Charged Lepton Flavor Violation at the EIC
Vincenzo Cirigliano, Kaori Fuyuto, Christopher Lee, Emanuele Mereghetti, Bin Yan
LLA-UR-21-20531
Prepared for submission to JHEP
Charged Lepton Flavor Violation at the EIC
Vincenzo Cirigliano, Kaori Fuyuto, Christopher Lee, Emanuele Mereghetti,and Bin Yan
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A.
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We present a comprehensive analysis of the potential sensitivity of the Electron-Ion Collider (EIC) to charged lepton flavor violation (CLFV) in the channel ep → τ X ,within the model-independent framework of the Standard Model Effective Field Theory(SMEFT). We compute the relevant cross sections to leading order in QCD and electroweakcorrections and perform simulations of signal and SM background events in various τ decaychannels, suggesting simple cuts to enhance the associated estimated efficiencies. To assessthe discovery potential of the EIC in τ - e transitions, we study the sensitivity of other probesof this physics across a broad range of energy scales, from pp → eτ X at the Large HadronCollider to decays of B mesons and τ leptons, such as τ → eγ , τ → e(cid:96) + (cid:96) − , and cruciallythe hadronic modes τ → eY with Y ∈ { π, K, ππ, Kπ, ... } . We find that electroweak dipoleand four-fermion semi-leptonic operators involving light quarks are already strongly con-strained by τ decays, while operators involving the c and b quarks present more promisingdiscovery potential for the EIC. An analysis of three models of leptoquarks confirms theexpectations based on the SMEFT results. We also identify future directions needed tomaximize the reach of the EIC in CLFV searches: these include an optimization of the τ tagger in hadronic channels, an exploration of background suppression through tagging b and c jets in the final state, and a global fit by turning on all SMEFT couplings, whichwill likely reveal new discovery windows for the EIC. a r X i v : . [ h e p - ph ] F e b ontents Z , Higgs and t decays 376.2 CLFV Drell-Yan 39
10 Leptoquark models 6711 Conclusions 72A Renormalization group equations and their solutions 76
A.1 Running between Λ and the electroweak scale 76A.2 Dipole contributions induced by the LFV Yukawa interaction 81A.3 Running below the electroweak scale 82
B Partonic cross sections for CLFV processes 86
B.1 DIS 86B.2 The squared amplitude of gg → e ± τ ∓ at the LHC 88– i – Conversion to a non-chiral basis of low-energy operators 88D Compendium of Decay rates 89
D.1 τ decay rates 89D.2 B decays 98 Processes involving charged lepton flavor violation (CLFV) are very powerful tools to searchfor new physics beyond the Standard Model (BSM) for a number of reasons. First, the ob-servation of CLFV at experiments in the foreseeable future would immediately point to newphysics beyond the minimal extension of the SM that only includes neutrino mass (so-called ν SM). This is because in the ν SM, CLFV amplitudes are proportional to ( m ν /m W ) [1, 2],leading to rates forty orders of magnitude below current sensitivity. Furthermore, currentand future CLFV searches are sensitive to new mediator particles with masses that canbe well above the scales directly accessible in current and near-future high-energy collid-ers. Classic examples include supersymmetric seesaw models [3] and supersymmetric grandunified models [4]. Finally, CLFV processes play a special role in probing extensions of theStandard Model (SM) connected to the generation of neutrino mass. Correlations betweenneutrino mass models and signatures in CLFV processes have been highlighted in the lit-erature (e.g. TeV see-saw mechanisms [5, 6] or in minimally flavor-violating GUT scalesee-saw models [7]). In a nutshell, CLFV processes offer a great discovery tool for BSMphysics as well as the possibility to “diagnose” the underlying new physics and its effecton neutrino mass generation. There is a vast literature on the subject, and for reviews werefer the reader to Refs. [8–11].Probes of CLFV exist across a broad spectrum of energy scales. Low-energy probesinclude decays of the µ and τ leptons, decays of the B and K mesons and quarkonia. High-energy probes include searches for SM-forbidden events such as pp → (cid:96) α ¯ (cid:96) β + X at the LargeHadron Collider (LHC) or ep → (cid:96) + X at electron-hadron colliders such as HERA and theupcoming Electron-Ion Collider (EIC). Currently, the most stringent limits on CLFV comefrom searches for µ ↔ e processes, e.g. the branching ratio BR( µ + → e + γ ) < . × − at 90% CL [12]. The constraints on τ ↔ e transitions, however, are much weaker, withBR( τ ± → e ± Y ) < few × − [13], with Y ∈ { γ, ππ, ... } . Although Belle-II [14] is expectedto improve these τ BR constraints, and High-Luminosity LHC [15] to extend its reach in pp → ¯ eτ + X , both by an order of magnitude, there remains nevertheless a competitiveopportunity for colliders to search for events ep → τ X , with hadronic final states X .In the recent past, HERA was able to put competitive constraints on τ ↔ e transi-tions [16]. The EIC will collide e ’s and p ’s at center-of-mass energy (cid:38)
100 GeV, smallerthan HERA, but at vastly higher luminosity, reaching 10–100 fb − per year [17]. Thusits reach to find CLFV may be a thousand times greater than HERA [16] and possiblycompetitive with improved searches for τ → eY at Belle-II [14]. The promise of the EIC– 1 –s a probe of CLFV was highlighted by the early study of Ref. [18], which estimated thatan EIC with a collision energy of 90 GeV could probe currently allowed CLFV interactionsin the context of leptoquark models.In this paper we perform a first comprehensive analysis of the CLFV physics reach ofthe EIC in the general framework of the Standard Model Effective Field Theory (SMEFT)[19–25], which captures new potential sources of CLFV in a model-independent way.SMEFT encodes new physics originating at energies higher than the electroweak scalein operators of dimension greater than four built out of SM fields. The SMEFT frameworkis applicable to processes in which the center-of-mass energy is well below the expectedscale of new physics. Given the null results so far for new physics searches at LHC, theSMEFT is perfectly applicable at an EIC with center-of-mass energy √ S < v ew ∼
200 GeV.In fact, the effect of any new physics model with particle masses above the electroweakscale will reduce to the SMEFT operators, with a model-specific pattern of effective cou-plings. Therefore, the SMEFT framework allows one to assess the discovery potential andmodel diagnosing power of the EIC in full generality, also allowing a consistent comparisonwith probes at lower energies, such as τ → eX and LFV B meson decays. Our workconsiderably improves on the current state of the art, in two ways. First, for EIC itself,we account for all leading (dimension-six) CLFV operators, including heavy quark oper-ators, in computing EIC’s reach in inclusive and differential ep → τ X searches. Second,we compare this reach with all existing CLFV probes today, at both high and low energy,within the model-independent framework of SMEFT. These include searches for pp → eτ at the LHC and decays of the τ lepton ( τ → eY ) and B meson. Concerning the τ decays,we will consider not only radiative ( τ → eγ ) and leptonic modes ( τ → e(cid:96) + (cid:96) − ), but alsohadronic modes such as τ → eπ , τ → eππ [26–29], which have so far not been consideredin studies of CLFV at EIC (e.g. [18]). The inclusion of hadronic channels is very relevantbecause (i) the current and prospective sensitivity in τ BRs for radiative and hadronicmodes are at the same level, namely ∼ − − − ; (ii) the hadronic modes provide thestrongest constraints on CLFV operators involving quarks and gluons [28, 29]. Throughthis analysis, we will also identify synergies and complementarity of CLFV searches at theEIC and in τ decays.In the recent literature, studies of e → τ transitions have appeared in various contexts.Ref. [30] discusses e → τ ( µ ) at a future LHeC, using a small subset of BSM operators,namely vector and scalar vertex corrections. Ref. [31] focuses on eN → τ N transitions ata fixed target experiment such as NA64 [32] within the SMEFT framework, performinga comparative study of this process with CLFV τ decays. Ref. [33] studies eN → τ N transitions at a fixed target experiment within minimal SM extensions with sterile fermions.Ref. [34] discusses e → τ transitions mediated by gluonic operators at both fixed targetexperiments and LHeC. In the context of this rich literature, our work introduces severalnew elements: the use of the full set of SMEFT operators, the study of a larger set ofprobes (including LHC and B meson decays besides all CLFV τ decays) and the focus onthe EIC sensitivity and reach.The paper is organized as follows. In Section 2 we present a high-level discussion ofthe relative sensitivity of collider and lepton decays in probing CLFV. This analysis will– 2 –rovide the minimum luminosity requirements for ep colliders to be competitive with CLFVlepton decays and will show that the EIC will be competitive only for e ↔ τ and not for e ↔ µ transitions. Specializing to e ↔ τ , in Section 3 we present the basis of relevantCLFV operators at dimension-six in the SMEFT. In Section 4 we present our results forthe CLFV deep inelastic scattering (DIS) process ep → τ X mediated by all dimension-sixoperators in SMEFT, and in Section 5 we discuss the EIC sensitivity to CLFV couplings.In Section 6 we discuss complementary high-energy probes of CLFV, such as CLFV decaysof the top quark, Higgs boson, Z boson and LFV Drell-Yan at the LHC. Going down inenergy scale, in Section 7 we discuss the connection between SMEFT and the low-energyeffective theory (LEFT) and study the constraints from CLFV decays of the τ leptonand B meson. Indirect low-energy probes of CLFV involving charged-current processesand neutrinos are discussed in Section 8. In Section 9 we summarize the single-couplingconstraints, and identify the classes of operators for which the EIC is competitive withother high- and low-energy probes. Finally, in Section 10 we apply our EFT formalismto the analysis of three different leptoquark models and compare our findings with theexisting literature. Our conclusions and outlook are given in Section 11. The appendicescontain technical details of our analysis. Historically, very strong constraints on CLFV couplings have been obtained by studyingdecays of µ and τ leptons, with current upper limits on the BRs in the 10 − and 10 − ballpark, respectively. Given an underlying LFV scenario (e.g. represented by one or moreCLFV operators in the SMEFT), the lepton decay BR limits translate into requirements onthe luminosity, energy, and efficiency for a collider search to be competitive. We formulatethe criterion as follows: for (cid:96) = τ, µ , we require that the number of expected signal eventsin a given decay channel (cid:96) → eY , denoted by N decayS , and in a collider process, denotedby N scattS , be comparable. For definiteness, we will phrase our discussion in terms of thecollider process ep → (cid:96)X , relevant for the EIC, but we will also consider pp → e(cid:96)X , relevantfor the LHC.Searches for (cid:96) → eY typically analyze a sample of N (cid:96) charged leptons produced eitherat e + e − machines or by hadronic decays in a fixed target experiment. These searches arealso characterized by a signal efficiency (cid:15) d , so that N decayS = (cid:15) d N (cid:96) BR (cid:96) → eY = (cid:15) d N (cid:96) Γ (cid:96) → eY τ (cid:96) , (2.1)where τ (cid:96) is the (cid:96) lepton lifetime. For example, in the case of both BaBar and Belle, N τ ∼ and (cid:15) d is in the 2 . →
6% range depending on the decay channel considered [35, 36].Currently, from experimental analyses one can infer only O (1) upper limits on N decayS ,from which one deduces upper limits (UL) on the BRsBR UL(cid:96) → eY ∼ (cid:15) d N (cid:96) , (2.2)where the symbol ∼ is used to indicate that analysis-dependent O (1) factors are missingon the RHS. – 3 –onversely, in a collider setup the relevant quantities are the integrated luminosity L ,the total signal efficiency (cid:15) s (including selection and reconstruction) and the cross section σ ep → (cid:96)X , leading to N scattS = (cid:15) s σ ep → (cid:96)X L . (2.3)Equating N scattS and N decayS one gets (cid:15) s L = ( (cid:15) d N (cid:96) ) τ (cid:96) Γ (cid:96) → eY σ ep → (cid:96)X ∼ UL(cid:96) → eY τ (cid:96) Γ (cid:96) → eY σ ep → (cid:96)X , (2.4)where in the last step we used (2.2). In Eq. (2.4) the ratio Γ (cid:96) → eY /σ ep → (cid:96)X depends inprinciple on the underlying new physics parameters. However, when considering a singledominant source of LFV (i.e. one SMEFT operator at a time), the dependence on newphysics parameters cancels completely in the ratio, which then depends only on the relevantmasses, collider energy, phase space factors and non-perturbative matrix elements. We willconsider below a few benchmark scenarios, in which the dominant new physics is either in (cid:96) → eγ dipole operators or in (cid:96)q ↔ eq four-fermion interactions.Denoting the new physics scale by Λ, for dipole operators dimensional considerationslead toΓ τ → eγ ∼ m τ v Λ , σ ep → τX ∼ v Λ , Γ τ → eγ σ ep → τX = κ D m τ = κ D · . · cm − s − , (2.5)where κ D ∼ O (1). Explicit calculations to be presented later in the manuscript showthat κ D = 0 .
33 for √ S = 100 GeV. Similarly, for pseudo-scalar and axial-vector operatorsinvolving first-generation quarks one can estimateΓ τ → eπ ∼ m τ Λ Λ , σ ep → τX ∼ S Λ , Γ τ → eπ σ ep → τX = κ A,P m τ Λ S = κ A,P · . · cm − s − , (2.6)where κ A,P ∼ O (1) and explicit calculation shows that κ P = 2 . κ A = 0 .
95 for √ S = 100 GeV. An analogous estimate for scalar and vector operators leads toΓ τ → eππ ∼ m τ Λ , σ ep → τX ∼ S Λ , Γ τ → eππ σ ep → τX = κ S,V m τ (2 π ) S = κ S,V · . · cm − s − , (2.7)where the extra (2 π ) in Γ /σ accounts for the mismatch in phase space factors betweendecay and collider process. Numerically we find κ S = 0 . κ V = 0 . • For the dipole operator, using the current limit BR
ULτ → eγ ∼ − [35], Eq. (2.4) im-plies that to match the τ → eγ sensitivity one would need an EIC with integratedluminosity satisfying (cid:15) s L D ∼ fb − . This is out of reach for the current EIC design. • For (pseudo)scalar and (axial) vector contact interactions involving first-generationquarks, using BR
ULτ → eππ ∼ − [36] one needs at √ S = 100 GeV an integratedluminosity of (cid:15) s L S,V ∼ fb − , which could be within reach of the current EIC– 4 –esign under optimal conditions after several years of running [17]. Therefore, the EICshould be competitive in constraining contact CLFV interactions and in probing themany directions in the SMEFT parameter space that are left unconstrained by low-energy probes of CLFV. It is also worth noting that in the case of contact interactionsEq. (2.6) implies that the constraining power of the EIC grows linearly with S . • From the above discussion one also sees that for new physics patterns that involvemore than a single dominant operator, the ratio Γ τ → eY /σ ep → τX could be suppresseddue to cancellations and therefore even for flavor-conserving light quark operatorsthe EIC could be more competitive than the simplest scenarios suggest. • Importantly, considering operators involving heavy quark flavors Q = c, b makes theanalysis more favorable for the EIC. As an example consider vector operators: thecross section σ ep → τX Q is suppressed with respect to the light flavor case by about oneorder of magnitude, due to the heavy flavor PDFs. On the other hand the heavy flavoroperators can contribute to τ decays such as τ → eππ only through loop amplitudessuppressed by a factor of a few × − . In turn, this implies a suppression of about ≈ − in the decay rate, much larger than the suppression in the cross section.Putting the ingredients together we find that Γ τ → eππ /σ ep → τX Q is suppressed by afactor of ≈ − compared to the light flavor case. Therefore, the requirement on theluminosity is only (cid:15) s L ∼ . − , well within the reach of the current EIC design,even with realistic (cid:15) s ∼ O (%). This analysis suggests that the largest discoverypotential at the EIC is in the DIS processes involving production of heavy quarkflavors in the final state. • For the LHC-relevant process pp → e(cid:96)X , the cross section scaling given in Eqs.(2.5)-(2.7) for ep → (cid:96)X is still valid, with √ S replaced by the τ - e invariant mass, m τe . Existing analyses reach m τe of a few TeV [37]. As a consequence, for dipoleoperators and vertex corrections one does not expect particularly great sensitivityat the LHC. On the other hand, for four-fermion operators the larger m τe bringsthe luminosity requirement for the LHC to the realistic levels (cid:15) s L S,V ∼
50 fb − .Taking into account the numerical factors and PDF integrations, this brings the LHCconstraints on dimension-six Wilson coefficients to within an order of magnitude ofthe constraints from τ decays. We also note the recent study [38] comparing LHC andEIC constraints on lepton flavor-conserving vector four-fermion operators, showingthe potential power of the EIC to lift degeneracies or flat directions in the space ofSMEFT operators that would remain using LHC alone.Finally, we note that one could repeat the above analysis for the case of e ↔ µ transitions, using BR ULµ → eY ∼ − and the appropriate changes m τ → m µ and τ τ → τ µ .Taking these effects into account we find that the integrated luminosity required for EICto be competitive in e → µ transitions would be eight orders of magnitude larger thanthe one required for e → τ transitions. This result implies that for these transitions theEIC cannot compete with low-energy muon processes, in agreement with the findings ofRef. [18]. Therefore, in what follows we will focus on e ↔ τ transitions.– 5 – The operator basis
We consider in this paper CLFV at the EIC, LHC and in low-energy τ and meson decays.At the center-of-mass energies reached at the EIC, it is appropriate to integrate out thedegrees of freedom that induce CLFV, and to work in the framework of the SMEFT. Wewill also frame the analysis of LHC data in the SMEFT, even though in this case the limitswe obtain should be interpreted with some care. The dimension-six SMEFT Lagrangian was constructed in Refs. [21, 22], and it containsthe most general set of operators that are invariant under the Lorentz group, the gaugegroup SU (3) c × SU (2) L × U (1) Y , and that have the same field content as the SM. Weconsider here the SM in its minimal version, with three families of leptons and quarks,and one scalar doublet. In particular, we do not introduce a light sterile neutrino ν R . Theleft-handed quarks and leptons transform as doublets under SU (2) L q L = (cid:32) u L d L (cid:33) , (cid:96) L = (cid:32) ν L e L (cid:33) , (3.1)while the right-handed quarks, u R and d R , and charged leptons, e R , are singlets under SU (2) L . The scalar field ϕ is a doublet under SU (2) L . In the unitary gauge we have ϕ = v √ U ( x ) (cid:32)
01 + hv (cid:33) , (3.2)where v = 246 GeV is the scalar vacuum expectation value (vev), h is the physical Higgsfield and U ( x ) is a unitary matrix that encodes the Goldstone bosons. We will denote by˜ ϕ the combination ˜ ϕ = iτ ϕ ∗ . The gauge interactions are determined by the covariantderivative D µ = ∂ µ + ig y B µ + i g τ I W Iµ + ig s G aµ t a (3.3)where B µ , W Iµ and G aµ are the U (1) Y , SU (2) L and SU (3) c gauge fields, respectively, and g , g , and g s are their gauge couplings. Furthermore, τ I / t a are the SU (2) L and SU (3) c generators, in the representation of the field on which the derivative acts. In the SM, thegauge couplings g and g are related to the electric charge and the Weinberg angle by g s w = g c w = e , where e > s w = sin θ W , c w = cos θ W .These relations are affected by SMEFT dimension-six operators, but these corrections aresubleading for the processes considered here, which have no SM background. Similarly, atthe order we are working, we can interchangeably use v or the Fermi constant G F , usingthe SM relation √ G F = v − . The values of the couplings g s , g , g and of the quarkmasses, and the hypercharge assignments of the SM fields are given in Table 19 and inEq. (A.2). – 6 –n the SM, lepton flavor is exactly conserved. There is a single, gauge-invariantdimension-five operator [19] L = C ( ˜ ϕ(cid:96) L ) T C ( ˜ ϕ † (cid:96) L ) , (3.4)where C is the charge conjugation matrix. When the Higgs takes its vev, L gives rise tothe neutrino Majorana masses and mixings, and thus to LFV in the neutral sector. Theoperator in Eq. (3.4) violates lepton number, and thus two insertions of C are neededto induce CLFV at the loop level. While formally dimension-six, the resulting CLFV isproportional to the masses of the light neutrinos and thus negligible [1, 2].CLFV processes are affected by many dimension-six operators. Following the notationof Ref. [22], we classify the relevant operators according to their gauge (denoted by X ),fermion ( ψ ), and scalar field ( ϕ ) content. The operators that contribute at tree level fallin the following four classes: L = L ψ ϕ D + L ψ Xϕ + L ψ ϕ + L ψ . (3.5)The first three classes contain fermion bilinear operators. ψ ϕ D contains corrections tothe SM couplings of quarks and leptons to the Z and W bosons, ψ Xϕ contains dipolecouplings to the U (1) Y , SU (2) L and SU (3) c gauge bosons, and ψ ϕ contains non-standardYukawa interactions. Focusing on purely leptonic operators, we consider L ψ ϕ D = − ϕ † i ↔ D µ ϕv (cid:16) ¯ (cid:96) L γ µ c (1) Lϕ (cid:96) L + ¯ e R γ µ c eϕ e R (cid:17) − ϕ † i ↔ D Iµ ϕv ¯ (cid:96) L τ I γ µ c (3) Lϕ (cid:96) L , (3.6) L ψ Xϕ = − √ (cid:96) L σ µν ( g Γ eB B µν + g Γ eW τ I W Iµν ) ϕv e R + h . c . , (3.7) L ψ ϕ = −√ ϕ † ϕv ¯ (cid:96) L Y (cid:48) e ϕe R + h.c. , (3.8)where ↔ D µ = D µ − ← D µ , ↔ D Iµ = τ I D µ − ← D µ τ I . The couplings c (1) Lϕ , c (3) Lϕ , c eϕ are hermitian, 3 × Z couplings, so that the Z vertices are given by L Z = − g c w Z µ (cid:26) (cid:18) z e L δ pr + 12 (cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) pr (cid:19) ¯ e pL γ µ e rL + (cid:18) z e R δ pr + 12 (cid:2) c eϕ (cid:3) pr (cid:19) ¯ e pR γ µ e rR + z u L ¯ u pL γ µ u pL + z u R ¯ u pR γ µ u pR + z d L ¯ d pL γ µ d pL + z d R ¯ d pR γ µ d pR (cid:27) , (3.9)with p, r being lepton flavor or quark family indices. The couplings z f L and z f R are z f L = T f − Q f s w , z f R = − Q f s w , (3.10)where T f and Q f are the fermion isospin and charge.Meanwhile, Γ eW and Γ eB in Eq. (3.7) are generic 3 × Z and photon fieldΓ eγ = Γ eB − Γ eW , Γ eZ = − c w Γ eW − s w Γ eB . (3.11) Here, ϕ † ←− D µ ϕ ≡ ( D µ ϕ ) † ϕ . – 7 –inally, Y (cid:48) e in Eq. (3.8) is a dimension-six Yukawa coupling, which corrects the dimension-four SM Yukawa L ψ ϕ = −√ (cid:96) L Y (0) e ϕe R + h.c. (3.12)When the Higgs gets its vev, we can write L yuk = − v ¯ e L Y e e R (cid:18) hv (cid:19) − ¯ e L Y (cid:48) e e R h + . . . + h.c. , Y e = Y (0) e + 12 Y (cid:48) e , (3.13)where the dots denote higher-order terms in h . We can always diagonalize the first term, sothat the charged lepton masses are given by M e = vY e . The second term can in general beoff-diagonal. For both quark and lepton SM Yukawa couplings we will use the convention M f = vY f . The quark Yukawa interactions are the same as the Y e term in Eq. (3.13) witheach e → q . L ψ includes four-fermion operators. The most relevant for collider searches are semilep-tonic four-fermion operators, L ψ = − G F √ (cid:26) C (1) LQ ¯ (cid:96) L γ µ (cid:96) L ¯ q L γ µ q L + C (3) LQ ¯ (cid:96) L τ I γ µ (cid:96) L ¯ q L τ I γ µ q L (3.14)+ C eu ¯ e R γ µ e R ¯ u R γ µ u R + C ed ¯ e R γ µ e R ¯ d R γ µ d R + C Lu ¯ (cid:96) L γ µ (cid:96) L ¯ u R γ µ u R + C Ld ¯ (cid:96) L γ µ (cid:96) L ¯ d R γ µ d R + C Qe ¯ e R γ µ e R ¯ q L γ µ q L (cid:27) − G F √ (cid:26) C LedQ ¯ (cid:96) iL e R ¯ d R q iL + C (1) LeQu ε ij ¯ (cid:96) iL e R ¯ q jL u R + C (3) LeQu ε ij ¯ (cid:96) iL σ µν e R ¯ q jL σ µν u R + h.c. (cid:27) . Here, i, j represent SU (2) L indices. Of these operators, only a few affect charged currents,introducing new Lorentz structures, such as scalar-scalar and tensor-tensor interactions.All of the above operators modify neutral currents and the couplings are, in general, four-index tensors in flavor. We allow the operators to have a generic structure in quark flavor.We follow the flavor conventions of Ref. [39] and assign operator labels to the neutralcurrent components with charged leptons, after rotating to the u and d quark mass basis.This induces factors of the SM CKM matrix V CKM in the charged-current and in the neutralcurrent neutrino components, which play a minimal role here. For example, introducing C LQ,U = ( U u ) † L (cid:16) C (1) LQ − C (3) LQ (cid:17) U uL , C LQ,D = (cid:16) U dL (cid:17) † (cid:16) C (1) LQ + C (3) LQ (cid:17) U dL , (3.15)where U u,dL,R are unitary matrices that diagonalize the quark mass matrices, the first twoterms in the four-fermion Lagrangian Eq. (3.14) become L = − G F √ (cid:26) (cid:2) C LQ,U (cid:3) prst ¯ e pL γ µ e rL ¯ u sL γ µ u tL + (cid:2) C LQ,D (cid:3) prst ¯ e pL γ µ e rL ¯ d sL γ µ d tL + (cid:104) V CKM C LQ,D V † CKM (cid:105) prst ¯ ν pL γ µ ν rL ¯ u sL γ µ u tL + (cid:104) V † CKM C LQ,U V CKM (cid:105) prst ¯ ν pL γ µ ν rL ¯ d sL γ µ d tL + (cid:18)(cid:104) C LQ,D V † CKM − V † CKM C LQ,U (cid:105) prst ¯ ν pL γ µ e rL ¯ d sL γ µ u tL + h.c. (cid:19) (cid:27) , (3.16)where p , r , s , t are flavor indices in quark/lepton mass bases. With these conventions,all semileptonic operators have naturally either u -type or d -type quark flavor indices. The– 8 –nly exception is C Qe , which we choose to be d -type, leading to neutral current vertices ofthe form L = − G F √ e R γ µ e R (cid:16) ¯ d L C Qe γ µ d L + ¯ u L V CKM C Qe V † CKM γ µ u L (cid:17) . (3.17)As we discuss in Appendix A, the renormalization group evolution of the operators inEq. (3.14) also induces purely leptonic operators L ψ = − G F √ (cid:26) C LL ¯ (cid:96) L γ µ (cid:96) L ¯ (cid:96) L γ µ (cid:96) L + C ee ¯ e R γ µ e R ¯ e R γ µ e R + C Le ¯ (cid:96) L γ µ (cid:96) L ¯ e R γ µ e R (cid:27) . (3.18)These operators could be probed at the EIC by looking for final states with multiple leptons.At low energy, they can be sensitively probed by the process τ → e ¯ (cid:96)(cid:96) . The Lagrangians in Eqs. (3.6), (3.7), (3.8) and (3.14) are defined just below the newphysics scale Λ (cid:29) v . For the study of DIS at the EIC, of LHC constraints and of low-energy processes we first evolve the Lagrangian to a scale µ close to the electroweak scale.The renormalization group equations (RGEs) in the SMEFT were derived in Refs. [23–25]and we report them for convenience in Appendix A, where we also provide the numericalsolutions of the RGEs at leading logarithmic accuracy.We comment here on the most important qualitative effects: • The scalar and tensor operator coefficients C (1 , LeQu and C LedQ run in QCD. The run-ning from Λ ∼ µ = m t increases (decreases) the coefficient of the scalar(tensor) operators by roughly 10% (5%). • Z dipoles, scalar and tensor operators mix into the photon dipole Γ eγ at leading log[23–25]. We show the relevant RGEs in Eqs. (A.7)–(A.11). The mixing of the Z dipole is at the 10 − level, as expected from a weak loop correction. The mixingof the tensor operator is proportional to the quark Yukawa coupling and thus it isparticularly important for the tt component of C (3) LeQu . The strong constraints onflavor-changing dipoles imply that this mixing is also non-negligible for the charmcomponent of the tensor operator. C (1) LeQu mixes with C (3) LeQu via an electroweak loop.For the tt component of the scalar operator, the resulting contribution to Γ eγ is sizable.The coefficients of photon and Z dipoles, scalar and tensor four-fermion operators atthe scale µ = m t , as a function of top, bottom and charm scalar and tensor operatorsat the scale µ = 1 TeV, are given in Table 20. • Vector-like four-fermion operators with heavy quarks mix onto Z -boson vertices andfour-fermion operators with light quarks and leptons via the penguin diagrams shownin Fig. 1. As shown in Eqs. (A.14) and (A.15), the mixing with the CLFV Z couplingshas a component proportional to the quark Yukawa coupling and one to the gaugecouplings. For top-quark operators, the Yukawa component dominates, and inducesa very sizable mixing (cid:16) c (1) Lϕ + c (3) Lϕ (cid:17) ( µ t ) ∼ . C Lu − C LQ,U ) tt ( µ ) , c eϕ ( µ t ) ∼ . C eu − C Qe ) tt ( µ ) , (3.19)– 9 – Z, g e t e t e e e t t t W W W n t Figure 1 . One loop diagrams contributing to the running of heavy flavor operators onto operatorsthat can be tested at the EIC and in τ decays. Plain lines denotes leptons and light quarks,double lines heavy quarks, a square an insertion of a CLFV operator and dots SM vertices. Thefirst two diagrams represent penguin contributions of heavy flavor vector operators to CLFV Zcouplings, leptonic operators and semileptonic operators with light quarks. The latter also receivecontributions from W exchanges, as shown in the last three diagrams. Tensor operators run intoΓ eγ,Z via a diagram with the same topology as the first. where ( C Qe ) tt = V tj ( C Qe ) jk V ∗ tk , µ t ∼ m t and µ ∼ Λ ∼ b and c quarks, the gauge component dominates, and gives percent level correctionsto the Z couplings. The mixing with light-quark and lepton four-fermion operators,driven by the RGEs in Eqs. (A.16)–(A.22) and (A.29)–(A.34), is the same for all theflavor components of u - or d -type operators, and these mixing coefficients are at the10 − level. The coefficients of Z couplings, leptonic and semileptonic four-fermionoperators at the scale µ = m t , as a function of heavy quark operators at the scale µ = 1 TeV, are given in Table 21. • The mixing of quark-flavor off-diagonal four-fermion operators onto flavor diagonalis suppressed by small CKM and/or Yukawa couplings, see Eqs. (A.23)–(A.28), andit is in most cases negligible.In addition to the running effects, integrating out heavy flavors induces gluonic oper-ators. The EIC is sensitive to the CLFV Yukawa Y (cid:48) e in Eq. (3.13) via the couplings of theHiggs bosons to quarks and the effective Higgs-gluon coupling induced at the top threshold, L hgg = α s πv h G aµν G a µν . (3.20)In addition, CLFV SMEFT operators with heavy quarks can induce dimension-seven glu-onic operators of the form L g = C GG v α s π (cid:0) G aµν G aµν (cid:1) ¯ e L e R + C G (cid:101) G v α s π (cid:16) G aµν (cid:101) G a µν (cid:17) ¯ e L e R + h.c. (3.21)where (cid:101) G aµν = 12 (cid:15) µναβ G aαβ . At the top threshold, C GG,G (cid:101) G are induced by the scalar operators with matching coefficients[ C GG ] τe = − vm t (cid:104) C (1) LeQu (cid:105) τett , [ C GG ] eτ = − vm t (cid:104) C (1) LeQu (cid:105) eτtt , (3.22) (cid:2) C G (cid:101) G (cid:3) τe = − i vm t (cid:104) C (1) LeQu (cid:105) τett , (cid:2) C G (cid:101) G (cid:3) eτ = − i vm t (cid:104) C (1) LeQu (cid:105) eτtt . (3.23)Notice that both sides of Eqs. (3.22) and (3.23) are renormalization-scale-independent, atone loop in QCD. – 10 – CLFV Deep Inelastic Scattering
We obtain in this Section the expressions for deep inelastic scattering (DIS) cross sectionsin the presence of CLFV SMEFT operators. In Sec. 4.1 we factorize the generic DIScross section into leptonic and hadronic structures, matching the latter onto partonic hardmatching coefficients convolved with parton distribution functions (PDFs), reviewing thestandard derivation in QCD, followed by generalization to contributions from arbitrarySMEFT operators. We simplify to tree-level cross sections for the remainder of the analysis,and in Sec. 4.2 we collect the tree-level partonic cross sections induced by all the CLFVSMEFT operators we consider. In Sec. 4.3, we provide numerical values of the crosssections multiplying the SMEFT operator coefficients, and obtain initial estimates of EICsensitivity to each coupling based on the partonic cross sections. In Sec. 5 we will go tothe more realistic case of detector-level cross sections.
