The Full Lepton Flavor of the Littlest Higgs Model with T-parity

We re-examine lepton flavor violation (LFV) in the Littlest Higgs model with T--parity (LHT) including the full T--odd (non-singlet) lepton and Goldstone sectors. The heavy leptons induce two independent sources of LFV associated with the couplings necessary to give masses to the T--odd mirror fermions and to their partners in right-handed $SO(5)$ multiplets, respectively. The latter, which have been neglected in the past, can be decoupled from gauge mediated processes but not from Higgs mediated ones and must therefore also be included in a general analysis of LFV in the LHT. We also further extend previous analyses by considering on-shell $Z$ and Higgs LFV decays together with the LFV processes at low momentum transfer. We show that current experimental limits can probe the LHT parameter space up to global symmetry breaking scales $f \sim 10$ TeV. For lower $f$ values $\gtrsim 1$ TeV, $\mu-e$ transitions require the misalignment between the heavy and the Standard Model charged leptons to be $\lesssim 1 \, \%$. Future LFV experiments using intense muon beams should be sensitive to misalignments below the per mille level. For $\tau$ LFV transitions, which could potentially be observed at Belle II and the LHC as well as future lepton colliders, we find that generically they can not discriminate between the LHT and supersymmetric models though in some regions of parameter space this may be possible.


I. INTRODUCTION
The discovery of a Higgs boson at the Large Hadron Collider (LHC) [1,2] with Standard Model (SM) like properties [3,4] appears to have settled the nature of electroweak symmetry breaking and the focus now turns to uncovering the mechanism responsible for stabilizing the electroweak scale. Generically, models which solve the 'hierarchy problem' predict, in accordance with the naturalness principle, new particles at ∼ TeV. 1 As the precision and energy of LHC measurements increases without uncovering signs of physics beyond the SM [6][7][8][9], combined with constraints from electroweak precision data and other low energy experiments [10][11][12][13][14][15][16][17][18][19], the apparent fine tuning of the electroweak scale is brought into sharper focus. In the coming years this will motivate a rigorous search program at the LHC and low energy experiments in order to ensure no stone is left unturned in the search for a solution to the hierarchy problem and reconciliation of the naturalness principle.
The two most strongly motivated contenders for a natural theory of electroweak symmetry breaking are Supersymmetry and composite Higgs models where the Higgs boson arises as a pseudo Goldstone boson. Among the latter, Little Higgs Models [20][21][22] with T-parity [23,24] provide an elegant and calculable framework at the lowest orders and up to scales ∼ 10 TeV thanks to the collective symmetry breaking mechanism which protects the Higgs mass from quadratic divergences at one loop. The discrete T-parity also ensures that new particles cannot be singly produced or contribute at tree level to observables involving only SM particles, thus significantly relaxing direct and indirect constraints [25]. We focus on the particular case of the Littlest Higgs Model with T-Parity (LHT) [26] which has received considerable attention in many phenomenological studies [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45].
The stringent experimental limits on Flavor Changing Neutral Currents (FCNC) also pose an additional 'flavor hierarchy problem' to any SM extension at the TeV scale. In the SM it is the Glashow-Iliopoulos-Maiani (GIM) mechanism which implies loop-suppressed and very small FCNC [46]. Thus the GIM mechanism naturally explains the observed suppression of rare flavor violating processes. 2 By the same token, such stringent bounds strongly constrain any TeV extension of the SM. Indeed, in the absence of any complementary mechanism 1 There are exceptions such as the proposed 'relaxion' mechanism [5]. 2 The SM does not explain the large ratios between fermion masses of different families nor the observed charged current mixings, but FCNC are naturally suppressed.
aligning the new physics with the SM Yukawa couplings, present bounds on FCNC restrict the scale of flavor violation to be greater than ∼ 10 3−5 TeV for couplings O(1) depending on the quark flavor transition [47], or even higher for some lepton transitions [48] (see [49] for a recent review). Thus any extension of the SM must allow for small mixings and ideally provide an explanation for the alignment of the new flavor violating sources with the SM Yukawa couplings [50] as for example in Planck scale supergravity theories [51] where this alignment results from the universal origin of the soft breaking sfermion masses. In general this is not the case for composite Higgs models although they can accommodate the required (small) misalignment (see, however, [52][53][54][55][56]). Nonetheless, models with T-parity allow for a lower scale of new physics and a lower scale of flavor violation, mitigating the flavor hierarchy problem. 3 As the bounds are particularly precise in the leptonic case and the lepton mixing is unrelated to the mixing in the quark sector, one can study the implications of lepton flavor constraints independently, as we do in the following.
A number of phenomenological studies have examined the possibility of lepton flavor violation (LFV) in low-energy experiments in the LHT [38][39][40][41]44]. These have focused on LFV induced by the misalignment between the mass eigenstates of the light SM-like leptons and the heavy T-odd 'mirror' leptons. However, as we emphasized recently [57], there is an additional independent source of LFV in the LHT. This LFV is induced by the misalignment between the T-odd mirror leptons and the T-odd right-handed 'partner' leptons required to maintain the SO(5) global symmetry protecting the Higgs mass from dangerous quartic divergences at higher orders [24]. This new source of LFV has been overlooked in the past because the mass of the partner leptons is independent of the SU (5) breaking and they can be decoupled in LFV processes mediated by gauge bosons. However, they do not decouple in Higgs-mediated LFV processes [57] which require that all T-odd non-singlet leptons (and full Goldstone sector) be included to obtain a finite amplitude at one loop and avoid reintroducing a fine-tuning in the Higgs mass. The need to consider the full set of new T-odd non-singlet particles opens the possibility that the contributions from the partner leptons can significantly alter the allowed parameter space in the LHT as compared to previous studies in which they were neglected. This motivates a re-examination of LFV in the LHT in order to determine the full parameter space which is allowed by the stringent experimental constraints.
