Exotic Leptons: Higgs, Flavor and Collider Phenomenology

We study extensions of the standard model by one generation of vector-like leptons with non-standard hypercharges, which allow for a sizable modification of the h ->gamma gamma decay rate for new lepton masses in the 300 GeV - 1 TeV range. We analyze vaccum stability implications for different hypercharges. Effects in h ->Z gamma are typically much smaller than in h ->gamma gamma, but distinct among the considered hypercharge assignments. Non-standard hypercharges constrain or entirely forbid possible mixing operators with standard model leptons. As a consequence, the leading contributions to the experimentally strongly constrained electric dipole moments of standard model fermions are only generated at the two loop level by the new CP violating sources of the considered setups. We derive the bounds from dipole moments, electro-weak precision observables and lepton flavor violating processes, and discuss their implications. Finally, we examine the production and decay channels of the vector-like leptons at the LHC, and find that signatures with multiple light leptons or taus are already probing interesting regions of parameter space.


I. INTRODUCTION
In 2012, the Large Hadron Collider (LHC) experiments ATLAS and CMS both reported the discovery of a Higgs-like boson at m h 125 GeV [1,2]. At this mass, many of its Standard Model (SM) couplings are experimentally accessible, which makes the question whether new physics manifests itself through modifications of these couplings one of the most interesting ones for the LHC to answer. The hints for a possible enhancement of the Higgs to diphoton decay rate [3][4][5][6] therefore triggered a lot of activity in the model building community (see e.g. ). In the latest analysis by ATLAS [38], this excess of events in the h → γγ channel leads to a best fit value of the signal strength of 1.65 ± 0.24 +0.25 −0.18 times the predicted SM value, while with the full 7+8 TeV data set, the CMS h → γγ signal has gone down to 0.78 +0. 28 −0.26 [39]. The h → γγ decay is loop induced in the SM and therefore highly sensitive to new physics effects. It will be extremely interesting to monitor if a significant deviation from the SM prediction can be established at the 13 TeV LHC run.
Although all other presently measured decay rates are compatible with the SM predictions, there is still room to consider NP effects, and it is also interesting to explore connections with modifications in these channels. Here we concentrate on the properties of models which can lead to a significant modification in h → γγ, without sizable effects in other channels. The properties of such models can be narrowed down considerably [14].
A promising class of models to this end are extensions of the SM by a set of new vector-like leptons, transforming as electro-weak doublets and singlets, respectively. Sizable Yukawa couplings between the Higgs and these new states allow for a modification of the h → γγ rate without modifying the main production process via gluon fusion. The price for this modification is a severe vacuum instability bound, because the new Yukawa couplings will drive the Higgs quartic coupling negative at a scale around 10 TeV [16,17,40]. In addition, possible mixing terms between the new vector-like leptons and the SM leptons can induce 1-loop contributions to electric and magnetic dipole moments (EDM/MDM), as well as tree level contributions to lepton flavor violating processes, which are strongly constrained experimentally [23,32,41]. While the vacuum instability bound calls for an extension of this model at a relatively low scale (see e.g. [29,36,42]), the latter constraints have been usually avoided in the literature by either assuming very small coefficients for the mixing operators or by invoking a discrete symmetry [16,17,22]. In this work, we argue that a different hypercharge assignment to the new vector-like leptons can in principle not only relax the vacuum instability bound, but simultaneously also ensures automatically that the leading contributions to dipole moments only arise at the 2-loop level. (1) The case n = 0 corresponds to the widely discussed scenario in which the quantum numbers are a copy of the ones of the SM leptons [16,17,20,22,23,32,48,49]. In addition to a strong vacuum stability constraint, the n = 0 setup allows for direct mass mixing operators with the SM leptons. This can lead to large corrections of the couplings of the SM leptons with the weak gauge bosons and the Higgs and also induce tree level FCNC couplings. In addition, large 1-loop contributions to EDMs and MDMs can be generated. The measurements of EDMs, MDMs, lepton flavor violating observables and Z pole observables therefore constrain the coefficients of these mass mixing operators to be extremely small. One way to account for these bounds is to invoke a discrete symmetry which forbids mixing operators [16]. In the models considered in this article however, these operators are either absent in the n = 2 case, due to the different hypercharges between the vector-like and the SM leptons, or only one operator is allowed that couples the new sector just with the right handed SM sector, limiting the impact of the mass mixing operators for n = 1. In particular for n = 1, we find the following mass terms and Yukawa couplings, in which SM fields are denoted by lower case letters, andh ≡ iσ 2 h † . Note that the hypercharge assignment only allows one mixing operator, that mixes the right-handed SM leptons with the lefthanded doublet of vector leptons. The Lagrangian for the general scenario n > 1 corresponds to (2) with the coefficient of the mixing operator set to zero, y L = 0. We only discuss the scenarios n = 1 and n = 2 here. Models with even higher hypercharges might lead to interesting phenomenology as well, but result in a Landau pole of the hypercharge gauge coupling at scales of ∼ 10 4 TeV or below, see Section III B.