The generic cross section differential in the momentum transfer q = k − k (cid:48) in the scattering (cid:96) ( k ) p ( P ) → (cid:96) (cid:48) ( k (cid:48) ) X ( p X ) is dσd q = 12 S (cid:90) d Φ L (cid:88) X (cid:12)(cid:12) M ( (cid:96)p → (cid:96) (cid:48) X ) (cid:12)(cid:12) (2 π ) δ ( P + q − p X ) δ ( q − k + k (cid:48) ) . (4.1)where S = ( k + P ) , Φ L is the outgoing lepton (cid:96) (cid:48) phase space, and the sum is over all otherfinal state particles X . We do not yet specify whether we sum over (cid:96), p spins, allowing forthe possibility of polarized beams. We sum over (cid:96) (cid:48) spins. We will use the standard DISkinematic variables, Q ≡ − q , x ≡ Q P · q , y ≡ P · q P · k , xyS = Q . (4.2)To form the cross section differential in the DIS variables x, y , we insert the delta functionsdefining these variables, dσdx dy = (cid:90) d q dσd q δ (cid:16) x + q P · q (cid:17) δ (cid:16) y − P · q P · k (cid:17) . (4.3)It is convenient to pick a particular frame to perform the integrals with the delta functions,though the result is still Lorentz-invariant. For example, in the Breit or CM frames, theproton can be put in the +ˆ z direction, and P, q take the forms P = ¯ n z · P n z , q = ¯ n z · q n z n z · q ¯ n z q T , (4.4)where n z = (1 , ˆ z ), ¯ n z = (1 , − ˆ z ). Then we use the delta functions in Eq. (4.3) to integrateover n z · q, ¯ n z · q . To do the q T integrals, we express the (cid:96) (cid:48) phase space integral to leadingorder in electroweak interactions: (cid:90) d Φ L = (cid:90) d k (cid:48) (2 π ) δ (cid:0) ( q − k ) (cid:1) , (4.5)– 11 – igure 2 . The DIS process induced by CLFV SMEFT operators. The gray blob representsarbitrary CLFV interactions mediating ep → τ X . which will let us do the q T integral (using also azimuthal symmetry). In the end, ourformula Eq. (4.3) becomes dσdx dy = y π (cid:88) X (cid:12)(cid:12) M ( (cid:96)p → (cid:96) (cid:48) X ) (cid:12)(cid:12) (2 π ) δ ( P + q − p X ) , (4.6)where the value of q has been fixed by the above delta function integrals, e.g. in frameswhere P takes the form in Eq. (4.4), we have q = yS ¯ n z · P ¯ n z − xy ¯ n z · P n z Q (cid:112) − y ˆ n T , (4.7)where ˆ n T is a unit vector in any direction transverse to n z (azimuthally symmetric).Eq. (4.6) is our basic starting formula for a DIS cross section.The bulk of our calculations will come in evaluating the squared amplitudes |M| inthe presence of arbitrary SMEFT operators that can mediate the process (cid:96)P → (cid:96) (cid:48) X , whereprimarily we shall be interested in (cid:96) = e and (cid:96) (cid:48) = τ as in Fig. 2. All of the operators orchannels we consider give amplitudes that can be expressed in a form, M ( (cid:96)p → (cid:96) (cid:48) X ) = (cid:88) I C I (cid:10) (cid:96) (cid:48) ( k (cid:48) ) (cid:12)(cid:12) O I lep | (cid:96) ( k ) (cid:105) (cid:104) X | O I had | p ( P ) (cid:105) , (4.8)where each operator is factored into a leptonic and hadronic part, the two parts containingthe relevant leptonic and hadronic fields: O lep ∼ ¯ (cid:96) (cid:48) Γ Il (cid:96) , O had ∼ ¯ q f (cid:48) Γ Ih ¯ q f , G αβ G µν , (4.9)and in general we will lump constant prefactors into O lep . Here Γ l,h are any allowed Diracmatrix structures, and the gluon field indices may be contracted in different ways, e.g. GG, G (cid:101) G . These effective operators may also arise from contractions of other operators, inwhich case relevant propagators or other factors are lumped into the coefficients. In thesum over operator structures I , any appropriate contractions over Dirac or flavor indicesare understood.With amplitudes of the form Eq. (4.8), the cross section Eq. (4.6) also factors intoleptonic and hadronic structures, dσdx dy = (cid:88) IJ L IJ ⊗ W IJ , (4.10)– 12 –here L IJ = y π C I C ∗ J (cid:104) (cid:96) ( k ) | ¯ (cid:96) ¯Γ Jl (cid:96) (cid:48) (cid:12)(cid:12) (cid:96) (cid:48) ( k (cid:48) ) (cid:11) (cid:10) (cid:96) (cid:48) ( k (cid:48) ) (cid:12)(cid:12) ¯ (cid:96) (cid:48) Γ Il (cid:96) | (cid:96) ( k ) (cid:105) (4.11a) W IJ = (cid:88) X (2 π ) δ ( P + q − p X ) (cid:104) p ( P ) | ¯ O J had | X (cid:105) (cid:104) X | O I had | p ( P ) (cid:105) , (4.11b)where ¯Γ = γ Γ † γ , ¯ O = O † , and the ⊗ in Eq. (4.10) represents any appropriate indexcontractions. We have assumed that the inclusive state X is purely hadronic, appropriatefor us working at tree level in electroweak interactions.At this point we have not yet specified whether the incoming lepton and proton arepolarized or spin-averaged. In the leptonic part, we can pick out right- or left-handedpolarizations by summing over spins but including projection operators in the leptonicDirac structures Γ I,Jl , i.e., again at tree level, L IJ = y π C I C ∗ J (cid:88) spins ¯ u ( k ) 1 − λ (cid:96) γ Jl u ( k (cid:48) )¯ u ( k (cid:48) )Γ Il λ (cid:96) γ u ( k ) (4.12)= y π C I C ∗ J Tr (cid:16) k/ ¯Γ Jl k/ (cid:48) Γ Il λ (cid:96) γ (cid:17) , where λ (cid:96) = ± R, L -handed incoming (cid:96) . For the case of SM electroweak interactions,with photon and Z boson exchanges, expressions for the traces in Eq. (4.12) can be foundin, e.g., [40]. In the simplest case of tree-level photon exchange in the SM, we will relabel I, J → γf f (cid:48) indicating the photon coupling to quark flavors f, f (cid:48) in the hadronic part, andthe tensor Eq. (4.12) takes the value L µνγff (cid:48) = − α e f e f (cid:48) xS ( g µνT − iλ (cid:96) (cid:15) µνT ) , (4.13)where e f is the electric charge of quark flavor f in units of e , and α em ≡ e / (4 π ). Thetensor structures appearing in Eq. (4.13) are: g µνT ≡ g µν − k µ k (cid:48) ν + k ν k (cid:48) µ Q , (cid:15) µνT = 2 Q (cid:15) αβµν k α k (cid:48) β . (4.14)When W IJ → W γff (cid:48) in Eq. (4.11) is evaluated for partonic initial states, at tree level, wewill simply obtain the Born cross section for Eq. (4.10). In general we need to match W IJ onto quark and gluon PDFs (polarized and unpolarized) in the proton state. We sketchthis matching procedure in the next subsection. The hadronic part of the amplitude W IJ in Eq. (4.11) can be expressed, as in usualDIS, as convolutions of perturbative matching coefficients and PDFs. Using the deltafunction to translate one of the operators, and summing over X , we obtain: W IJ = (cid:90) d x e iq · x (cid:104) p ( P ) | ¯ O J had ( x ) O I had (0) | p ( P ) (cid:105) . (4.15)– 13 –his forward matrix element of the product of operators can be related to twice the imag-inary part or the discontinuity of the matrix element of the time-ordered product of theoperators (e.g. [41, 42]): W IJ = Disc T IJ , T IJ ≡ i (cid:90) d x e iq · x (cid:104) p ( P ) | T (cid:8) ¯ O J had ( x ) O I had (0) (cid:9) | p ( P ) (cid:105) , (4.16)which can be evaluated from ordinary Feynman diagrams. This operator product typicallycontains two pairs of quark or gluon bilinears, separated by x . We will perform an operatorproduct expansion (OPE) to match onto products of a single bilinear operator containingquark or gluon fields, separated only along the light-cone direction n z conjugate to theproton momentum P . In general, the product of operators in Eq. (4.16) will match, atleading power (twist) onto: W IJ −→ (cid:90) dr (cid:2) C IJq ( r ) O q ( r ) + C IJ ( r ) O ( r ) + C IJg O g ( r ) + C IJ ˜ g O ˜ g ( r ) (cid:3) , (4.17)where O q, are quark bilinear operators: O q ( r ) = (cid:90) dz π e − izr ¯ q ( z ¯ n z ) ¯ n/ z W ( z ¯ n z , q (0) (4.18a) O ( r ) = (cid:90) dz π e − izr ¯ q ( z ¯ n z ) ¯ n/ z γ W ( z ¯ n z , q (0) (4.18b)and O g, ˜ g are gluon bilinear operators: O g ( r ) = − (cid:90) dz πr e − izr ¯ n µz ¯ n αz G µλ ( z ¯ n z ) Y ( z ¯ n z , G λα (0) − ( r → − r ) (4.19a) O ˜ g ( r ) = i (cid:90) dz πr e − izr ¯ n µz ¯ n αz G µλ ( z ¯ n z ) Y ( z ¯ n z , (cid:101) G λα (0) + ( r → − r ) . (4.19b)In Eqs. (4.18) and (4.19), each pair of quark or gluon fields are separated only along thelight-cone direction ¯ n z conjugate to the large proton momentum along n z , and the W, Y are fundamental or adjoint Wilson line gauge links along ¯ n z ensuring gauge invariance (inthis paper, we can take W = Y = 1). Matrix elements of these bilinear operators in theproton state give the unpolarized and polarized PDFs [43–46]: f q ( ξ ) = 12 (cid:88) λ (cid:104) p, λ | O q ( ξ ¯ n z · P ) | p, λ (cid:105) , λ ∆ f q ( ξ ) = (cid:104) p, λ | O ( ξ ¯ n z · P ) | p, λ (cid:105) , (4.20a) f g ( ξ ) = 12 (cid:88) λ (cid:104) p, λ | O g ( ξ ¯ n z · P ) | p, λ (cid:105) , λ ∆ f g ( ξ ) = (cid:104) p, λ | O ˜ g ( ξ ¯ n z · P ) | p, λ (cid:105) . (4.20b)The matching coefficients C IJq, ,g, ˜ g in Eq. (4.17) are computed by matching partonic matrixelements of the operators on either side of the equation, with hard propagators betweenextra fields on the left-hand side contracted or integrated out. This procedure is illustratedin Fig. 3. At tree level we will not encounter mixing of quark and gluon operators, but athigher orders they will mix. – 14 – ¯ n z ξP ξP ⊗ ⊗ n z r ¯ n z 𝒪 q ,5 q qξP ξPξP + q ¯𝒪 J 𝒪 I q qξP ξPξP + q ¯𝒪 J 𝒪 I ⊗ ⊗ n z 𝒪 g ,˜ g r ¯ n z r ¯ n z ξP ξP Figure 3 . Matching products of operators in hadronic tensor Eq. (4.15) onto quark or gluon bilinearoperators in Eq. (4.18) or Eq. (4.19) at tree level, using partonic matrix elements in external quarkor gluon states with momentum ξP . The operators O I,J are the hadronic part of generic SM orSMEFT operators or amplitudes, and proton matrix elements of O q, or O g, ˜ g give (un)polarizedquark and gluon PDFs in the proton, see Eqs. (4.20) and (4.20b). SM QCD:
In the usual case of QCD in the SM, for photon exchange diagrams, we obtainfor the hadronic tensor in Eq. (4.11) that contracts with the leptonic tensor in Eq. (4.13), W µνγff (cid:48) = Disc i (cid:90) d x e iq · x (cid:104) p ( P ) | T { ¯ q f (cid:48) γ ν q f (cid:48) ( x )¯ q f γ µ q f (0) } | p ( P ) (cid:105) . (4.21)To match the operator in this matrix element onto those on the RHS of Eq. (4.17), wecompute matrix elements of each in a quark state (see Fig. 3): W µν ( q ) γff (cid:48) = Disc i (cid:90) d x e iq · x (cid:104) q ( ξP ; λ ) | T { ¯ q f (cid:48) γ ν q f (cid:48) ( x )¯ q f γ µ q f (0) | q ( ξP ; λ ) (cid:105) (4.22)= − πxδ ( ξ − x ) (cid:104) g µν ⊥ − x P µ P ν Q + iλ(cid:15) µν ⊥ (cid:105) δ ff (cid:48) , with momentum ξP and spin λ , and where the transverse tensor structures are: g µν ⊥ ≡ g µν − P µ q ν + P ν q µ P · q , (cid:15) µν ⊥ ≡ P · q (cid:15) µναβ P α q β . (4.23)Meanwhile, the quark matrix elements of O q , O in Eq. (4.18) are: (cid:104) q ( ξP ; λ ) | O q ( r ) | q ( ξP ; λ ) (cid:105) = δ (cid:16) rξ ¯ n z · P − (cid:17) (4.24a) (cid:104) q ( ξP ; λ ) | O ( r ) | q ( ξP ; λ ) (cid:105) = λδ (cid:16) rξ ¯ n z · P − (cid:17) . (4.24b)This tells us that the matching coefficients in Eq. (4.17) for the operator in Eq. (4.21) are: C µνq,γff (cid:48) ( r ) = − πδ ( r − x ¯ n z · P ) (cid:104) g µν ⊥ − x P µ P ν Q (cid:105) δ ff (cid:48) (4.25a) C µν ,γff (cid:48) = − πδ ( r − x ¯ n z · P ) i(cid:15) µν ⊥ δ ff (cid:48) . (4.25b)– 15 –sing these matching conditions in Eq. (4.21) and the PDF operator definitions in Eq. (4.20),and contracting the perturbative matching coefficients in W µνγff (cid:48) with the leptonic tensor inEq. (4.13), we obtain for the cross section Eq. (4.11) for the photon channel in SM QCDDIS at LO: dσ LO λ e λ T dx dy = 2 σ (cid:88) q e q (cid:26) [1 + (1 − y ) ] f q ( x ) − λ e λ T y (2 − y )∆ f q ( x ) (cid:27) , (4.26)for incoming e and proton target spins λ e,T , and where σ ≡ α πSxQ . (4.27)Averaged over incoming spins, we obtain the familiar unpolarized DIS cross section at LOin QCD: dσ LOun dxdy = 2 σ (cid:88) q e q f q ( x ) (cid:104) − y ) (cid:105) . (4.28) SMEFT four-fermion operators:
We can generalize the above derivation in the SMfor generic four-fermion operators in SMEFT. The leptonic tensor for a given operator stilltakes the form Eq. (4.12), and the hadronic tensor Eq. (4.15) will take the form: W IJψ = Disc i (cid:90) d x e iq · x (cid:104) p ( P ) | T { ¯ q f ¯Γ Jh q f (cid:48) ( x )¯ q f (cid:48) Γ Ih q f (0) } | p ( P ) (cid:105) , (4.29)where f, f (cid:48) are the particular quark flavors appearing in a given operator from, e.g.,Eq. (3.14). The operator in the matrix element matches onto O q, in Eq. (4.18) in similarmanner as in the SM above and illustrated in Fig. 3, with the partonic matrix elementanalogous to Eq. (4.22) now given by: W IJ ( q ) ψ = Disc i (cid:90) d x e iq · x (cid:104) q f ( ξP ; λ ) | T { ¯ q f ¯Γ Jh q f (cid:48) ( x )¯ q f (cid:48) Γ Ih q f (0) | q f ( ξP ; λ ) (cid:105) (4.30)= 2 πδ ( ξ − x ) 12 P · q Tr (cid:104) ξP/ ¯Γ Jh ( ξP/ + q/ )Γ Ih λγ (cid:105) , where any L, R projections of the quark fields q f,f (cid:48) are understood to be contained in Γ I,Jh .The matching coefficients in Eq. (4.17) for these operators onto O q, are: C IJq,ψ ( r ) = 2 πδ ( r − xn z · P ) 14 P · q Tr (cid:104) xP/ ¯Γ Jh ( xP/ + q/ )Γ Ih (cid:105) (4.31a) C IJ ,ψ ( r ) = 2 πδ ( r − xn z · P ) 14 P · q Tr (cid:104) xP/ ¯Γ Jh ( xP/ + q/ )Γ Ih γ (cid:105) . (4.31b)These perturbative coefficients contracted with the leptonic tensor in Eq. (4.12) will givethe partonic cross sections when plugged into Eq. (4.10), and we can write, similar to theSM formula Eq. (4.26), the four-fermion operator contribution to the full cross section: dσ ψ λ e λ T dx dy = L λ e IJ (cid:90) dr (cid:104) C IJq,ψ ( r ) f q ( x ) + λ T C IJ ,ψ ( r )∆ f q ( x ) (cid:105) , (4.32)– 16 –sing Eq. (4.24) for the PDFs. At tree level the integral over r here just removes the deltafunction in Eq. (4.31). The contraction of L IJ with each of C IJq, gives the tree-level partoniccross section from each operator.For example, for the scalar LeQu operator in Eq. (3.14), the operator contributing to ep → τ X is( C (1) LeQu ) ij ( O (1) LeQu ) ij ≡ O S lep O S had , where O S lep = 4 G F √ C (1) LeQu ) ij ¯ τ L e R , O S had = ¯ u iL u jR = ¯ u i γ u j , (4.33)where here i, j = u, c label the quark flavors. So the leptonic “tensor” Eq. (4.12) for initialelectron spin λ e and hadronic “tensor” in Eq. (4.30) in a quark state of momentum ξP andspin λ q are: L ( ij ) S = G F yQ π λ e (cid:12)(cid:12)(cid:12) ( C (1) LeQu ) ij (cid:12)(cid:12)(cid:12) , W ( ij ) S = 2 πxδ ( ξ − x ) 1 + λ q . (4.34)The matching coefficients Eq. (4.31) for the hadronic tensor are then C Sq ( r ) = C S ( r ) = πδ ( r − xn z · P ) , (4.35)and at tree level the gluon coefficients are zero. Thus the contribution of the operatorsEq. (4.33) to the cross section is dσ Sλ e λ T dx dy = G F yQ π λ e (cid:88) i,j = u,c (cid:12)(cid:12)(cid:12) ( C (1) LeQu ) ij (cid:12)(cid:12)(cid:12) (cid:2) f u j ( x ) + λ T ∆ f u j ( x ) (cid:3) . (4.36)and similarly for ¯ u j antiquark contibutions. The procedure for other four-fermion operatorcontributions is also similar. (Many of the resulting cross sections have been given recentlyin Ref. [38].) The contribution of dipole and Higgs operators follows substantially the sameprocedure as SM QCD or SMEFT four-fermion operator matching as well. We collect allrelevant partonic cross sections in Sec. 4.2. SMEFT gluon operators:
The matching procedure for products of gluon operatorsfrom Eq. (3.21) is similar: L g = O G lep O G had + O (cid:101) G lep O (cid:101) G had + h.c. , (4.37)where O G lep = (cid:2) C GG (cid:3) τe v α s π ¯ τ L e R , O (cid:101) G lep = (cid:2) C G (cid:101) G (cid:3) τe v α s π ¯ τ L e R , O G had = G aµν G aµν , O (cid:101) G had = G aµν (cid:101) G aµν . The cross section in Eq. (4.10) then takes the form dσ G dx dy = L G W G + L (cid:101) G W (cid:101) G + L G (cid:101) G W G (cid:101) G + L (cid:101) GG W (cid:101) GG , (4.38)where L G = yQ π v (cid:16) α s π (cid:17) (cid:110) λ e (cid:12)(cid:12)(cid:2) C GG (cid:3) τe (cid:12)(cid:12) + 1 − λ e (cid:12)(cid:12)(cid:2) C GG (cid:3) eτ (cid:12)(cid:12) (cid:111) , (4.39)– 17 – (cid:101) G = yQ π v (cid:16) α s π (cid:17) (cid:110) λ e (cid:12)(cid:12)(cid:2) C G (cid:101) G (cid:3) τe (cid:12)(cid:12) + 1 − λ e (cid:12)(cid:12)(cid:2) C G (cid:101) G (cid:3) eτ (cid:12)(cid:12) (cid:111) L G (cid:101) G = L ∗ (cid:101) GG = yQ π v (cid:16) α s π (cid:17) (cid:110) λ e (cid:2) C GG (cid:3) ∗ τe (cid:2) C G (cid:101) G (cid:3) τe + 1 − λ e (cid:2) C GG (cid:3) ∗ eτ (cid:2) C G (cid:101) G (cid:3) eτ (cid:111) , and W G = (cid:90) d x e iq · x (cid:104) p ( P ) | G aµν G aµν ( x ) G bαβ G bαβ (0) | p ( P ) (cid:105) (4.40) W (cid:101) G = (cid:90) d x e iq · x (cid:104) p ( P ) | G aµν (cid:101) G aµν ( x ) G bαβ (cid:101) G bαβ (0) | p ( P ) (cid:105) W G (cid:101) G = (cid:90) d x e iq · x (cid:104) p ( P ) | G aµν G aµν ( x ) G bαβ (cid:101) G bαβ (0) | p ( P ) (cid:105) = W (cid:101) GG ( q → − q ) . The matrix elements of the gluon PDF operators in Eq. (4.20b) in a partonic gluon statewtih polarization λ (Fig. 3) are (cid:104) g ( ξP ; λ ) | O g ( r ) | g ( ξP ; λ ) (cid:105) = r [ δ ( r − ξ ¯ n z · P ) + δ ( r + ξ ¯ n z · P )] ε ∗ αλ ε λα , (4.41) (cid:104) g ( ξP ; λ ) | O ˜ g ( r ) | g ( ξP ; λ ) (cid:105) = r [ δ ( r − ξ ¯ n z · P ) − δ ( r + ξ ¯ n z · P )] i (cid:15) µναβ ¯ n µz n νz ε ∗ αλ ε βλ , where ε λ is the polarization vector for the gluon in polarization state λ . Meanwhile, thetree level matrix elements of the operators in Eq. (4.40) in a gluon state (Fig. 3) are W ( g ) G = W ( g ) (cid:101) G = 8 πxQ δ ( ξ − x ) ε ∗ αλ ε λα , W ( g ) G (cid:101) G = 0 . (4.42)Thus the gluon matching coefficients in Eq. (4.17) are: C Gg ( r ) = C (cid:101) Gg ( r ) = 8 πQ δ ( r − x ¯ n z · P ) , C G, (cid:101) G ˜ g = 0 , (4.43)and at tree level the quark coefficients are zero. The contribution of the gluon operatorsEq. (4.37) to the cross section Eq. (4.10) is then dσ Gλ e λ T dx dy = yQ πv (cid:16) α s π (cid:17) f g ( x ) (cid:110) λ e (cid:12)(cid:12)(cid:2) C GG (cid:3) τe (cid:12)(cid:12) + (cid:12)(cid:12)(cid:2) C G (cid:101) G (cid:3) τe (cid:12)(cid:12) ) (4.44)+ 1 − λ e (cid:12)(cid:12)(cid:2) C GG (cid:3) eτ (cid:12)(cid:12) + (cid:12)(cid:12)(cid:2) C G (cid:101) G (cid:3) eτ (cid:12)(cid:12) ) (cid:111) For other possible SMEFT operator channels
I, J in the hadronic tensor Eq. (4.15),we can compute the matching of the T -ordered products of operators in Eq. (4.16) ontoquark and gluon bilinears Eq. (4.20) in the same way as we hve illustrated above. Withmore exclusive measurements on final states we may even become sensitive to more generalparton distributions in the proton. In this paper, we shall limit ourselves to tree-levelresults in QCD, which will always yield the na¨ıve Born-level parton model prediction,which we collect in Sec. 4.2 for all the SMEFT operators we consider. At LO in QCD, following the steps in the previous subsection at tree level in thematching onto PDFs in hadronic tensor, we obtain the DIS cross sections induced by the– 18 –LFV operators introduced in Section 3 in terms of the partonic cross sections σ aij where i, j ∈ { L, R } denote the helicity of the electron and quark/gluon, respectively, and a = q, g (where q = u, d, s, c, b or their antiquarks) denotes the partonic species. For beams withelectron and proton polarizations λ e,T , we obtain the generic cross section1 σ dσ λ e λ T dx dy = 12 (cid:88) a (cid:20) − λ e σ a LL + ˆ σ a LR ) + 1 + λ e σ a RL + ˆ σ a RR ) (cid:21) f a ( x, Q ) . (4.45)+ 12 (cid:88) a (cid:20) − λ e − ˆ σ a LL + ˆ σ a LR ) + 1 + λ e − ˆ σ a RL + ˆ σ a RR ) (cid:21) λ T ∆ f a ( x, Q ) , where λ e,T = ± R, L polarizations, respectively. Each individual ˆ σ aij on the right-hand side is the cross section for the specified incoming polarizations, normalized by σ inEq. (4.28), i.e.ˆ σ aij = y Q π α (cid:88) a (cid:48) (cid:12)(cid:12) M (cid:0) (cid:96) ( k, λ i ) a ( xP, λ j ) → (cid:96) (cid:48) ( k (cid:48) ) a (cid:48) (cid:1)(cid:12)(cid:12) (2 π ) δ ( xP + q − p a (cid:48) ) , (4.46)with the incoming parton a = q, g having the momentum fraction x of the proton mo-mentum P . The spin and flavor of the outgoing parton a (cid:48) are determined by the SMEFToperator(s) mediating the amplitude M .In the case of the unpolarized cross section, the dependence on polarized PDFs ∆ f inEq. (4.45) drops out, and we obtain the familiar spin-averaged unpolarized cross section: dσ un σ dx dy = 14 (cid:88) i (cid:0) ˆ σ i LL + ˆ σ i LR + ˆ σ i RL + ˆ σ i RR (cid:1) f i ( x, Q ) . (4.47)Since the absolute value of polarized PDFs are always smaller and have a larger uncertaintythan their unpolarized counterparts [38, 47, 48], we will focus on unpolarized targets inthis work and defer the impact of nonzero λ T to future work. For example, single-spinasymmetries could be used to study the polarized beam effects since the PDF uncertaintiescancel to a good degree. In the next subsection we give the expressions for the partoniccross sections corresponding to different operators. The Z couplings c (1 , Lϕ and c eϕ , and the four-fermion operators C LQ,U , C LQ,D , C Lu , C Ld , C eu , C ed and C Qe , which are the product of a quark and lepton left- or right-handed vectorcurrent, induce DIS cross sections whose x and y dependence are similar to neutral currentDIS in the SM. For example, defining the prefactor F Z as F Z = 1 c w s w Q ( Q + m Z ) , (4.48)we find that the partonic cross sections for u -type quarks are given byˆ σ u i LL = F Z (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)(cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe z u L + Q + m Z m Z (cid:2) C LQ, U (cid:3) τeu i u i (cid:12)(cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12)(cid:12) Q + m Z m Z (cid:2) C LQ, U (cid:3) τeu j u i (cid:12)(cid:12)(cid:12)(cid:12) (cid:27) – 19 – σ u i RR = F Z (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) [ c eϕ ] τe z u R + Q + m Z m Z [ C eu ] τeu i u i (cid:12)(cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12)(cid:12) Q + m Z m Z [ C eu ] τeu j u i (cid:12)(cid:12)(cid:12)(cid:12) (cid:27) (4.49)ˆ σ u i LR = F Z (1 − y ) (cid:26) (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe z u R + Q + m Z m Z [ C Lu ] τeu i u i (cid:12)(cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12)(cid:12) Q + m Z m Z [ C Lu ] τeu j u i (cid:12)(cid:12)(cid:12)(cid:12) (cid:27) ˆ σ u i RL = F Z (1 − y ) (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) [ c eϕ ] τe z u L + Q + m Z m Z [ C Qe ] τeu i u i (cid:12)(cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12)(cid:12) Q + m Z m Z [ C Qe ] τeu j u i (cid:12)(cid:12)(cid:12)(cid:12) (cid:27) , where u i = { u, c } , and [ C Qe ] u j u i includes factors of the CKM matrix as in Eq. (3.17). Thepartonic cross sections for ¯ u, d, ¯ d -type (anti)quarks are given in Appendix B.The Z couplings c (1) Lϕ + c (3) Lϕ and c eϕ induce contributions that are diagonal in quarkflavor, and, as seen in the relevant terms of Eq. (4.49), can interfere with the quark-flavor-diagonal components of the semileptonic operators in Eq. (3.14). Note the Z couplingcontributions and four-fermion operator contributions have different dependences on Q ,which, as we will discuss in Section 5, will lead to different transverse momentum andrapidity distributions for the τ decay products, and thus to different efficiencies. In the case of dipole operators given by Eqs. (3.7) and (3.11), we factor out the prefactor F dip = 4 Q v . (4.50)For the τ e coefficient of the dipole operators, the electron is right-handed, and the u -typequark contribution to the cross section isˆ σ uLL = ˆ σ uLR = 0 , ˆ σ uRR = F dip (1 − y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) Γ eγ (cid:3) τe Q u + z u R c w s w Q ( Q + m Z ) [Γ eZ ] τe (cid:12)(cid:12)(cid:12)(cid:12) , ˆ σ uRL = F dip (1 − y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) Γ eγ (cid:3) τe Q u + z u L c w s w Q ( Q + m Z ) [Γ eZ ] τe (cid:12)(cid:12)(cid:12)(cid:12) . (4.51)The d -type quark contribution is obtained by the following replacements z u L → z d L , z u R → z d R , Q u → Q d , (4.52)and, since the helicity of massless antiparticles is opposite to their chirality, the antiquarkcontributions can be obtained from the quarks by the replacementˆ σ ¯ u, ¯ dRR ↔ ˆ σ u,dRL . (4.53)The expressions for (cid:2) Γ eγ,Z (cid:3) eτ are identical, upon replacing the lepton helicity label R → L .For completeness, we give the expressions in Appendix B. Notice that for the photon dipole– 20 – eγ , the power of Q in F dip is not sufficient to cancel the divergence at Q → | Γ eZ | and to the Γ eγ –Γ eZ interference are, on theother hand, finite at Q → In the case of Higgs Yukawa operators Eq. (3.13) and scalar and tensor operators in thelast line of Eq. (3.14), we define the prefactor F S = Q c w s w m Z . (4.54)Starting from the τ e component of the operator coefficients, the e is right-handed. Thepartonic cross sections initiated by u -type quarks receive contributions from both scalar andtensor operators. In both, the u is right-handed. In addition, Higgs exchanges contributeto this channel, and the Higgs couples to both right- and left-handed u quarks. The totalcontributions of all these operators to the partonic cross sections are:ˆ σ uLL = ˆ σ uLR = 0 , ˆ σ u i RR = F S y (cid:26) (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C (1) LeQu (cid:105) τeu i u i + 4 (cid:18) − y (cid:19) (cid:104) C (3) LeQu (cid:105) τeu i u i + Y u i (cid:2) Y (cid:48) e (cid:3) τe v m H + Q (cid:12)(cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C (1) LeQu (cid:105) τeu j u i + 4 (cid:18) − y (cid:19) (cid:104) C (3) LeQu (cid:105) τeu j u i (cid:12)(cid:12)(cid:12)(cid:12) (cid:27) ˆ σ u i RL = F S y (cid:12)(cid:12)(cid:12)(cid:12) Y u i (cid:2) Y (cid:48) e (cid:3) τe v m H + Q (cid:12)(cid:12)(cid:12)(cid:12) . (4.55)The partonic cross sections for the ¯ u -type antiquarks are given in Appendix B. For d -type quarks, the main difference is the absence of a tensor operator, and the chirality ofthe incoming d quark, which is now left-handed. The relevant expressions are given inAppendix B. For eτ operators, the results are the same, but the electron is left-handed. We finally consider gluonic operators. These come from two sources, first, through Eq. (3.20),which talks to eτ through the Yukawa interaction Eq. (3.13); and second, from Eq. (3.21),induced by scalar and tensor operators below the top threshold. The left-handed andright-handed gluon will give same results,ˆ σ gLL = ˆ σ gLR = F G y (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) [ C GG ] eτ − v Q + m H (cid:2) Y (cid:48) eτ (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:2) C G ˜ G (cid:3) eτ (cid:12)(cid:12) (cid:27) , ˆ σ gRR = ˆ σ gRL = F G y (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) [ C GG ] τe − v Q + m H (cid:8) Y (cid:48) τe (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:2) C G ˜ G (cid:3) τe (cid:12)(cid:12) (cid:27) (4.56)where here the factor F G is F G = 14 (cid:16) α s π (cid:17) c w s w Q m Z v . (4.57)– 21 – S
63 GeV 100 GeV 141 GeV √ S
63 GeV 100 GeV 141 GeV τ L σ (pb) σ (pb) σ (pb) τ R σ (pb) σ (pb) σ (pb) c (1) Lϕ + c (3) Lϕ . . . c eϕ . . . τ L σ (pb) σ (pb) σ (pb) τ L σ (pb) σ (pb) σ (pb)( C LQ, U ) uu C Lu ) uu C LQ,U ) cu C Lu ) cu C LQ, U ) uc C Lu ) uc C LQ,U ) cc C Lu ) cc C LQ, D ) dd C Ld ) dd C LQ, D ) sd C Ld ) sd C LQ,D ) bd C Ld ) bd C LQ,D ) ds C Ld ) ds C LQ,D ) ss C Ld ) ss C LQ, D ) bs C Ld ) bs C LQ, D ) db C Ld ) db C LQ,D ) sb C Ld ) sb C LQ, D ) bb C Ld ) bb τ R σ (pb) σ (pb) σ (pb) τ R σ (pb) σ (pb) σ (pb)( C Qe ) dd C Qe ) ds C Qe ) sd C Qe ) ss C Qe ) bd C Qe ) bs C Qe ) db C Qe ) sb C Qe ) bb Table 1 . Numerical coefficients a iJ that control the cross sections σ i = a iJ | C J | for the CLFV pro-cess ep → τ X induced by CLFV Z couplings, vector and axial four-fermion operators. The subscript i = { , , } denotes each of the three benchmark points discussed in the text, at √ S = 63 , , J is the operator label. Here we omit interference terms between Z cou-plings and four-fermion operators. The labels τ L,R denote the polarization of the τ lepton emittedby the effective operators. The cross section is computed with the NNPDF31 lo as 0118
PDF set[49]. The error estimates includes PDF and scale uncertainties. Terms quadratic in C eu and C ed are identical to C LQ, U and C LQ,D , respectively, and are not given explictly. – 22 – S
63 GeV 100 GeV 141 GeV √ S
63 GeV 100 GeV 141 GeV τ L σ (pb) σ (pb) σ (pb) τ L σ (pb) σ (pb) σ (pb) Y (cid:48) τe (10 − ) 0.22(2) 0 . . C GG (10 − ) 0.103(5) 0.32(1) 1.77(7)Γ eγ . .