In this paper we revise previous studies in the LHT on lepton transitions with small momentum transfer. We extend previous analyses in two ways; first we include the full set of T-odd non-singlet leptons which introduces the new source of LFV; second we include highenergy LFV observables from LEP and the LHC as they can become competitive in certain regions of parameter space and in particular once the LHC enters a high luminosity phase. 4 We also discuss the implications of these new contributions to the (flavor conserving) muon magnetic moment, a µ . 5 (For a discussion on the interplay between the muon anomalous magnetic moment and LFV see [60].) Analogous considerations hold in the quark sector, but we leave a study of this to ongoing work.
The right-handed multiplet containing heavy leptons transforming under the SO(5) global symmetry also includes an SM singlet whose T-parity can be chosen to be odd, as originally assumed [23,24], or even [38]. Although the quantum effects of these singlets depend on this choice [61], they can be safely neglected in the processes considered here since, as emphasized in [57] and demonstrated below, the gauge and Higgs mediated amplitudes involving two SM fermions are one-loop finite in their absence. This allows for a consistent and quantitative phenomenological analysis, at the order to which we work, which leaves out the heavy SM singlet leptons and hence, it is independent of their T-parity assignment.
The phenomenological implications of the LHT are largely dictated by the global symmetry breaking scale f with the new physics effects becoming negligible in the large f limit. Thus, electroweak precision data and LHC limits primarily constrain this parameter in the LHT as well as the magnitude of the Yukawa couplings κ which generate masses for the T-odd mirror fermions (e.g. [43,45] and [62,63] for recent reviews). The global symmetry breaking scale is constrained to be f 1 TeV while for the mirror quark masses we have m q H = √ 2κ q f few TeV with less stringent limits on the mirror lepton masses m l H = √ 2κ l f . Our focus here is on rare LFV processes which provide complementary constraints and in particular imply quite stringent limits on the amount of 'misalignment' 4 High-Q 2 Drell-Yan LFV production at the LHC will be considered elsewhere. 5 The typical size of these contributions is more than two orders of magnitude smaller than the present experimental error. Moreover, the apparent discrepancy between the measured value and the SM prediction of a µ is coming under further scrutiny [58,59]. On the other hand, the electric dipole moment (EDM) e of the electron cancels at the order we consider.
between the heavy lepton flavor sector and the SM one. In our quantitative analysis we will take as benchmark values f = 1.5 TeV and m q Hi = 2 TeV [64], allowing to vary the heavy lepton masses and mixings. While the LHT does not incorporate any specific mechanism to explain the number of families or contain any flavor symmetries, these could in principle be incorporated in a UV complete model. 6 With this in mind, the LHT can easily accommodate the SM spectrum at low energies and the small mixings necessary to satisfy constraints over a range of parameters. Below we show that in order to satisfy present limits on rare LFV processes, at least some alignment is needed if f is below ∼ 10 TeV while strong alignment is needed for f 1 TeV. In particular, µ − e transitions require the alignment between the heavy LFV sources and the SM Yukawa couplings of the first two families to be 1 % while branching ratios for LFV Z and Higgs decays into τ µ, τ e can be as large as ∼ 10 −7 for ∼ TeV masses and sizable mixing values of the T-odd particles. These limits also serve as guidance for constructing UV models which contain the flavor symmetries needed to naturally explain the alignment.
Dedicated LFV experiments plan to exploit intense muon beams to increase their sensitivity by several orders of magnitude [67]. Should no signal be observed, this will improve the current limit on the misalignment between the heavy leptons and two lightest SM lepton families by an order of magnitude to ∼ 1%. Although LFV processes involving the third family do not significantly constrain the LHT at present, Belle II [68], the LHC [69], and/or future lepton colliders [70,71] may eventually be sensitive. However, as we discuss in more detail below, when the effects of both the mirror and partner leptons are included, three-body τ decays will not be able to unambiguously discriminate between the LHT and supersymmetric models, in contrast to the case in which the partner leptons are decoupled [39,72].
In the next section we summarize the most restrictive LFV processes and the LHT parameter space to be confronted with the corresponding experimental limits. In Section III various amplitudes for LFV processes are computed including two and three body τ and µ decays as well as µ → e transitions and Z →¯ decays. The relevant details of the LHT are reviewed in Ref. [57] where one can find the Feynman rules necessary for calculating 6 An exception is the top quark sector, which must be extended to implement the collective breaking protecting the Higgs mass from the one-loop quadratic divergence due to the top quark Yukawa. Large effects induced by the quark sector will be discussed elsewhere [65,66].
the amplitude for LFV Higgs decays while those needed for computing amplitudes of gauge mediated processes are completed here. The predictions of the LHT are then confronted with experimental data in Section IV. Prospects at future facilities using intense muon beams as well as Belle II, LHC and future lepton colliders are reviewed and examined in Section V.
Finally, Section VI is devoted to discussion and summary. A study of the muon magnetic moment is also included in the Appendix.

II. LEPTON FLAVOR VIOLATING LIMITS AND LHT PARAMETRIZATION
LFV processes provide a powerful probe of the flavor structure of new physics models. We gather in Table I the present limits on lepton flavor changing transitions mediated by electroweak gauge and Higgs bosons. As we discuss in detail below, the corresponding LHT amplitudes are one-loop suppressed, as required by T-parity, and result from effective operators of dimension 6. They are therefore universally suppressed [38,40,57] by a factor of (1/4 π) 2 × (v/f ) 2 . Thus, although they are quite stringent, this suppression factor alone largely accounts for many of the bounds in Table I to be easily satisfied for f 1 TeV. The strongest constraint results from the most recent measurements of µ → eγ [73] with bounds involving τ leptons in general being less restrictive and not very sensitive to the T-odd masses under consideration. Only in the case of LFV Higgs decays, which are proportional to the light SM lepton masses, are channels involving τ leptons the more sensitive ones.
