorbars with center and leftmost plots in Figs. 2 and 3, however we show the maximum allowed coupling value (labeled) for a note that the decoupled parameter planes do not reflect the particular mass and coupling ratio. For S1, these narrow influence of interplay between electron and muon cou- rapidly with increased mass mϕ to a preferred coupling ratio, plings for relevant constraints. Note that patterns emergent in the couplings within these −1 y23=y12 ∼ Oð10 Þ; ð45Þ datasets may motivate an algebraic relationship between LQ couplings of the quarks to charged-lepton generations. for both right- and left-handed couplings. These plots for R2 Known theoretical approaches can be explored to generate show a much less rapid convergence to a coupling ratio with such hierarchical coupling structures in UV-complete NP increased mass mϕ, but to the same approximate ratios. For models: for example, Froggatt-Nielson mechanisms [55]. smaller masses a wider range of coupling ratios are permit- We refrain from discussing this concept further here, but ted. These ratios are consistent with the relations in rather we identify it as an avenue for future exploration.

075037-11 INNES BIGARAN and RAYMOND R. VOLKAS PHYS. REV. D 102, 075037 (2020)

C. Future of predicting ðg − 2Þτ Hierarchical Yukawa coupling relationships may emerge There are proposed measurement techniques for the in the allowed points for S1 and R2, depending on the particular choice of mϕ. This motivates further study into quantity ðg − 2Þτ [56]; however, up to the present there are only rough bounds on its experimental value [44]: possible texture generation mechanisms and their extension to models including effects in ðg − 2Þτ. −0.052

[1] G. F. Giudice, P. Paradisi, and M. Passera, J. High Energy [3] T. Blum, A. Denig, I. Logashenko, E. de Rafael, B. L. Phys. 11 (2012) 113. Roberts, T. Teubner, and G. Venanzoni, arXiv:1311.2198. [2] F. Campanario, H. Czy, J. Gluza, T. Jeliski, G. Rodrigo, S. [4] A. Chapelain (Muon g-2 Collaboration), EPJ Web Conf. Tracz, and D. Zhuridov, Phys. Rev. D 100, 076004 (2019). 137, 08001 (2017).

075037-12 GETTING CHIRALITY RIGHT: SINGLE SCALAR LEPTOQUARK … PHYS. REV. D 102, 075037 (2020)