5) Γ eZ τ L σ (pb) σ (pb) σ (pb) τ L σ (pb) σ (pb) σ (pb)( C (1) LeQu ) uu C (3) LeQu ) uu C (1) LeQu ) cu C (3) LeQu ) cu C (1) LeQu ) uc C (3) LeQu ) uc C (1) LeQu ) cc C (3) LeQu ) cc C (1) LedQ ) dd C (1) LedQ ) ds C (1) LedQ ) sd C (1) LedQ ) ss C (1) LedQ ) bd C (1) LedQ ) bs C (1) LedQ ) db C (1) LedQ ) sb C (1) LedQ ) bb Table 2 . Numerical coefficients a iJ that control the cross sections σ i = a iJ | C J | for the CLFVprocess ep → τ X , induced by CLFV Higgs couplings, photon and Z dipoles and scalar and tensorfour-fermion operators. The subscript i = { , , } denotes each of the three benchmark pointsdiscussed in the text, at √ S = 63 , ,
141 GeV, respectively, while J is the operator label. Herewe omit interference terms between photon and Z dipoles and between Higgs couplings, scalar andtensor four-fermion operators. The cross section is computed with the NNPDF31 lo as 0118
PDFset. The error estimates includes PDF and scale uncertainties. We give here the cross section forthe τ e component of the operators, in which the τ lepton is left-handed. The results are identicalfor the eτ components, with the difference that a right-handed τ is emitted. To get an idea of the number of CLFV events that can be produced at the EIC, we calculatehere the total DIS cross section from different SMEFT operators, obtained by integratingEq. (4.47) over x and y in the range x, y ∈ [0 , S dependence of theSMEFT cross sections, we use a few benchmark points,1. E e = 20 GeV, E p = 50 GeV, √ S = 63 GeV,2. E e = 10 GeV, E p = 250 GeV, √ S = 100 GeV,3. E e = 20 GeV, E p = 250 GeV, √ S = 141 GeV.These are typical beam energies of EIC [17, 50], with the last point corresponding to themaximum √ S the EIC plans to achieve. The renormalization and factorization scales arechosen as µ F = µ R = Q , and we assess the scale uncertainty by varying µ F = µ R between– 23 – / Q . We use the NNPDF31 lo as 0118
PDF set [49], and we evaluate the PDFerrors by calculating the cross section for the 100 members of this PDF set. Furthermore,we have compared the results of our numerical calculations with those obtained using
MadGraph5 [51] and found excellent agreement. We show the cross section from variousCLFV operators with λ e = 0 in Tables 1 and 2. It is straightforward to include thepolarization of the electron beam, see Eq. (4.45).The cross section for SMEFT operators grows as √ S increases, with more markedincrease for the dimension-7 gluonic operators. The CLFV Z couplings and four-fermionoperators induce cross sections that are comparable to the Z boson contributions to stan-dard DIS, multiplied by the square of the operator coefficients, scaling as ( v/ Λ) . Operatorswith a sea quark in the initial state are suppressed by the PDF of the s , c or b quark. Thesuppression is not too severe, but notice that the PDF and scale errors become sizable,especially in the case of operators dominated by the s and b contribution. For these op-erators, it will be important to extend the analysis beyond leading order. We stress thatwe use the PDF and scale errors only as a rough estimate of the theoretical error, a morerobust assessment requires extending the calculation to next-to-leading order (NLO).The scalar and tensor four-fermion operators induce contributions of similar size asvector operators, with some enhancement in the case of the C (3) LeQu . The photon dipoleΓ eγ gives a large contribution to the cross section, but, as we will discuss in Section 5,the divergence at Q → p T distributions of the τ decayproducts is hardly distinguishable from the SM backgrounds. The Yukawa operator Y (cid:48) e contributes to DIS via the Higgs coupling to light quarks and the effective gluon-Higgscoupling induced by top loops. At the EIC, the dominant contribution arises from theHiggs coupling to b quarks. The cross section is however too small to provide bounds on Y (cid:48) e that are competitive with the LHC or low-energy probes.We can use the cross sections in Tables 1 and 2 to provide a first estimate of the EICsensitivity to CLFV operators, as a function of a selection efficiency (cid:15) n b , defined as thenumber of signal events that pass the cuts required to reduce the SM background to n b events. We consider separately the three decays channel τ − → e − ¯ ν e ν τ , τ − → µ − ¯ ν µ ν τ and τ − → X h ν τ , where X h denotes an hadronic final state. The branching ratios in thesechannels are [52] BR( τ − → e − ¯ ν e ν τ ) = 17 . ± . , (4.58)BR( τ − → µ − ¯ ν µ ν τ ) = 17 . ± . , (4.59)BR( τ − → X h ν τ ) = 64 . . (4.60)Assuming the backgrounds are known with negligible errors, we can estimate the upperlimit on the CLFV coefficients at the 1 − α credibility level, when n events have beenobserved and n b events are expected, by solving the equation [52]1 − α = 1 − Γ (1 + n, n b + n s )Γ (1 + n, n b ) , (4.61)where n s is a function of the SMEFT coefficient, of the decay channel and of the selection– 24 – → e ¯ ν e ν τ or τ → µ ¯ ν µ ν τ τ → X h ν τ n b = 0 n b = 100 n b = 0 n b = 100 (cid:12)(cid:12)(cid:12) ( c (1) Lϕ + c (3) Lϕ ) √ (cid:15) n b (cid:12)(cid:12)(cid:12) . · − . · − . · − . · − | c eϕ √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ, U ) uu √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ,U ) cu √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ, U ) uc √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ,U ) cc √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ, D ) dd √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ, D ) sd √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ,D ) bd √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ,D ) ds √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ,D ) ss √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ, D ) bs √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ, D ) db √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ,D ) sb √ (cid:15) n b | . · − . · − . · − . · − | ( C LQ, D ) bb √ (cid:15) n b | . · − . · − . · − . · − Table 3 . EIC sensitivity to CLFV Z couplings and vector four-fermion operators with left-handedquark and leptons, from the τ electronic, muonic and hadronic decay channels. We assume √ S =141 GeV and L = 100 fb − . The two sets of 90% CL bounds are obtained assuming that the EICwill observe n = n b = 0 and n = n b = 100 events. (cid:15) and (cid:15) accounts for the selection cuts thatensure 0 and 100 background events, respectively, and are functions of the decay channel and ofthe operator structure. Bounds on the right-handed operators C eu and C ed are the same as C LQ,U and C LQ,D . efficiency n s ( C i , (cid:15) n b , X j ) = L × ( σ(cid:15) n b | C i | ) × BR( τ → X j ν τ ) , (4.62)with L the integrated luminosity. For the cross section σ we use the central values givenin Tables 1 and 2. We however notice that processes initated by sea quarks have largetheoretical uncertainties, which can significantly shift the upper bound on the SMEFTcoefficients.In Tables 3, 4 and 5 we give the 90% CL bounds on the product of the operatorcoefficients and the efficiency (cid:15) , assuming n = n b and for two choices, n b = 0 and n b = 100.We consider a center of mass energy of √ S = 141 GeV, and assume an integrated luminosity– 25 – → e ¯ ν e ν τ or τ → µ ¯ ν µ ν τ τ → X h ν τ n b = 0 n b = 100 n b = 0 n b = 100 | ( C Lu ) uu √ (cid:15) n b | . · − . · − . · − . · − | ( C Lu ) cu √ (cid:15) n b | . · − . · − . · − . · − | ( C Lu ) uc √ (cid:15) n b | . · − . · − . · − . · − | ( C Lu ) cc √ (cid:15) n b | . · − . · − . · − . · − | ( C Ld ) dd √ (cid:15) n b | . · − . · − . · − . · − | ( C Ld ) sd √ (cid:15) n b | . · − . · − . · − . · − | ( C Ld ) bd √ (cid:15) n b | . · − . · − . · − . · − | ( C Ld ) ds √ (cid:15) n b | . · − . · − . · − . · − | ( C Ld ) ss √ (cid:15) n b | . · − . · − . · − . · − | ( C Ld ) bs √ (cid:15) n b | . · − . · − . · − . · − | ( C Ld ) db √ (cid:15) n b | . · − . · − . · − . · − | ( C Ld ) sb √ (cid:15) n b | . · − . · − . · − . · − | ( C Ld ) bb √ (cid:15) n b | . · − . · − . · − . · − | ( C Qe ) dd √ (cid:15) n b | . · − . · − . · − . · − | ( C Qe ) sd √ (cid:15) n b | . · − . · − . · − . · − | ( C Qe ) bd √ (cid:15) n b | . · − . · − . · − . · − | ( C Qe ) ds √ (cid:15) n b | . · − . · − . · − . · − | ( C Qe ) ss √ (cid:15) n b | . · − . · − . · − . · − | ( C Qe ) bs √ (cid:15) n b | . · − . · − . · − . · − | ( C Qe ) db √ (cid:15) n b | . · − . · − . · − . · − | ( C Qe ) sb √ (cid:15) n b | . · − . · − . · − . · − | ( C Qe ) bb √ (cid:15) n b | . · − . · − . · − . · − Table 4 . EIC sensitivity to CLFV four-fermion operators with left(right)-handed leptons andright(left)-handed quarks, from the τ electronic, muonic and hadronic decay channels. The two setsof 90% CL bounds are obtained assuming that the EIC will observe n = n b = 0 and n = n b = 100events. (cid:15) and (cid:15) accounts for the selection cuts that ensure 0 and 100 background events,respectively, and are functions of the decay channel and of the operator structure. – 26 – → e ¯ ν e ν τ or τ → µ ¯ ν µ ν τ τ → X h ν τ n b = 0 n b = 100 n b = 0 n b = 100 (cid:12)(cid:12) Γ eγ √ (cid:15) n b (cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12) Γ eZ √ (cid:15) n b (cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12) Y (cid:48) te √ (cid:15) n b (cid:12)(cid:12) .
90 2 . .
47 1 . (cid:12)(cid:12) C GG √ (cid:15) n b (cid:12)(cid:12) . . . . (cid:12)(cid:12)(cid:12) ( C (1) LeQu ) uu √ (cid:15) n b (cid:12)(cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12)(cid:12) ( C (1) LeQu ) cu √ (cid:15) n b (cid:12)(cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12)(cid:12) ( C (1) LeQu ) uc √ (cid:15) n b (cid:12)(cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12)(cid:12) ( C (1) LeQu ) cc √ (cid:15) n b (cid:12)(cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12) ( C LedQ ) dd √ (cid:15) n b (cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12) ( C LedQ ) sd √ (cid:15) n b (cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12) ( C LedQ ) bd √ (cid:15) n b (cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12) ( C LedQ ) ds √ (cid:15) n b (cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12) ( C LedQ ) ss √ (cid:15) n b (cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12) ( C LedQ ) bs √ (cid:15) n b (cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12) ( C LedQ ) db √ (cid:15) n b (cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12) ( C LedQ ) sb √ (cid:15) n b (cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12) ( C LedQ ) bb √ (cid:15) n b (cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12)(cid:12) ( C (3) LeQu ) uu √ (cid:15) n b (cid:12)(cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12)(cid:12) ( C (3) LeQu ) cu √ (cid:15) n b (cid:12)(cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12)(cid:12) ( C (3) LeQu ) uc √ (cid:15) n b (cid:12)(cid:12)(cid:12) . · − . · − . · − . · − (cid:12)(cid:12)(cid:12) ( C (3) LeQu ) cc √ (cid:15) n b (cid:12)(cid:12)(cid:12) . · − . · − . · − . · − Table 5 . EIC sensitivity to CLFV γ and Z dipole couplings, Higgs couplings, gluon couplingsand scalar and tensor four-fermion operators, from the τ electronic, muonic and hadronic decaychannels. We assume √ S = 141 GeV and L = 100 fb − . The two sets of 90% CL bounds areobtained assuming that the EIC will observe n = n b = 0 and n = n b = 100 events. (cid:15) and (cid:15) accounts for the selection cuts that ensure 0 and 100 background events, respectively, and arefunctions of the decay channel and of the operator structure. – 27 –f 100 fb − . In the case of Z couplings and four-fermion operators with valence quarks,the EIC could reach better than percent sensitivities with (cid:15) ∼
10% in the τ leptonic orhadronic decay channels. Flavor-changing operators and operators with heavy quarks couldalso be probed at the few percent level. In these cases, however, theoretical uncertaintiescannot be neglected. Considering, e.g., the extreme case of the operator [ C Ld ] bb , varyingthe cross section in the uncertainty range given in Table 1 causes the 90% CL upper limitto vary between 5 . · − and 12 · − . This large range can be narrowed by includingNLO QCD corrections. We will present a detailed comparison of sensitivities of EIC withother collider and low-energy probes in Section 9. Here we anticipate that the EIC can bequite competitive for four-fermion semileptonic operators, both diagonal and non-diagonalin quark flavor. We will present an estimate of the selection efficiencies (cid:15) n b in Section 5. Next we perform a detailed Monte Carlo simulation to explore the potential of probingCLFV effects via e − p → τ − X at the EIC with collider energy E e = 20 GeV and E p =250 GeV (benchmark point 3 at √ S = 141 GeV in Sec. 4.3). It is straightforward togeneralize our analysis to other collider energies. The main challenges for the identificationof τ CLFV at the EIC are, first of all, that, differently from muons, the τ leptons decayvery quickly inside the detector and, secondly, that all decay channels involve missingenergy, complicating the reconstruction of the τ momentum and thus of the DIS variables x and y . It is therefore necessary to identify distinctive features of the signal events, inorder to disentangle them from the SM background. Based on the τ decay modes, there arethree classes of final states: (1) e − p → τ − X → e − ¯ ν e ν τ X ; (2) e − p → τ − X → µ − ¯ ν µ ν τ X ; (3) e − p → τ − X → ν τ X h X . In the first case, signal events are characterized by an electron andmissing energy recoiling against at least one jet. In the second case, the electron is replacedby a muon, which, as we will see, largely suppresses the SM background. Finally, in thehadronic channels the signal events have missing energy, at least two jets and no chargedleptons. The major backgrounds from SM processes include neutral current ( e − p → e − j )and charged current ( e − p → ν e j ) DIS. Other backgrounds, such as lepton pair production( e − p → e − (cid:96) + (cid:96) − j ) and real W boson production ( e − p → e − W ± j ), can at this stage beignored due to the small cross sections.We use Pythia8 [53] to generate 10 and 10 events for the background and signals,respectively. A transverse momentum cut on the final states transverse momentum p T >
10 GeV is applied to the DIS background generation. The
Delphes package is used tosimulate the detector smearing effects [54]. We use in this analysis the EIC input carddeveloped by M. Arratia and S. Sekula, based on parameters in Ref. [55] and used andprovided in [56, 57]. As the EIC handbook does not specify muon identification parameters[55], we assumed the same performance for muons and electrons, and we modified the EICDelphes card accordingly. This assumption relies on having a dedicated muon detector inthe EIC design, which is currently being discussed . The anti- k t jet algorithm with jetcone size R = 1 and p T > We thank M. Arratia for clarifying this point. – 28 – C LQ, U ) uu ( C LQ, U ) cc ( C LQ, D ) dd ( C LQ, D ) ss ( C LQ, D ) bb � �� ���� - � �� - � �� - � �� � � �� [ ��� ] � � � � � �� �� �� � � � � � �� �� �� � �� ���� - � �� - � �� - � �� � � � μ [ ��� ] � � � � � �� �� �� � � � � � �� �� �� �� �� ���� - � �� - � �� - � �� � � � �� [ ��� ] � � � � � �� �� �� � � � � � �� �� �� �� �� ���� - � �� - � �� - � �� � � � �� [ ��� ] � � � � � �� �� �� � � � � � �� �� �� � �� ���� - � �� - � �� - � �� � � / � [ ��� ] � � � � � �� �� �� � � � � � �� �� �� Figure 4 . Electron, muon, leading and subleading jet p T distributions and missing energydistribution induced by four-fermion operators with different flavor components at the EIC, with E e = 20 GeV and E p = 250 GeV ( √ S = 141 GeV). In Figs. 4 and 5 we show the transverse momentum distributions of the hardest electron( p eT ), muon ( p µT ) and of the leading ( p j T ), sub-leading ( p j T ) jets and the missing energy( (cid:54) E T ) distribution induced by various four-fermion SMEFT operators. The distributionsare normalized by the total cross section for each individual contribution, i.e. normalizedto a total integral of 1. (Thus these figures compare the shapes but not relative sizes ofindividual cross sections.). We note that these distributions are very sensitive to the flavorof the quark in the initial state, while they do not strongly depend on the polarization of the τ lepton. In Fig. 4 we consider the purely left-handed operators ( C LQ,U ) ii and ( C LQ,D ) jj ,where i = u, c and j = d, s, b . In the massless limit, these operators create a left-handed– 29 – C LQ, U ) uu ( C LQ, D ) dd ( C eu ) uu ( C ed ) dd � �� ���� - � �� - � � �� [ ��� ] � � � � � �� �� �� � � � � � �� �� �� � �� ���� - � �� - � � � μ [ ��� ] � � � � � �� �� �� � � � � � �� �� �� �� �� ���� - � �� - � � � �� [ ��� ] � � � � � �� �� �� � � � � � �� �� �� �� �� ���� - � �� - � � � �� [ ��� ] � � � � � �� �� �� � � � � � �� �� �� � �� ���� - � �� - � � / � [ ��� ] � � � � � �� �� �� � � � � � �� �� �� Figure 5 . Electron, muon, leading and subleading jet p T distributions and missing energy distribu-tion induced by four-fermion operators with different τ polarization at the EIC, with E e = 20 GeVand E p = 250 GeV ( √ S = 141 GeV). τ and the different kinematic behaviors in Fig. 4 are solely due to the flavor of the quarkin the initial state. The strange and heavy quark components ( C LQ,D ) ss , ( C LQ,U ) cc and( C LQ,D ) bb would favor small p T or (cid:54) E T , due to the suppression of the sea quark PDFs atlarge transverse momenta, while the valence components ( C LQ,U ) uu and ( C LQ,D ) dd havesignificant tails at large p T and (cid:54) E T . Fig. 5 shows the same distributions for the left-handedoperators ( C LQ,U ) uu and ( C LQ,D ) dd , and the right-handed operators ( C eu ) uu and ( C ed ) dd .In the massless limit, the τ lepton is left-handed polarized for ( C LQ,U ) uu , and ( C LQ,D ) dd (solid lines), and right-handed polarized for ( C eu ) uu and ( C ed ) dd (dashed lines). Fig. 5shows that the kinematical distributions we are considering in this work are not sensitive– 30 – ISBG ( C LQ, U ) uu ( C L φ ) τ e ( C GG ) τ e ( Γ γ e ) τ e ( Γ Ze ) τ e ( C LQ, D ) bb � �� �� �� �� ���� - � �� - � �� - � �� - � �� � � �� [ ��� ] � � � � � �� �� �� � � � � � �� �� �� � �� �� �� �� ���� - � �� - � �� - � �� - � �� � � � μ [ ��� ] � � � � � �� �� �� � � � � � �� �� �� �� �� �� �� ���� - � �� - � �� - � �� - � �� � � � �� [ ��� ] � � � � � �� �� �� � � � � � �� �� �� �� �� �� �� ���� - � �� - � �� - � �� - � �� � � � �� [ ��� ] � � � � � �� �� �� � � � � � �� �� �� � �� �� �� �� ���� - � �� - � �� - � �� - � �� � � / � [ ��� ] � � � � � �� �� �� � � � � � �� �� �� Figure 6 . Electron, muon, leading and subleading jet p T distributions and missing energy distribu-tion induced by SMEFT operators with left-handed τ leptons and by the SM background (DISBG)at the EIC, with E e = 20 GeV and E p = 250 GeV ( √ S = 141 GeV). to the τ polarization. This is true in particular for the p T of the leading jet, which, beingproduced in the hard scattering e − p → τ − j , does not depend on the τ polarization. In Figs.4 and 5 we only show vector and axial operators. We verified that scalar, pseudoscalar andtensor four-fermion operators give rise to similar distributions.As discussed in Section 4, flavor-changing Z couplings, photon and Z dipoles, andgluonic operators induce DIS cross sections with different dependence on Q with respectto four-fermion operators. As a consequence, also the p T and (cid:54) E T distributions showdifferent features. In Fig. 6, we show kinematic distributions for the SM background andSMEFT operators with left-handed τ leptons. The distributions induced by operators with– 31 –ight-handed τ are similar to those with left-handed τ , and will not be shown here. Weuse ( C LQ,U ) uu and ( C LQ,D ) bb as examples of four-fermion operators, and, in addition, weshow the signal from the left-handed Z coupling c Lϕ ≡ c (1) Lϕ + c (3) Lϕ , from the photon and Z dipoles Γ eγ and Γ eZ , and from the CP-even gluonic operator C GG . All distributions areagain normalized to area one.With the results depicted in Fig. 6, several comments are in order: • The SM distributions tend to peak/grow at small values of p T and (cid:54) E T . In the case ofthe electron and leading jet p T distributions, we begin plotting the DIS backgroundonly at 10 GeV, in order to limit the number of events we had to simulate, as theSM cross section blows up rapidly as these p T → • The electron p T distribution induced by valence four-fermion operators, Z couplings,Γ eZ and gluonic operators shows a slower decrease at high p T compared to the SM.Still the very large SM background implies that even imposing hard cuts on theelectron p T is not sufficient to fully suppress the SM background. • Muons in the background sample are generated by the parton shower and by thedecay of hadrons. Therefore, most background muons have very small p T . For signalevents, the muon spectrum is similar to the electron p T spectrum. • The p T spectra of the two leading jets induced by four-fermion operators with va-lence quarks, Z couplings, Γ eZ and gluonic operators are harder than for the SMbackground. For heavy-quark operators, the shape of the signal is similar to the SMbackground. • (cid:54) E T in the background sample is generated by charged-current DIS, by the partonshower and by the decay of hadrons. The background distribution is peaked at small (cid:54) E T , but, differently from the muon p T distributions, charged-current DIS causes asizable tail at larger values of (cid:54) E T (cid:38)
20 GeV. • There is a collinear enhancement for the p T of leptons and jets from the photondipole operator (Γ eγ ) τe . Consequently, the distributions from Γ eγ are similar to theDIS background.These observations are summarized in Fig. 7, where we show the cut efficiency asa function of the kinematic cut for both the signals and background. In these plots,we consider one observable at a time. Fig. 7 shows that the cut efficiencies for the SMbackground and for the γ dipole operator Γ eγ drop quickly as we increase the p T or (cid:54) E T cut. This is particularly true for the muon channel. Here, asking for a muon in the finalstate already suppresses the SM background by a factor of about 10 − , and requiring that p µT >
10 GeV brings the suppression to 10 − . The same p T cut reduces the signal eventsby about ∼ τ branching ratio in this channel. We also notethat the Z boson dipole operator Γ eZ typically has the largest cut efficiency. Althoughthe cross section is small compared to other SMEFT operators, the large cut efficiencyimplies that the EIC will impose relatively strong constraints on the Z dipoles. C GG – 32 – �� �� �� �� ���� - � �� - � �� - � �� - � �� - � �� � � ���� � [ ��� ] � � � � � � � � � � � � ( Γ γ e ) τ e ( Γ Ze ) τ e ( C L φ ) τ e ( C GG ) τ e DISBG ( C LQ, U ) uu ( C LQ, D ) bb � �� �� �� �� ���� - � �� - � �� - � �� - � �� - � �� � � ���� μ [ ��� ] � � � � � � � � � � � � � �� �� �� �� ���� - � �� - � �� - � �� � � ������ [ ��� ] � � � � � � � � � � � � � �� �� �� �� ���� - � �� - � �� - � �� � � ������ [ ��� ] � � � � � � � � � � � � � �� �� �� �� ���� - � �� - � �� - � �� � � / ���� [ ��� ] � � � � � � � � � � � � Figure 7 . Cut efficiency for the SM background (DISBG) and the signal induced by SMEFToperators with left-handed τ leptons, as a function of the cut on the electron, muon, leading andsubleading jet p T or on the missing energy. We only implement one cut at a time. and c Lϕ ≡ c (1) Lϕ + c (3) Lϕ show a comparable cut efficiency. However, the cross section fromthe gluonic operators is very small, O (10 − ) pb, so that we do not expect very strongconstraints on these operators. Based on Figs. 4–7, we suggest the following kinematicacceptance cuts to suppress the background for the three classes of decay modes: • τ − → e − ¯ ν e ν τ : at least one electron, one jet and p eT >
10 GeV , p j T >
20 GeV , (cid:54) E T >
15 GeV , | η e | , | η j | < . (5.1)– 33 – ISBG ( C LQ, U ) uu ( C L φ ) τ e ( C GG ) τ e ( Γ γ e ) τ e ( Γ Ze ) τ e ( C LQ, D ) bb � �� �� �� �� ���� � �� � �� � �� � �� � �� � � � μ [ ��� ] � � � � � � � � �� �� �� �� ���� � �� � �� � �� � �� � �� � �� � �� � � � �� [ ��� ] � � � � � � � � � �� �� �� �� ���� � �� � �� � �� � �� � �� � �� � �� � � / � [ ��� ] � � � � � � � � Figure 8 . The distributions of p µT , p j T and (cid:54) E T from ep → τ ( → µ ¯ ν µ ν τ ) + X at the EIC with E e = 20 GeV, E p = 250 GeV ( √ S = 141 GeV) and L = 100 fb − . The Wilson coefficients of theCLFV operators is set to C i = 1. - � - � - � � � � ��� - � �� - � �� - � �� - � η μ � � � � � �� �� �� � � � � � �� �� �� DISBG ( C LQ, U ) uu ( C L φ ) τ e ( C GG ) τ e ( Γ γ e ) τ e ( Γ Ze ) τ e ( C LQ, D ) bb Figure 9 . Muon η distribution induced by SMEFT operators with left-handed τ leptons and bythe SM background (DISBG) at the EIC, with E e = 20 GeV and E p = 250 GeV ( √ S = 141 GeV). • τ − → µ − ¯ ν µ ν τ : at least one muon, one jet and p µT >
10 GeV , p j T >
20 GeV , (cid:54) E T >
15 GeV , | η µ | , | η j | < . (5.2)A rejection on electrons is also applied if p eT >
10 GeV. • τ − → ν τ + X h : no leptons and at least two jets with, p j T >
20 GeV , p j T >
15 GeV , (cid:54) E T >
15 GeV , | η j | , | η j | < . (5.3)– 34 – C LQ,U ) uu ( C LQ,U ) cc ( C LQ,D ) dd ( C LQ,D ) ss ( C LQ,D ) bb c (1) Lϕ + c (3) Lϕ (cid:15) cut (%) 9 . . . . .
91 4 . C eu ) uu ( C eu ) cc ( C ed ) dd ( C ed ) ss ( C ed ) bb C eϕ (cid:15) cut (%) 9 . . . . .
85 3 . C GG ) τe (Γ eγ ) τe (Γ eZ ) τe ( C GG ) eτ (Γ eγ ) eτ (Γ eZ ) eτ (cid:15) cut (%) 6 . .
15 19 6 . .
15 18
Table 6 . Cut efficiency in the muonic channel, in units of 10 − , for various SMEFT operators atthe EIC with energy E e = 20 GeV and E p = 250 GeV ( √ S = 141 GeV). There is no backgroundafter including the kinematic cuts. Here η i = − ln tan( θ i /
2) is the pseudorapidity of the particle i with respect to the p direction, with i = e, µ, j , .In the electronic and hadronic modes, the typical cut efficiency of the SM background afterwe include the cuts in Eqs. (5.1) and (5.3) is O (10 − ). Combining the inclusive productioncross section with the background cut efficiency, the background cross section after the cutsis around O (1) pb, which is still much larger than the signals. To get sensitive bounds inthese channels, it is therefore necessary to further refine the analysis. In the hadronic mode,this could be done by including jet-substructure information to single out the jet emergingfrom τ decay, which is expected to be displaced from the primary vertex, have small hadronmultiplicity and to be correlated with the missing energy [58]. We will pursue this directionin future work. Here we will focus on the muonic channel, which is essentially background-free and thus allows for strong constraints on the CLFV coefficients. The distributions of p µT , p j T and (cid:54) E T from ep → τ ( → µ ¯ ν µ ν τ ) + X are shown in Fig. 8, with Wilson coefficientsset to one. Most of our results do not change notably if we extend the rapidity cuts inEqs. (5.1)–(5.3) into the more forward/backward regions | η | < η µ distributions for several possible signals and the DIS background in Fig. 9. We note that η µ from most of the signals would favor the central rapidity region, although the backgroundfalls a bit faster for forward rapidity than most of the signals. This is especially trueof the dipole Γ eγ signal, which peaks significantly in the forward region. The distinct η µ distributions between signals and background will be interesting in future studies to furtheroptimize EIC sensitivity, although we may need to consider smaller p T triggers, especiallyif we want to consider forward jets.The cut efficiency (cid:15) cut (i.e. percentage of events left intact by the cuts) for differentSMEFT operators is shown in Table 6. Notice that, as in Eq. (4.62), (cid:15) cut is defined after– 35 – ( C LQ, U ) uu | | ( C Lu ) uu | (cid:12)(cid:12)(cid:12) ( C (1) LeQu ) uu (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ( C (3) LeQu ) uu (cid:12)(cid:12)(cid:12) | ( C LQ,U ) cu | | ( C Lu ) cu | (cid:12)(cid:12)(cid:12) ( C (1) LeQu ) cu (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ( C (3) LeQu ) cu (cid:12)(cid:12)(cid:12) | ( C LQ, U ) uc | | ( C Lu ) uc | (cid:12)(cid:12)(cid:12) ( C (1) LeQu ) uc (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ( C (3) LeQu ) uc (cid:12)(cid:12)(cid:12) | ( C LQ,U ) cc | | ( C Lu ) cc | (cid:12)(cid:12)(cid:12) ( C (1) LeQu ) cc (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ( C (3) LeQu ) cc (cid:12)(cid:12)(cid:12) | ( C LQ, D ) dd | | ( C Ld ) dd | | ( C LedQ ) dd | | ( C Qe ) dd | | ( C LQ, D ) sd | | ( C Ld ) sd | | ( C LedQ ) sd | | ( C Qe ) sd | | ( C LQ,D ) bd | | ( C Ld ) bd | | ( C LedQ ) bd | | ( C Qe ) bd | | ( C LQ,D ) ds | | ( C Ld ) ds | | ( C LedQ ) ds | | ( C Qe ) ds | | ( C LQ,D ) ss | | ( C Ld ) ss | | ( C LedQ ) ss | | ( C Qe ) ss | | ( C LQ, D ) bs | | ( C Ld ) bs | | ( C LedQ ) bs | | ( C Qe ) bs | | ( C LQ, D ) db | | ( C Ld ) db | | ( C LedQ ) db | | ( C Qe ) db | | ( C LQ,D ) sb | | ( C Ld ) sb | | ( C LedQ ) sb | | ( C Qe ) sb | | ( C LQ, D ) bb | | ( C Ld ) bb | | ( C LedQ ) bb | | ( C Qe ) bb | (cid:12)(cid:12)(cid:12) ( c (1) Lϕ + c (3) Lϕ ) (cid:12)(cid:12)(cid:12) | c eϕ | (cid:12)(cid:12) (Γ eγ ) τe/eτ (cid:12)(cid:12) (cid:12)(cid:12) (Γ eZ ) τe/eτ (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) Y (cid:48) τe/eτ (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) ( C GG ) τe/eτ (cid:12)(cid:12) (cid:12)(cid:12) ( C G ˜ G ) τe/eτ (cid:12)(cid:12) Table 7 . EIC sensitivity, in units of 10 − , to CLFV operators at 90% CL with E e = 20 GeV, E p = 250 GeV ( √ S = 141 GeV) and L = 100 fb − . Bounds on the right-handed operators C eu and C ed are almost the same as C LQ,U and C LQ,D . factoring out the branching ratio in a specific channel. For four-fermion operators, (cid:15) cut isonly sensitive to the flavor of the initial state quark, and does not depend on the Lorentzstructure and on the flavor of the quark in the final state. We can therefore use the (cid:15) cut shown in Table 6 for C LQ,U , C LQ,D , C eu and C ed for the other four-fermion operatorsin our basis. In the muonic channel, after combining all the cuts, (cid:15) BGcut = 0, that is, weobtain a background-free process. The typical (cid:15) cut for four-fermion operators with valencequarks is around ∼ ∼ b tagging inthe final state, which could further suppress the background with more moderate cuts,thus increasing (cid:15) cut for heavy quarks. c eϕ , c (1) Lϕ + c (3) Lϕ and the gluonic operators also havea sizable (cid:15) cut , from 3% to 7%. Γ eZ has the biggest efficiency, around 20%. For the photondipole, on the other hand, (cid:15) cut is very small, (cid:15) cut ∼ . τ polarization, the difference between operators withleft-handed and right-handed τ , such as C LQ,U and C eu , being about few percent.For the background-free channels, we can use the Bayesian posterior probability method– 36 –o determine the upper limits on the CLFV coefficients; see Eq. (4.61) with n b = 0. The 90%CL upper limits on the CLFV operators at the EIC, assuming E e = 20 GeV, E p = 250 GeVand L = 100 fb − , are given in Table 7. The EIC can put very strong constraints on thelight quark components of four-fermion operators, ranging from 0.2% to few percent independence of the Lorentz and quark-flavor structures of the operators. With our cuts, thesmall (cid:15) cut causes the heavy quark components to be relatively less well constrained, at the10% level. The limits on Z boson CLFV couplings and dipole operators are comparableto the four-fermion operators. Finally, it will be difficult to give useful constraints on theYukawa and gluonic operators, because of the small production cross sections at the EIC.The polarization of the electron beam will be very useful to single out the chiral struc-ture of SMEFT operators. Since the cut efficiencies of CLFV operators are not sensitiveto the τ polarization, the limits on CLFV coefficients with λ e (cid:54) = 0 can be written as | C i ( e L , λ e ) | = 1 √ − λ e | C i ( e L , λ e = 0) | , | C i ( e R , λ e ) | = 1 √ λ e | C i ( e R , λ e = 0) | . (5.4)Here e L,R is the helicity of the incoming electron in e − p → τ − X . It is clear that a negative λ e would improve the limits of the operators with left-handed electron, while it wouldweaken the results for the right-handed electron operators and vice versa. CLFV interactions have been probed at other high-energy collider experiments. In partic-ular, LEP and the LHC have searched for CLFV decays of the Higgs boson [59], Z boson[60, 61], and t quark [62–64]. The relevant scales for these processes are the decaying par-ticles’ masses, well within the regime of validity of SMEFT. The ATLAS experiment hasalso looked for the process pp → τ e [37]. In this case, the invariant mass of the eτ pair canreach values larger than 3 TeV, and the comparison of the LHC and projected EIC limitsrequires to make sure that one is working in the regime of validity of the EFT. Z , Higgs and t decays The OPAL collaboration at the LEP experiment constrained the branching ratio of the Z boson into τ e to be BR( Z → eτ ) < . · − (95% CL) [60]. This limit was recentlysuperseded by the ATLAS collaboration [61], which foundBR( Z → eτ ) < . · − (95% CL) . (6.1)This branching ratio is mostly sensitive to the operators c eϕ and c (1 , Lϕ , which induce CLFV Z vertices, and to the dipole operator Γ eZ . Their contributions to the branching ratio areBR( Z → eτ ) = 1 (cid:98) Γ Z (cid:18) (cid:12)(cid:12)(cid:12)(cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe (cid:12)(cid:12)(cid:12) + 14 | [ c eϕ ] τe | + m Z v (cid:0) | [Γ eZ ] eτ | + [Γ eZ ] τe | (cid:1)(cid:19) , (6.2)where the branching ratio includes both e + τ − and e − τ + channels, and we usedΓ Z = G F m Z √ π (cid:98) Γ Z . (6.3)– 37 –he dimensionless number (cid:98) Γ Z is, at leading order in QCD and EW corrections, (cid:98) Γ Z = (cid:88) f N fc ( z f L + z f R ) , (6.4)with N fc = 1 for leptons and N fc = N c for quarks. In terms of the observed Z width, (cid:98) Γ Z = 3 .