The LHT has recently been reviewed in [57] to which we refer the reader for details. Here we give the expressions for the LFV amplitudes mediated by gauge bosons including the full T-odd sector while those for h → decays are given in [57] and used here for our phenomenological analysis. These can all be written in terms of form factors which, apart from the common suppression factor mentioned above, have a generic dependence on the different mixing matrix elements similar to those mediated by Higgs bosons [57]. As we will discuss more explicitly below, we can write the two-body decays → γ, Z → , and h → in terms of three-point form factors with the generic form, Branching Ratio 90% C.L. Bound Ref.
where V i are the matrix elements of the 3 × 3 unitary mixing matrix parametrizing the misalignment between the SM left-handed charged leptons with the heavy mirror ones l H . The W jk are the matrix elements of the 3 × 3 unitary mixing matrix parametrizing the misalignment between the mirror leptons and their partnersl c in the SO(5) (right-handed) multiplets [57]. Note that the mass eigenstates l H andl c are both SU ( where we have defined the (flavor) mixing coefficients  (4), because lepton and baryon number are preserved in the model. 8 We perform various scans on these mixings and masses parametrizing the LHT and present results in Section IV. Before doing so we first present the amplitudes for LFV processes mediated by gauge bosons including for the first time the full T-odd sector.

III. LEPTON FLAVOR VIOLATING PROCESSES IN THE LHT
The contributions to LFV processes of T-odd particles in the LHT, which can be mediated by gauge or Higgs bosons, have been discussed a number of times in the past. In the former 7 As we discuss below, unlike the single flavor violating decays → , , the double flavor changing transitions → only receive contributions at 1-loop from box diagrams while Z and γ penguin diagrams (see Figure 2) do not contribute until 2-loops. 8 SM neutrinos can be assumed to be massless throughout this work. case only the contributions from the T-odd sector without the heavy partner leptons have been considered [38,40] and furthermore, only at low Q 2 ∼ 0. This is justified because these contributions alone sum to a finite amplitude while the contributions from the partner leptons decouple when their masses are taken to be very large. In contrast, for Higgs mediated processes, without including the partner leptons these amplitudes are not finite due to divergences introduced by the mirror leptons [57]. As a consequence, the partner leptons can not be considered infinitely heavy and with their presence in the spectrum an additional source of LFV is introduced. In particular, the LFV processes mediated by gauge bosons at low Q 2 must be recalculated adding the contributions of the T-odd partner leptons which in turn introduces new contributions involving the T-odd electroweak triplet Φ. We recalculate here these amplitudes in the low Q 2 limit and in addition at Q 2 = M 2 Z when we consider LFV on-shell Z decays, which have not been previously examined.
The LHT Lagrangian and our conventions are presented in [57] to which we refer the reader for details. The Feynman rules necessary to calculate LFV amplitudes mediated by gauge bosons are collected in Tables II and III and include the extra Feynman rules for the new partner lepton and Φ contributions which were not included in previous studies [38].
They are given in terms of generic couplings defining the vertices for scalars (S), fermions (F) and/or gauge bosons (V): where all momenta are assumed incoming. The conjugate vertices are obtained replacing: We use the conventions in [57] which differ from those in [38] by the inclusion of the electromagnetic coupling constant e in the generic coupling definition. 9 The g and g couplings are the SU (2) L and U (1) Y usual gauge couplings respectively and e = gs W = g c W (with s W and c W the sine and cosine of the electro-weak mixing or Weinberg angle, respectively, ). The Scalar-Fermion-Fermion vertices are given generically by 9 These conventions coincide with those in [33] up to a sign in the definition of the abelian gauge couplings. where the couplings can be directly read off from Table 2 in [57] with conjugate couplings c L,R ↔ c * R,L which also enter in the calculation. There are also triple gauge boson vertices with the various couplings given in Table IV.
Finally, for the calculation of muon to electron conversion in nuclei µ N → e N the corresponding quark couplings to gauge and scalar bosons are also necessary. The relevant Vector-Quark-Quark couplings are collected in Table V. We do the same for the Scalar-Quark-Quark couplings in Table VI and include the partner quark couplings which are also  [38] and of [57] in order to include the couplings to the scalar triplet Φ. Note the different vertex normalization used in [38], where e is factored out. needed to evaluate the µ N → e N amplitude. 10 We provide further details elsewhere when discussing flavor changing top decays in the LHT. Here we only note that the three generations of T-odd partner quarks in the right-handed SO(5) multiplets are SU (2) L doublets q Ri = (x RiũRi ) T (i = 1, 2, 3) with hypercharge 7/6 which get their vector-like masses by combining with three left-handed SO(5) quark multiplets, analogously to the heavy lepton 10 Note that the flavor index i should read j in the couplings c R of Table 2 in [40].