For n = 2, the new leptons carry electric charges Q E = −3 and Q N = −2. After electro-weak symmetry breaking (EWSB), the Lagrangian (2) leads to the following mass terms with the masses of the SM leptons m = y SM v and the Higgs vev, v = 174 GeV. In the absence of a doubly charged singlet, the mass of the charge two component of L will only be given by its vector mass, while the charge three leptons mix, n = 2 : While the parameters M L , M E , y L and y E can all be complex, three phases can be absorbed by re-phasing the vector lepton fields, leaving one physical CP violating phaseφ = Arg(M L M E y * L y E ). In the following we will work in a convention where the vector masses are real and positive and parametrize the physical phase by the relative phase of the Yukawa couplings, i.e.φ = Arg(y * L y E ). For the case of n = 1, the new leptons carry electric charges Q E = −2 and Q N = −1, and the mass Lagrangian reads Mixing with the SM leptons is generated proportional to the Yukawa couplings y L , so that n = 1 : in which only mixing with one SM lepton generation is considered for simplicity. The extension to the 3 generation case is straightforward. The phases of the mixing Yukawas y L are additional physical sources of CP violation.
Both in the n = 1 and n = 2 case, we can diagonalize the mass matrix M E by a bi- T to describe the light and heavy mass eigenstates with masses In the n = 2 scenario, the charge two lepton N = N L +Ñ R is its own mass eigenstate with m N = M L . In the n = 1 scenario however, this is only true up to corrections proportional to the mixing coefficients y L . As we will see in the following, the size of the mixing between the SM leptons and the vector leptons is constrained to be small. We therefore treat this mixing perturbatively and find the following leading corrections to the masses of N and the SM leptons In the n = 1 case, the mixing terms also lead to modifications of the couplings of the SM leptons, once one rotates into the mass eigenstate basis. In particular, the flavor diagonal couplings of the Higgs to SM leptons and of the Z boson to right-handed SM leptons are modified at the order Based on the mass matrices given in eqns. (4) and (6), one can obtain the contribution to the h → γγ decay rate at leading order in the electro-weak scale over the vector masses, using low energy theorems [49][50][51]. Notice, that in contrast to the n = 0 scenario, for n = 1 there is only one off-diagonal mixing term between the vector-like and the SM leptons. As a consequence, both in the n = 2 and n = 1 scenarios, at leading order, the only non-SM contributions to the h → γγ decay rate are generated by the mass matrix M E . In contrast to the case of chiral fermions, the effective interaction of the Higgs with photons contains both a CP-even and a CP-odd part [49] As corrections to the Higgs production cross section are negligible in our framework, the ratio of the Higgs diphoton rate normalized to the respective SM rate is to an excellent approximation given by the ratio of the h → γγ partial decay widths. We find which is valid for both scenarios. Here, τ i = 4m 2 i /m 2 h and we neglected the tiny bottom quark contribution to the SM width. To a good approximation, one has for the SM W and top loops Expressions for the loop functions A 1 and A 1/2 are collected in Appendix A. The explicit form of the derivatives of the mass matrix read Note that the CP-odd contribution does not interfere with the SM amplitude. Even though the CP-odd part will therefore always enhance the h → γγ cross section, it will typically amount to at most a percent correction for all phenomenologically viable parameters of the considered model.