76. From Eqs. (6.1) and (6.2) we get the 90% CL limits | c eϕ | < . · − , | c (1) Lϕ + c (3) Lϕ | < . · − , | [Γ eZ ] eτ, τe | < . · − . (6.5)The Higgs decay width into τ e is given by [65]Γ( H → e − τ + + τ − e + ) = m H π (cid:16)(cid:2) Y (cid:48) e (cid:3) τe + (cid:2) Y (cid:48) e (cid:3) eτ (cid:17) . (6.6)Using the bounds on the branching ratio [59]BR( H → e − τ + + τ − e + ) ≡ B e < . · − (95%CL) , (6.7)and the relation: (cid:16)(cid:2) Y (cid:48) e (cid:3) τe + (cid:2) Y (cid:48) e (cid:3) eτ (cid:17) = 8 πm H B e − B e Γ SM , (6.8)where the SM Higgs width is Γ SM = 4 . · − GeV, one gets the strong constraint [59] (cid:2) Y (cid:48) e (cid:3) τe, eτ < . · − . (6.9)The ATLAS experiment has put bounds on the top branching ratio BR( t → q(cid:96)(cid:96) (cid:48) ) < . · − (95% CL) [62]. The analysis is sensitive to the eτ , µτ and eµ channels, puttingthe strongest constraints on the latter. To obtain a constraint on the eτ channel, we first ofall get the yield and shape of the t → qeτ and t → qµτ signal distributions by subtractingthe signal histograms with and without τ vetos in Fig. 3 of Ref. [62]. We then estimatethe t → qeτ fraction of signal events by accounting for the different electron versus muonacceptance, obtained from the yields of the two validation regions given in Ref. [63]. Wethen used signal and background events in a likelihood analysis using pyhf [66], obtaining BR( t → qeτ ) ≤ . · − . (6.10)Dedicated analyses in the τ channels are in progress, and preliminary results for BR( t → qµτ ) show bounds at the 10 − level [64]. The BR for the decay t → qe + τ − is [67]BR( t → qe + τ − ) = 16 (cid:98) Γ t (cid:16) m t πv (cid:17) (cid:20) (cid:16)(cid:12)(cid:12)(cid:12) [ C LQ,U ] τeqt (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) [ C Lu ] τeqt (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) [ C Qe ] τeqt (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) [ C eu ] τeqt (cid:12)(cid:12)(cid:12) (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C (1) LeQu (cid:105) τeqt (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C (1) LeQu (cid:105) eτtq (cid:12)(cid:12)(cid:12)(cid:12) + 48 (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C (3) LeQu (cid:105) τeqt (cid:12)(cid:12)(cid:12)(cid:12) + 48 (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C (3) LeQu (cid:105) eτtq (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) , (6.11)where we expressed the SM top width asΓ( t → W b ) = m t πv (cid:98) Γ t , (6.12) We thank C. A. Gottardo for illustrating the procedure for the extraction of bounds on t → qeτ fromRef. [62], and for checking the limit in Eq. (6.10). – 38 – m e (TeV) e v e n t s data[ C LQ , U ] uu = 1 10 [ C LQ , U ] cc = 1 10 [ C LQ , D ] bb = 3 10 backgrounduncertainty Figure 10 . Observed and background events in pp → eτ , at √ S = 13 TeV with luminosityof 36.1 fb − , as a function of the τ -electron invariant mass m τe [37]. The dashed area denotesthe uncertainty on the backgrounds. The red, magenta and green lines denote the sum of thebackground and signal events induced by SMEFT operators. with (cid:98) Γ t a dimensionless function of V tb , m t and m W . In terms of the measured top width, (cid:98) Γ t = 1 . +0 . − . [52]. The resulting constraints on top CLFV operators are[ C LQ,U ] τeqt < . , (cid:104) C (1) LeQu (cid:105) τeqt < . , (cid:104) C (3) LeQu (cid:105) τeqt < . , (6.13)where the limit on C Lu , C Qe and C eu is the same as the one on C LQ,U . The SMEFT operators in Eqs. (3.7) and (3.14) can also affect the process pp → eτ , whichhas been studied in Refs. [37, 68]. These analyses look for eτ , eµ and µτ pairs in severalinvariant mass bins, and they provide the strongest constraints at high invariant mass,where the SM background is highly suppressed. They are thus most sensitive to four-fermion operators [69]. In the eτ channel, Ref. [37] considered 6 invariant mass bins, from m eτ <
300 GeV to m eτ > POWHEG implementation of Ref. [39]. We include NLO QCD corrections, which, asshown in Ref. [39], can give a ∼
30% correction in the high invariant mass bins, and theparton-level events are showered with
Pythia8 , which we also use for the decays of the τ lepton. We apply the selection cuts described in Ref. [37], in particular the electronand the jet from hadronic τ decays are required to have p T >
65 GeV and | η | < . τ tagging with Delphes . The effect of selecting hadronic– 39 –0% CL 90% CL 90% CL C LQ, U , C eu uu . · − uc . · − cc . · − C Lu uu . · − uc . · − cc . · − C LQ, D , C ed dd . · − ds . · − db . · − ss . · − sb . · − bb . · − C Ld dd . · − ds . · − db . · − ss . · − sb . · − bb . · − C Qe dd . · − ds . · − db . · − ss . · − sb . · − bb . · − C LedQ dd . · − ds . · − db . · − ss . · − sb . · − bb . · − C (1) LeQu uu . · − uc . · − cc . · − tt . · − C (3) LeQu uu . · − uc . · − cc . · − Γ eγ Table 8 . 90% CL bounds on four-fermion LFV operators from the LHC. The coefficients areevaluated at the scale µ = 1 TeV. For quark-flavor-changing operators, the bounds on the q j q i and q i q j components are identical. decays, of the cut on the electron and jet p T , and of the efficiency of τ -tagging combineto give a selection efficiency between (cid:15) = 0 .
24 and (cid:15) = 0 .
27 in the four different invariantmass bins. The efficiencies do not show a strong dependence on the Lorentz or flavorstructure of the four-fermion operators. We also simulated Z vertex corrections and dipoleoperators, but, for coefficients compatible with the bounds in Eq. (6.5), they are negligible.The top scalar operators (cid:104) C (1) LeQu (cid:105) τett and (cid:104) C (1) LeQu (cid:105) eτtt contribute to CLFV Drell-Yan viathe gluon fusion process gg → τ e at the loop level. We parametrize the finite one-loopcorrections as form factors that are function of the external momenta. The form factors areimplemented as effective new vertices in a dedicated UFO model file, which is then used in MadGraph5 . We have compared the cross section with the amplitude in Appendix B.2 to the
MadGraph5 code and found excellent agreement. QCD corrections are taken into accountby introducing a constant κ factor in our simulation, i.e. κ (cid:39) MCFM [71, 72]. Non-standard Yukawa couplings wouldcontribute via the same mechanism, but the constraints from off-shell Higgs production– 40 –re much weaker than those shown in Section 6.1.In Table 8 we show the 90% CL bounds on the coefficients of effective operators,evaluated at the renormalization scale µ = 1 TeV. To obtain the bounds, we use a gener-alization of Eq. (4.61) to multiple bins [73]. Since the uncertainties on the background arenon-negligible, we generate a large number of pseudoexperiments, assuming the numberof signal and background events in each bin to follow a Poisson distribution. The mean µ i of the distributions of signal events is given by Eq. (4.62), generalized to several bins.For each value of the operator coefficient C , the mean µ b i is picked randomly in the 1 σ intervals shown in Figure 10. Each pseudoexperiment is characterized by a number ofsignal and background events, n s i and n b i . We consider only the pseudoexperiments with n b i ≤ n i , where n i is the number of observed events, and we construct the confidence levelby counting the ratio of pseudoexperiments for which n s i + n b i < n i . If this ratio is lessthan 10%, C is excluded.The bounds in Table 8 are dominated by the last two bins, and our results agree wellwith Ref. [74], which also recasts the analysis of Ref. [37] in terms of SMEFT operators.The LHC puts very strong constraints on operators with two u or two d quarks. Thebounds deteriorate to the few percent level in the case of operators with heavy flavors.Converting into a new physics scale, vector operators with valence quarks give Λ (cid:38) . (cid:38) m eτ , and the SMEFT analysis is thus justified. For operatorswith two sea quarks Λ (cid:38) . eγ is at the 10% level, much weaker than from τ decays.We have so far assumed that the SMEFT is valid up to scales of a few TeV. For BSMphysics contributing at tree level in the s -channel, Ref. [37] found comparable limits on themasses of new CLFV degrees of freedom, in the range of 4–5 TeV. The limits in Table 8 canbe weakened if BSM particles are exchanged in the t -channel, as for example in the case ofscalar leptoquarks discussed in Section 10. At LO in QCD, we can study this scenario byreplacing the coefficients of SMEFT four-fermion operators with C → C M M − t , (6.14)where M denotes the mass of the exchanged particle. We find that the bounds on thelight-quark components of the four-fermion operators in Table 8 worsen by a factor of 5(2) for t -channel exchange of a particle of mass M = 1 TeV (2 TeV). We next discuss CLFV low-energy observables. The relatively heavy mass of the τ leptoncompared to light hadrons offers a rich array of channels to search for CLFV τ decaysincluding τ → eγ , the purely leptonic channels τ → e and τ → eµµ , and semileptonicdecays such as τ → eπ , η ( (cid:48) ) and τ → eπ + π − . Table 9 summarizes the LFV decay modes– 41 –ecay mode Upper limit on BR (90% C.L.) τ − → e − γ < . × − τ − → e − e + e + < . × − τ − → e − µ + µ − < . × − τ − → e − π < . × − τ − → e − η < . × − τ − → e − η (cid:48) < . × − τ − → e − K S < . × − τ − → e − π + π − < . × − τ − → e − π + K − < . × − τ − → e − π − K + < . × − B → e ± τ ∓ < . × − B + → π + e + τ − < . × − B + → π + e − τ + < . × − B + → K + e + τ − < . × − B + → K + e − τ + < . × − Table 9 . Summary of the low-energy decay modes and current experimental limits on their branch-ing ratios [52]. that we consider and the current experimental upper limits on each branching ratio (BR) at90% C.L. While most of the τ decays are associated with the CLFV quark-flavor-conservinginteractions, the decay modes τ → eK S and τ → eK ± π ∓ can probe the LFV quark-flavor-violating interactions. In Table 10, we present a tabulation of which operators contributeto each decay channel. The parentheses indicate decays that are induced only at 1- and/or2-loop level. For example, the LFV Yukawa interaction originating from ψ ϕ can induce τ → eγ through 1- and 2-loop diagrams. The semileptonic four-fermion operators denotedas ψ contribute to the leptonic τ decays via renormalization group running.Heavy D and B mesons, J/ψ and Υ and other quarkonia can decay into electrons and τ leptons, offering additional handles on CLFV interactions. D and B decays probe flavor-changing couplings. At the moment, there are no bounds on D → τ ± e ∓ . This decay wouldput interesting constraints on the uc and cu components of the flavor matrices introducedin Section 3, which, as we will see, are otherwise unconstrained at low energy. B decaysput strong constraints on the bd , db , bs and sb elements. Quarkonium decays constrain the cc and bb components, but the limits are weaker than those from τ decays.– 42 –ecay mode ψ Xϕ ψ ϕ D ψ ϕ ψ τ → eγ (cid:88) ( (cid:88) ) ( (cid:88) ) τ → ee + e − (cid:88) (cid:88) (cid:88) ( (cid:88) ) τ → eµ + µ − (cid:88) (cid:88) (cid:88) ( (cid:88) ) τ → eπ (cid:88) (cid:88) τ → eη (cid:88) (cid:88) τ → eη (cid:48) (cid:88) (cid:88) τ → eK S (cid:88) τ → eπ + π − (cid:88) (cid:88) (cid:88) (cid:88) τ → eK ± π ∓ (cid:88) B → e ± τ ∓ (cid:88) B + → π + e ± τ ∓ (cid:88) B + → K + e ± τ ∓ (cid:88) Table 10 . Illustration of the contributions from six different types of gauge-invariant CLFV oper-ators to low-energy decay modes. The parentheses imply that the operator induces the decay onlyat 1- or 2-loop level.
We start this section by introducing the low-energy basis in Section 7.1. We thendiscuss quark-flavor-conserving τ and quarkonium decays in Section 7.2 and quark-flavor-violating observables in Section 7.3. Additional low-energy observables that indirectlyprobe CLFV interactions are studied in Section 8. In order to study the low-energy observables, we first map the LFV operators listed inSection 3 onto a low-energy SU (3) c × U (1) em EFT (LEFT). The matching can be donemore immediately in the basis of Ref. [75–77], from which we differ only in the fact that wefactorize dimensionful parameters so that the Wilson coefficients of the LEFT operatorsbecome dimensionless.At dimension five, we consider leptonic dipole operators L = − e v ¯ e pL σ µν (cid:2) Γ eγ (cid:3) pr e rR F µν + h . c ., (7.1)where p, r are leptonic flavor indices.At dimension six, there are several semileptonic four-fermion operators. Those relevantfor direct LFV probes have two charged leptons. There are eight vector-type operators L = − G F √ (cid:18) C eu VLL ¯ e L γ µ e L ¯ u L γ µ u L + C ed VLL ¯ e L γ µ e L ¯ d L γ µ d L + C eu VRR ¯ e R γ µ e R ¯ u R γ µ u R – 43 – C ed VRR ¯ e R γ µ e R ¯ d R γ µ d R + C ue VLR ¯ e R γ µ e R ¯ u L γ µ u L + C de VLR ¯ e R γ µ e R ¯ d L γ µ d L + C eu VLR ¯ e L γ µ e L ¯ u R γ µ u R + C ed VLR ¯ e L γ µ e L ¯ d R γ µ d R (cid:19) , (7.2)and six scalar-tensor type operators L = − G F √ (cid:18) C eu SRR ¯ e L e R ¯ u L u R + C ed SRR ¯ e L e R ¯ d L d R + C eu TRR ¯ e L σ µν e R ¯ u L σ µν u R (7.3)+ C ed TRR ¯ e L σ µν e R ¯ d L σ µν d R + C eu SRL ¯ e L e R ¯ u R u L + C ed SRL ¯ e L e R ¯ d R d L (cid:19) + h.c.There are in addition four purely leptonic operators L = − G F √ (cid:104) C ee VLL ¯ e L γ µ e L ¯ e L γ µ e L + C ee VRR ¯ e R γ µ e R ¯ e R γ µ e R + C ee VLR ¯ e L γ µ e L ¯ e R γ µ e R + (cid:0) C ee SRR ¯ e L e R ¯ e L e R + h . c . (cid:1)(cid:105) . (7.4)LFV operators can also affect probes with one or two neutrinos, in which the neutrinoflavor is not observed. There are four operators with two neutrinos, which will affect raremeson decays, L = − G F √ (cid:18) C νu VLL ¯ ν L γ µ ν L ¯ u L γ µ u L + C νd VLL ¯ ν L γ µ ν L ¯ d L γ µ d L + C νu VLR ¯ ν L γ µ ν L ¯ u R γ µ u R + C νd VLR ¯ ν L γ µ ν L ¯ d R γ µ d R (cid:19) , (7.5)and five charged-current operators L = − G F √ (cid:18) C νedu VLL ¯ ν L γ µ e L ¯ d L γ µ u L + C νedu VLR ¯ ν L γ µ e L ¯ d R γ µ u R (7.6)+ C νedu TRR ¯ ν L σ µν e R ¯ d L σ µν u R + C νedu SRR ¯ ν L e R ¯ d L u R + C νedu SRL ¯ ν L e R ¯ d R u L (cid:19) + h . c . The coefficients of the operators in Eqs. (7.2), (7.3), (7.5) and (7.6) are not all independent,if one matches from SMEFT. For example, the four-fermion contributions to the semi-leptonic vector operators with charged leptons in Eq. (7.2) are given by: (cid:104) C eu VLL (cid:105) τeji = (cid:104) C LQ,U (cid:105) τeji + δ ij (cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe z u L , (7.7a) (cid:104) C ed VLL (cid:105) τeji = (cid:104) C LQ,D (cid:105) τeji + δ ij (cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe z d L , (7.7b) (cid:104) C eu VRR (cid:105) τeji = (cid:104) C eu (cid:105) τeji + δ ij (cid:104) c eϕ (cid:105) τe z u R , (7.7c) (cid:104) C ed VRR (cid:105) τeji = (cid:104) C ed (cid:105) τeji + δ ij (cid:104) c eϕ (cid:105) τe z d R , (7.7d) (cid:104) C eu VLR (cid:105) τeji = (cid:104) C Lu (cid:105) τeji + δ ij (cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe z u R , (7.7e) (cid:104) C ed VLR (cid:105) τeji = (cid:104) C Ld (cid:105) τeji + δ ij (cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe z d R , (7.7f)– 44 – C ue VLR (cid:105) τeji = (cid:104) V CKM C Qe V † CKM (cid:105) τeji + δ ij (cid:104) c eϕ (cid:105) τe z u L , (7.7g) (cid:104) C de VLR (cid:105) τeji = (cid:104) C Qe (cid:105) τeji + δ ij (cid:104) c eϕ (cid:105) τe z d L . (7.7h)The coefficients of the leptonic operators in Eq. (7.4) are given by: (cid:104) C ee VLL (cid:105) prst = [ C LL ] prst + z e L (cid:20)(cid:16) c (1) Lϕ + c (3) Lϕ (cid:17) pr δ st + (cid:16) c (1) Lϕ + c (3) Lϕ (cid:17) pt δ sr (cid:21) (7.8a)+ z e L (cid:104)(cid:16) c (1) Lϕ + c (3) Lϕ (cid:17) st δ pr + (cid:16) c (1) Lϕ + c (3) Lϕ (cid:17) sr δ pt (cid:105) , (cid:104) C ee VRR (cid:105) prst = [ C ee ] prst + z e R (cid:104) ( c eϕ ) pr δ st + ( c eϕ ) pt δ sr (cid:105) (7.8b)+ z e R (cid:2) ( c eϕ ) st δ pr + ( c eϕ ) sr δ pt (cid:3)(cid:104) C ee VLR (cid:105) prst = [ C Le ] prst + z e R (cid:16) c (1) Lϕ + c (3) Lϕ (cid:17) pr δ st + z e L ( c eϕ ) st δ pr , (7.8c) (cid:104) C ee SRR (cid:105) prst = − v m h (cid:16)(cid:0) Y (cid:48) e (cid:1) pr ( Y e ) st δ st + (cid:0) Y (cid:48) e (cid:1) st ( Y e ) pr δ pr (cid:17) . (7.8d)The coefficients of the vector charged-current operators in Eq. (7.6) are given by: (cid:104) C νedu VLL (cid:105) ν τ eji = (cid:104) C LQ,D V † CKM − V † CKM C LQ,U (cid:105) τeji − (cid:104) c (3) Lϕ (cid:105) ν τ e (cid:104) V † CKM (cid:105) ji , (7.9a) (cid:104) C νedu VLR (cid:105) ν τ eji = 0 , (7.9b)while the neutrino operators in Eq. (7.5) are (cid:104) C νu VLL (cid:105) ν τ ν e ji = (cid:104) V CKM C LQ,D V † CKM (cid:105) τeji + δ ij (cid:104) c (1) Lϕ − c (3) Lϕ (cid:105) τe z u L , (7.10a) (cid:104) C νd VLL (cid:105) ν τ ν e ji = (cid:104) V † CKM C LQ,U V CKM (cid:105) τeji + δ ij (cid:104) c (1) Lϕ − c (3) Lϕ (cid:105) τe z d L , (7.10b) (cid:104) C νu VLR (cid:105) ν τ ν e ji = (cid:104) C Lu (cid:105) τeji + δ ij (cid:104) c (1) Lϕ − c (3) Lϕ (cid:105) τe z u R , (7.10c) (cid:104) C νd VLR (cid:105) ν τ ν e ji = (cid:104) C Ld (cid:105) τeji + δ ij (cid:104) c (1) Lϕ − c (3) Lϕ (cid:105) τe z d R . (7.10d)The scalar and tensor operators, C (1) LeQu , C (3) LeQu and C LedQ , and the LFV Yukawa Y (cid:48) e match onto scalar and tensor operators Eq. (7.3) at low energy. In the neutral currentsector one finds (cid:104) C eu SRR (cid:105) τeji = − (cid:104) C (1) LeQu (cid:105) τeji − δ ij v m H (cid:104) Y (cid:48) e (cid:105) τe Y u , (7.11a) (cid:104) C ed SRR (cid:105) τeji = − δ ij v m H (cid:104) Y (cid:48) e (cid:105) τe Y d , (7.11b) (cid:104) C eu SRL (cid:105) τeji = − δ ij v m H (cid:104) Y (cid:48) e (cid:105) τe Y u (7.11c) (cid:104) C ed SRL (cid:105) τeji = + (cid:104) C LedQ (cid:105) τeji − δ ij v m H (cid:104) Y (cid:48) e (cid:105) τe Y d (7.11d)– 45 – C eu TRR (cid:105) τeji = − (cid:104) C (3) LeQu (cid:105) τeji , (7.11e) (cid:104) C ed TRR (cid:105) τeji = 0 , (7.11f)while the charged-current operators in Eq. (7.6) are (cid:104) C νedu TRR (cid:105) ν τ eji = (cid:104) V † CKM C (3) LeQu (cid:105) τeji , (7.12a) (cid:104) C νedu SRR (cid:105) ν τ eji = (cid:104) V † CKM C (1) LeQu (cid:105) τeji (7.12b) (cid:104) C νedu SRL (cid:105) ν τ eji = (cid:104) C LedQ V † CKM (cid:105) τeji . (7.12c)At the b and c thresholds, the scalar operators also induce corrections to the gluonicoperators in Eq. (3.21), yielding[ C GG ] τe = 13 (cid:88) q = b,c vm q (cid:2) C eq SRR + C eq SRL (cid:3) τeqq , [ C GG ] eτ = 13 (cid:88) q = b,c vm q (cid:2) C eq SRR + C eq SRL (cid:3) eτqq (7.13a) (cid:2) C G (cid:101) G (cid:3) τe = i (cid:88) q = b,c vm q (cid:2) C eq SRR − C eq SRL (cid:3) τeqq , (cid:2) C G (cid:101) G (cid:3) eτ = i (cid:88) q = b,c vm q (cid:2) C eq SRR − C eq SRL (cid:3) eτqq (7.13b)The running of the LEFT operators between the electroweak scale and the scalesrelevant for τ and B decays was computed in Ref. [76] and is summarized in AppendixA.3. The most important effects are the QCD running of the scalar and tensor operators,and the penguin contributions from operators with b and c quarks onto purely leptonicoperators and operators with light quarks. The coefficients of LEFT operators, evaluatedat the scale µ = 2 GeV, as a function of SMEFT operators at the scale µ = 1 TeVare given in Tables 22, 23 and 24. In the computation of τ decay rates we follow veryclosely Ref. [29], which adopts a different basis for the low-scale operators. We provide theappropriate conversion formulae in Appendix C. We first discuss bounds on Γ eγ , c (1 , Lϕ , c eϕ and quark-flavor-conserving four-fermion oper-ators from τ decays. In this section, we give explicitly the full expressions for the decayrates of two decay modes, τ → eγ and τ → eπ + π − , which lead to many of the strongestlimits on these operators. Expressions for all other τ decay rates we consider, along withrelevant input parameters, are collected in Appendix D. All branching ratios are expressedin terms of LEFT operator coefficients evaluated at the scale µ = 2 GeV. These can beexpressed in terms of SMEFT coefficients at the high-energy scale µ ∼ Λ via the matchingformulae given in Section 7.1 and the RGEs discussed in Sections A.1 and A.3.The branching ratio for τ → eγ is given byBR ( τ → eγ ) = τ τ m τ α em v (cid:104)(cid:12)(cid:12)(cid:0) Γ eγ (cid:1) eτ (cid:12)(cid:12) + (cid:12)(cid:12)(cid:0) Γ eγ (cid:1) τe (cid:12)(cid:12) (cid:105) , (7.14)where τ τ is the τ lifetime, given in Table 25. Writing τ τ = (cid:16) G F m τ π (cid:98) Γ τ (cid:17) − , (7.15)– 46 –ith the dimensionless factor (cid:98) Γ τ = 1 .