[5] R. Parker, C. Yu, W. Zhong, B. Estey, and H. Mãeller, [33] I. Bigaran, J. Gargalionis, and R. R. Volkas, J. High Energy Science 360, 191 (2018). Phys. 10 (2019) 106. [6] J. Liu, C. E. M. Wagner, and X.-P. Wang, J. High Energy [34] A. Crivellin, D. Müller, and F. Saturnino, arXiv:1912 Phys. 03 (2019) 008. .04224. [7] A. Crivellin, M. Hoferichter, and P. Schmidt-Wellenburg, [35] K. Kowalska, E. M. Sessolo, and Y. Yamamoto, Phys. Rev. Phys. Rev. D 98, 113002 (2018). D 99, 055007 (2019). [8] M. Abdullah, B. Dutta, S. Ghosh, and T. Li, Phys. Rev. D [36] W. Dekens, J. de Vries, M. Jung, and K. K. Vos, J. High 100, 115006 (2019). Energy Phys. 01 (2019) 069. [9] A. E. C`arcamo Hern`andez, S. F. King, H. Lee, and S. J. [37] A. Djouadi, T. Kohler, M. Spira, and J. Tutas, Z. Phys. C 46, Rowley, Phys. Rev. D 101, 115016 (2020). 679 (1990). [10] M. Hoferichter, in 54th Rencontres de Moriond on Electro- [38] E. C. Leskow, G. D’Ambrosio, A. Crivellin, and D. Müller, weak Interactions and Unified Theories (2019). Phys. Rev. D 95, 055018 (2017). [11] M. Endo and W. Yin, J. High Energy Phys. 08 (2019) 122. [39] D. Das, C. Hati, G. Kumar, and N. Mahajan, Phys. Rev. D [12] M. Badziak and K. Sakurai, J. High Energy Phys. 10 (2019) 94, 055034 (2016). 024. [40] P. Arnan, D. Becirevic, F. Mescia, and O. Sumensari, J. High [13] G. Hiller, C. Hormigos-Feliu, D. F. Litim, and T. Steudtner, Energy Phys. 02 (2019) 109. arXiv:1910.14062. [41] J. Fuentes-Martin, A. Greljo, J. Martin Camalich, and J. D. [14] C. Cornella, P. Paradisi, and O. Sumensari, J. High Energy Ruiz-Alvarez, arXiv:2003.12421. Phys. 01 (2020) 158. [42] J. Aebischer, J. Kumar, and D. M. Straub, Eur. Phys. J. C 78, [15] N. Haba, Y. Shimizu, and T. Yamada, Prog. Theor. Exp. 1026 (2018). Phys. 2020, 093B05 (2020). [43] D. M. Straub, arXiv:1810.08132. [16] I. Doršner, S. Fajfer, and S. Saad, arXiv:2006.11624. [44] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, [17] I. Doršner, S. Fajfer, A. Greljo, J. F. Kamenik, and N. 030001 (2018). Košnik, Phys. Rep. 641, 1 (2016). [45] A. Greljo and D. Marzocca, Eur. Phys. J. C 77, 548 (2017). [18] D. Bečirević,N.Košnik, O. Sumensari, and R. Zukanovich [46] J. de Blas, J. Criado, M. Perez-Victoria, and J. Santiago, Funchal, J. High Energy Phys. 11 (2016) 035. J. High Energy Phys. 03 (2018) 109. [19] O. Popov and G. A. White, Nucl. Phys. B923, 324 (2017). [47] M. Aaboud et al. (ATLAS Collaboration), J. High Energy [20] D. Buttazzo, A. Greljo, G. Isidori, and D. Marzocca, J. High Phys. 10 (2017) 182. Energy Phys. 11 (2017) 044. [48] D. A. Faroughy, A. Greljo, and J. F. Kamenik, Phys. Lett. B [21] A. Angelescu, D. Bečirević, D. A. Faroughy, and O. 764, 126 (2017). Sumensari, J. High Energy Phys. 10 (2018) 183. [49] G. Aad et al. (ATLAS Collaboration), Phys. Rev. D 100, [22] Y. Cai, J. Herrero-García, M. A. Schmidt, A. Vicente, and 052013 (2019). R. R. Volkas, Front. Phys. 5, 63 (2017). [50] A. M. Sirunyan et al. (CMS Collaboration), J. High Energy [23] J. C. Pati and A. Salam, Phys. Rev. D 8, 1240 (1973). Phys. 06 (2018) 128. [24] J. Pati, A. Salam, and J. Strathdee, Il Nuovo Cimento A 26, [51] R. Mandal and A. Pich, J. High Energy Phys. 12 (2019) 72 (1975). 089. [25] J. L. Hewett and T. G. Rizzo, Phys. Rev. D 56, 5709 (1997). [52] G. Kumar, Phys. Rev. D 94, 014022 (2016). [26] L. Lavoura, Eur. Phys. J. C 29, 191 (2003). [53] G. Aad et al. (ATLAS Collaboration), arXiv:2006.05872. [27] I. Doršner, S. Fajfer, and O. Sumensari, J. High Energy [54] A. M. Sirunyan et al. (CMS Collaboration), Phys. Rev. Lett. Phys. 06 (2020) 089. 121, 241802 (2018). [28] A. Baldini et al. (MEG Collaboration), Eur. Phys. J. C 76, [55] C. Froggatt and H. Nielsen, Nucl. Phys. B147, 277 434 (2016). (1979). [29] M. Bauer and M. Neubert, Phys. Rev. Lett. 116, 141802 [56] G. Gonzlez-Sprinberg and J. Vidal, J. Phys. Conf. Ser. 912, (2016). 012001 (2017). [30] A. Crivellin, D. Müller, and T. Ota, J. High Energy Phys. 09 [57] S. Borsanyi et al., arXiv:2002.12347. (2017) 040. [58] F. Jegerlehner, EPJ Web Conf. 166, 00022 (2018). [31] Y. Cai, J. Gargalionis, M. A. Schmidt, and R. R. Volkas, [59] A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D J. High Energy Phys. 10 (2017) 047. 101, 014029 (2020). [32] B. Chauhan, B. Kindra, and A. Narang, Phys. Rev. D 97, [60] M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Eur. 095007 (2018). Phys. J. C 80, 241 (2020).

075037-13