In Section IV, we will see that even this is very optimistic, given the very stringent bounds on the new physics phase,φ, coming from the electron EDM. The CP-even part in (10) interferes with the SM contribution and therefore allows for significantly larger corrections. Depending on the overall sign of the numerator in (11), this can lead to an enhancement or decrease of the h → γγ cross section. The term ∼ |y E y L | 2 in (11) always leads to a decreased cross section compared to the SM, but can be neglected to a first approximation. The term could only become relevant for very small vector masses M L and M E , that are strongly constrained by direct searches, or for large Yukawa couplings, that are theoretically unattractive, as they imply large corrections to the running of the Higgs quartic coupling, forcing it to become negative at very low scales. The sign of the interference is therefore mainly determined by the sign of Re(y * L y E ) = |y L ||y E | cosφ. While the expression (10) captures the leading contributions in an expansion in the ratio of the electro-weak scale over the vector masses, one can easily go beyond this approximation working with mass eigenstates of the new leptons. Doing so, the corrections to the Higgs diphoton decay rate can be written as The expressions for the couplings g hχ i χ i of the Higgs with the new lepton mass eigenstates are given in Appendix B. In the n = 1 case, there are in principle also contributions from the charge 1 leptons that are formally of higher order in v 2 /M 2 L . Working with mass eigenstates, they can be taken into account in a straight forward way. However, given the constraints on the mixing Yukawas that will be discussed in Section V, we find that contributions from the new charge 1 states are negligible even for very light masses M L = O(v). Expanding (13) in v/M we recover the approximate expression in (10). We find that (10) is accurate at the one percent level as long as the vector masses are M L , M E 300 GeV. In our numerical analysis, we work with mass eigenstates, though.
Due to their large charges, the new leptons can lead to sizable effects in h → γγ even for moderate values of the Yukawa couplings. In particular, for fixed vector masses M L and M E , the Yukawa couplings can be smaller by a factor while keeping the decay rate constant compared to the Q χ = 1 (n = 0) scenario. Conversely, for fixed Yukawa couplings, higher charges allow for heavier vector masses. For all three plots, y E = y L = 0 corresponds to R γγ = 1 and the effects become larger for larger absolute values of y E and y L .
contribute at 1-loop to the running of the Higgs quartic coupling through the box diagram on the right hand side of Figure 1. We find a correction to the SM beta function of 1 The scale at which the quartic coupling runs negative is plotted in Figure 2 versus the absolute value of the new Yukawa couplings |y| = |y E | = |y L |. For y = 0 one recovers the SM limit that, for values of α s = 0.1184 [52] and m t = 173.2 GeV [53], and considering a two loop renormalization group running, yield a vanishing value of λ at a UV scale of Λ UV 10 10 GeV (see e.g. [54,55]).
1 In the n=1 case there are additional contributions to the beta function coming from the mixing Yukawas y L . In regions of parameter space where the vector leptons can lead to sizable modifications of the h → γγ rate, they are bound to be small from indirect constraints (see Section V). Therefore their impact on the running of the Higgs quartic is negligible.
The effect of non-zero Yukawa couplings |y| = |y E | = |y L | on the vacuum stability of the Higgs potential has to be compared with the effects in R γγ , which are shown in Figure 3  If one requires an enhancement of the Higgs diphoton rate by 30%, one finds in the n = 0 case that the Higgs quartic coupling runs negative at Λ UV ≈ 10 − 100 TeV, even for the most optimistic assumptions like the lightest mass eigenstate close to the LEP bound m ∼ 100 GeV , in agreement with Ref. [17]. As a consequence, such models would require a UV completion at or below the 10- Therefore, given that the vector leptons have to be considerably heavy in the minimal setups we have investigated, it turns out that the UV scale where the quartic Higgs coupling becomes negative is actually comparable to the n = 0 case, namely around ∼ 10 − 100 TeV in the n = 1 case and even lower in the n = 2 case. As will be discussed in Section VI, in extensions of the setups with an additional massive neutral state and with additional interactions parametrized by higher dimensional operators, lighter vector-like leptons can become viable also for n = 1 and n = 2.
For the numerical calculation of the running we take into account the Higgs quartic, the SM gauge couplings, the top Yukawa and the contributions from the new Yukawas y E and y L . We use 2-loop expressions for the SM beta functions [56][57][58][59][60] and add the 1-loop contributions from the new leptons. The running of the Higgs quartic coupling was already given in (15). For the gauge and The beta function of the strong gauge coupling is not affected by the new uncolored states and we use SU (5) normalization for the weak couplings g 2 1 = 5 3 g 2 and g 2 2 = g 2 . To first order in the ratio of the electro-weak scale over the vector masses, there is a direct correlation of contributions to the QED beta function and the CP-even coupling of the Higgs to two photons [14,50,51]. Therefore, a modification of R γγ is necessarily correlated with a positive contribution to the running of the SU (2) L × U (1) Y gauge couplings. In particular, both in the n = 1 and n = 2 case, the running of the hypercharge leads to a Landau pole below the Planck scale, but for both scenarios, this Landau pole is orders of magnitude above the UV scale extracted from vacuum stability considerations in regions of parameter space with a sizable modification of R γγ . It should be mentioned, that this is not necessarily the case for scenarios with new leptons that carry even larger hypercharges. For example in the n = 3 case, the Landau pole arises already at a scale of ∼ 10 4 TeV.