12, we obtainBR ( τ → eγ ) = 96 π α em (cid:98) Γ τ v m τ (cid:104)(cid:12)(cid:12)(cid:0) Γ eγ (cid:1) eτ (cid:12)(cid:12) + (cid:12)(cid:12)(cid:0) Γ eγ (cid:1) τe (cid:12)(cid:12) (cid:105) (cid:39) . × (cid:104)(cid:12)(cid:12)(cid:0) Γ eγ (cid:1) eτ (cid:12)(cid:12) + (cid:12)(cid:12)(cid:0) Γ eγ (cid:1) τe (cid:12)(cid:12) (cid:105) . (7.16)The τ → eγ branching ratio is thus enhanced with respect to other modes by the two-bodyphase space, and by the dipole operator appearing at dimension five at low energy. Wenotice that τ → eγ also receives contributions from the tensor operators [78], which shiftthe original contribution as(Γ eγ ) eτ → (Γ eγ ) eτ − (cid:18) i Π V T (0) v (cid:19) [ C eu TRR ] eτuu (7.17a)(Γ eγ ) ∗ τe → (Γ eγ ) ∗ τe − (cid:18) i Π V T (0) v (cid:19) [ C eu TRR ] ∗ τeuu , (7.17b)with the non-perturbative parameter ( i Π V T (0) /v ) ≈ . × − at µ = 2 GeV (see Ap-pendix D.1 for details). As we will show, this is mostly relevant for global analyses, becausein a single operator analysis τ → eππ provides a bound on the tensor Wilson coefficientthat is four times stronger than the one from τ → eγ .In the case of τ → eπ + π − , the differential decay width is given by τ τ d Γ d ˆ s = 140 (cid:98) Γ τ (cid:16) − ρ π ˆ s (cid:17) / (1 − ˆ s ) (cid:26) m τ (cid:16)(cid:12)(cid:12) Q (cid:48) L (cid:12)(cid:12) + (cid:12)(cid:12) Q (cid:48) R (cid:12)(cid:12) (cid:17) (7.18)+8 (cid:16) − ρ π ˆ s (cid:17) | F V ( s ) | (cid:20) s s (cid:16) | A L | + | A R | (cid:17) + ( B L + B R ) (cid:21)(cid:27) , where s is the invariant mass of the charged pions, s = ( p π + + p π − ) and we define thedimensionless quantities ˆ s = s/m τ and ρ π = 4 m π /m τ . ˆ s is kinematically allowed to bein the range ρ π ≤ ˆ s ≤
1. Here we follow the expression in Ref. [29], where the Wilsoncoefficients are assumed to be real. Q (cid:48) L,R , A L,R and B L,R are combinations of Wilsoncoefficients and form factors Q (cid:48) L = 29 v ( θ π ( s ) − Γ π ( s ) − ∆ π ( s )) ( C GG ) τe + ∆ π m s (cid:16) C ed SRR + C ed SRL (cid:17) τess (7.19a)+ 12 Γ π ( s ) (cid:26) m ( C eu SRR + C eu SRL ) τeuu + 1ˆ m (cid:16) C ed SRR + C ed SRL (cid:17) τedd (cid:27) ,Q (cid:48) R = 29 v ( θ π ( s ) − Γ π ( s ) − ∆ π ( s )) ( C GG ) ∗ eτ + ∆ π m s (cid:16) C ed SRR + C ed SRL (cid:17) ∗ eτss (7.19b)+ 12 Γ π ( s ) (cid:26) m ( C eu SRR + C eu SRL ) ∗ eτuu + 1ˆ m (cid:16) C ed SRR + C ed SRL (cid:17) ∗ eτdd (cid:27) ,A L = 4 πα em vm τ (cid:0) Γ eγ (cid:1) τe + ˆ s √ ρ π B π,uT (0) (cid:110)(cid:16) C ed TRR (cid:17) τedd − ( C eu TRR ) τeuu (cid:111) , (7.20a) A R = 4 πα em vm τ (cid:0) Γ eγ (cid:1) ∗ eτ + ˆ s √ ρ π B π,uT (0) (cid:110)(cid:16) C ed ∗ TRR (cid:17) eτdd − ( C eu ∗ TRR ) eτuu (cid:111) , (7.20b)– 47 – igure 11 . Vector contributions to the differential decay rate of τ → eπ + π − (left) and τ → eπ ± K ∓ (right). The solid lines include vector form factors as a function of ˆ s , while they are fixed at 1 inthe dashed lines. The Wilson coefficients are set to unity. B L = (cid:110)(cid:16) C ed VLL + C ed VLR (cid:17) τedd − ( C eu VLL + C eu VLR ) τeuu (cid:111) (7.21a) × (cid:20) A L + (2ˆ s + 1) (cid:110)(cid:16) C ed VLL + C ed VLR (cid:17) τedd − ( C eu VLL + C eu VLR ) τeuu (cid:111) (cid:21) ,B R = (cid:110)(cid:16) C ed VRR (cid:17) τedd + (cid:16) C de VLR (cid:17) ddτe − ( C eu VRR ) τeuu − ( C ue VLR ) uuτe (cid:111) (7.21b) × (cid:20) A R + (2ˆ s + 1) (cid:110)(cid:16) C ed VRR (cid:17) τedd + (cid:16) C de VLR (cid:17) ddτe − ( C eu VRR ) τeuu − ( C ue VLR ) uuτe (cid:111) (cid:21) , and ˆ m = ( m u + m d ) / Q (cid:48) L,R depend on the Wilson coefficients of the scalar and gluonicoperators, and on the scalar form factors Γ π ( s ) , ∆ π ( s ) , θ π ( s ), for which we follow theconventions and determinations in Ref. [28] (for related work see [26, 27, 79]). A L,R encodethe contributions of dipole and tensor operators, with the value of B π,uT (0) taken from Ref.[80]. Finally, B L,R encode the contributions of vector operators and their interference withdipole and tensor operators. For the vector form factor F V ( s ) we use the extraction inRef. [28]. In the left panel of Fig. 11, the solid line depicts the vector contributions to thedifferential decay rate of τ → eπ + π − . Compared to the dashed line that assumes F V ( s ) = 1,the blue line has a peak around ˆ s ∼ . ρ (770) resonance. Analogousenhancements are also seen in the scalar contributions as discussed in [29]. Because of theresonance contribution, the branching ratio in this mode is relatively large. Neglecting theinterference terms, the following useful expression for the BR can be obtainedBR (cid:0) τ → eπ + π − (cid:1) (cid:39) . × (cid:12)(cid:12) Γ eγ (cid:12)(cid:12) τe + 1 . × − (cid:12)(cid:12) C GG (cid:12)(cid:12) + 0 . (cid:12)(cid:12) C ed SRR + C ed SRL (cid:12)(cid:12) ss (7.22)+ (cid:16) . (cid:12)(cid:12)(cid:12)(cid:2) C eq SRR + C eq SRL (cid:3) τe ( qq ) (0) (cid:12)(cid:12)(cid:12) + 0 . (cid:12)(cid:12)(cid:12)(cid:2) C eq VLL + C eq VLR (cid:3) τe ( qq ) (1) (cid:12)(cid:12)(cid:12) (cid:17) + 1 . (cid:12)(cid:12)(cid:12)(cid:2) C ed TRR (cid:3) τedd − (cid:2) C eu TRR (cid:3) τeuu (cid:12)(cid:12)(cid:12) , – 48 – Γ eγ (cid:1) τe (Γ eZ ) τe ( Y (cid:48) e ) τe (cid:16) c (1) Lϕ (cid:17) (cid:16) c (3) Lϕ (cid:17) τe ( c eϕ ) τe . × − . × − . × − . × − . × − . × − Table 11 . 90% C.L. upper limits on lepton bilinear operators, assuming a single operator is turnedon at the scale Λ = 1 TeV. The bounds on the the dipole and Yukawa operators are dominated by τ → eγ , while those on the Z couplings c (1 , Lϕ and c eϕ by τ → eπ + π − . where the notation ( qq ) (0) , (1) indicates that the isoscalar or isovector ( uu ± dd ) combinationof Wilson coefficients has to be taken. A similarly large branching ratio, due to the φ (1020)resonance, can be seen in the τ → eK + K − mode [31, 81], which we discuss in AppendixD.1. Because the scalar, tensor and gluonic contributions are affected by larger theoreticaluncertainties [28, 31], we do not use this channel in our main analysis, and remark on itsimpact in the multiple operator scenario discussed in Section 9.2.In order to compare sensitivities across various CLFV τ decays, we present the numer-ical results for the remaining decay modes, neglecting interference between operators withdifferent Lorentz structure. For leptonic τ decays we have BR (cid:0) τ → ee + e − (cid:1) (cid:39) . × (cid:12)(cid:12) Γ eγ (cid:12)(cid:12) τe + 0 . (cid:16) | C ee VLL | τeee + 0 . (cid:12)(cid:12) C ee VLR (cid:12)(cid:12) τeee (cid:17) , (7.23a)BR (cid:0) τ → eµ + µ − (cid:1) (cid:39) (cid:12)(cid:12) Γ eγ (cid:12)(cid:12) τe + 0 . (cid:16)(cid:12)(cid:12) C ee VLL (cid:12)(cid:12) τeµµ + (cid:12)(cid:12) C ee VLR (cid:12)(cid:12) τeµµ (cid:17) , (7.23b)while for semileptonic τ decays we haveBR (cid:0) τ → eπ (cid:1) (cid:39) (cid:16) . × − (cid:12)(cid:12)(cid:12)(cid:2) C eq VLR − C eq VLL (cid:3) τe ( qq ) (1) (cid:12)(cid:12)(cid:12) + 0 . (cid:12)(cid:12)(cid:12)(cid:2) C eq SRR − C eq SRL (cid:3) eτ ( qq ) (1) (cid:12)(cid:12)(cid:12) (cid:17) , (7.24a)BR ( τ → eη ) (cid:39) (cid:16) [3 . × − (cid:12)(cid:12)(cid:12)(cid:2) C eq VLR − C eq VLL (cid:3) τe ( qq ) (0) (cid:12)(cid:12)(cid:12) + 1 . × − (cid:12)(cid:12)(cid:12)(cid:2) C eq SRR − C eq SRL (cid:3) eτ ( qq ) (0) (cid:12)(cid:12)(cid:12) (cid:17) + 6 . × − (cid:12)(cid:12)(cid:12) C ed VLR − C ed VLL (cid:12)(cid:12)(cid:12) τess + 0 . (cid:12)(cid:12)(cid:12) C ed SRR − C ed SRL (cid:12)(cid:12)(cid:12) eτss + 1 . × − (cid:12)(cid:12) C G ˜ G (cid:12)(cid:12) τe , (7.24b)BR (cid:0) τ → eη (cid:48) (cid:1) (cid:39) (cid:16) [1 . × − (cid:12)(cid:12)(cid:12)(cid:2) C eq VLR − C eq VLL (cid:3) τe ( qq ) (0) (cid:12)(cid:12)(cid:12) + 1 . × − (cid:12)(cid:12)(cid:12)(cid:2) C eq SRR − C eq SRL (cid:3) eτ ( qq ) (0) (cid:12)(cid:12)(cid:12) (cid:17) + 6 . × − (cid:12)(cid:12)(cid:12) C ed VLR − C ed VLL (cid:12)(cid:12)(cid:12) τess + 0 . (cid:12)(cid:12)(cid:12) C ed SRR − C ed SRL (cid:12)(cid:12)(cid:12) eτss + 5 . × − (cid:12)(cid:12) C G ˜ G (cid:12)(cid:12) τe . (7.24c)Note that Wilson coefficients corresponding to operators with opposite chiralities ( L ↔ R )contribute to each decay mode with the same prefactors.With the above results at hand, we can get a reasonable picture of the constraintsimposed by τ decays on various CLFV operators. Table 11 shows the upper limits onlepton bilinear operators. Starting with photon-dipole operator, we see that τ → eγ gives Here, we neglect contributions from scalar four-lepton operators to leptonic decays as they do not giverelevant limits on CLFV operators of our interest. – 49 –
LQ,U uu . × − ∗ cc . × − ∗ tt . × − ∗ C eu uu . × − ∗ cc . × − ∗ tt . × − ∗ C Lu uu . × − ∗ cc . × − ∗ tt . × − ∗ C LQ,D dd . × − ∗ ss . × − § bb . × − ∗ C ed dd . × − ∗ ss . × − § bb . × − ∗ C Ld dd . × − ∗ ss . × − § bb . × − ∗ C Qe dd . × − § ss . × − § bb . × − ∗ C LedQ dd . × − ∗ ss . × − ∗ bb . × − (cid:93) C (1) LeQu uu . × − ∗ cc . × − (cid:93) tt . × − † C (3) LeQu uu . × − ∗ cc . × − † tt . × − † Table 12 . 90% C.L. upper limits on the quark-flavor-conserving semileptonic operators, assuminga single operator is turned on at the scale Λ = 1 TeV. The superscripts represent that the strongestlimit is imposed by decay modes ( ∗ ) τ → eπ + π − , ( † ) τ → eγ , ( § ) τ → eη and ( (cid:93) ) τ → eη (cid:48) . For thescalar and tensor operators, the bounds apply to both the τ e and eτ components. the strongest limit. The bound on Γ eZ is obtained by considering operator mixing betweenthe Z - and γ -dipole operators. The running effect is given by Γ eγ ( m t ) = − . × − Γ eZ (Λ)with Λ = 1 TeV, yielding (Γ eZ ) τe < . × − . Moreover, the τ - e LFV Yukawa couplinginduces the photon-dipole operator at 1- and 2-loop level (the expressions are given inAppendix A.2). The resulting limit is 1 . × − , which is consistent with the result in[65].A noteworthy feature of CLFV τ decay phenomenology is a somewhat large contri-bution of the vector operators to τ → eπ + π − compared to other τ decay channels. Thisis caused by a resonant effect in the pion vector form factor as seen from the left panelof Fig. 11. The bounds on the ψ ϕ D -type operators, c (1 , Lϕ and c eϕ , in Table 11 are pre-dominantly given by the τ → eπ + π − channel. The contributions stem from the tree-level Z -exchange process as listed in Section 7.1. Similarly, in a single operator analysis, most ofthe semileptonic vector operators receive the strongest bounds from the τ → eπ + π − mode.In Table 12, we show the upper limits on the four-fermion operators, where the symbol “ ∗ ”indicates that τ → eπ + π − provides the most stringent bound. For the vector operators, weconsider the RGEs for the heavy quarks ( q = t, b, c ) from 1 TeV to 2 GeV. For light-quarkoperators, running effects are negligible. The details of the RGEs are given in Appendix A.The isoscalar ( C Qe ) τedd and the strange components of vector operators are not constrainedby τ → eππ . In this case, among the observables in Table 9, the strongest bounds arisefrom τ → eη , marked with “ § ” in Table 12. As discussed in Appendix D.1, τ → eK + K − imposes stronger constraints on strange operators, | [ C LQ, D ] τess | < . × − , and verysimilar constraints on ss components of C Ld , C ed and C Qe .– 50 –he last three rows in Table 12 correspond to the limits on the quark-flavor-conservingscalar and tensor operators. Here, we take into account the QCD self-running of these op-erators. In addition, there are several paths in the RGEs: (1) threshold corrections of theheavy quarks to C GG and C G (cid:101) G as in Eqs (3.22) and (3.23); (2) mixing between C (1) LeQu and C (3) LeQu ; and (3) mixing from C (3) LeQu to Γ eγ . These paths enable us to constrain the operatorsfrom τ → eγ , yielding the predominant bounds on the top-quark operators. The induced C GG and C G (cid:101) G are not large enough to compensate for the suppression factor of roughly O (10 − ) as seen in Eqs. (7.23a), (7.24b) and (7.24c). The mixing to the dipole operatoris proportional to the Yukawa coupling, while the threshold corrections are enhanced bythe inverse of the coupling in the lighter-quark case. For the charm-quark scalar operator,although each related decay channel gives the comparable limit of O (10 − ), τ → eη (cid:48) pro-vides the slightly stronger bound. Apart from the heavy up-type quarks, since no mixingis present, it is straightforward to examine ( C LedQ ) bb , whose bound results from the contri-bution of C G ˜ G to τ → eη (cid:48) . The rest of the light-quark operators are primarily constrainedby τ → eπ + π − .Finally, we comment on LFV quarkonium decays such as Υ(2 S ) → τ e and J/ψ → τ e .The current experimental bounds on BRs of these decay modes are O (10 − ). Based on theanalysis in [82], we find that the resulting limit on the four-fermion operators is roughly O (0 . τ decays. Therefore, we do not include thequarkonium decays in our analysis. We now turn to the quark-flavor-violating operators that can be constrained by B mesondecays as well as τ decay involving strange mesons. As in the previous section, below weonly give a rough sketch of each BR to have an idea of which decay modes are relevant.All the expressions of BRs are listed in Appendix D.2.The channels τ − → e − K S and τ − → e − π ± K ∓ put bounds on the sd and ds compo-nents of the LFV down-type operators:BR( τ − → e − K S ) (cid:39) . × − (cid:12)(cid:12)(cid:12)(cid:104) C ed VLR − C ed VLL (cid:105) τeds − ( d ↔ s ) (cid:12)(cid:12)(cid:12) (7.25a)+ 0 . (cid:12)(cid:12)(cid:12)(cid:104) C ed SRR − C ed SRL (cid:105) τeds − ( d ↔ s ) (cid:12)(cid:12)(cid:12) , BR (cid:0) τ − → e − π + K − (cid:1) (cid:39) . (cid:12)(cid:12)(cid:12) C ed VLL + C ed VLR (cid:12)(cid:12)(cid:12) τeds + 0 . (cid:12)(cid:12)(cid:12) C ed SRR + C ed SRL (cid:12)(cid:12)(cid:12) τeds . (7.25b)Wilson coefficients with opposite lepton chirality contribute to each decay mode with thesame prefactors. Compared to τ − → e − K S , the τ − → e − π + K − decay has a strongersensitivity to Wilson coefficients of the vector semi-leptonic operators. This enhancementstems from the K ∗ (892) resonance, which is seen in the right panel of Fig. 11. On theother hand, the scalar contribution is comparable between the two decay modes. In this panel, we only plot the vector contribution from V ( s ) in Eq. (D.35). – 51 – LQ,D ds . × − ♦ sb . × − (cid:92) db . × − (cid:91) sd . × − ♦ bs . × − (cid:92) bd . × − (cid:63) C ed ds . × − ♦ sb . × − (cid:92) db . × − (cid:91) sd . × − ♦ bs . × − (cid:92) bd . × − (cid:63) C Ld ds . × − ♦ sb . × − (cid:92) db . × − (cid:91) sd . × − ♦ bs . × − (cid:92) bd . × − (cid:63) C Qe ds . × − ♦ sb . × − (cid:92) db . × − (cid:91) sd . × − ♦ bs . × − (cid:92) bd . × − (cid:63) C LedQ ds . × − ‡ sb . × − (cid:92) db . × − (cid:63) sd . × − ‡ bs . × − (cid:92) bd . × − (cid:63) Table 13 . 90% C.L. upper limits on the down-type quark-flavor-violating semileptonic operators,assuming a single operator is turned on at the scale Λ = 1 TeV. The superscripts denote that thelimits come from τ decay modes ( ♦ ) τ → eπK or ( ‡ ) τ → eK S , or B meson decay modes ( (cid:63) ) B d → τ e , ( (cid:91) ) B + → π + τ e or ( (cid:92) ) B + → K + τ e . The limit on the scalar operators is applicable toboth the τ e and eτ elements. The bd and db elements of the LFV down-type operators contribute to B d → τ e and B + → π + τ e modes:BR (cid:0) B d → τ − e + (cid:1) (cid:39) . (cid:12)(cid:12)(cid:12)(cid:104) C ed VLR − C ed VLL (cid:105) τebd (cid:12)(cid:12)(cid:12) + 84 . (cid:12)(cid:12)(cid:12)(cid:104) C ed SRR − (cid:104) C ed SRL (cid:105) τebd (cid:12)(cid:12)(cid:12) (7.26a)BR (cid:0) B + → π + τ − e + (cid:1) (cid:39) . (cid:12)(cid:12)(cid:12)(cid:104) C ed VLL + C ed VLR (cid:105) τebd (cid:12)(cid:12)(cid:12) + 8 . (cid:12)(cid:12)(cid:12)(cid:104) C ed SRR + C ed SRL (cid:105) τebd (cid:12)(cid:12)(cid:12) . (7.26b)Similarly, B d → τ + e − and B + → π + τ + e − are described via the interchange of b ↔ d , and,as usual, we are showing only one lepton chirality. For the bd components of the vectoroperators, although both decay modes give similar bounds, B d → τ − e + gives the somewhatstronger bound due to its slightly stronger experimental limit. The opposite situation canbe seen in the db components, which are most strongly restricted by B + → π + τ + e − . Onthe other hand, in the case of the scalar operator, the prefactor in B d → τ − e + is enhancedby roughly ( m B /m τ ) compared to the vector operator, making this the most restrictivedecay channel.The last elements are bs and sb , which are restricted by B + → K + e ± τ ∓ :BR (cid:0) B + → K + e + τ − (cid:1) (cid:39) . (cid:12)(cid:12)(cid:12)(cid:2) C ed VLL + C ed VLR (cid:3) τebs (cid:12)(cid:12)(cid:12) + 12 . (cid:12)(cid:12)(cid:12)(cid:2) C ed SRR + C ed SRL (cid:3) τebs (cid:12)(cid:12)(cid:12) . (7.27)The resulting upper limits on the four-fermion operators are summarized in Table 13.Overall, these limits are less than or equal to O (10 − ). The third column represents thoseof ds and sd components and their bounds originate from τ → eπK for the vector operatorsand τ → eK S for the scalar operators. These decay modes are represented by “ ♦ ” and “ ‡ ”,– 52 –R (90% CL) BR (90% CL) π → eν (1 . ± . × − K + → π + ν ¯ ν < . × − K → eν (1 . ± . × − K L → π ν ¯ ν < . × − D → eν < . × − B + → π + ν ¯ ν < . × − D → τ ν (1 . ± . × − B + → K + ν ¯ ν < . × − D s → eν < . × − D s → τ ν (5 . ± . × − B → eν < . × − B → µν (6 . ± . × − B → τ ν (1 . ± . × − Table 14 . Charged current and neutrino processes sensitive to CLFV operators. All limits aretaken from Ref. [52], with the exception of K + → π + ν ¯ ν , for which we use the more recent resultin Ref. [83]. respectively. The fifth column corresponds to the bounds on the bs and sb elements fromthe B + → K + τ e channel symbolized by “ (cid:92) ”. The constraints on the bd and db elementsfrom B d → τ e ( (cid:63) ) and B + → π + τ e ( (cid:91) ) are in the rightmost column. Invariance under the SU (2) L gauge group implies that some of the SMEFT four-fermionoperators in Eq. (3.14) induce LFV operators with one or two neutrinos rather than chargedleptons. These can mediate meson or nuclear β decays with a eν τ or τ ν e in the finalstate, or flavor-changing-neutral-current meson decays such as K + → π + ν e ¯ ν τ . Theseobservables probe LFV indirectly, since the flavor of the neutrino is not identified. However,the agreement between experiment and SM predictions for these processes can put severeconstraints on the coefficient of LFV four-fermion operators and provide useful informationon their flavor structure. The branching ratios that we use in this section are summarizedin Table 14.Leptonic decays of charged pseudoscalar mesons are particularly sensitive to new scalarinteractions. CLFV interactions can contribute to these processes, since the flavor of theoutgoing neutrino is not determined. The branching ratio is given byBR( P + u i d j → (cid:96) + ν (cid:96) (cid:48) ) = G F | V ij | π τ P f P m P m (cid:96) (cid:18) − m (cid:96) m P (cid:19) (8.1) × (cid:32) δ (cid:96)(cid:96) (cid:48) + 1 | V ij | (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C νedu VLL (cid:105) ν (cid:96) (cid:48) (cid:96)ji + m P m (cid:96) ( m u i + m d j ) (cid:104) C νedu SRR − C νedu SRL (cid:105) ν (cid:96) (cid:48) (cid:96)ji (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) , – 53 –here the CLFV operators do not interfere with the SM contribution. Here, the indices u i and d j correspond to constituent quarks of pseudoscalar meson P .In the case of light pseudoscalar mesons, the ratios R P = Γ( P → eν ) / Γ( P → µν ),with P = π, K , are very well determined. The general expression in the SMEFT (extendedwith light sterile neutrinos) is given in Ref. [84]. Neglecting flavor-conserving operators,and considering only CLFV in the τ - e sector, the ratios R π and R K are R π R SMπ = 1 + 1 | V ud | (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C νedu VLL (cid:105) ν τ edu + m π m e ( m u + m d ) (cid:104) C νedu SRR − C νedu SRL (cid:105) ν τ edu (cid:12)(cid:12)(cid:12)(cid:12) , (8.2) R K R SMπ = 1 + 1 | V us | (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C νedu VLL (cid:105) ν τ esu + m K m e ( m u + m s ) (cid:104) C νedu SRR − C νedu SRL (cid:105) ν τ esu (cid:12)(cid:12)(cid:12)(cid:12) . (8.3)Comparing theory and experiment (see [84] and reference therein) one obtains R π R SMπ = 0 . ± . , R K R SMK = 1 . ± . , (8.4)which, because of the enhancement of 1 /m e , leads to strong bounds on the scalar operators.For the D mesons, we can look at the ratio between the τ and µ or e leptonic decays.Using the input in Table 14, we obtain R µD = Γ( D → µν )Γ( D → τ ν ) = 0 . ± . , R eD = Γ( D → eν )Γ( D → τ ν ) < . , (8.5) R µD s = Γ( D s → µν )Γ( D s → τ ν ) = 0 . ± . , R eD s = Γ( D s → eν )Γ( D s → τ ν ) < . × − . (8.6)Similarly, for B mesons the ratio of B → eν and B → τ ν is constrained to be R eB = Γ( B → eν )Γ( B → τ ν ) < . × − . (8.7)The expressions for these ratios are R µD = (cid:32) m D − m µ m D − m τ (cid:33) m µ m τ
11 + | V cd | − (cid:12)(cid:12)(cid:12)(cid:2) C νedu VLL (cid:3) ν e τdc + m D m τ ( m c + m d ) (cid:2) C νedu SRR − C νedu SRL (cid:3) ν e τdc (cid:12)(cid:12)(cid:12) (8.8) R eD = (cid:18) m D − m e m D − m τ (cid:19) m e m τ | V cd | − (cid:12)(cid:12)(cid:12)(cid:2) C νedu VLL (cid:3) ν τ edc + m D m e ( m c + m d ) (cid:2) C νedu SRR − C νedu SRL (cid:3) ν τ edc (cid:12)(cid:12)(cid:12) | V cd | − (cid:12)(cid:12)(cid:12)(cid:2) C νedu VLL (cid:3) ν e τdc + m D m τ ( m c + m d ) (cid:2) C νedu SRR − C νedu SRL (cid:3) ν e τdc (cid:12)(cid:12)(cid:12) , (8.9)with d → s for D s decays, and m D → m B , d → b , c → u for B decays.The operators C LQ, U , C LQ, D , C Lu and C Ld also induce effective interactions with twoneutrinos of different flavor. In these cases, strong constraints can arise from bounds on K → πν ¯ ν , B → Kν ¯ ν and B → πν ¯ ν . For kaon decays, the differential decay rate can beexpressed as [85, 86] d Γ( K + → π + ν i ¯ ν j ) dzdy = G F m K π (cid:12)(cid:12) f Kπ + (0) (cid:12)(cid:12) ρ ( y, z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C νd VLL + C νd VLR (cid:105) ν i ν j sd (cid:12)(cid:12)(cid:12)(cid:12) (8.10)– 54 – C (1) LeQu ) τe uu . × − ∗ uc . × − § cu . × − † cc . × − § tu . × − (cid:93) ( C (1) LeQu ) eτ cu . × − (cid:93) tu . × − (cid:93) ( C LedQ ) τe dd . × − ∗ ds . × − ∗ db . × − ∗ sd . × − † ss . × − † sb . × − † bd . × − (cid:93) bs . × − (cid:93) bb . × − (cid:93) ( C LedQ ) eτ bd . × − (cid:93) bs . × − (cid:93) C LQ,U uu . × − (cid:91) uc . × − (cid:91) ut . × − (cid:63) cc . × − (cid:91) ct . × − (cid:63) C Ld ds . × − (cid:91) sb . × − (cid:63) db . × − (cid:63) Table 15 . 90% C.L. limits from charged-current leptonic decays and neutrino processes, assuminga single operator is turned on at the scale Λ = 1 TeV. The superscripts denote limits from decayratios ( ∗ ) R π /R SMπ , ( † ) R K /R SMK , ( § ) R eD and R eD s , and ( (cid:93) ) R eB and R µB , and from decay modes( (cid:91) ) K → πνν and ( (cid:63) ) B → ( K, π ) νν . Purely leptonic decays of pseudoscalar mesons constrainthe τ e component of the scalar operators C (1) LeQu and C LedQ . Limits on the eτ components ofscalar operators are much weaker. For example, | ( C (1) LeQu ) eτcc | (cid:46) .
22. On these components, weonly quote bounds that are better than 0.1. C LQ, U and C Ld are constrained by K → πν ¯ ν and B → ( K, π ) νν . In this case, the bounds on the ji and ij components are the same, and we onlyshow one flavor combination. d Γ( K L → π ν i ¯ ν j ) dzdy = G F m K π (cid:12)(cid:12) f Kπ + (0) (cid:12)(cid:12) ρ ( y, z ) 12 × (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C νd VLL + C νd VLR (cid:105) ν i ν j sd − (cid:104) C νd VLL + C νd VLR (cid:105) ν i ν j ds (cid:12)(cid:12)(cid:12)(cid:12) , (8.11)where y = 2 p ν i · p K /m K , z = p π · p K /m K . The function ρ is given by ρ ( y, z ) = 4( z + y − − y ) − r π , (8.12)with r π = m π /m K , and the limits of integrations0 < y < − r π , − y + r π − y < z < r π . (8.13)Integrating over the phase space, we getBR( K + → π + ν i ¯ ν j ) = 1 . (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C νd VLL + C νd VLR (cid:105) ν i ν j sd (cid:12)(cid:12)(cid:12)(cid:12) , (8.14)BR( K L → π ν i ¯ ν j ) = 3 . (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C νd VLL + C νd VLR (cid:105) ν i ν j sd − (cid:104) C νd VLL + C νd VLR (cid:105) ν i ν j ds (cid:12)(cid:12)(cid:12)(cid:12) . (8.15)– 55 –rom B → Kν i ¯ ν j and B → πν i ¯ ν j we can use the expressions for B → Keτ reported inAppendix D.2, and take the limit of zero lepton masses.The limits on the C LQ,U and C Ld operators from processes with two neutrinos areshown in Table 15. Here we neglect the SM contributions and assume that the bound issaturated by SMEFT operators. For K → πν ¯ ν , this approximation leads to a weaker, andhence more conservative, bound. In this Section we summarize and display our results so far on constraints from low- andhigh-energy experiments on coefficients of CLFV SMEFT operators. First in Sec. 9.1 wesummarize results for turning on one SMEFT operator at a time. In Sec. 9.2 we previewwhat a more global analysis might look like by considering two scenarios of constraints onmultiple CLFV operators that contribute simultaneously. A full global analysis is deferredto future work.
We summarize here the upper limits on the LFV couplings discussed in Sections 5, 6 and7, obtained by assuming that a single operator at a time is turned on at the high scale Λ.While not necessarily reflecting the pattern of Wilson coefficients in concrete extensions ofthe SM, this analysis nonetheless provides a good guidance on the relative sensitivity ofvarious probes of e - τ CLFV. Our findings are summarized in Figs. 12–22. The leftmostand rightmost vertical axes in Figs. 12–22 present the bounds on the dimensionless Wilsoncoefficient C ( µ = 1 TeV) and the scale Λ, respectively. The value of Λ is obtained bytaking 4 G F C/ √ ≡ / Λ . The blue and pink bars represent existing 90% C.L. limitsfrom the LHC and low-energy observables, respectively. The pink bars are labeled by thedecay mode that gives the strongest limit as in Tables 12 and 13. The green bars show theEIC sensitivity, assuming √ S = 141 GeV and an integrated luminosity L =100 fb − (thebound on the Wilson coefficient scales as 1 / √L ). The light green bars are based on theanalysis with muonic τ decay, for which the cuts discussed in Section 5 allow to reduce theSM background to a negligible level. The cut efficiencies are given in Table 6, and varybetween 10% and 1%, depending on whether the SMEFT operators include valence or seaquarks. The darker green bar overlaid on the lighter one depicts the maximally optimisticscenario utilizing hadronic τ decay channels, and assuming that the SM background canbe reduced to n b = 0 with (cid:15) n b = 1. The indirect bounds discussed in Sec. 8 are indicatedby a mark “ ∗ ” in orange.The bar charts in Figs. 12–22 contain several interesting messages. For the ‘vertexcorrection’ operators (dipoles, gauge-fermion, Higgs-fermion) the bounds are depicted inFig. 12. The main take-away points are: • The photon dipole Γ eγ receives by far the strongest constraint from τ → eγ , corre-sponding to the effective new physics scale Λ (cid:38)
200 TeV. This is the highest scalecurrently probed by e - τ LFV transitions. High-invariant-mass Drell-Yan is not very– 56 – igure 12 . Upper limits on Γ eγ,Z , Y (cid:48) e , c (1 , Lϕ and c eϕ from the EIC (light green, left), LHC (blue,middle) and low-energy observables (pink, right). The rightmost vertical axis depicts the lowerlimit on the scale of new physics. The darker green bar overlaid on the light green one is theexpected sensitivity in hadronic τ decays at the EIC assuming the efficiency is 100% with no SMbackgrounds. sensitive to this operator, leading to weak limits from the LHC. The EIC can inprinciple provide better constraints, but, even in the most optimistic scenario, theywould be three orders of magnitude weaker than from τ → eγ . • Similarly, the Z dipole Γ eZ is most strongly constrained by τ → eγ , via RGE running.The second best limit is currently from Z → eτ at the LHC. To be competitive with τ → eγ , however, the branching ratio BR( Z → eτ ) needs to reach the prohibitivelevel of 2 · − . • The most severe limit on non-standard Yukawa couplings [ Y (cid:48) e ] τe originates from theATLAS search for h → τ e [59]. The strongest low-energy limit on [ Y (cid:48) e ] τe comes from τ → eγ , which is roughly a factor of five weaker than the LHC. The EIC can at bestprobe Yukawa couplings of order one. • The constraints on the Z couplings c (1) Lϕ + c (3) Lϕ and c eϕ are dominated by τ → eπ + π − ,which limits these couplings to be less than 4 · − , corresponding to a new physicsscale of 10 TeV. High-invariant mass Drell-Yan is not sensitive to these couplings,since the cross section shows the same dependence on √ S as the SM. The best LHClimit therefore comes from Z → eτ . A measurement of the Z → eτ branching ratio– 57 – igure 13 . Upper limit on C Ld (leftmost axis) and lower limit on new physics scale Λ (rightmostaxis) from the EIC (left), LHC (middle) and low-energy observables (right). The symbol “ ∗ ”indicates indirect bounds discussed in Sec. 8. For the EIC expected sensitivity, the light green barcorresponds to the result in Table 7, while the dark green one represents the case in hadronic taudecay mode assuming (cid:15) n b = 1 with n b = 0 in Tables 3 – 5. Figure 14 . Upper limit on C Lu (leftmost axis) and lower limit on new physics scale Λ (rightmostaxis). For the EIC expected sensitivity, the light green bar corresponds to the result in Table 7,while the dark green one represents the case in hadronic tau decay mode assuming (cid:15) n b = 1 with n b = 0 in Tables. 3 – 5. – 58 – igure 15 . Upper limit on C LQ,D (leftmost axis) and lower limit on new physics scale Λ (rightmostaxis). For the EIC expected sensitivity, the light green bar corresponds to the result in Table 7,while the dark green one represents the case in hadronic tau decay mode assuming (cid:15) n b = 1 with n b = 0 in Tables 3 – 5. Figure 16 . Upper limit on C LQ,U (leftmost axis) and lower limit on new physics scale Λ (rightmostaxis). For the EIC expected sensitivity, the light green bar corresponds to the result in Table 7,while the dark green one represents the case in hadronic tau decay mode assuming (cid:15) n b = 1 with n b = 0 in Tables 3 – 5. – 59 – igure 17 . Upper limit on C ed (leftmost axis) and lower limit on new physics scale Λ (rightmostaxis). For the EIC expected sensitivity, the light green bar corresponds to the result in Table 7,while the dark green one represents the case in hadronic tau decay mode assuming (cid:15) n b = 1 with n b = 0 in Tables 3 – 5. Figure 18 . Upper limit on C eu (leftmost axis) and lower limit on new physics scale Λ (rightmostaxis). For the EIC expected sensitivity, the light green bar corresponds to the result in Table 7,while the dark green one represents the case in hadronic tau decay mode assuming (cid:15) n b = 1 with n b = 0 in Tables 3 – 5. – 60 – igure 19 . Upper limit on C Qe (leftmost axis) and lower limit on new physics scale Λ (rightmostaxis). For the EIC expected sensitivity, the light green bar corresponds to the result in Table 7,while the dark green one represents the case in hadronic tau decay mode assuming (cid:15) n b = 1 with n b = 0 in Tables 3 – 5. Figure 20 . Upper limit on C LedQ (leftmost axis) and lower limit on new physics scale Λ (rightmostaxis). For the EIC expected sensitivity, the light green bar corresponds to the result in Table 7,while the dark green one represents the case in hadronic tau decay mode assuming (cid:15) n b = 1 with n b = 0 in Tables 3 – 5. – 61 – igure 21 . Upper limit on C (1) LeQu (leftmost axis) and lower limit on new physics scale Λ (rightmostaxis). For the EIC expected sensitivity, the light green bar corresponds to the result in Table 7,while the dark green one represents the case in hadronic tau decay mode assuming (cid:15) n b = 1 with n b = 0 in Tables. 3 – 5. Figure 22 . Upper limit on C (3) LeQu (leftmost axis) and lower limit on new physics scale Λ (rightmostaxis). For the EIC expected sensitivity, the light green bar corresponds to the result in Table 7,while the dark green one represents the case in hadronic tau decay mode assuming (cid:15) n b = 1 with n b = 0 in Tables. 3 – 5. – 62 –t the 10 − level will be competitive with low-energy constraints. At the EIC, thesecouplings can be probed at the few permill level.Bounds on the four-fermion operators with vector/axial and scalar/tensor Lorentzstructure are reported in Figs. 13–19 and 20–22, respectively. We note that: • The uu component of the vector-like operators C LQ,U , C eu and C Lu , and the dd component of the C LQ,D , C ed , C Ld are very well constrained by hadronic τ decays,in particular τ → eπ + π − . The LHC limits are currently weaker by a factor of five,and the EIC, especially with improvements in the hadronic channel, can reach levelscomparable to the LHC. • The dd component of the isoscalar operator C Qe is constrained at low-energy by τ → eη . Current LHC limits are already comparable with low-energy. The dominantconstraint on the ss components of vector-like operators is also from τ → eη . • The sensitivities to the cc and bb elements of C Ld/Lu , C
LQ,U/D and C ed/eu are com-parable among the EIC, LHC and low-energy observables. At low energy, theseoperators are constrained via mixing with leptonic operators and semileptonic op-erators with light quarks, with the weak loop causing a ∼ − suppression in theamplitude. High energy processes are relatively less suppressed. • The top component of the vector operators has a large mixing with c (1) Lϕ + c (3) Lϕ and c eϕ . As a consequence, these operators are constrained by τ → eπ + π − at the fewpermill level. • The uu , dd and ss components of the scalar operators C (1) LeQu and C LedQ receive theirdominant direct constraints from τ → eπ + π − . LHC limits are a factor of five toten weaker. The cc and bb components are equally well constrained by high- andlow-energy experiments, while the top component runs at two loop onto Γ eγ , whichdominates the bound. Low energy and collider constraints on the uu component ofthe tensor operator C (3) LeQu are similar, while low-energy dominates on the cc and tt components, due to the mixing of the tensor operator onto the dipole. • Concerning quark-flavor-changing decays, τ → K S e , τ → eKπ , B → τ e , B → πτ e and B → Kτ e allow to constrain the off-diagonal components of d -type vector andscalar operators. A bound on B s → eτ at the same level as the recent LHCb limiton B s → µτ [87, 88] would provide complementary information.Finally, we note that the results from the indirect observables (orange “ ∗ ” in theplots), when available, provide limits that are comparable to or stronger than those fromthe direct observables.We conclude this survey with some considerations on the current and future impact ofLHC and EIC searches for e - τ CLFV: – 63 –
Collider searches play a crucial role in bounding off-diagonal elements of up-typefour-fermion operators, while the low-energy observables are insensitive to them. Forthe t - q components, weak running onto flavor-diagonal operators is very suppressedby small Yukawa and CKM elements, so that top decays provide the only sensitiveprobe. The uc and cu components could be constrained by D → eτ , which will beinvestigated at LHCb [89] . For both pseudoscalar and axial operators, however, theDrell-Yan limits imply the prohibitive BR( D → eτ ) ∼ − − − . • Inclusion of hadronic τ decays in the EIC analysis provides a great opportunity toimprove the sensitivity by a factor of 10 depending on LFV operators. The discussion has so far focused on a single coupling analysis. In most extensions of theSM this is not a realistic scenario, as several operators are generated at the matching scaleΛ. ‘Switching on’ more than one coupling at the high scale could in principle result incancellations that weaken the bounds reported in previous sections. We next discuss theextent to which this is possible, showing that complementary information from colliders ingeneral and EIC in particular becomes very relevant. Our discussion below is exploratoryand we refrain from a global analysis that is beyond the scope of this work.To facilitate the identification of directions in parameter space that are unconstrainedby low-energy probes, in Table 16 we summarize the dependence of the τ and B branchingratios used in our analysis on LEFT semileptonic operators, defined at the matching scalebetween the SMEFT and the LEFT. For exclusive channels, the contributions are moreeasily organized by constructing combinations in which the quark bilinears have well definedparity transformations [28, 29]. Since the interference between operators with left- andright-handed electrons is suppressed by the electron mass and always negligible, in Table16 we only show operators with left-handed electrons, similar conclusions can be drawn foroperators with right-handed electrons.For down-type operators, assuming the presence of a single operator structure with allthe flavor entries simultaneously turned on does not entail a significant weakening of theconstraints. Consider for example the operator C Ld . From the summary in Fig. 13 wecan see that only two components receive the strongest constraint from the same process,namely dd and bb , which are both limited by τ → eπ + π − (the sd and ds componentscontribute to τ − → e − K − π + and τ − → e − K + π − , respectively, for which there are twoindependent constraints; similar considerations apply to the sb and bs components). Whenwe simultaneously turn on [ C Ld ] dd and [ C Ld ] bb , τ → eπ and τ → e(cid:96)(cid:96) become relevant andthe limits only slightly deteriorate, | [ C Ld ] dd | < . · − and | [ C Ld ] bb | < . · − , at the90% CL.The situation changes if we simultaneously turn on two or more operators at the sametime. As an example, we consider two scenarios, in which we turn on: A) the left-handed Z -coupling operator c (1) Lϕ + c (3) Lϕ and the light-quark components of all operators with two We thank M. Fontana, D. Mitzel and M. Williams for communications on this point. – 64 – ecay mode C eq VLL+VLR C eq VLL − VLR C eq ∗ SRR+SRL C eq ∗ SRR − SRL C eq ∗ TRR τ → eγ ( uu ), ( cc ) τ → e(cid:96) + (cid:96) − ( cc ), ( bb ) τ → eπ uu − dd uu − ddτ → eη ( (cid:48) ) uu + dd , ss uu + dd , ss ( cc ), ( bb ) τ → eπ + π − uu − dd , uu + dd , ss uu ( cc ), ( bb ) ( cc ), ( bb ) τ → eK + K − uu + dd , uu − dd uu + dd , uu − dd uuss , ( cc ), ( bb ) ss , ( cc ), ( bb ) τ → eK S sd − ds sd − dsτ − → e − K + π + sd dsτ − → e − K − π + ds sdB → e ± τ ∓ db , bd bd , dbB + → π + e − τ + db bdB + → π + e + τ − bd dbB + → K + e − τ + sb bsB + → K + e + τ − bs sb Table 16 . Dependence of low-energy decay channels on the coefficients of semileptonic LEFToperators at the matching scale µ ∼ v . We focus here on operators with left-handed electrons, ananalogous table can be made for operators with right-handed electrons. The parentheses imply thatthe operator induces the decay mode at the loop level, either in perturbation theory, e.g. via theRGE running of (cid:0) C ed VLL+VLR (cid:1) τebb onto four-lepton operators or the matching of (cid:0) C ed ∗ SRR − SRL (cid:1) eτbb onto C G (cid:101) G at the m b threshold, or via hadronic loops, e.g. the contribution of ( C eu ∗ TRR ) eτuu to τ → eγ .We ignore d -type tensor operators, which are not induced by matching onto SMEFT. left-handed leptons and B) c (1) Lϕ + c (3) Lϕ and all flavor-diagonal components of the two down-type operators with two left-handed leptons, [ C Ld ] and [ C LQ, D ]. The nonzero coefficients inthe two scenarios are summarized in Table 17. Including operators with two right-handedleptons would not further weaken the limits, since, as we already noted, the interference ofvector operators with leptons with different chirality is suppressed by the electron mass.In the left panel of Fig. 23, we show the 90% C.L. limits on [ C LQ,U ] τeuu and [ C LQ,U ] τedd in the scenario A, marginalized over the six remaining couplings. The region between thetwo pink lines is allowed by low-energy experiments. The blue solid and dashed linescorrespond to the limits from the LHC. The solid line corresponds to the EFT analysisof Section 6, while the dashed line is obtained by assuming that the effective operators– 65 –cenario OperatorsA c (1 , Lϕ , [ C LQ,U ] uu , [ C LQ,D ] dd,ss , [ C Lu ] uu , [ C Ld ] dd,ss B c (1 , Lϕ , [ C LQ,D ] dd,ss,bb , [ C Ld ] dd,ss,bb Table 17 . Multi-operator scenarios A and B. are induced by the t -channel exchange of a new particle with mass M = 1 TeV, seeEq. (6.14). The green solid (dash-dotted) line represents the projected EIC sensitivity inhadronic τ decay mode assuming the efficiency is 1 (0.2) with zero SM background. Wesee that now there are enough couplings to engineer cancellations in the leading hadronicchannel, τ → eπ + π − . In the axial direction there are still enough constraints from τ → eπ , τ → eη and τ → eη (cid:48) . The isoscalar combination of vector couplings, [ C LQ,U + C Lu ] τeuu +[ C LQ,D + C Ld ] τedd , is however unconstrained by the observables we consider in Sec. 7,leading to the appearance of a free direction. Including the τ → eK + K − channel closesthis free direction, since the process receives contributions from both isovector and isoscalaroperators. The fit including the τ → eK + K − mode is presented by the pink dotted contour.Even with the inclusion of this mode, colliders are very competitive with low-energy.The right panel of Fig. 23 presents the bounds on [ C LQ,D ] τebb and [ C Ld ] τebb in thescenario B, where the rest of the operators are marginalized in the same way as in sce-nario A. As can be seen from Table 23, modulo a small component induced by the b Yukawa, the purely leptonic and the semileptonic operators with light quarks receive acontribution that is proportional to the vector combination [ C LQ, D + C Ld ] τebb , leaving theaxial combination [ C LQ, D − C Ld ] τebb unconstrained by low-energy processes. The free di-rection can be closed using LHC data, which currently impose percent level constraints.In this case, assuming that the effective operators are induced by the t -channel exchangeof a mediator with M = 1 TeV (dashed blue line) only weakens the bound by a fac-tor of two. The EIC can potentially do much better and improve the bounds by a fac-tor of five. In addition, while high-invariant-mass Drell-Yan is sensitive to the sum ofall quark flavors, the EIC could clearly identify the CLFV mechanism by tagging the b quark in the final state. Similar considerations hold for off-diagonal couplings. While B → τ + e − + τ − e + and B → πτ e are sufficient to constrain both the vector and ax-ial combinations [ C LQ, D ± C Ld ] τebd, db , B → Kτ e and τ → eKπ only constrain the vec-tor combinations [ C LQ, D + C Ld ] τebs, sb and [ C LQ, D + C Ld ] τesd,ds , and τ → eK S the linearcombination [ C LQ, D − C Ld ] τesd − [ C LQ, D − C Ld ] τeds . Collider information is thus alwaysnecessary to complement the strong constraints from low-energy.For scalar and pseudoscalar operators, Table 16 suggests that the isovector uu − dd component of scalar operators and the bs and sb components of pseudoscalar operators areunconstrained at low energy. In this case, however, the SU (2) L × U (1) Y invariance of theSMEFT implies that the scalar and pseudoscalar linear combinations are not independent,and the observables included in our analysis are sufficient to fully constrain all the diagonalcomponents of C (1) LeQu and all the components of C LedQ .– 66 – igure 23 . The 90% C.L. limits in [ C LQ,U ] uu − [ C LQ,D ] dd (scenario A, left) and [ C LQ,D ] bb − [ C Ld ] bb planes (scenario B, right). The pink lines are limits from τ decays, while the pink dotted contourpresents the case incorporating τ → eK + K − channel. The blue and green solid lines are boundsfrom the LHC and EIC ( (cid:15) n b = 1 with n b = 0), respectively. The blue dashed line assumes a t -channel exchange of a particle with M = 1 TeV at the LHC, and the green dash-dotted line assumes (cid:15) n b = 0 .