sector. Also in analogy to the lepton sector, the misalignment between the partner quark mass eigenstates and the mirror quarks as well as those between the mirror and SM quarks are parametrized by the corresponding 3 × 3 unitary matrices, where the products are of the unitary matrices rotating left and right handed fields in order to diagonalize the various mass matrices. 11 The only physical combination is the Cabibbo-Kobayashi-Maskawa matrix V CKM = V u † V d relating the SM up and down quark sectors. In the following we make use of these Feynman rules to evaluate various LFV processes. 11 The mass matrices M are in general diagonalized by two 3 × 3 unitary matrices V L,R which we write The indices u, d, q H ,q stand for the SM up and down quarks and for the (heavy) mirror and partner quarks, respectively. For charged leptons we omit above some of these indices, following the conventions in Ref. [57]. Note also the different W subscript convention for diagonalizing the partner leptons in order to take into account that these lepton fields enter conjugated in the corresponding (right-handed) SO(5) multiplets. [SFF] A. Two or three-body lepton decays and µ → e conversion in nuclei As the contributions of mirror fermions and heavy gauge sector have been discussed in detail [38,40] in the low Q 2 limit, we will only quote their final expressions in the low Q 2 LFV processes studied in the following. 12 → γ: This process has garnered much attention in the past [83][84][85][86][87] due to the stringent experimental bound on µ → eγ. Gauge invariance reduces this vertex for an on-shell photon to a dipole transition, where where α = e 2 4π . The contributions to these two dipole form factors from the T-odd mirror leptons are presented in detail in [38,72] where the notation used here is also introduced (we rename some functions for easy comparison between processes). The different one-loop topologies contributing to them in the 't Hooft-Feynman gauge are depicted in Figure 1. 13 The contributions from the partner leptonsl c = (ν c˜ c ) only involve topologies III, IV, IX and X in Figure 1 because they do not couple to one T-odd gauge boson and a SM charged lepton at the order considered (see Table II). Their total contribution is finite as expected and the cancellation of the infinities holds for the contributions exchangingν c and c independently in F γ M |νc and F γ M |˜ c respectively. In addition, these contributions decouple from the amplitude in the heavyl c limit. In summary, neglecting m ( m ) and defining 12 SM contributions mediated by W bosons are negligible due to the tiny neutrino masses. 13 There are two other new topologies with non-renormalizable Scalar-Scalar-Fermion-Fermion couplings in the corresponding Higgs decays, h → , compared with the analogous gauge transitions [57]. 14 We have validated our calculations with FeynCalc [88,89] and utilized the scalar integrals in [90] using the conventions in Appendix C of [38].
Note that these four loop functions F  can be taken to be degenerate when substituted in F . 15 This is also the case for the components of the scalar electroweak triplet Φ within Fν ,˜ M and in the denominator multiplying the last line in Eq. (12).
These processes involve photon and Z penguin diagrams as well as box contributions which are depicted in Figure 2. The discrete T-parity guarantees that T-odd particles only run in the loop. The penguin contributions with a photon include the two dipole form factors calculated above (see Eq. (12)) and the left-handed vector form factor F γ L : with Q ν = (p − p ) ν and P L,R = 1 2 (1 ∓ γ 5 ). The corresponding right-handed vector form factor vanishes with m and is further suppressed as are the corresponding scalar form factors. The left-handed vector form factor receives contributions from mirror and partner leptons in analogy with the dipole form factors above. Keeping with the notation above, with the loop functions defined as 15 This means that m ν Hi = m Hi when used within these finite loop functions while the masses ofν i and˜ i are the same.
The form factor F γ L | W H also has a universal infinite contribution which cancels due to the unitarity of the mixing matrices multiplying it. The new contributions proportional to Fν ,˜ L decouple when the masses of the partner leptonsl c i are taken to infinity. Summarizing, the form factors entering in photon penguin diagrams satisfy F γ M = −iF γ E while current conservation implies that the vector form factors (in particular, F γ L ) must be proportional to Q 2 and vanish for on-shell photons. Of course they contribute to photon penguin diagrams which include a photon propagator proportional to ∼ Q −2 .
In contrast, Z penguin diagrams involve the Z boson propagator which for small momentum transfer processes is proportional to M −2 Z . The dipole form factors, which flip chirality, are proportional to SM lepton masses and negligible when compared to the vector ones while the scalar form factors are also negligible as they are proportional to SM lepton masses. Thus, at leading order the Z vertex reduces to, The corresponding right-handed vector form factor F Z R is O(m 2 /f 2 ) in the LHT and thus negligible at the order we work. As in the case of the photon, the contributions of the T-odd fermions to F Z L result from the running of the mirror and partner leptons inside the loops in Figure 1. Using the Feynman rules introduced above and splitting these contributions as before, we obtain: Again F Z L | W H has a universal infinite loop contribution which cancels due to the unitarity of the mixing matrices and there are similar relations among the finite loop functions  16) and (19). The form factor F Z L | W H has two different finite contributions, one which is proportional to H W (0) L and independent of Q 2 , so absent in the photon case, and another linear in Q 2 and proportional to H W L (second line in Eq. (18)). Only the first contributes to threebody lepton decays and to µ → e conversion in nuclei. The second one as well as all the other contributions which are proportional to Q 2 in F Z L (see Eq. (18)) are negligible as long as Q 2 v 2 . We give the complete result here because we will make use of it when discussing leptonic on-shell Z decays below where we also provide further details of the calculation.
The (p ) → (p 1 ) (p 2 ) (p 3 ) amplitude M → 1 2 3 then gets contributions from the photon and Z vertices in Eqs. (14) and (17) after contracting them with the corresponding gauge boson propagators and the tree-level SM leptonic vertices. It also receives contributions from the box diagrams shown in Figure 3. Following the analysis in [38] (see also [84,87]) we write the full → 1 2 3 amplitude as, where we have defined the individual amplitudes as, exchanged must be added in the three-body leptonic decays.
The photon magnetic and Z left-handed vector form factors, F γ M (0) and F Z L (0) respectively, are evaluated at Q 2 = (p 1 − p ) 2 = 0 because their leading terms are momentum independent for small momentum transfer Q 2 ∼ m 2 while the photon left-handed vector form factor, F γ L ((p 1 − p ) 2 ), is linear in Q 2 . The g Z L and g Z R couplings of the Z boson to charged leptons are given in Table II. Moving on to the box diagrams shown in Figure 3, these all reduce to the same product of currents in Eq. (23) in the limit of zero external momenta [38] (all internal masses are much larger than the external ones). The box loop-integrals are finite by power counting and in this limit their contributions can all be absorbed into the form factor e 2 B L (0). In addition to the contributions from mirror leptons accounted for in [38], those from partner leptons must also be included. Bearing in mind the degenerate mass limit for different components of the heavy T-odd SU (2) L multiplets we can group them according to the bosonic particle masses running in the loop, 16 where we have defined the functions, The mixing coefficients are defined in Eqs. (3) and (4) with = = respectively while the loop functions read, withd(x, y) =d (x, y, 1). The new contributions from the partner leptonsν c and˜ c are equal (neglecting small mass differences within the scalar triplet Φ components) and included in B ΦΦ L . After integrating the three-body phase space the decay width reads: 17 16 We use the Fierz identity 1|γ µ P L | 3|γ µ P L |2 = − 3|γ µ P L | 1|γ µ P L |2 to relate crossed diagrams with The phase space factor for |A R | 2 is singular when the second and third final lepton masses with m 2 = m 3 vanish, and it must be carefully calculated. We obtain the same result as in Ref. [91], Eq. (C.4).
where we have defined in order to simplify the expression: (28) with g L,R the corresponding Z couplings to the charged lepton (see Table II). With these results for → , the other LFV three-body lepton decays are easily obtained.