C. The h → Zγ Rate
The new vector-like leptons do not only contribute at the 1-loop level to the h → γγ decay, but they also modify the h → Zγ rate. In the scenario where the new leptons have the same hypercharges as the SM leptons, their effect in h → Zγ is accidentally suppressed by 1−4s 2 W 0.08 and h → Zγ is to an excellent approximation SM-like [14]. This strong suppression does not arise for our non-standard hypercharge assignments, and larger effects can in principle be expected.
The corrections to the h → Zγ rate can be written in the following generic form Here, F SM is the SM amplitude and F NP (F NP ) is the CP conserving (CP violating) part of the NP amplitude. As in the case of h → γγ, the by far dominant NP contributions come from loops involving the charge 2 states (for n = 1) or the charge 3 states (for n = 2), respectively. Working with mass eigenstates, we find In the SM amplitude, we neglected the tiny contribution from the bottom quark loop. The W and top contributions, F W and F t , can be found for example in [61]. Numerically, we find approximately The loop functions in the NP amplitudes are given by with f and g given in [61]. The relevant couplings of the Higgs and the Z boson to the new lepton mass eigenstates are collected in Appendix B. Note that (23) and (24) contain contributions from loops where both mass eigenstates enter simultaneously. These contributions are parametrically of the same order as the contributions from loops that contain only one mass eigenstate.
In order to obtain an analytical understanding of the NP contributions to h → Zγ, we expand the corrections to R Zγ to leading order in the electro-weak scale over the vector masses. We find The functions h 1 , h 2 , and h 3 depend on the ratio of the vector masses x = M 2 E /M 2 L and for degenerate masses we have h 1 (1) = h 2 (1) = h 3 (1) = 0. The explicit expressions for the h i functions are given in the appendix. Even for large splittings of the vector masses, we find that the effects of the h i is typically small. Therefore, we indeed observe that in the n = 0 case, the corrections to h → Zγ are accidentally suppressed by 1 − 4s 2 W , while such a suppression is absent in the n = 1 and n = 2 cases.  most values between −5% to +10% for a strongly enhanced h → γγ rate. For the scenarios with the larger hypercharges, the effects in h → Zγ can be slightly larger, but still typically do not exceed ±10%, due to the fact that we have considered in each case values of the vector masses M that we expect could be compatible with direct LHC constraints on the vector-like fermions. The correlation of R Zγ and R γγ is markedly distinct in the 3 cases, but NP effects in h → Zγ at the 10% level will be very challenging to probe at the LHC.

ELECTRO-WEAK PRECISION OBSERVABLES
In addition to the need for a low UV cut-off, models in which the vector leptons share all quantum numbers with the SM leptons induce 1-loop contributions to SM fermion EDMs and MDMs.
Measurements of these quantities result in very constraining limits, especially EDM measurements, which already probe electro-weak 2-loop contributions [62][63][64][65]. As a consequence, the mixing operators in these models must have very small coefficients or must be forbidden by an additional symmetry. Remarkably, in both scenarios discussed in this work, the leading contributions to EDMs and MDMs are automatically lifted to the 2-loop level.

A. Electric Dipole Moments
For both n = 1 and n = 2, we can estimate contributions to the EDM of a SM fermion f by considering the 2-loop Barr-Zee type diagram in Figure 5, which contains the h → γγ loop as a sub-diagram. Given that the Higgs couplings to the SM fermions are proportional to The loop function g can be found in Appendix A. The source of this 2-loop EDM is the same as the CP violating contribution to h → γγ, namely the irreducible phase in the mass matrix M E .
In the limit m i v and for M L = M E = M we can write thus making the correlation with the CP-odd contribution to the h → γγ decay rate in (10) manifest. The explicit expression for the derivative was already given in (12). Note that the Barr-Zee contributions to the EDMs scale in the same way with the charge of the vector leptons, Q χ , as the NP amplitude in h → γγ does.
As we will show, bringing the 2-loop contributions in agreement with the most recent measurements of the electron EDM [62,63], still requires a fine-tuning of the phase of about 10%, in regions of parameter space that allow for a sizable modification of the CP conserving part of the h → γγ amplitude, see e.g. Figure 7.