10 Leptoquark models
To illustrate the EFT framework we consider three simplified models involving scalar lepto-quarks (LQ). In the notation of Ref. [90], we consider the leptoquarks S / , and (cid:101) S / , whichare color (anti)triplets and weak isospin doublets, with weak hypercharge − / − / . We further restrict the interactions of S / by requiring that it couples onlyto L-handed leptons ( S L / ) or R-handed leptons ( S R / ). Apart from the LQ gauge-kineticterm and mass terms, the SM Lagrangian density is extended by: L S L / = λ αaL ¯ u αR (cid:96) aL S L † / + h . c . , (10.1) L S R / = λ αaR ¯ q αR iτ e aR S R † / + h . c . , (10.2) L (cid:101) S / = ˜ λ αa ¯ d αR (cid:96) aL (cid:101) S † / + h . c . . (10.3)In the above equations we have denoted by α and a the quark and lepton generation indices,respectively. In what follows we will continue to use greek letters for quark generation andlatin letters for lepton generation indices.Assuming the LQ masses to be considerably above the electroweak scale (consistentlywith LHC phenomenology [92–96]), we integrate out the LQ and match onto the SMEFTeffective Lagrangian. Each of the above models matches at tree level onto a single four- These fields correspond via charge conjugation to R and ˜ R in the notation of Ref. [91]. – 67 –ermion operator in SMEFT at dimension six. At loop level one can generate more oper-ators. However, for the purposes of studying lepton flavor violation the most relevant oneis the photon dipole operator mediating τ → eγ . For the one-loop matching coefficient wewill use the results of Refs. [90] and [18]. For the three models we find: • Integrating out S L / generates O Lu and the dipole, with coefficients[ C Lu ] abαβ = v M LQ ( λ † L ) aβ ( λ L ) αb , (10.4a) (cid:2) Γ eγ (cid:3) eτ = − π vm τ M LQ (cid:88) α ( λ † L ) eα ( λ L ) ατ = − π m τ v (cid:88) α (cid:104) C Lu (cid:105) τeαα , (10.4b) (cid:2) Γ eγ (cid:3) ∗ τe ∝ Y e ≈ , (10.4c)where Y (cid:96) = m (cid:96) /v is the charged lepton Yukawa coupling and we have set the electronmass to zero in the last equation. Hermiticity implies (cid:104) C Lu (cid:105) abαβ = (cid:104) C Lu (cid:105) ∗ baβα . (10.5)LFV τ decays probe (cid:2) Γ eγ (cid:3) eτ and [ C Lu ] eταβ , while EIC processes probe the complexconjugate of these coefficients. • Integrating out S R / generates O Qe and the dipole, with coefficients (cid:104) C Qe (cid:105) αβab = v M LQ ( λ † R ) aβ ( λ R ) αb , (10.6a) (cid:2) Γ eγ (cid:3) ∗ τe = − π vm τ M LQ (cid:88) α ( λ † R ) eα ( λ R ) ατ = − π m τ v (cid:88) α (cid:104) C Qe (cid:105) αατe , (10.6b) (cid:2) Γ eγ (cid:3) eτ ∝ Y e ≈ . (10.6c) • Integrating out (cid:101) S / generates O Ld and no dipole operator due to a cancellationbetween the photon emission from internal quark and LQ lines [18, 90]. For thefour-fermion operator we find (cid:104) C Ld (cid:105) αβab = v M LQ (˜ λ † ) aβ (˜ λ ) αb . (10.7)Introducing two vectors that express the LQ couplings, e.g. ( v τ ) α ≡ λ ατ and ( v e ) α ≡ λ αe (the index α runs over the three quark generations), we can express the induced LFVcouplings as an outer product of the two vectors, [ C M ] τeαβ = v ατ v βe , where M labels four-fermion operator. Our analysis assumes that each Wilson coefficient is real, which enablesall the coefficients to be expressed by only five independent parameters. For example, if[ C M ] τe , , are chosen as three independent parameters, the rest of the components aredescribed by a product of one of the three elements and a ratio, r = v e /v e or r = v e /v e .In what follows, we determine the allowed regions in parameter space by minimizing a χ function which includes LFV τ decays, B meson decays and LHC searches. We present– 68 – igure 24 . The region of ∆ χ < .
71 in [ C Lu ] tt − [ C Lu ] uu (left) and [ C Lu ] cc − [ C Lu ] cu (right)planes. While the purple contour represents existing limits from low-energy experiments and theLHC, the dashed green line corresponds to the EIC expected sensitivity in hadronic τ decays underthe assumption of (cid:15) n b = 1 with n b = 0. The projection of these regions onto each axis correspondsto the 90% C.L. allowed region for that coupling. our results in terms of two-dimensional plots marginalizing over the remaining three freeparameters in each model. The regions we obtain correspond to ∆ χ < .
71, which givesa 90% C.L. limit on single operator couplings when we project the obtained confidenceregions onto one dimension. Below, we present our fitting results in several scenarios.In the case of the S L / LQ model, since the induced operators are those of up-typequarks, the LFV τ decays can only restrict quark-flavor-conserving elements, namely,[ C Lu ] uu,cc,tt . On the other hand, the LFV searches at the LHC play a significant rolein bounding off-diagonal components [ C Lu ] uc/cu and [ C Lu ] tu/ut as well as the first- andsecond-diagonal elements. Figure 24 shows the results of χ fitting in [ C Lu ] tt − [ C Lu ] uu and [ C Lu ] cc − [ C Lu ] cu planes. In the left panel, the bound on [ C Lu ] tt is determined by thelow-energy observables via the RGEs as the collider searches cannot constrain the flavor-diagonal top-quark operator. Conversely, in the right panel, the width of the contour alongthe vertical direction is controlled by the LHC limit. While the single-operator analysespresented in Tables 7, 8 and 12 show the constraint on [ C Lu ] cc is O (10 − ), the contourindicates the relatively strong limit ∼ × − , which originates from τ → eγ contribution.This happens because in this particular model there exists a correlation between four-quarkand dipole operators, as shown by the matching conditions in Eqs. (10.4).Unlike the S L / LQ case, in the S R / and (cid:101) S / models, all the elements of the inducedoperators can be constrained by both low-energy observables and LHC searches. Theallowed regions in these models are depicted in Fig. 25. In the upper two panels, thecontours in the vertical direction are controlled by τ → eπ ± K ∓ and τ → eK S . On theother hand, the LHC search contributes to the bound on [ C Qe ] dd due to the comparable The resulting contour in two dimensions corresponds to the allowed region at 74 .
2% C.L. – 69 – igure 25 . [Upper] the contour that satisfies ∆ χ < .
71 in [ C Qe ] dd − [ C Qe ] sd (left) and [ C Ld ] dd − [ C Ld ] sd (right) planes. [Lower] the same contour as the top two panels but in [ C Qe ] bd − [ C Qe ] bb (left)and [ C Ld ] bd − [ C Ld ] bb (right) planes. Current limits from low-energy experiments and the LHC aredepicted by the purple contour. The EIC expected sensitivity in hadronic τ decays is described bythe dashed green line under the assumption of n b = 0 and (cid:15) n b = 1. limit to that from τ decays as seen from the single-operator analyses in Table 8 and 12. Inthe lower two panels, an order of magnitude difference between the S R / and (cid:101) S / models isfound in the width of the contours in the [ C M ] bb direction. This is due to the fact that theRGEs of [ C Qe ] bb involve top-quark Yukawa coupling, resulting in relatively large correctionsto LEFT operators as seen from Table 23.In Figs. 24 and 25, we also report the prospective reach of the EIC with L =100 fb − ,in the ideal scenario in which the τ is reconstructed through the hadronic decay channels,– 70 – C Lu ] uu [ − . , .
37] [ C Lu ] uc [ − . , .
0] [ C Lu ] ut [ − , S L / [ C Lu ] cu [ − . , .
0] [ C Lu ] cc [ − . , .
6] [ C Lu ] ct [ − , C Lu ] tu [ − , C Lu ] tc [ − , C Lu ] tt [ − . , . C Qe ] dd [ − . , .
66] [ C Qe ] ds [ − . , .
44] [ C Qe ] db [ − . , . S R / [ C Qe ] sd [ − . , .
40] [ C Qe ] ss [ − . , .
1] [ C Qe ] sb [ − . , . C Qe ] bd [ − . , .
0] [ C Qe ] bs [ − . , .
1] [ C Qe ] bb [ − . , . C Ld ] dd [ − . , .
23] [ C Ld ] ds [ − . , .
44] [ C Ld ] db [ − . , . S / [ C Ld ] sd [ − . , .
41] [ C Ld ] ss [ − . , .
1] [ C Ld ] sb [ − . , . C Ld ] bd [ − . , .
0] [ C Ld ] bs [ − . , .
1] [ C Ld ] bb [ − . , . Table 18 . 90% C.L. ranges for the Wilson coefficients (in units of 10 − ), in the three leptoquarkmodels considered here. The ranges are obtained after marginalizing over all other couplings. and the SM background can be reduced to n b = 0 with (cid:15) n b = 1 (this corresponds to thedark green bands in Figs. 12–22). While for the couplings involving light quarks and topquark the EIC is not competitive (almost the entire plotted region is allowed), the EICcan be quite competitive for couplings involving the charm and beauty quarks, both flavordiagonal and off-diagonal. These simple models illustrate a general lesson emerging fromour study: the discovery window for CLFV at the EIC comes mostly from semileptonicinteractions that involve one or two heavy flavors.The currently allowed 90% C.L. ranges for each coupling are summarized in Table 18.When comparing our leptoquark analysis to previous studies in Refs. [18, 97, 98], severalremarks are in order: • We improve the bounds on the first-generation quark-flavor diagonal couplings byincluding τ → eπ + π − and the LHC searches. This leads to constraints that are anorder of magnitude stronger than the expected sensitivity at the EIC. • As discussed in [18], τ → eγ constrains the quark-flavor diagonal components of thefour-fermion operators in the S L / and S R / models, yielding a somewhat strongerlimit on [ C Lu ] cc than those from the LHC and other low-energy decay channels. Forthis coupling, prospective EIC limits are quite competitive. On the other hand, τ → eη, τ → eπ + π − and the LHC searches, which are newly incorporated intoour analyses, give the most stringent bounds on the rest of the second- and third-generation diagonal elements. • Concerning the quark-flavor changing couplings, the LHC searches currently providethe strongest bounds on the uc and cu elements , but the EIC can be quite competitivein the future. In addition, the recent ATLAS search for LFV top-quark decays enablesus to put bounds of O (0 .
1) on the flavor-violating operators involving top quark.– 71 –
For the strangeness-changing couplings, the inclusion of τ → eπ ± K ∓ improves thelimits on sd and ds elements by a factor 10 compared to previous analyses. Thebounds on the sb and bs components are improved by incorporating the latest ex-perimental results of B ± → K ± τ e , yielding stronger limits than the LHC and futureEIC sensitivities. • As illustrated by Figs. 24 and 25, we find that after inclusion of low-energy constraintsthe CLFV discovery potential at the EIC arises mostly for LQ couplings involvingthe charm and beauty quarks, both flavor diagonal and off-diagonal.Although our analysis focuses on LFV τ − e couplings, with emphasis on the EICdiscovery potential, the above LQ models have several intriguing connections to otherinteresting phenomenology, such as neutrino mass [99, 100] and B physics [101–103]. Thiswould open a number of additional observables to probe LQ couplings. We defer theanalysis to future work.
11 Conclusions
It has been long recognized that searches for CLFV processes are a very promising toolto probe new physics beyond the SM. In this paper we have performed, in the frameworkof the SMEFT, a first comprehensive analysis of the CLFV sensitivity at the EIC in thechannel ep → τ X . The SMEFT is particularly appealing because it captures a large classof new physics models originating at energies above the electroweak scale and allows fora systematic comparison of all probes of CLFV in the τ - e sector. We considered all thedimension-six CLFV operators in the SMEFT, including CLFV Z and Higgs couplings,photon and Z dipole interactions, and ten semileptonic four-fermion operators, with dif-ferent Lorentz and completely general quark-flavor structures.For the DIS cross section ep → τ X we found that, for all operators except Yukawaand electron-gluon operators, the unpolarized cross sections at √ S = 141 GeV are in the1–10 pb range for SMEFT coefficients of order one (see Tables 1–2). Operators withsea quarks in the initial state give rise to somewhat smaller cross sections, as expectedfrom the suppression of the corresponding PDFs. In order to account for the decay ofthe τ lepton, and to realistically assess the sensitivity of the EIC, we simulated SMEFTevents in Pythia8 , using the
Delphes package to simulate the detector smearing effects (seeFigs. 4–8). We found the muonic reconstruction channel τ → µ ¯ ν µ ν τ to be very promising,since moderate cuts on the muon p T and on the missing energy allow one to eliminate allSM background without excessively suppressing the signal. The signal efficiency dependsstrongly on the flavor of the SMEFT operators, since operators with heavy quarks in theinitial state give rise to distributions peaked at smaller p T , which are more affected by thecuts to suppress the SM background. The efficiency is on the other hand rather insensitive For [ C Ld ] sd/ds , if we include indirect bounds from kaon decays, they are superior to other low-energylimits as also discussed in [97, 98]. Recall we have written the dimensionful couplings for the dimension-six operators as ∼ C/v , where v = 246 GeV is the electroweak scale and C are dimensionless SMEFT coefficients. – 72 –o the Lorentz structure of the SMEFT operators. In the electron channel τ → e ¯ ν e ν τ , thebackground from neutral and charged-current DIS is always very large. In the hadronicchannels τ → X h ν τ , the naive cuts we imposed in Section 5, where we vetoed leptons with p T >
10 GeV and asked for two jets with p T larger than 15 and 20 GeV, are not sufficientto fully suppress the SM background. We however did not use additional information onthe jet that emerges from τ decay, such as the presence of a secondary vertex, the hadronmultiplicity or the correlation with (cid:54) E T , which could provide more efficient ways to taghadronic τ events [58]. At √ S = 141 GeV and L = 100 fb − , the EIC expected sensi-tivity for the dimensionless SMEFT coefficients reaches C ∼ O (10 − (3 − ) for light-quarkfour-fermion, dipole and Z -coupling operators. Bounds on heavy-quark operators result in C ∼ O (10 − (2 − ), while it is more challenging to constrain LFV Yukawa and electron-gluonoperators as their cross sections are strongly suppressed.To assess the discovery potential of the EIC in τ - e transitions, we have compared itssensitivity to other probes of the same interactions, across a broad range of energy scales,ranging from other collider processes to decays of τ lepton and B meson. In Sec. 2 wehave provided simple order of magnitude estimates, substantiated by a detailed analysis inSecs. 6 , 7 , 8, and 9. We summarize our main findings below, starting with the LHC andgoing down in energy.The LHC can probe LFV by studying the decays of the Z and Higgs bosons andof the top quark. In addition, if the scale of new physics is larger than a few TeV, thesame semileptonic four-fermion operators that induce CLFV DIS can be studied in high-invariant mass Drell-Yan pp → eτ X . The bounds we obtain are discussed in Sec. 6.1 andTable 8. While the LHC has a clear edge in measuring Higgs and top quark-flavor-changingcouplings, we found that the EIC could competitively probe Z couplings and four-fermioninteractions with light quarks, especially if the efficiency in the hadronic channel can be im-proved with respect to our simple analysis. Four-fermion operators with two heavy quarksare somewhat more suppressed in Drell-Yan compared to DIS, because of the presenceof two heavy quark PDFs. Here the EIC could have a larger impact, provided analysisstrategies are devised in order to improve the signal efficiency. When comparing the EICwith the LHC, it is worth keeping in mind that the formalism of the SMEFT might notbe applicable at LHC energies. The two colliders could thus be probing complementaryregions in parameter space, and are both necessary to fully constrain CLFV.We then carried out a comprehensive comparison of the EIC and LHC sensitivitywith current bounds from τ and B decays, including the radiative decay τ → eγ , purelyleptonic channels, τ → e(cid:96) + (cid:96) − , and semileptonic channels. The limits obtained under thehypothesis that a single SMEFT operator is present at the new physics scale are shown inFigs. 12–22. τ → eγ gives the most severe limits, at least a factor of 100 stronger than thoseexpected from the EIC, on dipole and Z -coupling operators. In the single coupling analysis,quark-flavor-diagonal four-fermion operators with light quarks are very well constrained by τ → eπ + π − , τ → eη , τ → eη (cid:48) and τ → eπ . In particular, the constraints from τ → eπ + π − on operators with valence quarks are currently a factor of five better than high-invariantmass Drell-Yan and a factor of fifty better than the EIC in the muonic channel. In the caseof heavy quarks, however, contributions to τ decay only arise at the loop level, and the EIC– 73 –ensitivity on these operators is very competitive with LHC and low-energy observables.While the muonic reconstruction channel is rather clean, the full potential of the EIC isbetter represented by the hadronic channel, whose BR is a factor of 4 larger than themuonic channel. Assuming, optimistically, that all the SM background can be suppressedwithout losing any signal event, we find that the EIC sensitivity to four-fermion operatorscan exceed that of the current LHC and low-energy experiments. It will therefore be veryimportant to more thoroughly explore the hadronic reconstruction channels and deviseanalysis strategies to maximize the signal/background ratio.Due to the prominence of τ → eππ in the single coupling analysis, it is rather easy toweaken the low-energy bounds by turning on several operators at the matching scale. Inthis paper we refrained from a global fit to CLFV observables, but explored the impact ofmultiple operators in two scenarios, turning on two down-type vector operators with genericquark flavors, left-handed leptons and left- or right-handed quarks, and turning on fiveoperators with left-handed leptons and couplings only to the light u , d and s quarks. In bothcases, low-energy observables are not sufficient to constrain all operator coefficients and freedirections appear. As illustrated in Fig. 23, collider experiments are crucial to close thesefree directions and discovery windows arise for the EIC in these more general scenarios.Our analysis applies to any new physics originating at energies higher than the elec-troweak scale. In Sec. 10 we have applied our framework to study three different leptoquarkscenarios, which yield more than one LFV operator at the matching scale. Leptoquarksprovide interesting extensions of the Standard Model motivated both by model building andby several phenomenological puzzles. We improve upon the current literature by includingstate-of-the art analyses of τ decays, in particular semi-leptonic modes. As expected fromthe single-operator analysis, most of the LFV leptoquark couplings are constrained quiteseverely by the LHC, τ and B decays. We find that after inclusion of low-energy constraintsthe CLFV discovery potential at the EIC arises mostly for LQ couplings involving the charmand beauty quarks (see Figs. 24 and 25). We leave to future work a more comprehensiveanalysis of leptoquark models and their implications for lepton flavor violation and beyond.To fully explore the EIC potential to probe CLFV physics, our analysis needs to beextended in several directions: • The LO DIS cross sections for processes initiated by a strange or heavy quark areaffected by significant theoretical errors, as shown in Tables 1 and 2. To reduce theerror, and to have a more robust assessment of the theory uncertainties, it is necessaryto consider NLO QCD corrections. These corrections for the SM DIS inclusive crosssection range from O (10%) for light flavor contributions to O (20 − • As shown in Figs. 12 to 22, improving the analysis in the hadronic channels ep → τ X → ν τ X h X , which has the largest branching ratio, could highly impact the EICreach. It will be important to take advantage of the distinctive features of the jetsemerging from the hadronic τ decay in order to devise a robust and efficient τ tagger,to suppress the SM background without losing signal events.– 74 – One of the most promising directions for the EIC is to probe CLFV operators withheavy c and b quarks, whose effects are suppressed by one electroweak loop in τ de-cays and by two heavy quark PDFs at the LHC. However, the missing energy, leptonand jet p T distributions induced by heavy-flavor operators are peaked at small p T ,and thus severely suppressed by the cuts imposed in Section 5. It will be importantto explore whether tagging b and c jets allows to achieve the same background sup-pression with looser p T cuts, thus boosting the efficiency for heavy flavor operators.Higher-order perturbative QCD corrections and resummation may be particularlyimportant to predict accurately the dependence of cross sections with jet p T cuts. • New data on τ , B and D meson decays at Belle II, LHCb and BESIII, combined withincreased luminosity at the LHC and, in the future, with data from the EIC, will helppaint a complete picture of CLFV in the τ sector. To fully exploit this wealth of data,a global analysis (beyond single-coupling) is highly desirable. In this context, the in-clusion of more observables will help eliminate flat directions that emerged already inour discussion. For light quarks, additional constraints can be obtained by includingthe decays τ → eK + K − , τ → eK S K S and τ → eπππ , whose branching ratios arebounded at the few × − level. For flavor changing interactions, D → eτ could bemeasured at BESIII and LHCb, while B s → eτ will be in reach of LHCb and Belle II.At colliders, heavy flavor tagging at the EIC (e.g. [104]) could provide unambiguousprobes of the operator flavor structure, while angular distributions in high-invariantmass Drell-Yan and helicity fractions in top decays can pinpoint the Lorentz struc-ture of the contributing operators. Because of the similar sensitivity, we expect that,in case of observation, by correlating observables at low and high energy it will bepossible to remove degeneracies and clearly identify the dominant CLFV mechanism. Acknowledgments
We thank Carlo Alberto Gottardo for illustrating the procedure for the extraction of boundson t → qeτ from Ref. [62], and for checking the limit on the branching ratio we obtained.We acknowledge interesting discussion with Marianna Fontana, Dominik Mitzel, and MarkWilliams on the LHCb sensitivity to τ CLFV in D and B s decays. We thank Miguel Arratiafor discussion on the EIC Delphes card used in the analysis of Section 5. We are indebtedto Emilie Passemar for sharing numerical data files on the Kπ form factors, and to Kon-stantin Beloborodov for providing the K + K − form factors. We thank Jure Zupan for earlydiscussions on a SMEFT analysis of CLFV at the EIC. We also acknowledge stimulatingdiscussions with Sacha Davidson and Krishna Kumar. This work is supported by the USDepartment of Energy through the Office of Nuclear Physics, an Early Career ResearchAward, and the LDRD program at Los Alamos National Laboratory. Los Alamos NationalLaboratory is operated by Triad National Security, LLC, for the National Nuclear SecurityAdministration of U.S. Department of Energy (Contract No. 89233218CNA000001).– 75 – s ( m Z ) 0.118 g g m u (2 GeV) 2 . +0 . − . MeV m d (2 GeV) 4 . +0 . − . MeV m s (2 GeV) 93 +11 − MeV m c ( m c ) 1 . ± .
02 GeV m b ( m b ) 4 . +0 . − . GeV m t ( m t ) 162 . +2 . − . GeV
Table 19 . Standard Model parameters used in the solution of the RGEs
A Renormalization group equations and their solutions
In this section we consider the renormalization group evolution of SMEFT operators be-tween Λ and the electroweak scale, and then between the EW scale and the low-energyscale µ ∼ A.1 Running between Λ and the electroweak scale The RGEs in the SMEFT can be found in Refs. [23–25]. We report them here for com-pleteness, in a slightly different choice of basis. We work in a basis in which both the u and d quark mass matrices are diagonal, and define the SM Yukawa coupling as Y f = m f v . (A.1)The values of the masses, in the MS scheme, are given in Table 19. The hyperchargeassignments are y q = 16 , y u = 23 , y d = − , y l = − , y e = − . (A.2)The running of the SM couplings is given by µ ddµ g s ( µ ) = − (cid:18) N c − T f n f (cid:19) ( g s ( µ )) (4 π ) , (A.3) µ ddµ g ( µ ) = − (cid:18) − n G (cid:19) ( g ( µ )) (4 π ) , (A.4) µ ddµ g ( µ ) = 53 (cid:18)
110 + 43 n G (cid:19) ( g ( µ )) (4 π ) , (A.5) µ ddµ Y t ( µ ) = 1(4 π ) Y t ( µ ) (cid:18) Y t ( µ ) − g s ( µ ) − g ( µ ) − g ( µ ) (cid:19) , (A.6)where N c = 3, T f = 1 / n f is the number of active quarks, n f = 6 above the EW scale,and n G the number of fermion generations. The RGEs in (A.3)-(A.6) are not affected bythe dimension-six CLFV SMEFT operators we are considering.– 76 – C (1) LeQu (cid:17) τett (cid:16) C (3) LeQu (cid:17) τett (cid:16) C (1) LeQu (cid:17) τecc (cid:16) C (3) LeQu (cid:17) τecc (cid:16) C LedQ (cid:17) τebb Γ eγ . · − − .
46 2 . · − − . · − –Γ eZ . · − − .
07 4 . · − − . · − – (cid:16) C (1) LeQu (cid:17) τecc – – 1 . − .
13 – (cid:16) C (3) LeQu (cid:17) τecc – – − . · − .
96 – (cid:16) C LedQ (cid:17) τebb – – – – 1 . C GG ) τe − .
57 0 . −
127 29 32 (cid:0) C G ˜ G (cid:1) τe − .