For the (p ) → (p 1 ) (p 2 ) (p 3 ) amplitude there are no crossed penguin diagram contributions due to swapping and because two gauge boson LFV transitions would be needed. This implies a higher order process both in loops and in v 2 /f 2 . This means that the (p ) → (p 1 ) (p 2 ) (p 3 ) amplitude M → 1 2 3 has no p 1 ↔ p 3 term in Eqs. (21) and (22). However, for the box amplitudes there are additional diagrams at this order for swapping and . These are automatically taken into account by the mixing coefficient definitions in Eqs. (3) and (4) with the same simplifying definitions as in Eq. (28) and corresponding changes to account for the Z couplings to the charged lepton . gives for the total decay width (with same phase space as for → ), µ N → e N: For this transition we follow Ref. [40] where the contributions of the (T-odd) mirror fermions to µ → e conversion in nuclei are discussed. 18 This process has penguin and box contributions as in Figure 2. As for the leptonic decay → , it has no crossed penguin diagrams because the lower fermionic line where the gauge boson is attached is now a coherent sum of quarks composing the probed nucleus. There is also no crossed box contributions due to the exchange of leptons. Putting everything together, including the new partner fermion contributions, we can write the µ → e conversion through the interaction with a quark q equal to u or d: with the amplitudes defined as, The form factors F γ M (0), F γ L , and F Z L (0) are given in Eqs. (12), (15), and (18) respectively while the couplings g γ(Z) L(R)q are gathered in Table II for q = u, d. The three form factors include the contributions from both mirror and partner leptons, the latter of which have not been previously computed. Analogously, B q L (0) can be read from Eqs. (24) and (25) and replacing the appropriate charges, masses, and mixings (and multiplying by a global factor of one-half to account for no crossed box diagrams for swapping leptons). The mirror fermion contribution is detailed in Ref. [40], and is contained in the corresponding sums exchanging Our definition of Q ν = (p e − p µ ) ν has opposite sign to that in [40], as well as our definition of A R .
Obviously, now the mixing coefficients involve mirror lepton mixing matrices as well as mirror quark ones, defined in Eq. (9): The (new) partner fermion contribution can be also read from Eq. (25). In this case the exchanged bosons are the charged scalar triplet components with the contributions from the different field sets running in the box being equal up to mixing coefficientsχ u,d ij : where the mixing coefficients are defined in terms of the mixing matrices as, The extra factor of 2 for B ΦΦ Ld accounts for the fact that for down quarks we can havex (of charge 5/3) and Φ ++ as well asũ (of charge 2/3) and Φ + circulating in the loop and keeping in mind the components of the heavy SU (2) L multiplets are degenerate at the order we work. Summing the different box contributions,  This gives for the corresponding conversion width in a nucleus N with Z protons and N neutrons [84] (with Z not to be confused with the Z gauge boson), where Z eff is the nucleus effective charge for the muon and F p the associated form factor. We also have m = m µ in the definition of A R in Eq. (28) while g Z L(R)q are the Z couplings to the quark q = u, d given in Table II and entering in the definition of F q LL(LR) of the same equation. In Table VII we gather the input parameters for Al and for Ti and Au, to be used below when comparing with future [94,95] and current limits, respectively.

B. One-loop contributions to Z → decays
Let us finally evaluate the Z boson decay into a pair of charged leptons of different flavor. The corresponding Z penguin contributing to rare processes with small transfer momentum Q 2 ∼ m 2 has been discussed in the previous subsection (see Eqs. (17), (18) and (19)). In that case only the first contribution to F Z L in Eq. (18), proportional to v 2 /f 2 , is relevant because Q 2 v 2 . However, in Z decays with Q 2 = M 2 Z ∼ v 2 all the contributions to F Z L in Eq. (18) are comparable and must be taken into account. Overall the Z vertex only receives significant contributions from the F Z L form factor (see Eq. (17)), as all others are suppressed by (light) SM lepton masses. Therefore, the corresponding amplitude for a Z boson with polarization s decaying into can be written in our case where µ s (p − p ) is the Z polarization vector, with Q = p − p and Q 2 = M 2 Z . Hence, the Z width reduces to In order to conclude this section we provide further details of the calculation of F Z L in Eqs. (18) and (19) where we obtained the leading contributions from the full T-odd spectrum in the LHT. It is instructive to explicitly check the cancellation of the divergent terms in order to compare with the corresponding Higgs decay and previous calculations of LFV Z decays which did not include the partner leptons and electroweak triplet Φ. The unitarity of the mixing matrices V and W ensures that one is left only with divergent terms for a given topology (see Figure 1) and T-odd fields running in the loop which have at least two m Hi insertions (see Eq. (18)). These terms can be O(1) or O( v 2 f 2 ), but in both cases must sum to zero. This can be ensured for the O(1) terms because they are generated by dimension four operators which can always be assumed (rotated) to be flavor diagonal. For the higher dimensional operators this is not the case and a more subtle cancellation between different divergences is needed.