Experimental results on EDMs of hadronic systems, e.g. the neutron EDM or mercury EDM [64,65], h h Figure 6. Example 1-loop diagrams giving rise to an electron EDM (left) and MDM (right) in the n = 1 scenario. The photon can be attached to all charged particles in the loops lead to constraints on quark EDMs that translate into comparable bounds on the model parameters, but they are subject to large hadronic uncertainties. Note, that additional diagrams with the internal hγ replaced by a hZ can be important for quark EDMs, but will play essentially no role for leptons because of the accidentally small vector coupling of the Z to SM leptons. 2-loop diagrams with W + W − in the loop turn out to be small for both quarks and leptons, see also [23]. Nonetheless, in our numerical analysis, we take into account the full set of hγ, hZ, and W + W − contributions.  Correspondingly, a CP-odd contribution to the h → γγ rate at the percent level would already be in conflict with EDMs, barring accidental cancellations with contributions induced by additional CP violating sources from beyond the models considered here. This agrees with the findings in [23,32].
Analogously, EDM bounds also strongly restrict possible CP violating effects in h → Zγ well below the percent level. Possible CP violation in the experimentally most favorable h → ZZ channel is even further suppressed below the 10 −4 level, because loop induced CP violating effects have to compete with the CP conserving tree level hZZ coupling. Since the imaginary part of the couplings is constrained to be very small, we will only work with real y L and y E couplings for the remainder of this paper.

B. Anomalous Magnetic Moments
The 2-loop Barr-Zee diagrams also give contributions to anomalous magnetic moments of leptons in both scenarios with the explicit form of the 2-loop function f given in Appendix A. In the limit m i v and for This shows clearly the correlation of the anomalous magnetic moments with the CP-even contributions to the h → γγ decay rate. The explicit expression for the derivative can be found in (11).
However, given the uncertainty of the current experimental results and the precision of the SM we find that the 2-loop contributions lead to effects that are one order of magnitude below the current sensitivities or even smaller, even for vector masses at the order of the electro-weak scale and Yukawa couplings of order 1.

C. S and T Parameter
Additional constraints on the discussed scenarios arise from electro-weak precision observables, in particular the S and T parameters. The latest constraints on S and T read [68] ∆S = 0.03 ± 0.10 , ∆T = 0.05 ± 0.12 ,  Yukawa couplings y E and y L . Even for sizable Yukawas, y E = y L = 1, vector masses as low as M L = M E = 300 GeV are allowed [20]. Contributions to the S parameter do depend on the hypercharge assignments. We calculate corrections to the S parameter in our scenarios by adapting the general expressions given in [69]. We find that despite the large hypercharges, corrections to the S parameter are typically also moderate. This is illustrated in Figure 8

V. CONSTRAINTS ON MIXING WITH THE STANDARD MODEL LEPTONS
In the n = 1 case, the mixing between the SM leptons and the new leptons is subject to strong indirect constraints from Z pole observables and lepton flavor violating processes. In this section, we discuss the most stringent constraints and their implications.
The couplings of the SM leptons to the Z boson have been precisely measured at LEP. As already mentioned at the end of Section II, the Yukawa couplings that mix the SM leptons with the new particles lead to corrections to the coupling of the Z with the right-handed SM leptons. Such corrections are constrained at the 10 −3 level and better [70]. Combining the experimental results with the SM predictions collected in [70] we find where the δg R are defined as the relative deviations of the coupling of the Z with the right-handed SM leptons The model predicts always positive corrections to the couplings g R . As the measured coupling of electrons is almost 2σ below the SM prediction, the derived constraints are particularly strong in the case of electrons. The constraints are illustrated in the plots of Figure 9 in the M Ly L planes.
Dark and light orange regions are excluded at the 3σ and 2σ level, respectively.
There can in principle be also corrections to the decay of the Higgs to leptons. We find at leading order the following modification of the h → τ τ signal strength Contours of constant R τ τ are superimposed in the bottom plot of Figure 9. Given the constraints from the Z pole measurements, this correction is unobservably small. This is in contrast to the n = 0 case where the additionally allowed mixing Yukawas and masses can lead to visible modifications of Higgs couplings to fermions [20].

B. Lepton Flavor Violation
Very stringent constraints on the coefficients of the mixing operators in the n = 1 Lagrangian also come from observables measuring the flavor changing couplings of the Z. The most severe bounds result from the tree-level induced µ → e conversion in nuclei, and flavor violating τ decays, like τ → 3e and τ → 3µ.
For the µ → e conversion in nuclei, the branching ratio can be written as [71] BR in which ω N cap. denotes the muon capture rate of the nucleus N , and V (p) and V (n) are nucleus dependent overlap integrals [71].
The coefficients C u and C d are defined by the effective Hamiltonian and are generated by off-diagonal Z couplings. We find The current most stringent experimental bounds are coming from measurements using Au and Ti atoms [72,73] BR(µ → e in Au) < 7 × 10 −13 @ 90% C.L. , BR(µ → e in Ti) < 1.7 × 10 −12 @ 90% C.L. , and can be translated into bounds on the combinations of couplings, which enter (43).