85 0 . −
190 44 − Table 20 . The Wilson coefficients of dipole, scalar and tensor operators at µ = m t induced by anonzero top-, bottom- and charm-quark scalar and tensor operators through operator mixing. Thestarting point of the running is taken at Λ = 1 TeV. The threshold corrections to C GG and C G ˜ G are given in Eqs. (3.22), (3.23), (7.13a), (7.13b). We evaluate them at µ = m t ( m b ) for the top(bottom) quark while µ = 2 GeV for the charm quark. We consider the QCD running of scalar and tensor operators. In addition, because ofthe very strong low-energy limits on the CLFV dipole operator Γ eγ , we take into accountthe mixing of the Z dipole and of the tensor operators onto Γ eγ . For the tensor operator,the mixing is proportional to the SM Yukawa coupling and thus is most relevant in thecase of operators involving the top quark. The bounds on Γ eγ are so stringent that theyalso constrain the top component of C (1) LeQu , which mixes onto the tensor operator C (3) LeQu at one loop. We thus solve the RGEs dd log µ (cid:2) Γ eγ (cid:3) τe = 6 g (4 π ) (cid:0) − tan θ W (cid:1) [Γ eZ ] τe + 32(4 π ) N c Q t [ Y u ] ii (cid:104) C (3) LeQu (cid:105) τeii , (A.7) dd log µ [Γ eZ ] τe = 16(4 π ) N C ( z u L + z u R ) [ Y u ] ii (cid:104) C (3) LeQu (cid:105) τeii , (A.8) dd log µ C (1) LeQu = α s π γ (0) S C (1) LeQu − π ) (cid:20)
24 (y q + y u ) (2y e − y q + y u ) g − g (cid:21) C (3) LeQu , (A.9) dd log µ C (3) LeQu = α s π γ (0) T C (3) LeQu + 18(4 π ) (cid:20) − q + y u ) (2y e − y q + y u ) g + 3 g (cid:21) C (1) LeQu , (A.10) dd log µ C LedQ = α s π γ (0) S C LedQ , (A.11)where γ (0) S = − C F , γ (0) T = 2 C F , (A.12)– 77 –ith C F = 4 /
3. The solutions of these RGEs take the form, C i low ( µ ) = A ij C j hi (Λ) , (A.13)where C low is a vector of the coefficients on the LHS of Eqs. (A.7)–(A.11), evaluated at µ = m t , and C hi is the vector of coefficients at the scale Λ = 1 TeV on the RHS. Thecoefficients A ij that solve these equations are given in Table 20.Considering now vector-like operators, four-quark operators involving heavy quarksrun into the Z couplings c (1 , Lϕ and c eϕ via the first diagram in Fig. 1. The RGE has apiece proportional to the quark Yukawas, and a gauge component dd log µ (cid:16) c (1) Lϕ + c (3) Lϕ (cid:17) = − N c (4 π ) (cid:40)
12 [ Y u ] ii ( − C LQ,U + C Lu ) ii −
12 [ Y d ] jj ( − C LQ,D + C Ld ) jj + 13 (cid:18) g c w (cid:19) (cid:16) ( z u L C LQ,U + z u R C Lu ) ii + ( z d L C LQ,D + z d R C Ld ) jj (cid:17) (cid:41) , (A.14) dd log µ c eϕ = − N c (4 π ) (cid:40)
12 [ Y u ] ii ( C eu ) ii −
12 [ Y d ] jj ( C ed ) jj + 12 (cid:16) [ Y d ] jj δ jk − V ∗ ik [ Y u ] ii V ij (cid:17) ( C Qe ) jk + 13 (cid:18) g c w (cid:19) (cid:16) z u R ( C eu ) ii + z d R ( C ed ) jj − q s w ( C Qe ) jj (cid:17) (cid:41) , (A.15)where V ij denotes elements of the CKM matrix. We use i to denote u -type indices, i ∈{ u, c, t } and j, k to denote d -type indices, j, k ∈ { d, s, b } , and a sum over repeated indicesis understood in Eqs. (A.14) and (A.15).The penguin contributions to the semileptonic four-fermion operators are dd log µ ( C LQ,U ) ll = 43 N c g (4 π ) (cid:40) y ( C LQ,U ) ii + y ( C LQ,D ) jj + y q y u ( C Lu ) ii + y q y d ( C Ld ) jj + g g (( C LQ,U ) ii − ( C LQ,D ) jj ) (cid:41) (A.16) dd log µ ( C LQ,D ) kk = 43 N c g (4 π ) (cid:40) y ( C LQ,U ) ii + y ( C LQ,D ) jj + y q y u ( C Lu ) ii + y q y d ( C Ld ) jj − g g (( C LQ,U ) ii − ( C LQ,D ) jj ) (cid:41) (A.17) dd log µ ( C Lu ) ll = 43 N c g (4 π ) (cid:40) y u y q (( C LQ,U ) ii + ( C LQ,D ) jj ) + y ( C Lu ) ii + y u y d ( C Ld ) jj (cid:41) (A.18) dd log µ ( C Ld ) kk = 43 N c g (4 π ) (cid:40) y d y q (( C LQ,U ) ii + ( C LQ,D ) jj ) + y d y u ( C Lu ) ii + y ( C Ld ) jj (cid:41) – 78 –A.19) dd log µ ( C eu ) ll = 43 N c g (4 π ) (cid:40) y ( C eu ) ii + y u y d ( C ed ) jj + 2y u y q ( C Qe ) jj (cid:41) (A.20) dd log µ ( C ed ) kk = 43 N c g (4 π ) (cid:40) y d y u ( C eu ) ii + y ( C ed ) jj + 2y d y q ( C Qe ) jj (cid:41) (A.21) dd log µ ( C Qe ) kk = 43 N c g (4 π ) (cid:40) y q y u ( C eu ) ii + y q y d ( C ed ) jj + 2y ( C Qe ) jj (cid:41) , (A.22)where, as before, summation over u and d -type flavor indices i and j on the r.h.s of Eqs.(A.16)–(A.22) is understood.In addition to the penguin diagrams, there are also current-current contributions shownin the last three diagrams in Fig. 1. Neglecting again the operator self-renormalization, wehave: [24] dd log µ C LQ, U = − π ) V CKM Y d C Ld Y d V † CKM , (A.23) dd log µ C LQ, D = − π ) V † CKM Y u C Lu Y u V CKM , (A.24) dd log µ C Lu = − π ) Y u V CKM C LQ, D V † CKM Y u , (A.25) dd log µ C Ld = − π ) Y d V † CKM C LQ, U V CKM Y d , (A.26) dd log µ C eu = − π ) Y u V CKM C Qe V † CKM Y u , (A.27) dd log µ C Qe = − π ) V † CKM Y u C eu Y u V CKM . (A.28)Because of the CKM and Yukawa factors, these RGE always induce negligible effects.The penguin diagrams also induce the following leptonic operators that contribute to τ → eµµ and τ → e [24, 25]. The RGEs for the left-handed operator C LL are dd log µ ( C LL ) τeµµ = − N c g (4 π ) ( − ( C LQ,U ) τeii + ( C LQ,D ) τejj ) + 23 N c g (4 π ) (A.29) × (cid:26) y l y q (( C LQ,U ) τeii + ( C LQ,D ) τejj ) + y l y u ( C Lu ) τeii + y l y d ( C Ld ) τejj (cid:27) dd log µ ( C LL ) τµµe = 13 N c g (4 π ) ( − ( C LQ,U ) τeii + ( C LQ,D ) τejj ) (A.30) dd log µ ( C LL ) τeee = + 16 N c g (4 π ) ( − ( C LQ,U ) τeii + ( C LQ,D ) τejj ) + 23 N c g (4 π ) (A.31) × (cid:26) y l y q (( C LQ,U ) τeii + ( C LQ,D ) τejj ) + y l y u ( C Lu ) τeii + y l y d ( C Ld ) τejj (cid:27) The RGEs for the (cid:96)(cid:96)τ e and (cid:96)eτ (cid:96) components are the same as Eqs. (A.29) – (A.31). dd log µ ( C ee ) τe(cid:96)(cid:96) = 23 N c g (4 π ) (cid:110) e y q ( C Qe ) τejj + y e y u ( C eu ) τeii + y e y d ( C ed ) τejj (cid:111) , (A.32)– 79 – C LQ,U ) tt ( C Lu ) tt ( C eu ) tt ( C LQ,D ) bb ( C Ld ) bb ( C ed ) bb ( C Qe ) bb c (1) Lϕ + c (3) Lϕ −
102 106 – −
10 1 . − – c eϕ – – 106 – − . − C LQ,U ) cc ( C Lu ) cc ( C eu ) cc c (1) Lϕ + c (3) Lϕ . − . c eϕ – – − . C LQ,U ) ll ( C Lu ) ll ( C eu ) ll ( C LQ,D ) kk ( C Ld ) kk ( C ed ) kk ( C Qe ) kk ( C LQ,U ) ii − . − .
65 – 4 . . − –( C LQ,D ) jj . − .
69 – − . . − –( C Lu ) ii − . − . − .
67 1 . − –( C Ld ) jj .
34 1 . . − . − –( C eu ) ii – – − . − − . − . C ed ) jj – – 1 . − − − .
67 0 . C Qe ) jj – – − . − − . − . C LL ) τeµµ − . . . − . − − ( C LL ) τµµe . − . − − ( C LL ) τeee . . − . − . − − ( C ee ) τe(cid:96)(cid:96) – – 2 . − − . . C Le ) τe(cid:96)(cid:96) . . . − . − –( C Le ) (cid:96)(cid:96)τe – – 2 . − − − . . Table 21 . The Wilson coefficients at µ = m t induced by nonzero top-, bottom- and charm-quarkoperators at the scale Λ = 1 TeV, in units of 10 − . The indices i , l and j , k denote, respectively, u - and d -type flavor indices, and we consider mixing onto operators of different flavor, i (cid:54) = l , j (cid:54) = k . and again the (cid:96)(cid:96)τ e component has the same RGE. Finally, the LR operator dd log µ ( C Le ) τe(cid:96)(cid:96) = 43 N c g (4 π ) (cid:40) y e y q (( C LQ,U ) τeii + ( C LQ,D ) τejj )+y e y u ( C Lu ) τeii + y e y d ( C Ld ) τejj (cid:41) , (A.33) dd log µ ( C Le ) (cid:96)(cid:96)τe = 43 N c g (4 π ) (cid:40) l y q ( C Qe ) τejj + y l y u ( C eu ) τeii + y l y d ( C ed ) τejj (cid:41) . (A.34)The solutions of the RGEs in Eqs. (A.14)–(A.34) take the form Eq. (A.13), and the solutioncoefficients are given in Table 21. – 80 – .2 Dipole contributions induced by the LFV Yukawa interaction The τ − e LFV Yukawa coupling contributes to τ → eγ through one- and two-loop diagrams.For the one-loop diagram, the expression is given by [65] (cid:2) Γ eγ (cid:3) eτ = − vm τ π m h (cid:0) Y (cid:48) e (cid:1) eτ ( Y e ) ττ (cid:18) log m h m τ − (cid:19) , (A.35) (cid:2) Γ eγ (cid:3) ∗ τe = − vm τ π m h (cid:16) Y (cid:48) e (cid:17) ∗ τe ( Y ∗ e ) ττ (cid:18) log m h m τ − (cid:19) , (A.36)with the Higgs mass m h = 125 GeV.The two-loop contribution consists of several diagrams, known as Barr-Zee diagrams[105, 106], where not only top quark but also W boson runs in the loop. They are givenby (cid:2) Γ eγ (cid:3) eτ = N c Q t α em π vm t (cid:0) Y (cid:48) e (cid:1) eτ (cid:20) Q t { Re ( Y u ) tt f ( x th ) + i Im ( Y u ) tt g ( x th ) }− s w c w ( z τ L + z τ R ) ( z t L + z t R ) (cid:110) Re ( Y u ) tt ˜ F H ( x th , x tZ ) + i Im ( Y u ) tt ˜ F A ( x th , x tZ ) (cid:111) (cid:21) − α em π (cid:0) Y (cid:48) e (cid:1) eτ (cid:20) J γW ( m h ) − s w ( z τ L + z τ R ) J ZW ( m h ) (cid:21) , (A.37) (cid:2) Γ eγ (cid:3) ∗ τe = N c Q t α em π vm t (cid:0) Y (cid:48) e (cid:1) ∗ τe (cid:20) Q t { Re ( Y u ) tt f ( x th ) − i Im ( Y u ) tt g ( x th ) }− s w c w ( z τ L + z τ R ) ( z t L + z t R ) (cid:110) Re ( Y u ) tt ˜ F H ( x th , x tZ ) − i Im ( Y u ) tt ˜ F A ( x th , x tZ ) (cid:111) (cid:21) − α em π (cid:0) Y (cid:48) e (cid:1) ∗ τe (cid:20) J γW ( m h ) − s w ( z τ L + z τ R ) J ZW ( m h ) (cid:21) , (A.38)where x ij = m i /m j , and the Z couplings z f L and z f R are given in Eq. (3.10). The loopfunctions are given by f ( x ) = x (cid:90) dy − y (1 − y ) y (1 − y ) − x ln (cid:18) y (1 − y ) x (cid:19) , (A.39) g ( x ) = x (cid:90) dy y (1 − y ) − x ln (cid:18) y (1 − y ) x (cid:19) , (A.40)˜ F A ( a, b ) = 1 a − b [ ag ( b ) − bg ( a )] , (A.41)˜ F H ( a, b ) = 1 a − b [ af ( b ) − bf ( a )] , (A.42) J VW ( m h ) = 2 m W m h − m V (cid:20) − (cid:26)(cid:18) − m V m W (cid:19) + (cid:18) − m V m W (cid:19) m h m W (cid:27) × ( I ( m W , m h ) − I ( m W , m V ))+ (cid:26)(cid:18) − m V m W (cid:19) + 14 (cid:18) − m V m W (cid:19) + 14 (cid:18) − m V m W (cid:19) m h m W (cid:27) – 81 – eZ ( Y (cid:48) e ) τe Γ eγ − . · − . · − ( C GG ) τe – − . (cid:0) C G ˜ G (cid:1) τe – 0 C eq SRR , C eq SRL – − . Y q C eq TRR – 7 . · − Q q Y q Table 22 . The Wilson coefficients of dipole, gluonic, scalar, and tensor operators at µ = 2 GeVinduced by a Z dipole or a CLFV Yukawa through operator mixing. The Yukawa contribution toΓ eγ does not include the numerically larger effects from top quarks and weak bosons, given in Eq.(A.46). q denotes a light quark, q ∈ { u, d, s } , and the quark Yukawa couplings Y q is evaluated atthe scale µ . × ( I ( m W , m h ) − I ( m W , m V )) (cid:21) , (A.43) I ( m , m ) = − m m f (cid:18) m m (cid:19) , (A.44) I ( m , m ) = − m m g (cid:18) m m (cid:19) . (A.45)In Eqs. (A.37) and (A.38) we consider only the top quark. At leading log, the contributionsof b and c quarks are accounted for by first matching onto LEFT scalar operators, C eq SRR and C eq SLR , which then run into tensor and dipole operators, as discussed in the next section.These contributions are shown in Table 22 and are negligible. Using the couplings andmasses of SM particles in Eqs. (A.35)–(A.38), we obtain (cid:2) Γ eγ (cid:3) − loop eτ/τe = − . × − (cid:0) Y (cid:48) e (cid:1) eτ/τe , (A.46) (cid:2) Γ eγ (cid:3) − loop eτ/τe = − . × − (cid:0) Y (cid:48) e (cid:1) eτ/τe . (A.47)One- and two-loop contributions are thus of similar size. A.3 Running below the electroweak scale
The RGEs below the electroweak scale are listed in Ref. [76]. For the QCD and QEDcouplings, the one-loop running is given by µ ddµ g s ( µ ) = − π ) (cid:20) N c −
23 ( n u + n d ) (cid:21) ( g s ( µ )) , (A.48) µ ddµ e ( µ ) = 43 1(4 π ) (cid:0) n e Q e + n d N c Q d + n u N c Q u (cid:1) ( e ( µ )) , (A.49)where n u , n d and n e are the number of active up-, down-type quarks and charged leptons.For example, up to m b scale, n u = 2 , n d = 3 and n e = 3.– 82 –he anomalous dimension of the dipole, scalar and tensor operators are µ ddµ (cid:0) Γ eγ (cid:1) τe = + 1(4 π ) (cid:20) Q e e (cid:0) Γ eγ (cid:1) τe − N c (cid:88) q Q q ( Y q ) ww (cid:0) C eq TRR (cid:1) τeww (cid:21) , (A.50) µ ddµ C eq SRR = − π ) (cid:20) (cid:0) e (cid:0) Q e + Q q (cid:1) + 6 g s C F (cid:1) C eq SRR + 96 e Q e Q q C eq TRR (cid:21) , (A.51) µ ddµ C eq TRR = + 1(4 π ) (cid:20) (cid:0) e (cid:0) Q e + Q q (cid:1) + 2 g s C F (cid:1) C eq TRR − e Q e Q q C eq SRR (cid:21) , (A.52) µ ddµ C eq SRL = − π ) (cid:20) e (cid:0) Q e + Q q (cid:1) + g s C F (cid:21) C eq SRL , (A.53)where, in the last three lines, we have omitted the quark and lepton flavor indices τ est or eτ st . q here denotes both u and d -type quark, q ∈ { u, d } , while w ∈ { u, c } for u -typeoperators and w ∈ { d, s, b } for d -type operators. A summation over repeated flavor indicesis understood. We integrate out the bottom quark at the scale µ = m b , while the charmquark at the scale µ = 2 GeV. The solutions of the RGEs for Γ eZ and ( Y (cid:48) e ) τe are given inTable 22.Purely leptonic operators at low-energy are induced by photon penguin diagrams. TheRGEs for the left-handed leptonic operators are µ ddµ ( C ee VLL ) τeµµ = e π ) Q e (cid:20) N c (cid:88) q Q q (cid:0) C eq VLL + C eq VLR (cid:1) + Q e (4 C ee VLL + C ee VLR ) (cid:21) τeww + 12(4 π ) e Q e ( C ee VLL ) τeµµ , (A.54) µ ddµ ( C ee VLL ) τµµe = e π ) Q e (cid:20) N c (cid:88) q Q q (cid:0) C eq VLL + C eq VLR (cid:1) + Q e (4 C ee VLL + C ee VLR ) (cid:21) τeww + 12(4 π ) e Q e ( C ee VLL ) τµµe , (A.55) µ ddµ ( C ee VLL ) τeee = 23(4 π ) e Q e (cid:20) N c (cid:88) q Q q (cid:0) C eq VLL + C eq VLR (cid:1) + Q e (4 C ee VLL + C ee VLR ) (cid:21) τeww + 12(4 π ) e Q e ( C ee VLL ) τeee , (A.56)while those for right-handed operators are µ ddµ ( C ee VRR ) τeµµ = 12(4 π ) e Q e ( C ee VRR ) τeµµ + e Q e π ) (cid:34) N c (cid:88) q Q q C qe VLR + Q e C ee VLR (cid:35) wwτe + (cid:34) N c (cid:88) q Q q C eq VRR + 4 Q e C ee VRR (cid:35) τeww (A.57) µ ddµ ( C ee VRR ) τµµe = 12(4 π ) e Q e ( C ee VRR ) τµµe + e Q e π ) (cid:34) N c (cid:88) q Q q C qe VLR + Q e C ee VLR (cid:35) wwτe – 83 – C LQ,U ) tt ( C Lu ) tt ( C LQ,D ) bb ( C Ld ) bb ( C eu ) tt ( C ed ) bb ( C Qe ) bb C eu VLL −
39 35 3 . . C eu VRR −
19 3 . C ed VLL − − . − . C ed VRR . − . − C eu VLR −
19 3 . . C ue VLR
35 3 . − C ed VLR − . . − . − . C de VLR − − . C LQ,U ) cc ( C Lu ) cc c (1) Lϕ + c (3) Lϕ ( C eu ) cc c eϕ C eu VLL − . − . C eu VRR − . − C ed VLL . . − C ed VRR . C eu VLR − . − . − C ue VLR − . C ed VLR . . C de VLR . − Table 23 . The Wilson coefficients (in units of 10 − ) at µ = 2 GeV induced by nonzero top-,bottom- and charm-quark vector-like operators at the scale Λ = 1 TeV, and by Z CLFV vector andaxial couplings. The u -type operators have flavor indices uu , while the d -type dd or ss . + (cid:34) N c (cid:88) q Q q C eq VRR + 4 Q e C ee VRR (cid:35) τeww (A.58) µ ddµ ( C ee VRR ) τeee = 12(4 π ) e Q e ( C ee VRR ) τµµe + 2 e Q e π ) (cid:34) N c (cid:88) q Q q C qe VLR + Q e C ee VLR (cid:35) wwτe + (cid:34) N c (cid:88) q Q q C eq VRR + 4 Q e C ee VRR (cid:35) τeww . (A.59)The RGEs for LR operators are given by µ ddµ ( C ee VLR ) τe(cid:96)(cid:96) = 43(4 π ) e Q e (cid:20) N c (cid:88) q Q q (cid:0) C eq VLL + C eq VLR (cid:1) + Q e (4 C ee VLL + C ee VLR ) (cid:21) τeww − π ) e Q e ( C ee VLR ) τell , (A.60) µ ddµ ( C ee VLR ) (cid:96)(cid:96)τe = − π ) e Q e ( C ee VLR ) llτe + 4 e Q e π ) (cid:34) N c (cid:88) q Q q C qe VLR + C ee VLR (cid:35) wwτe + (cid:34) N c (cid:88) q Q q C eq VRR + 4 C ee VRR (cid:35) τeww (A.61)– 84 – C LQ,U ) tt ( C Lu ) tt ( C LQ,D ) bb ( C Ld ) bb ( C eu ) tt ( C ed ) bb ( C Qe ) bb [ C ee VLL ] τe ee − − . − . C ee VRR ] τe ee − . − C ee VLL ] τe µµ − . − . C ee VRR ] τe µµ − . − C ee VLL ] τµ µe − − . − . C ee VRR ] τµ µe − . − C ee VLR ] τe (cid:96)(cid:96) − . − . − . C ee VLR ] (cid:96)(cid:96) τe − − . C LQ,U ) cc ( C Lu ) cc c (1) Lϕ + c (3) Lϕ ( C eu ) cc c eϕ [ C ee VLL ] τe ee . . −
132 [ C ee VRR ] τe ee . C ee VLL ] τe µµ − . . −
66 [ C ee VRR ] τe µµ . C ee VLL ] τµ µe . . −
66 [ C ee VRR ] τµ µe . C ee VLR ] τe (cid:96)(cid:96)
10 10 237 [ C ee VLR ] (cid:96)(cid:96) τe − Table 24 . The Wilson coefficients of purely leptonic operators (in units of 10 − ) at µ = 2 GeVinduced by nonzero top-, bottom- and charm-quark vector-like operators at the scale Λ = 1 TeV,and by Z CLFV vector and axial couplings. Note that [ C ee VLL ] eeτe ,[ C ee VLL ] µµτe and [ C ee VLL ] µeτµ arealso generated with the same contributions as the operators listed above. Finally, the anomalous dimensions of the semileptonic operators are µ ddµ (cid:0) C eq VLL (cid:1) τest = 43(4 π ) e Q q δ st (cid:20) N c (cid:88) q (cid:48) Q q (cid:48) (cid:16) C eq (cid:48) VLL + C eq (cid:48) VLR (cid:17) + Q e (4 C ee VLL + C ee VLR ) (cid:21) τeww + 12(4 π ) e Q e Q q (cid:0) C eq VLL (cid:1) τest , (A.62) µ ddµ (cid:0) C eq VRR (cid:1) τest = 43(4 π ) e Q q δ st (cid:20) N c (cid:88) q (cid:48) Q q (cid:48) C q (cid:48) e VLR + Q e C ee VLR (cid:21) wwτe (A.63)+ (cid:20) N c (cid:88) q (cid:48) Q q (cid:48) C eq (cid:48) VRR + 4 Q e C ee VRR (cid:21) τeww + 12(4 π ) e Q e Q u ( C eu VRR ) τest ,µ ddµ (cid:0) C eq VLR (cid:1) τest = 43(4 π ) e Q q δ st (cid:20) N c (cid:88) q (cid:48) Q q (cid:48) (cid:16) C eq (cid:48) VLL + C eq (cid:48) VLR (cid:17) + Q e (4 C ee VLL + C ee VLR ) (cid:21) τeww − π ) e Q e Q q (cid:0) C eq VLR (cid:1) τest , (A.64) µ ddµ (cid:0) C qe VLR (cid:1) stτe = 43(4 π ) e Q q δ st (cid:20) N c (cid:88) q (cid:48) Q q (cid:48) C q (cid:48) e VLR + Q e C ee VLR (cid:21) wwτe (A.65)+ (cid:20) N c (cid:88) q (cid:48) Q q (cid:48) C eq (cid:48) VRR + 4 Q e C ee VRR (cid:21) τeww − π ) e Q e Q q ( C ue VLR ) stτe , We give the solution of the RGEs in Eqs. (A.54)-(A.66) in Tables 23 and 24.– 85 –
Partonic cross sections for CLFV processes
B.1 DIS
Here we complete the collection of expressions for the partonic cross sections, as defined inEq. (4.47), induced by the CLFV SMEFT operators, some of which we gave in Sec. 4.2.The prefactors F Z , F dip , F S and F G are defined in Eqs. (4.48), (4.50), (4.54) and (4.57). Vertex corrections and vector-axial four-fermion operators
In Eq. (4.49) we gavethe u -type quark contributions to the partonic cross sections for the Z coupling and vector-axial four-fermion operators. Here we give corresponding results for ¯ u, d, ¯ d -type quark andantiquark contributions.For ¯ u antiquarks:ˆ σ ¯ u i LR = F Z (1 − y ) (cid:12)(cid:12)(cid:12)(cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe z u L + ˆ ρ Z (cid:2) C LQ, U (cid:3) τeu i u i (cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12) ˆ ρ Z (cid:2) C LQ, U (cid:3) τeu i u j (cid:12)(cid:12)(cid:12) ˆ σ ¯ u i RL = F Z (1 − y ) (cid:12)(cid:12) [ c eϕ ] τe z u R + ˆ ρ Z [ C eu ] τeu i u i (cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12) ˆ ρ Z [ C eu ] τeu i u j (cid:12)(cid:12)(cid:12) ˆ σ ¯ u i LL = F Z (cid:12)(cid:12)(cid:12)(cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe z u R + ˆ ρ Z [ C Lu ] τeu i u i (cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12) ˆ ρ Z [ C Lu ] τeu i u j (cid:12)(cid:12)(cid:12) ˆ σ ¯ u i RR = F Z (cid:12)(cid:12)(cid:12) [ c eϕ ] τe z u L + ˆ ρ Z [ C Qe ] τeu i u i (cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12) ˆ ρ Z [ C Qe ] τeu i u j (cid:12)(cid:12)(cid:12) , (B.1)where ˆ ρ Z = ( m Z + Q ) /m Z , u i = { u, c } , and [ C Qe ] u j u i includes factors of the CKM matrixas in Eq. (3.17).For d type quarks, the partonic cross sections areˆ σ d i LL = F Z (cid:12)(cid:12)(cid:12)(cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe z d L + ˆ ρ Z (cid:2) C LQ, D (cid:3) τed i d i (cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12) ˆ ρ Z (cid:2) C LQ, D (cid:3) τed j d i (cid:12)(cid:12)(cid:12) ˆ σ d i RR = F Z (cid:12)(cid:12) [ c eϕ ] τe z d R + ˆ ρ Z [ C ed ] τed i d i (cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12) ˆ ρ Z [ C ed ] τed j d i (cid:12)(cid:12)(cid:12) ˆ σ d i LR = F Z (1 − y ) (cid:12)(cid:12)(cid:12)(cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe z d R + ˆ ρ Z [ C Ld ] τed i d i (cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12) ˆ ρ Z [ C Ld ] τed j d i (cid:12)(cid:12)(cid:12) ˆ σ d i RL = F Z (1 − y ) (cid:12)(cid:12)(cid:12) [ c eϕ ] τe z d L + ˆ ρ Z [ C Qe ] τed i d i (cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12) ˆ ρ Z [ C Qe ] τed j d i (cid:12)(cid:12)(cid:12) , (B.2)while for d antiquarks they are:ˆ σ ¯ d i LR = F Z (1 − y ) (cid:12)(cid:12)(cid:12)(cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe z d L + ˆ ρ Z (cid:2) C LQ, D (cid:3) τed i d i (cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12) ˆ ρ Z (cid:2) C LQ, D (cid:3) τed i d j (cid:12)(cid:12)(cid:12) – 86 – σ ¯ d i RL = F Z (1 − y ) (cid:12)(cid:12) [ c eϕ ] τe z d R + ˆ ρ Z [ C ed ] τed i d i (cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12) ˆ ρ Z [ C ed ] τed i d j (cid:12)(cid:12)(cid:12) ˆ σ ¯ d i LL = F Z (cid:12)(cid:12)(cid:12)(cid:104) c (1) Lϕ + c (3) Lϕ (cid:105) τe z d R + ˆ ρ Z [ C Ld ] τed i d i (cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12) ˆ ρ Z [ C Ld ] τed i d j (cid:12)(cid:12)(cid:12) ˆ σ ¯ d i RR = F Z (cid:12)(cid:12)(cid:12) [ c eϕ ] τe z d L + ˆ ρ Z [ C Qe ] τed i d i (cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12) ˆ ρ Z [ C Qe ] τed i d j (cid:12)(cid:12)(cid:12) , (B.3)where d i = { d, s, b } . Dipole operators
For dipole operators, the u -type quark contributions were given inEq. (4.51). The ¯ u quark contribution is obtained by the replacement,ˆ σ ¯ uRR ↔ ˆ σ uRL , (B.4)while the down-type contribution is obtained by Eq. (4.52). The eτ component, meanwhile,where the electron is left-handed, is given by:ˆ σ uLL = F dip (1 − y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) Γ eγ (cid:3) eτ Q u + z u L c w s w Q ( Q + m Z ) [Γ eZ ] eτ (cid:12)(cid:12)(cid:12)(cid:12) , ˆ σ uLR = F dip (1 − y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) Γ eγ (cid:3) eτ Q u + z u R c w s w Q ( Q + m Z ) [Γ eZ ] eτ (cid:12)(cid:12)(cid:12)(cid:12) , ˆ σ uRR = ˆ σ uRL = 0 . (B.5)The antiquark and d type components are obtained as before. Higgs, scalar and tensor four-fermion operators
The u -type quark partonic crosssections induced by the τ e component of Yukawa, scalar and tensor operators were givenin Eq. (4.55), while the ¯ u -type antiquark contributions are given by:ˆ σ ¯ uLL = ˆ σ ¯ uLR = 0 , ˆ σ ¯ u i RL = F S y (cid:40) (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C (1) LeQu (cid:105) τeu i u i − (cid:18) − y (cid:19) (cid:104) C (3) LeQu (cid:105) τeu i u i + Y u i (cid:2) Y (cid:48) e (cid:3) τe v m H + Q (cid:12)(cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) C (1) LeQu (cid:105) τeu i u j + 4 (cid:18) − y (cid:19) (cid:104) C (3) LeQu (cid:105) τeu i u j (cid:12)(cid:12)(cid:12)(cid:12) (cid:41) , ˆ σ ¯ u i RR = F S y (cid:12)(cid:12)(cid:12)(cid:12) Y u i (cid:2) Y (cid:48) e (cid:3) τe v m H + Q (cid:12)(cid:12)(cid:12)(cid:12) . (B.6)For down-type quarks, there is no tensor operator and the incoming d is left-handedˆ σ dLL = ˆ σ dLR = ˆ σ ¯ dLL = ˆ σ ¯ dLR = 0 , ˆ σ d i RL = F S y (cid:40) (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) C LedQ (cid:3) τed i d i − Y d i (cid:2) Y (cid:48) e (cid:3) τe v m H + Q (cid:12)(cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12)(cid:2) C LedQ (cid:3) τed j d i (cid:12)(cid:12)(cid:12) (cid:41) – 87 – σ d i RR = F S y (cid:12)(cid:12)(cid:12)(cid:12) Y d i (cid:2) Y (cid:48) e (cid:3) τe v m H + Q (cid:12)(cid:12)(cid:12)(cid:12) ,, ˆ σ ¯ d i RR = F S y (cid:40) (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) C LedQ (cid:3) τed i d i − Y d i (cid:2) Y (cid:48) e (cid:3) τe v m H + Q (cid:12)(cid:12)(cid:12)(cid:12) + (cid:88) j (cid:54) = i (cid:12)(cid:12)(cid:12)(cid:2) C LedQ (cid:3) τed i d j (cid:12)(cid:12)(cid:12) (cid:41) ˆ σ ¯ d i RL = F S y (cid:12)(cid:12)(cid:12)(cid:12) Y d i (cid:2) Y (cid:48) e (cid:3) τe v m H + Q (cid:12)(cid:12)(cid:12)(cid:12) . (B.7)For eτ operators, the results are the same, but the electron is left-handed. B.2 The squared amplitude of gg → e ± τ ∓ at the LHC The top component of scalar operator (cid:104) C (1) LeQu (cid:105) eτtt contributes to the pp → eτ cross sectionat one loop, via the partonic process gg → e ± τ ∓ . Since in the analysis of Ref. [37] m eτ ranges from about m Z to m eτ (cid:29) m t , it is here necessary to use the full one-loop resultsrather than the heavy top quark mass expansion in Eqs. 3.22 and 3.23. With a slightabuse of notation, we denote the squared amplitude of gg → e ± τ ∓ , averaged over gluonpolarizations and colors, by |M| = α s s π v (cid:16) | C GG | τe + | C GG | eτ + (cid:12)(cid:12) C G (cid:101) G (cid:12)(cid:12) τe + (cid:12)(cid:12) C G (cid:101) G (cid:12)(cid:12) eτ (cid:17) , (B.8)where s = m eτ and the functions [ C GG ] τe/eτ and [ C G (cid:101) G ] τe/eτ are defined, for s > m t , as[ C GG ] τe/eτ = − m t v s (cid:34) (4 m t − s ) ln (cid:112) s ( s − m t ) + 2 m t − s m t + 4 s (cid:35) (cid:104) C (1) LeQu (cid:105) τett/eτtt , (B.9)[ C G (cid:101) G ] τe/eτ = im t v s ln (cid:112) s ( s − m t ) + 2 m t − s m t (cid:104) C (1) LeQu (cid:105) τett/eτtt . (B.10)The dependence of C GG and C G (cid:101) G on s/ (2 m t ) is the same as for gluon fusion into a scalaror pseudoscalar Higgs, see for example Ref. [107]. C Conversion to a non-chiral basis of low-energy operators
Here we make contact with the basis used in Ref. [29], which employs non-chiral quarkbillinears with good parity quantum number, more convenient for the analysis of hadronic τ decays. For dipole operators one has C DR = Λ vm τ e [Γ e ] ∗ τe , C DL = Λ vm τ e [Γ e ] eτ . (C.1)The vector/axial couplings to the u quark are given by C u VL = Λ v ( C eu VLR + C eu VLL ) , C u AL = Λ v ( C eu VLR − C eu VLL ) , (C.2)– 88 – u VR = Λ v ( C eu VRR + C ue VLR ) , C u AR = Λ v ( C eu VRR − C ue VLR ) . (C.3)The matching for down-type operators is simply obtained by replacing u → d .For scalar and tensor operators, the conversion is C u SR = √ m τ m u [ C eu SRL + C eu SRR ] ∗ τe , C u PR = √ m τ m u [ C eu SRL − C eu SRR ] ∗ τe , (C.4) C u SL = √ m τ m u [ C eu SRR + C eu SRL ] eτ , C u PL = √ m τ m u [ C eu SRR − C eu SRL ] eτ , (C.5)and u → d yields the results for the d quark. At tree-level, the tensor operator is C u TL = 2 √ m u m τ ( C eu TRR ) eτ , C u TR = 2 √ m u m τ ( C eu TRR ) ∗ τe . (C.6) D Compendium of Decay rates
D.1 τ decay rates Below we report the expressions for LFV τ -decay rates. Most of these results are taken fromthe existing literature. We devote separate subsections to original results on τ → eKπ , thetensor operator contribution to τ → eγ , and τ → eK + K − . • τ → eγ [29] Γ ( τ → eγ ) = m τ α em v (cid:20) (cid:12)(cid:12)(cid:12)(cid:0) Γ eγ (cid:1) eτ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:0) Γ eγ (cid:1) τe (cid:12)(cid:12)(cid:12) (cid:21) . (D.1)Besides, contributions from nonzero tensor semileptonic operators are given by(Γ eγ ) eτ → (Γ eγ ) eτ − (cid:18) c + c √ (cid:19) i Π V T (0) v (D.2)(Γ eγ ) ∗ τe → (Γ eγ ) ∗ τe − (cid:18) ˜ c + ˜ c √ (cid:19) i Π V T (0) v . (D.3)The expressions of c , and ˜ c , in terms of the tensor semileptonic couplings and thenon-perturbative parameter Π V T (0) are given in Section D.1. • τ → e [109]Γ ( τ → e ) = α m τ πv (cid:20) (cid:12)(cid:12)(cid:12)(cid:0) Γ γ (cid:1) eτ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:0) Γ γ (cid:1) τe (cid:12)(cid:12)(cid:12) (cid:21) (cid:26) (cid:18) m τ m e (cid:19) − (cid:27) + X γ + m τ G F π (cid:20) | ( C ee SRR ) eτee | + 16 | ( C ee VLL ) τeee | + 8 | ( C ee VLR ) τeee | + | ( C ee SRR ) τeee | + 16 | ( C ee VRR ) τeee | + 8 | ( C ee VLR ) eeτe | (cid:21) (D.4)– 89 – τ . × − s [52] α em / .