These cancellations can be seen explicitly in Tables VIII and IX. 19 The columns label the contributing topologies listed in Figure 1 while the rows label the different field sets running in the loop. As pointed out, the leading divergences C  Table VIII cancel, as do the corresponding finite parts, for each field set when adding all the diagrams. This is represented by the bullets in the last column of this table. The same happens for the next to leading ones C Table IX, but now the corresponding finite parts do not cancel. This is indicated by the zeroes in the last column of this second table. Hence, as noted in the previous subsection all the contributions to F Z L in Eqs. (18) and (19) are finite. In particular, the contributions from the mirror leptons    Finally, we comment that the partner lepton contributions to the Z left-handed vector form factor F Z L cancel for Q 2 → 0, but this is not the case for mirror lepton contributions. Thus, although they are more restrictive, LFV processes with small momentum transfer (Q 2 ∼ m 2 ) such as lepton decays and transitions, probe a different kinematic regime than the one probed by on-shell Z decays which have Q 2 = M 2 Z . This implies low energy LFV processes and LFV Z (and Higgs) decays probe different form factor combinations making them sensitive to different regions of parameter space. However, as we discuss in the next section, limits on LFV Z decays are not yet sensitive to much of the presently allowed LHT parameter space.

IV. CONFRONTING LFV PROCESSES WITH EXPERIMENT
In this section we study the qualitative behavior of the different contributions to the where θ W must not be confused with the electro-weak mixing or Weinberg angle. The physical range for the mixing angles is between [0, π/2) since the amplitudes depend only on sin(2θ V,W ). Except otherwise stated, in the following we will use as default values θ V,W = π/4 to maximize the LFV effects. For the first two mirror lepton families we will take as default where we have also defined the mass dimension squared variablesx,ỹ parametrizing the product of heavy lepton masses and the dimensionless variables δ H , δνc parametrizing the mass (squared) splittings between the heavy leptons. The scalar triplet mass is related to the mass of the Higgs from the Coleman-Weinberg potential and fixed at leading order [96,97] TeV. The evaluation of the µ → e conversion rate also requires fixing the masses and mixings of T-odd quarks. We will assume no extra quark mixing and degenerate heavy quarks. Hence, V q H L and W q in Eq. (9) will be equal to the identity. We will use as default value m d Hi = mũ i = 2 TeV, fulfilling the current bound on pair-production of vector-like quarks [98] although it does not directly apply here because T-odd quark decays must involve lighter T-odd particles.
For τ − µ mixing we use the same notation but with the 2 × 2 rotation matrices in the bottom-right corner in Eq. (43) (see Eq. (4.8) in Ref. [57]). We also use the same   default masses and mixings with the understanding that the first and second T-odd lepton families, 1 and 2, stand for the second and third ones, 2 and 3, respectively. Similarly for the evaluation of processes with τ − e mixing but the 2 × 2 rotation matrices now involve the first and third T-odd lepton families, and 1 and 2 stand for 1 and 3. In both cases the mass of the remaining T-odd lepton family must be also fixed because it enters in the calculation of τ → µee and τ → eµµ, respectively. In either case we will take this to be equal to the largest one of the other two T-odd lepton family masses.
In Table X we collect the LHT contributions to the LFV processes in Table I calculated in previous sections assuming the default values for the model parameters above. For processes  involving τ leptons we assume τ −µ or τ −e mixing depending on which final flavor enters an odd number of times. This is because the branching ratio vanishes when this flavor coincides with the unmixed one since unmixed fermions must be joined pairwise and there will always be one left unmatched. Moreover, the processes with double-flavor violation τ → µeµ and τ → eµe also vanish when only τ − e or τ − µ mixing is assumed. As is apparent comparing the two tables, the LHT predictions for the τ sector are similar to the µ sector, but do not constrain the model appreciably though they could do so in the future [99]. 20 On the other 20 The µ and τ sector predictions differ due to the different lepton masses involved in the process as well as the different branching ratio normalization, where Br( → ν ν ) stands for the corresponding experimental value [77] and Γ( → ν ν ) for the SM prediction. In particular, the prediction for τ − → e − e + e − is larger than for τ − → µ − µ + µ − mainly due to the logarithmic mass dependent term in Eq. (27), and similarly for τ − → µ − e + e − relative to τ − → e − µ + µ − . The branching ratio for the radiative decay Br( → γ) is defined analogously but replacing Γ( → ) by Γ( → γ) while the prediction for the µ → e conversion rate in nuclei is   Table I which illustrates the sensitivity of the different processes. The first obvious observation about the behavior of the predictions of the LHT is that if the T-odd leptons are aligned with the SM ones, there is no LFV. For vanishing mixing θ V,W = 0, π/2, all of the LFV transitions go to zero. We see in Figure 4 the largest branching ratios correspond to µ − e conversion in nuclei and to µ → eγ (left panel) and at the same time they are also the best measured (right panel). In contrast, limits from on-shell Z and Higgs decays are much less restrictive. In particular, Higgs decays are outside of the left panel and omitted in the following. Furthermore, the anomalous magnetic moment a µ , which is flavor conserving and not sensitive to the mixing entering through V and W , is obtained dividing the conversion width by the corresponding capture width in Table VII, R = Γ(µN → eN)/Γ capture . Finally, the Z and h branching ratios into + are normalized to the SM total widths.
x [TeV 2 ] Ratio to current limits more than two orders of magnitude below present sensitivity. As can be observed in both panels, muon decay into three electrons shows an asymmetric dependence on the θ V mixing angle. This is in fact the generic behavior of all the observables when we vary the default values. In Figure 5 we plot the same decays as in Figure 4 but for τ − µ (being similar the plots for processes involving τ − e mixing). Comparing both figures, it is apparent how much less restrictive τ data is as all predictions for τ decays are below their current experimental limits (see right panel in Figure 5). Thus, in what follows we concentrate on µ−e transitions.