The corresponding parameter space is shown in Figure 10, with the excluded region shaded in orange. Generically, for y Le y Lµ , couplings at the order of 10 −3 are probed. However, as only the product of these two couplings is constrained, either of the couplings can be as large as the bound obtained from Z pole observables in the previous section, as long as the other coupling is strongly suppressed. The expected sensitivity of the Mu2e experiment to µ → e conversion in Al, BR(µ → e in Al) 6 × 10 −17 [74], will probe large regions of the presently allowed parameter space.
For → 3 decays, the branching ratio can be written as where the coefficients C i are defined by the effective Hamiltonian The dominant contribution comes again from the tree level exchange of the Z boson with its flavor violating coupling to right-handed leptons. We have and contributions to C LL and C LR are negligible, see eq. (B10). The resulting branching ratio for µ → 3e gives a weaker bound than µ → e conversion, while the τ → 3e and τ → 3µ branching ratios allow to constrain the mixing of the vector-like leptons with the τ .
The current bounds on the τ branching ratios are [75] BR(τ → 3e) < 2.7 × 10 −8 @ 90% C.L. , The allowed parameter space is shown in Figure 11, again with the experimentally excluded region shaded in orange. Other flavor violating leptonic tau decays like τ + → e + µ + µ − , τ + → µ + e + e − , or lepton flavor violating semi-leptonic tau decays lead to very similar constraints. Bounds from the loop induced → γ decays constrain the same combination of couplings as the observables discussed previously, but -due to the loop suppression -result in much weaker constraints, so that we refrain from presenting a detailed discussion of these bounds.
Note that due to the strong constraint from µ → e conversion either τ → 3e or τ → 3µ can be close to the current bound, but not both simultaneously. Indeed, combining the expressions for BR(τ → 3e), BR(τ → 3µ), BR(µ → e in Au), and δg Rτ , we arrive at the following relation that is independent of any model parameters The proportionality constant is purely given by known SM parameters, and we find: const.
1.2 × 10 −4 . The constraint from µ → e conversion on possible NP effects in BR(τ → 3e) and BR(τ → 3µ) is illustrated in Figure 12. Shown in orange is the region in the BR(τ → 3e) vs.
BR(τ → 3µ) plane that is excluded by the current bound on BR(µ → e in Au), allowing a correction to δg Rτ that saturates Finding both BR(τ → 3e) and BR(τ → 3µ) close to the current bounds would clearly rule out the studied framework.
In conclusion, observables measuring deviations of the Z couplings to SM leptons lead to constraints on the mixing Yukawas in the n = 1 case. The strongest bounds are summarized in Table I.
The analysis in this section has to be contrasted with the results from studies of models of new vector-like leptons, which have the exact same quantum numbers as their SM cousins. In these models, highly non-generic CP and flavor structures are necessary in order to satisfy the constraints |y Lτ | 2 < 3.9 × 10 −3 discussed in this section, see for example [66] and references therein.

A. Production of the Vector-Like Leptons
In both scenarios, the new vector leptons will dominantly be pair produced in Drell-Yan processes due to their large hypercharges. Sub-dominant channels are Higgs mediated pair production or the production of a pair of vector-like leptons with different charges through a W .
In the n = 1 scenario, the W channel does also allow for a charge 2 vector-like lepton to be produced together with a charged SM lepton, or for the charge 1 vector-like lepton to be produced together with a SM neutrino. For n = 1, there is also Drell-Yan production of a charge 1 vector-like lepton together with a charged SM lepton. The single production channels are however suppressed by the Yukawa matrix y L , which parametrizes the mixing of the vector leptons with the SM leptons, as well as by powers of the electro-weak scale over the vector masses. They turn out to be two to three orders of magnitude smaller compared to the Drell-Yan production at the current LHC where a combinatorial factor n = 27, which counts the different SM lepton flavor variations that can appear in the operators (52), has been taken into account. If the scale of new physics Λ is sufficiently large, Λ 10 3 TeV, the new states can behave as stable particles within the collider.
In such a case, bounds from the searches of long-lived multi-charged particles apply, which are approximately m χ 800 GeV [76,77]. This translates into a bound on the vector mass of conversion, flavor constraints are typically also satisfied in this parameter region.