036 [52] G F . × − GeV − [52] m τ .
78 GeV [52] m π .
98 MeV [52] m π ± .
57 MeV [52] m η .
862 MeV [52] m η (cid:48) .
78 MeV [52] f π . B π,uT (0) 0 .
195 [80] f qη .
11 GeV [29] f sη − .
11 GeV [29] h qη .
001 GeV [29] h sη − .
055 GeV [29] f qη (cid:48) .
087 GeV [29] f sη (cid:48) .
135 GeV [29] h qη (cid:48) .
001 GeV [29] h sη (cid:48) .
068 GeV [29] a η .
022 GeV [29] a η (cid:48) .
056 GeV [29] f K . m K .
611 MeV [52]
Table 25 . Input parameters for τ decays. where X γ is the interference term with the dipole operator X γ = − √ α em π ) m τ G F vm τ Re (cid:20) (cid:0) Γ eγ (cid:1) ∗ eτ { ( C ee VLR ) ∗ eeτe + 2 ( C ee VRR ) ∗ τeee } + (cid:0) Γ eγ (cid:1) τe { ( C ee VLR ) ∗ τeee + 2 ( C ee VLL ) ∗ τeee } . (cid:21) . (D.5)Notice that in Eq. (D.4) we use a single symbol to denote the contributions of boththe τ eee and eeτ e components of LEFT operators, for example( C ee VLL ) τeee ≡ ( C ee VLL ) τeee + ( C ee VLL ) eeτe . • τ → eµ + µ − [110]Γ (cid:0) τ → eµ + µ − (cid:1) = (cid:90) X max X min dX (cid:90) Y max Y min dY × (cid:20) G F π m τ (cid:26)(cid:12)(cid:12)(cid:12) ( C ee VRR ) τeµµ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( C ee VLL ) τeµµ (cid:12)(cid:12)(cid:12) (cid:27) (cid:8) m τ − (2 X − m τ − m µ ) (cid:9) + G F π m τ (cid:26)(cid:12)(cid:12)(cid:12) ( C ee VLR ) τeµµ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( C ee VLR ) µµτe (cid:12)(cid:12)(cid:12) (cid:27) (cid:8) m τ − (2 Z − m τ − m µ ) (cid:9) + α πm τ v (cid:26)(cid:12)(cid:12)(cid:12)(cid:0) Γ eγ (cid:1) τe (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:0) Γ eγ (cid:1) eτ (cid:12)(cid:12)(cid:12) (cid:27) × (cid:26) m µ Y (cid:0) Y − m τ (cid:1) + 12 Y (cid:0) X + m − m µ (cid:1) + 12 (cid:0) m τ − Y (cid:1) (cid:27) + X µµγ (cid:21) , (D.6)– 90 –here the interference term is expressed by X µµγ = − α em π v (cid:20) Re (cid:110) ( C ee VLL ) τeµµ (cid:0) Γ eγ (cid:1) ∗ τe + ( C ee VRR ) τeµµ (cid:0) Γ eγ (cid:1) eτ (cid:111) (cid:32) m τ (cid:0) X − m µ (cid:1) + m µ Y (cid:33) + Re (cid:110) ( C ee VLR ) τeµµ (cid:0) Γ eγ (cid:1) ∗ τe + ( C ee VLR ) µµτe (cid:0) Γ eγ (cid:1) eτ (cid:111) (cid:32) m τ (cid:0) Z − m µ (cid:1) + m µ Y (cid:33) (cid:21) . (D.7)As for τ → eee , in Eq. (D.6) we use a single symbol to denote the sum of equivalentcontributions, for example( C ee VLL ) τeµµ ≡ ( C ee VLL ) τeµµ + ( C ee VLL ) µµτe + ( C ee VLL ) τµµe + ( C ee VLL ) µeτµ , with the coefficients on the right hand side given in Table 24. The parameters, X, Y and Z , denote invariant masses m ij as X = m = (cid:0) p e + p µ − (cid:1) , (D.8) Y = m = (cid:0) p µ − + p µ + (cid:1) , (D.9) Z = m = m τ + 2 m µ − X − Y, (D.10)which are kinematically limited by( m e + m µ ) ≤ X ≤ ( m τ − m µ ) , (D.11) Y min , max = (cid:0) E µ − + E µ + (cid:1) − (cid:20) (cid:16) E µ − − m µ (cid:17) ± (cid:16) E µ + − m µ (cid:17) (cid:21) , (D.12)with E µ − = X − m e + m µ m , E µ + = m τ − m µ − X m . (D.13) • τ + → e + M, ( M = π , K S )Γ (cid:0) τ + → e + M (cid:1) = m τ π (cid:18) − m M m τ (cid:19) G F f M (cid:20) (cid:12)(cid:12) A ML (cid:12)(cid:12) + (cid:12)(cid:12) A MR (cid:12)(cid:12) (cid:21) , (D.14)where f M is the decay constant. A ML,R is expressed by A πL = (cid:16) C eu VLR − C eu VLL (cid:17) τeuu − (cid:16) C ed VLR − C ed VLL (cid:17) τedd + m π m τ ( m u + m d ) (cid:104)(cid:16) C eu ∗ SRR − C eu ∗ SRL (cid:17) eτuu − (cid:16) C ed ∗ SRR − C ed ∗ SRL (cid:17) eτdd (cid:105) , (D.15) A πR = (cid:16) C ue VLR (cid:17) uuτe − (cid:16) C eu VRR (cid:17) τeuu − (cid:104)(cid:16) C de VLR (cid:17) ddτe − (cid:16) C ed VRR (cid:17) τedd (cid:105) + m π m τ ( m u + m d ) (cid:104)(cid:16) C eu SRR − C eu SRL (cid:17) τeuu − (cid:16) C ed SRR − C ed SRL (cid:17) τedd (cid:105) , (D.16)– 91 –or τ + → e + π , A KL = (cid:16) C ed VLR − C ed VLL (cid:17) τeds + m K m τ ( m d + m s ) (cid:16) C ed ∗ SRR − C ed ∗ SRL (cid:17) eτsd − ( d ↔ s ) , (D.17) A LR = (cid:16) C de VLR (cid:17) dsτe − (cid:16) C ed VRR (cid:17) τeds + m K m τ ( m d + m s ) (cid:16) C ed SRR − C ed SRL (cid:17) τeds − ( d ↔ s ) , (D.18)for τ + → e + K S . • τ → eη ( (cid:48) ) [29]Γ ( τ → eη )= m τ π (cid:32) − m η m τ (cid:33) G F (cid:20) (cid:32) √ a η m τ v (cid:33) (cid:16) (cid:12)(cid:12) C G ˜ G (cid:12)(cid:12) τe + (cid:12)(cid:12) C G ˜ G (cid:12)(cid:12) eτ (cid:17) + (cid:12)(cid:12) A ηL (cid:12)(cid:12) + (cid:12)(cid:12) A ηR (cid:12)(cid:12) (cid:21) , (D.19)with A ηL = (cid:88) q = u,d (cid:104) f qη (cid:16) C eq VLR − C eq VLL (cid:17) τeqq + h qη m τ ( m u + m d ) (cid:16) C eq ∗ SRR − C eq ∗ SRL (cid:17) eτqq (cid:105) + √ (cid:104) f sη (cid:16) C ed VLR − C ed VLL (cid:17) τess + h sη m τ m s (cid:16) C ed ∗ SRR − C ed ∗ SRL (cid:17) eτss (cid:105) , (D.20) A ηR = (cid:88) q = u,d (cid:104) f qη (cid:110)(cid:16) C qe VLR (cid:17) qqτe − (cid:16) C eq VRR (cid:17) τeqq (cid:111) + h qη m τ ( m u + m d ) (cid:16) C eq SRR − C eq SRL (cid:17) τeqq (cid:105) + √ (cid:104) f sη (cid:110)(cid:16) C de VLR (cid:17) ssτe − (cid:16) C ed VRR (cid:17) τess (cid:111) + h sη m τ m s (cid:16) C ed SRR − C ed SRL (cid:17) τess (cid:105) . (D.21)Here, f q,sη , h q,sη and a η denote decay constants. The BR for τ → eη (cid:48) can be expressedby the replacement of η → η (cid:48) . • τ → eπ + π − [29] d Γ ds = (cid:0) s − m π (cid:1) / (cid:0) m τ − s (cid:1) π m τ s / × (cid:20) m τ s G F (cid:16)(cid:12)(cid:12) Q (cid:48) L (cid:12)(cid:12) + (cid:12)(cid:12) Q (cid:48) R (cid:12)(cid:12) (cid:17) − (cid:0) m π − s (cid:1) | F V ( s ) | × (cid:26) m τ + s m τ (cid:16) | A L | + | A R | (cid:17) + √ sG F ( B L + B R ) (cid:27)(cid:21) . (D.22) Q (cid:48) L,R , A L,R and B L,R are combinations of Wilson coefficients and form factors, andare given in Eqs. (7.19a)–(7.21b).All the related input parameters are listed in Table 25.– 92 – .1.1 τ → eπK modes We provide below a detailed expression for the decay rate in the channel τ − → e − π + K − , mediated by operators with structure ¯ e Γ τ ¯ s Γ d . The decay τ − → e − π − K + has a completely analogous expression, in terms of the Wilson Coefficients of the operators¯ e Γ τ ¯ d Γ s . Similar considerations apply to the decay of τ + . Finally, note that while thePDG does not provide a bound on the mode τ − → e − π ¯ K , its theoretical prediction isrelated τ → eπ + K − by isospin symmetry,Γ( τ → eπ ¯ K ) = 12 Γ( τ → eπ + K − ) . (D.23)To obtain an expression for Γ( τ → eπ + K − ) we note that isospin symmetry gives (cid:104) π − ¯ K | ¯ s Γ u | (cid:105) = (cid:104) π + K − | ¯ s Γ d | (cid:105) , (D.24)which in turn implies that for our LFV decay we can use the form factors f Kπ + , ( s ) and B KπT ( s ) ( s = ( p K + p π ) ) appearing in the decay τ − → ν τ π − ¯ K : (cid:104) ¯ K ( p K ) π − ( p π ) | ¯ sγ µ u | (cid:105) = ( p K − p π ) µ f Kπ + ( s ) + ( p K + p π ) µ f Kπ − ( s ) , (cid:104) ¯ K ( p K ) π − ( p π ) | ¯ su | (cid:105) = M K − M π m s − m u f Kπ ( s ) , (cid:104) ¯ K ( p K ) π − ( p π ) | ¯ sσ µν u | (cid:105) = i p µK p νπ − p νK p µπ M K B KπT ( s ) , (D.25)where f Kπ − ( s ) = M K − M π s (cid:0) f Kπ ( s ) − f Kπ + ( s ) (cid:1) . (D.26)Moreover, in the limit m e → τ → eπ + K − ) can be read off theexpressions for Γ( τ → ν τ π − ¯ K ) given in Ref. [111]. In terms of the effective couplings c V = (cid:104) C ed VLL (cid:105) eτsd + (cid:104) C ed VLR (cid:105) eτsd + (cid:104) C de VLR (cid:105) sdτe + (cid:104) C ed VRR (cid:105) eτsd , (D.27) c A = − (cid:104) C ed VLL (cid:105) eτsd + (cid:104) C de VLR (cid:105) sdτe + (cid:104) C ed VRR (cid:105) eτsd − (cid:104) C ed VLR (cid:105) eτsd , (D.28) c S = (cid:104) C ed SRR (cid:105) eτsd + (cid:104) C ed SRL (cid:105) eτsd + (cid:104) C ed ∗ SRR (cid:105) τeds + (cid:104) C ed ∗ SRL (cid:105) τeds , (D.29) c P = − i (cid:110)(cid:104) C ed SRR (cid:105) eτsd + (cid:104) C ed SRL (cid:105) eτsd − (cid:104) C ed ∗ SRR (cid:105) τeds − (cid:104) C ed ∗ SRL (cid:105) τeds (cid:111) , (D.30) c T R = 2 (cid:104) C ed TRR (cid:105) eτsd , (D.31) c T L = 2 (cid:104) C ed ∗ TRR (cid:105) τeds , (D.32)we find d Γ ds ( τ → eπ + K − ) = G F λ / πK ( s )( m τ − s ) ( M K − M π ) π m τ s × (cid:20) ξ ( s ) (cid:26) | V ( s ) | + | A ( s ) | + 2( m τ − s ) sm τ (cid:16) | T + ( s ) | + | T − ( s ) | (cid:17)(cid:27) + | S ( s ) | + | P ( s ) | (cid:21) , (D.33)– 93 –here λ πK ( s ) = λ ( s, M π , M K ), λ ( a, b, c ) = a + b + c − ab + ac + bc ), ξ ( s ) = ( m τ + 2 s ) λ πK ( s )3 m τ ( M K − M π ) , (D.34)and V ( s ) = f Kπ + ( s ) c V − T + ( s ) , A ( s ) = f Kπ + ( s ) c A + T − ( s ) ,S ( s ) = f Kπ ( s ) (cid:18) c V + sm τ ( m s − m u ) c S (cid:19) ,P ( s ) = f Kπ ( s ) (cid:18) c A − sm τ ( m s − m u ) ic P (cid:19) ,T ± ( s ) = 3 sm τ + 2 s m τ m K ( c T R ± c T L ) B KπT ( s ) . (D.35)Finally, for the vector and scalar form factors f Kπ + , ( s ) /f Kπ + , (0) we use the numericalresults from Ref. [112] and for the normalization we use the lattice QCD input f Kπ + (0) = f Kπ (0) = 0 . B KπT ( s ) we use the elastic unitarityrelation (accurate in the dominant region of phase space) [80, 111] B KπT ( s ) = B KπT (0) f Kπ + (0) × f Kπ + ( s ) (D.36)with the lattice QCD input B KπT (0) = 0 . D.1.2 Tensor operator contribution to τ → eγ In order to derive the tensor operator contribution to τ → eγ , we write the relevantpart of the low-scale effective Lagrangian as follows L ⊃ − G F √ e L σ µν τ R ¯ qσ µν (cid:2) c I + c T + c T (cid:3) q + { L ↔ R, c , , → ˜ c , , } + eA µ J EMµ . (D.37)Here q = ( u, d, s ) T , the electromagnetic current is given by J EMµ = ¯ q γ µ (cid:20) √ T + T (cid:21) q , (D.38)and the matrices T a are SU(3) flavor generators. The tensor couplings are given by c = 13 (cid:16)(cid:104) C eu TRR (cid:105) eτuu + (cid:104) C ed TRR (cid:105) eτdd + (cid:104) C ed TRR (cid:105) eτss (cid:17) (D.39) c = (cid:104) C eu TRR (cid:105) eτuu − (cid:104) C ed TRR (cid:105) eτdd (D.40) c = 1 √ (cid:16)(cid:104) C eu TRR (cid:105) eτuu + (cid:104) C ed TRR (cid:105) eτdd − (cid:104) C ed TRR (cid:105) eτss (cid:17) (D.41)and ˜ c = 13 (cid:16)(cid:104) C eu TRR (cid:105) ∗ τeuu + (cid:104) C ed TRR (cid:105) ∗ τedd + (cid:104) C ed TRR (cid:105) ∗ τess (cid:17) (D.42)– 94 – c = (cid:104) C eu TRR (cid:105) ∗ τeuu − (cid:104) C ed TRR (cid:105) ∗ τedd (D.43)˜ c = 1 √ (cid:16)(cid:104) C eu TRR (cid:105) ∗ τeuu + (cid:104) C ed TRR (cid:105) ∗ τedd − (cid:104) C ed TRR (cid:105) ∗ τess (cid:17) . (D.44)The S-matrix element for the process τ ( p τ ) → e ( p e ) γ ( q ) is obtained in second-orderperturbation theory, by simultaneously inserting the tensor and electromagnetic interactionfrom (D.37) S = − i eG F √ u L ( p e ) σ µν u R ( p τ ) (cid:90) d y e iy · ( q + p e − p τ ) × (cid:90) d x e iq · x (cid:104) T (cid:16) J EMσ ( x ) ¯ q (0) σ µν (cid:2) c I + c T + c T (cid:3) q (0) (cid:17) (cid:105) . (D.45)The non-perturbative hadronic contribution to the amplitude is contained in the correlationfunction of the vector and tensor densities V aµ ( x ) = ¯ q ( x ) γ µ T a q ( x ) T aµν ( x ) = ¯ q ( x ) σ µν T a q ( x ) . (D.46)Using the decomposition (cid:90) d x e iq · x (cid:104) T (cid:16) V aσ ( x ) T bµν (0) (cid:17) (cid:105) = − iδ ab Π V T ( q ) ( g νσ q µ − g µσ q ν ) , (D.47)and the definition of the amplitude S ≡ i (2 π ) δ ( p τ − p e − q ) A , one arrives at A = (cid:104) ie ¯ u L ( p e ) σ µν u R ( p τ ) ( q µ (cid:15) ∗ ν ( q ) − q ν (cid:15) ∗ µ ( q )) (cid:105) × √ G F (cid:18) c + c √ (cid:19) i Π V T (0) . (D.48)The term in the square brackets coincides with the matrix element of the dipole operator,namely (cid:104) eγ | ¯ e L σ µν τ R ( eF µν ) | τ (cid:105) . Following Refs. [114–116] we estimate the non-perturbativeparameter Π V T (0) by using a large- N C inspired lowest resonance saturation for the V T correlation function, Π
V T ( q ) = i (cid:104) ¯ qq (cid:105) M V − q = − i B F π M V − q , (D.49)which is also consistent with the high- q behavior dictated by the OPE. In the MS schemeat µ = 2 GeV one has (cid:104) ¯ qq (cid:105) = − (286(23)MeV) or equivalently B (cid:39) . . The piondecay constant F π is 92 . ρ meson mass M V = 770 MeV.Based on the above results, the formulae for the τ → eγ decay rate are modified bythe substitutions: (Γ eγ ) eτ → (Γ eγ ) eτ − (cid:18) c + c √ (cid:19) i Π V T (0) v (D.50)(Γ eγ ) ∗ τe → (Γ eγ ) ∗ τe − (cid:18) ˜ c + ˜ c √ (cid:19) i Π V T (0) v . (D.51) In the case of matching to SMEFT only [ C eu TRR ] eτuu (cid:54) = 0 and one has c + c / √ /
3) [ C eu TRR ] eτuu . These numbers are from the FLAG 2019 review [108], using 2+1+1 dynamical quarks. – 95 –he interference between dipole and tensor couplings is controlled by the non-perturbativeparameter i Π V T (0) v = B v F π M V (D.52)which takes the numerical value ≈ . × − at µ = 2 GeV. Since the above estimateis based on large- N C considerations and a truncation of the spectrum to the lowest lyingresonance, we assign to it a 50% uncertainty. Lattice QCD calculations of Π V T ( q ) canreduce the uncertainty in the future. Finally, we note that our result is consistent with asimilar analysis of the tensor operator to µ → eγ [78]. D.1.3 τ → eK + K − We discuss here the contribution of vector operators to τ → eK + K − . Since the scalarand gluonic contributions are affected by large theoretical errors, we do not use this processin the analysis of Section 7. As discussed in Section 9.2, τ → eK + K − can play an importantrole in global analyses, since it receives contributions from isoscalar combinations of vectorcouplings, which are otherwise unconstrained at low energy. In the case of τ → eK + K − ,the differential decay width induced by vector operators is τ τ d Γ d ˆ s = 15¯Γ τ (cid:16) − ρ K ˆ s (cid:17) / (1 − ˆ s ) (2ˆ s + 1) (cid:16) | B L | + | B R | (cid:17) , (D.53)where s is the invariant mass of the charged kaons, and we define the dimensionless quanti-ties ˆ s = s/m τ and ρ K = 4 m K /m τ . The kinematically allowed region is ρ K ≤ ˆ s ≤ B L,R are combinations of Wilson coefficients and form factors B L = (cid:110) ( C eu VLL + C eu VLR ) τeuu (cid:16) F (8) V ( s ) + F (3) V ( s ) + F (0) V ( s ) (cid:17) + (cid:16) C ed VLL + C ed VLR (cid:17) τedd (cid:16) F (8) V ( s ) − F (3) V ( s ) + F (0) V ( s ) (cid:17) + (cid:16) C ed VLL + C ed VLR (cid:17) τess (cid:16) − F (8) V ( s ) + F (0) V ( s ) (cid:17)(cid:111) , (D.54) B R = (cid:110) ( C eu VRR + C ue VLR ) τeuu (cid:16) F (8) V ( s ) + F (3) V ( s ) + F (0) V ( s ) (cid:17) + (cid:16) C ed VRR + C de VRL (cid:17) τedd (cid:16) F (8) V ( s ) − F (3) V ( s ) + F (0) V ( s ) (cid:17) + (cid:16) C ed VRR + C de VRL (cid:17) τess (cid:16) − F (8) V ( s ) + F (0) V ( s ) (cid:17)(cid:111) , (D.55)with the form factors defined as12 (cid:104) K + ( p ) K − ( p ) | (cid:0) ¯ uγ µ u − ¯ dγ µ d (cid:1) | (cid:105) = ( p − p ) µ F (3) V ( s ) (D.56)16 (cid:104) K + ( p ) K − ( p ) | (cid:0) ¯ uγ µ u + ¯ dγ µ d − sγ µ s (cid:1) | (cid:105) = ( p − p ) µ F (8) V ( s ) (D.57)13 (cid:104) K + ( p ) K − ( p ) | (cid:0) ¯ uγ µ u + ¯ dγ µ d + ¯ sγ µ s (cid:1) | (cid:105) = ( p − p ) µ F (0) V ( s ) (D.58)The isoscalar and isovector form factors F (8) V and F (3) V have been extracted in Ref. [81] fromdata on e + e − → K + K − , e + e − → K L K S and τ → K + K ν τ . Ref. [81] used a parametriza-tion in terms of resonances, with the ρ resonance and its excitations contributing to F (3) V – 96 –nd the ω and φ resonances to the F (8) V , F (3) V ( s ) = 12 (cid:88) V = ρ,ρ (cid:48) ,... c V BW V ( s ) (D.59) F (8) V ( s ) = 16 (cid:88) V = ω,ω (cid:48) ,... c V BW V ( s ) + 13 (cid:88) V = φ,φ (cid:48) ,... c V BW V ( s ) , (D.60)with BW V ( s ) = M V M V − s − iM V Γ V ( s ) . (D.61)We thank K. Beloborodov for providing the energy-dependent widths Γ V ( s ). The fit coef-ficients c V are given in Ref. [81], in two scenarios, Model I and II, with the latter achievinga better description of the data. The case c ω = c φ = 1, with coefficients of the ω and φ excitations set to zero, corresponds to the case of single-resonance dominance and “idealmixing”, with φ coupling only to ¯ sγ µ s and ω to ¯ uγ µ u + ¯ dγ µ d . The actual fits coeffi-cients c ω = 1 . ± .
14 and c φ = 1 . ± .
001 do not deviate from this expectation verysignificantly.The isosinglet component F (0) V cannot be directly extracted from data. We will hereassume that the tower of φ resonances couple only to the s quarks, while the ω , ω (cid:48) , . . . tolight u and d . This assumption is well justified for the ω (782) and φ (1020), which are veryclose to ideal mixing, and lattice QCD calculations of the meson spectrum find very smallmixing between the ¯ ss and ¯ uu + ¯ dd components also for other vector isoscalar excitations[117, 118]. We thus write F (0) V ( s ) = 13 (cid:88) V = ω,ω (cid:48) ,... c V BW V ( s ) − (cid:88) V = φ,φ (cid:48) ,... c V BW V ( s ) . (D.62)The coefficients from Ref. [81] are compatible with F (0) V (0) = 0, as expected at NLO in χ PT. An alternative model for F (0) V is provided in Ref. [31], and corresponds to consideringonly the contribution of the lowest resonances.Using Eqs. (D.59), (D.59) and (D.62), the fit coefficients from Model II in Ref. [81],and assuming all coefficients to be real, we obtainBR( τ → eK + K − ) = 0 . (cid:12)(cid:12)(cid:12)(cid:16) C ed VLL + C ed VLR (cid:17) τess (cid:12)(cid:12)(cid:12) + (1 . ± . · − (cid:12)(cid:12)(cid:12)(cid:16) C eu VLL + C eu VLR (cid:17) τeuu (cid:12)(cid:12)(cid:12) +(0 . ± . · − (cid:12)(cid:12)(cid:12)(cid:16) C ed VLL + C ed VLR (cid:17) τedd (cid:12)(cid:12)(cid:12) − (4 . ± . · − (cid:16) C ed VLL + C ed VLR (cid:17) τess (cid:16) C eu VLL + C eu VLR (cid:17) τeuu − (4 . ± . · − (cid:16) C ed VLL + C ed VLR (cid:17) τess (cid:16) C ed VLL + C ed VLR (cid:17) τedd +(3 . ± . · − (cid:16) C ed VLL + C ed VLR (cid:17) τedd (cid:16) C eu VLL + C eu VLR (cid:17) τeuu , (D.63)where the error is obtained by propagating the errors in the fit parameters in Ref. [81].We can assess at least part of the theoretical error by using the extraction of c V with– 97 – B . × − s [52] f B . m B .
65 MeV [52] τ B ± . × − s [52] m B ± .
34 MeV [52] m K ± .
677 MeV [52]
Table 26 . Input parameters for B meson decays. Model I in Ref.[81], and the one-resonance model for F (0) V discussed in Ref. [31]. While theprefactor of the (cid:12)(cid:12)(cid:12)(cid:16) C ed VLL + C ed VLR (cid:17) τess (cid:12)(cid:12)(cid:12) Wilson coefficient in Eq. (D.63) barely changes, theprefactors of (cid:12)(cid:12)(cid:12)(cid:16) C eu VLL + C eu VLR (cid:17) τeuu (cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:16) C eu VLL + C eu VLR (cid:17) τedd (cid:12)(cid:12)(cid:12) show a ∼
40% and > ss component of thevector current to the branching ratio is enhanced, resulting in very strong single-couplinglimits on (cid:12)(cid:12) [ C LQ,D , C Ld , C ed , C Qe ] τess (cid:12)(cid:12) < . · − . The limits on the uu component areweaker by approximately a factor of ten, and affected by larger theoretical uncertainties. D.2 B decays The input parameters relevant for B decays are listed in Table 26. D.2.1 B d → τ − e + The BR of B d → eτ is expressed byBR (cid:0) B d → τ − e + (cid:1) = τ B G F π f B m B λ (cid:18) , m τ m B , m e m B (cid:19) × (cid:20) (cid:0) m B − ( m τ + m e ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ( m τ − m e ) A + m B m b + m d C (cid:12)(cid:12)(cid:12)(cid:12) + (cid:0) m B − ( m τ − m e ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ( m τ + m e ) B + m B m b + m d D (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) , (D.64)with A = (cid:16) C ed VLR (cid:17) τebd − (cid:16) C ed VLL (cid:17) τebd + (cid:16) C ed VRR (cid:17) τebd − (cid:16) C de VLR (cid:17) bdτe , (D.65) B = − (cid:16) C ed VLR (cid:17) τebd + (cid:16) C ed VLL (cid:17) τebd + (cid:16) C ed VRR (cid:17) τebd − (cid:16) C de VLR (cid:17) bdτe , (D.66) C = (cid:16) C ed SRR (cid:17) τebd − (cid:16) C ed SRL (cid:17) τebd + (cid:16) C ed ∗ SRL (cid:17) eτdb − (cid:16) C ed ∗ SRR (cid:17) eτdb , (D.67) D = (cid:16) C ed SRR (cid:17) τebd − (cid:16) C ed SRL (cid:17) τebd − (cid:16) C ed ∗ SRL (cid:17) eτdb − (cid:16) C ed ∗ SRR (cid:17) eτdb . (D.68) D.2.2 B + → K + ( π + ) τ ± e ∓ For the estimation of B + → K + τ ± e ∓ , we follow the discussion in [119] where the effectiveLagrangian for b → sl − i l + j is defined as L eff = N F (cid:20) C V ¯ sγ µ P L b ¯ l i γ µ l j + C A ¯ sγ µ P L b ¯ l i γ µ γ l j – 98 – C (cid:48) V ¯ sγ µ P R b ¯ l i γ µ l j + C (cid:48) A ¯ sγ µ P R b ¯ l i γ µ γ l j + C S ¯ sP R b ¯ l i l j + C P ¯ sP R b ¯ l i γ l j + C (cid:48) S ¯ sP L b ¯ l i l j + C (cid:48) P ¯ sP L b ¯ l i γ l j (cid:21) , (D.69)with N F = G F α em V tb V ∗ ts / ( √ π ). The Wilson coefficients are converted into those in ourbasis N F C V = −√ G F (cid:20) (cid:16) C ed VLL (cid:17) τesb + (cid:16) C de VLR (cid:17) sbτe (cid:21) , (D.70) N F C (cid:48) V = −√ G F (cid:20) (cid:16) C ed VRR (cid:17) τesb + (cid:16) C ed VLR (cid:17) τesb (cid:21) , (D.71) N F C A = −√ G F (cid:20) − (cid:16) C ed VLL (cid:17) τesb + (cid:16) C de VLR (cid:17) sbτe (cid:21) , (D.72) N F C (cid:48) A = −√ G F (cid:20) (cid:16) C ed VRR (cid:17) τesb − (cid:16) C ed VLR (cid:17) τesb (cid:21) , (D.73) N F C S = −√ G F (cid:20) (cid:16) C ed SRR (cid:17) τesb + (cid:16) C ed ∗ SRL (cid:17) eτbs (cid:21) , (D.74) N F C (cid:48) S = −√ G F (cid:20) (cid:16) C ed ∗ SRR (cid:17) eτbs + (cid:16) C ed SRL (cid:17) τesb (cid:21) , (D.75) N F C P = −√ G F (cid:20) (cid:16) C ed SRR (cid:17) τesb − (cid:16) C ed ∗ SRL (cid:17) eτbs (cid:21) , (D.76) N F C (cid:48) P = −√ G F (cid:20) − (cid:16) C ed ∗ SRR (cid:17) eτbs + (cid:16) C ed SRL (cid:17) τesb (cid:21) . (D.77)The related form factors are analyzed in [108], in which the following N = 3 BCLparametrization is used: f + ,T ( q ) = 1 P + ,T ( q ) N − (cid:88) n =0 a + ,Tn (cid:104) z n − ( − n − N nN z N (cid:105) , (D.78) f ( q ) = 1 P ( q ) N − (cid:88) n =0 a n z n , (D.79)where P + , ,T ( q ) = 1 − q M , ,T , (D.80) z ( q , t ) = (cid:112) t + − q − √ t + − t (cid:112) t + − q + √ t + − t , (D.81) t + = ( m B ± + m P ) , (D.82) t = ( m B ± + m P ) ( √ m B ± − √ m P ) , (D.83)where m P = m K ± or m π ± . The pole mass is M + ,T = 5 . M = 5 .
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