Once f is fixed, the size of the LFV processes is largely determined by the masses of the mirror and partner mirror leptons. This is apparent from Figure 6   Finally, in Figure 8 we plot contours in the mixing angle plane which saturate the current experimental limits for LFV processes having sufficient sensitivity. We show two cases corresponding to expanding the misalignment around zero (left) or (θ V , θ W ) = (π/2, π/2) (right). The other model parameters are fixed to the default values in Eq. (44). As can be seen, µ → eγ decay and µ−e conversion in Au provide the most stringent constraints. We also see that current limits require the misalignment between the SM and the mirror and partner mirror leptons to be ∼ 1 % as can be inferred from the θ V,W values when the curves flatten near the axes. For µ → e conversion in nuclei we see similar shaped contours for Au and Ti, with Ti being less restrictive, while µ → eēe allows the misalignment to be generically ∼ 10 times larger. For misalignment around zero (left), the mixing angles can be larger if θ V,W are correlated to equally high precision. For misalignment around (θ V , θ W ) = (π/2, π/2) (right) the mixing angles are always constrained to be small, in particular by µ → eγ.
Similar comments apply when studying the dependence on the mirror and partner lepton masses and a similar precision is needed to fit current experimental data from LFV processes. In Figure 9 we show the corresponding contours in the δ H − δνc (see Eq. (44)) plane which saturate current experimental limits. With the default values in Eq. (44) chosen for the remaining parameters we see in the left hand panel that there is essentially no dependence on δνc but δ H must be tiny, ∼ 1 % to fulfill the bound from µ → eγ. This asymmetric behavior is due to the choice of mixing angles θ V,W = π/4 since for θ V + θ W = π/2 the partner fermion contribution to the relevant LFV processes is much smaller than the mirror one. This is made apparent in the right hand panel where we plot the corresponding contours for θ V,W = π/8. As we saw for the mixing angles, the µ → eγ decay requires the mass (squared) splitting parameter to be tuned within 1 % with analogous behavior in µ → e conversion while again for µ → eēe limits are ∼ 10 times weaker.
The tuning of the mass splittings and mixing angles required by µ to e transitions can also be quantified using the measure defined in [100] in order to compare with other new physics scenarios. Applying it to µ → eγ in Figure 7, and COMET experiments aim to improve by two to more than four orders of magnitude in two phases. We summarize these prospects for future LFV experiments using intense muon beams in Table XI. The current precision is estimated by dividing the predictions in Table   I by the limits in Table X In the case of τ decays experimental bounds are less restrictive but constrain other mix- 21 The current precision estimate for µ → e conversion in Al is 5 times larger than in Au because this is the ratio of their rates in the LHT for the default values of the model parameters assumed in Table X, R(µAu → eAu)/R(µAl → eAl) = 3.8 × 10 −9 /8.4 × 10 −10 . 22 We use a loose meaning of mixing to illustrate the stringent bounds on the first two lepton families in this summary because although there are several mixing angles involved (see e.g. Eq. (43)), the corresponding estimate roughly applies to all of them in large regions of parameter space. For instance, in Figure 8 the limit estimates for θ V,W can be directly read from the curves when they flatten.
ings. Belle II [99] aims to improve the precision to 10 −9 − 10 −10 in a variety of LFV processes [68] while LHCb also expects to reach a similar sensitivity [69]. The upper bound on τ → µµµ from LHCb [106] is already within a factor of 2 of its current best limit [77] (see also [107] for the LHC collaborations). For the default values in Eq. (44) we find that τ → µγ could constrain the corresponding mixing angles in the LHT (see Table X for an estimate) but τ → µμµ will likely not reach the necessary precision to test the LHT prediction of Γ(τ → µμµ)/Γ(τ → µγ) ∼ 2 × 10 −3 which we note is similar to predictions in supersymmetric models [108][109][110] (see also [87] and [111]). 23 This is in contrast to previous studies [39,72] which found that three-body τ decays could distinguish clearly between the LHT and supersymmetric models. However, these studies only included effects from the mirror leptons as they worked in the limit of decoupled partner leptons. When the contributions from both the mirror and partner leptons are included one finds that there are regions of parameter space where the LHT and supersymmetric predictions are similar. Thus these decays cannot unambiguously distinguish between these two models.
To understand this further we recall that the supersymmetric prediction [108][109][110] relies on the observation that when the photon dipole contribution dominates the three-body τ decay, the ratio to the corresponding radiative two-body τ decay in Eq. (11) is fixed by kinematics, 24 In the LHT the photon dipole term is the one proportional to |A R | 2 in Eqs. (27) and (29) so in regions of parameter space where this term dominates, the LHT prediction will also be fixed by kinematics and thus similar to the supersymmetric prediction. In the left panel of Figure   10 we plot the ratio Γ(τ → µee)/Γ(τ → µµµ) (see Eqs. (29) and (27), respectively) as a function of the mass product parameterx for the default values of the other LHT parameters fixed in Eqs. (44). We show the prediction when all T-odd (non-singlet) leptons, l H and 23 We must emphasize that the predictions in Table X for τ decays can be increased by several orders of magnitude for largerx values. Moreover, in the region where τ → µγ, eγ are suppressed three-body τ decays can easily saturate current experimental bounds. 24 These relations, which follow from the corresponding LHT expressions above when only F γ M = −iF γ E is taken into account, differ slightly from those often given for supersymmetric models where the constant terms in parenthesis are −11/4 and −8/3, respectively. These coefficients should be the same since they are phase space factors, hence model independent, and must be carefully evaluated as emphasized in footnote 17.   Table XII for Br(τ → µµµ)/Br(τ → µγ) and Br(τ → µee)/Br(τ → µγ) in supersymmetric models assuming the photon dipole dominates  supersymmetric models when the mirror and partner leptons are included while they differ significantly when only mirror leptons are taken into account. As can be deduced from Figure   10  Finally, it is also important to emphasize that future leptonic colliders will produce a large number of Drell-Yan lepton pairs and in particular τ τ . 25 A potentially promising probe of LFV in the LHT at ee colliders is searching for production (for a recent review of LFV τ decays at the FCC-ee see [112]). For illustration, in Fig. 11 we show the prediction for ee → τ µ (left panel) and its ratio to ee → τ τ (right panel) as a function of the center of mass energy squared (s = Q 2 ) for different mixing angles and for the default values of the 25 We thank A. Blondel for stressing to us the potential of the FCC-ee [70].