The production and decay topology of the lightest charge two mass eigenstate in the n = 1 scenario is shown in Figure 13. Right: expected number of signal events after all cuts (see text for details) for the n = 1 scenario with y Lµ = 0.1, y Lτ = y Le = 0 denoted by N µ signal as well as y Lτ = 0.1, y Lµ = y Le = 0, denoted by N τ signal , for different masses of the lightest mass eigenstate.
we consider an ATLAS search for at least one hadronic tau and three light leptons [82].
In order to study the signal cross section, we implemented our model in FeynRules [87] and generated events using MadGraph 5 [88]. In the case of tau decays, PYTHIA-PGS was used for hadronization and detector simulation [89]. All cuts have been applied after the detector simulation.
For the model parameters, M = M L = M E and y = y L = y E ∈ R has been assumed. Fiducial tau efficiency tables for the ATLAS detector are publicly available, and we find that for the hard taus required in the searches considered here, the PGS simulation yields efficiencies roughly within 10% of the numbers listed in Table V in [86]. Since we are only interested in an estimate on the bound on the masses of the new resonances, we will not correct for these differences here.

Light Leptons in the Final State
The n = 1 scenario with couplings to light leptons leads to the same signature expected from the electro-weak production of charginos and neutralinos, which subsequently decay into three light leptons, neutrinos and the lightest neutralino (LSP). In the model presented here, there is no massive neutral final state, so that the strong exclusion bounds for a mass-less LSP apply. If the new resonances are assumed to couple only to taus, this final state will also be a promising channel in the case that the taus decay leptonically.  Table II. We cross-checked our simulation by reproducing the irreducible ZZ background within its errors. The simulated number of signal events after cuts, depending on the mass of the lightest mass eigenstate of the model discussed here, is shown on the right-hand side of Table II. Since in our model a third lepton will always be the product of a W decay, in the low mass region m χ < 200 GeV, the E miss T cut is the most efficient cut on our signal, while the requirement on m SFOS represents the strongest cut for higher masses. In the case of only direct couplings to muons, the limits on the mass of the lightest charge 2 mass eigenstate are roughly m χ 460 GeV. The scenario in which only direct couplings to taus are assumed allows for the weaker bound m χ 320 GeV.

Hadronic Taus in the Final State
In addition to the bound derived from the decay into light leptons, we considered three different searches for hadronic taus in the final state in order to further constrain the n = 1 scenario in which the doubly charged leptons only decay into taus and W s. We studied searches for opposite sign hadronic taus (+ p miss T ) [81] or one hadronic tau together with a same-sign lepton (+ jets and E miss T ) [84] in the final state, as well as searches for hadronic tau pairs, jets and large E miss The CMS search [84] for two same sign leptons requires two jets and missing transverse energy.
In order to reduce the high trigger rates, a significant bound on E miss GeV, in order to suppress Z+jets events, and large E T miss > 150 GeV is required. At least two jets have to be present, with p j 1 T > 130 GeV and p j 2 T > 30 GeV, as well as a large scalar sum of the transverse momenta of these jets and taus H T = p j i T + p τ i T > 900 GeV. This H T cut, the requirement for large missing energy as well as the cut on the sum of the transverse tau masses strongly reduce our signal cross section, so that the experimental bounds do not lead to constraints throughout the scanned mass range.
Finally, we consider a search from ATLAS for three light leptons and (at least) one hadronic tau in the final state [82]. In the scenario discussed here, this final state requires both W s to decay leptonically as well as one leptonic tau and one hadronic tau. In the considered search region an "extended Z-veto" has been employed, which means, that events with pairs, triplets or quadruplets of light leptons with an invariant mass within 10 GeV of M Z = 91.2 GeV are vetoed. In addition, selected events are required to either have missing energy of E T miss > 100 GeV or an effective mass In the tau case, these bounds imply that a 30% enhancement of the Higgs di-photon rate is possible with y E , y L 0.7, M L , M E 440 GeV and the Higgs quartic runs negative at a scale Λ 100 TeV.
In the muon case instead, a 30% enhancement requires y E , y L 1, with M L , M E 630 GeV and the Higgs quartic runs negative already at a scale of a few TeV.
Finally, we want to mention, that a dedicated search based on existing data, looking for two hadronic W s and two light leptons in the final state might lead to stronger bounds on the parameter space.

Extended Scenario
The bounds in (54) and (55) can be relaxed by extending the model, such that the charge two leptons predominantly decay into a stable, neutral state with a mass close to the lightest charge two state. This can be arranged e.g. by adding to the model a SM singlet fermion χ 0 and coupling it to the hypercharge 2 singletẼ L and right-handed SM leptons i R = e R , µ R , τ R via a dimension six operator In order for the new decay mode ofẼ L to dominate over the decay into a W boson and SM lepton, the UV scale Λ where this operator is generated has to be sufficiently small, parametrically of order Λ 4 M 6 /(v 2 |y L | 2 ). For mixing Yukawas close to the bounds in Table I  Due to the exotic hypercharge assignments in the n = 1 and n = 2 cases, possible modifications of the h → Zγ rate can be larger compared to the n = 0 case. Still, we find that corrections to the h → Zγ rate typically do not exceed 10%. Precision measurements of the h → Zγ and h → γγ rates can in principle distinguish between the considered cases, but it will be very challenging to achieve the required precision at the LHC.
We further discussed the new physics contributions to electric and magnetic dipole moments.
The non-standard hypercharge assignments strongly restrict the possible mixing operators with SM leptons, so that the leading contribution to the electron and quark EDMs only appear at 2-loop for both the n = 1 and n = 2 scenario. The corresponding Barr-Zee diagrams contain the h → γγ loop as a sub-diagram, and a modification of R γγ is therefore correlated with a 2-loop contribution to EDMs and MDMs. This correlation allows in principle to constrain the imaginary part (EDMs) and the real part (MDMs) of the Yukawa couplings between the new leptons and the Higgs using the very precise measurements of these observables. We find, that the single new phase entering the contributions to EDMs in our setups has to be below the order of 10% in regions of parameter space Finally, we discussed the collider signals of the two models, that we already utilized to evaluate the possible modifications to h → γγ and h → Zγ. The dominant production cross section is the pair production of the lightest charge two (three) state in the case of n = 1 (n = 2). In the n = 2 scenario, the decay of the charge three state can only be mediated through higher dimensional operators. Possible dimension six operators violate the SM lepton number and generically also violate lepton flavor. If we assume that these operators are suppressed by a scale sufficiently high such that the new leptons are metastable at collider scales, bounds from searches for stable charged particles apply and the lightest mass eigenstate has to be heavier than about m χ 800 GeV. It is possible that this bound could be softened in a modified scenario, where the higher dimensional operators arise at scales low enough, such that the lightest charge three states decay promptly inside the detector. Further studies would be necessary to explore this scenario.
For n = 1, the pair produced charge two leptons can lead to final states with two or more leptons and missing energy. We studied the leading production of the lightest charge two mass eigenstate and the subsequent decay into W s and SM leptons. We assume only couplings to one lepton family in order to avoid bounds from lepton flavor violation. If the new vector leptons couple only to muons we find that searches for multiple light leptons and missing energy in the final state constrain the mass of this lightest state to be heavier than about m χ 460 GeV. The same analysis for a scenario in which only couplings to taus are assumed yields a weaker bound of m χ 320 GeV. A search for one hadronic tau and three light leptons leads to very similar bounds for the scenario in which only tau couplings are present.
We also studied the possibility of hadronic tau pairs in the final state and conclude that the present searches are not sensitive to our model. Future multi-lepton searches at LHC 13 as well as dedicated searches for 2 leptons and 2 hadronic W s should offer excellent opportunities to probe the considered model.
We briefly considered a modified n = 1 model, where the charge two leptons predominantly decay into an additional stable, neutral state with a mass close to the lightest charge two state. In this case searches for the charge 2 lepton are more challenging and the current bounds get relaxed.
In summary, the bounds from direct searches on the new vector leptons in the various scenarios with m 1 < m 2 real and positive. The most general parametrization of the Z L , Z R matrices reads with s 2 L + c 2 L = s 2 R + c 2 R = 1. We then denote the left-and right-handed components of the mass eigenstates by in which P L,R = 1 2 (1 ± γ 5 ) are the chiral projection operators. We collect the remaining vector-like lepton N together with the charged SM leptons into vectors η L = (P L e, P L µ, P L τ, P L N ) T , η R = (P R e, P R µ, P R τ, P R N ) T , even though they can only mix in the n = 1 scenario.
In the couplings of the photons we have Q η = −1 for η = and Q η = Q N for η = N . For the Higgs couplings with the mass eigenstates χ we find the following expressions For the Higgs couplings with η we expand in first order in v 2 /M 2 L and find For the couplings of the Z boson with χ we find, The couplings of the Z boson with η read In the case of the left-handed couplings, in principle also flavor changing couplings among the SM leptons are generated at the first order in v 2 /M 2 L . However, they are additionally suppressed by tiny factors y y and therefore completely irrelevant for all practical purposes, and set to 0 in (B10).
The couplings of the W boson with χ and η read, Finally also W couplings between N and the SM neutrinos ν are induced where we neglected neutrino mixing, which is irrelevant for our study.