.  remaining LHT parameters in Eqs. (44). For our estimate we use the amplitude in Eq. (20) which after crossing reads: 26 resulting in a cross section 26 The dipole term is suppressed because the total momentum factor Q ν = (p + p ) ν = (p + p ) ν results in light lepton masses m , ( √ s ) when acting on the external legs. There are no cross-terms either because the final leptons are assumed to be different from the initial ones. The flavor dependence is taken into account by the mixing coefficients within the form factors.
with χ Z (s) = s(s − M 2 Z + iM Z Γ Z ) −1 . As is apparent from Fig. 11 (left panel) where we plot the total cross-section for ee → τ µ and ee → τ τ , the LFV cross-section grows with energy. When the box diagram contributions are large, no Z peak emerges (dashed-dotted line corresponding tox = 10 TeV 2 ) in the invariant mass spectrum which opens up the possibility that a large enough LFV production cross-section can be observable without being excluded by Z-pole measurements. On the right hand panel we show the variation of this cross-section with the mixing angles which we see can be almost two orders of magnitude for O(1) mixing. To summarize, although for the default values of the parameters it may be difficult to observe LFV events, in other regions of parameter space it will be possible to constrain the τ sector in the LHT at a future FCC-ee [70] or ILC [71] machine assuming O(10 11 ) τ τ pairs are collected. As noted in footnote 23 these regions can be compatible with current constraints on LFV τ decays.
Thus, considering the huge statistics available at future lepton colliders, it will be worthwhile to study the detectability of such signals in detail.

VI. CONCLUSIONS
The LHT is an elegant and phenomenologically viable example of a composite Higgs model with improved radiative behavior due to its T-parity which helps mitigate the flavor hierarchy problem significantly. This discrete symmetry also requires the heavy T-odd spectrum to be pair-produced allowing their masses to be O(TeV) without violating present direct and indirect constraints. In this study we have revised the constraints on the T-odd spectrum of the LHT implied by present bounds on LFV processes. We have completed previous phenomenological analyses in two ways: First, we have included for the first time all contributions from the full electroweak charged T-odd sector and in particular, the T-odd partner leptons previously neglected in phenomenological studies of low momentum transfer LFV processes. As we have discussed, lepton electroweak singlets can be safely omitted in this phenomenological analysis. Second, we have computed on-shell LFV Z decays which, along with calculations of on-shell LFV Higgs decays [57], completes the set of observables at both small and large momentum transfers needed to analyze LFV in the LHT. We have compared these predictions with experiment for all available LFV processes in Table I. The µ − e and τ − µ and τ − e sectors are studied separately assuming a two-family description for each. The general case can be obtained combining the parameters of the three analyses, noting that the restrictions on the τ − e sector are virtually identical to those on τ − µ (see Table I).
For the τ − µ LFV observables we find that low energy observables, namely τ → µγ and τ → µµµ, are much more restrictive than on-shell LFV decays of the Z or Higgs bosons which are typically orders of magnitude below the current limits. We also find that even when allowing low energy observables to saturate their current experimental limits, on-shell LFV Z and Higgs decays can be at most O(10 −7 ). The pattern of LFV is quite similar for τ − e transitions with small differences due to the slightly different experimental bounds.
On the other hand, µ − e transitions present significant differences. First, Higgs decays are proportional to the masses of the SM leptons in the final state and are therefore strongly suppressed for decays not involving the τ lepton. In general constraints on µ − e transitions from low energy LFV observables are very strong and render on-shell Z or Higgs decays to µe too small to be observable even in future experiments. Furthermore, there is very little correlation between the different low-energy LFV observables so the only way of generically satisfying µ − e constraints is by reducing the global suppression factor by means of a relatively large f and/or small mixing angles since even with the new source of flavor violation, cancellations are far from generic. This implies large global symmetry breaking scales f > 10 TeV or small mixing angles θ V,W < 10 −2 .
Motivated by considerations of naturalness, we have focused our phenomenological studies on the case of small mixing angles and/or accidental cancellations with f = 1.5 TeV. We have explicitly shown that although all current bounds can be satisfied with small mixing angles, some of them can be the result of accidental cancellations between the new contributions to these processes. Furthermore, we have quantified the amount of fine tuning necessary and found it to be ∼ 1 % for particular parameter points. While the LHT can accommodate limits from searches of FCNC, a flavor completion of the model with a natural suppression of the leptonic FCNC would be welcome.
Future LFV experiments using intense muon beams will typically improve the current sensitivity by two to four orders of magnitude, constraining the effective µ − e mixing to 1 per mille for f TeV in the absence of a signal. In this case, future LFV experiments will strongly challenge the LHT scenario probing scales f ∼ 10 TeV even for mixings of order 1 %, thus pushing the concept of naturalness in the LHT to its limits. We have also shown, in contrast to previous studies, that in certain regions of parameter space the LHT model can give similar predictions to supersymmetric models for LFV processes. In other LHT parameter regions τ LFV transitions can saturate current experimental bounds and hence, can be observed at Belle II as well as the LHC and/or future lepton colliders. As the current sensitivity on τ LFV processes does not favor any definite pattern of τ mixing, a systematic study of τ µ and τ e production is worthwhile.

Acknowledgments
We thank useful discussions and comments by A. Blondel instance, [113]). The contribution of the T-odd spectrum in the LHT to F γ M (0) and hence, its contribution to a just defined, can be read from Eq. (12). It is real by inspection: the loop functions are real because only heavy particles far away from any physical threshold run in the loops, and the mixing matrix elements as well as their complex conjugates enter symmetrically. On the other hand the contribution of these T-odd particles to the real part of F γ E (0) (and then to d ) vanishes because the form factors satisfy F γ E (0) = iF γ M (0) and