The Lepton Flavour Violating Higgs Decays at the HL-LHC and the ILC

Run-I results from the CMS collaboration show an excess of events in the decay $h \to \mu \tau_e$ with a local significances of 2.4$\sigma$. This could be the first hint of flavour violation in the Higgs sector. We summarise the bounds on the flavour violating Yukawa couplings from direct searches, low energy measurements and projected future experiments. We discuss the sensitivity of upcoming HL-LHC runs and future lepton colliders in measuring lepton-flavour violating couplings using an effective field theory framework. For the HL-LHC we find limits on BR($h\to \mu \tau$) and BR$(h\to e \tau) \lesssim \mathcal{O}(0.5)\%$ and on BR($h \to e \mu) \lesssim \mathcal{O}(0.02)\%$. For an ILC with center-of-mass energy of 1 TeV we expect BR($h \to e \tau$) and BR($h\to \mu \tau$) to be measurable down to $\mathcal{O}(0.2)\%$.


Introduction
The Large Hadron Collider (LHC) has successfully discovered a scalar resonance of mass around 125 GeV [1,2], with properties in close agreement with the Standard Model (SM) Higgs boson. Already now, both ATLAS and CMS collaborations have established its couplings to massive gauge bosons and photons to a high degree of precision and found no significant deviations from SM predictions [3][4][5].
Direct limits on rather complex Higgs-fermion interactions were instead much less probed during initial LHC runs. Only fairly weak limits were obtained on Higgs couplings to b-quarks [3], τ -leptons [6,7] and t-quarks [8][9][10][11][12]. Albeit the good overall agreement between theory and experiment, i.e. the total signal strength measured from production and decay modes is µ = 1.09 ± 0.11 [3], more conclusive evidence is required to establish that the observed scalar resonance is indeed the SM Higgs boson.
In particular the Higgs decays to the first and second generation fermions are yet to be observed. While searches have been performed by ATLAS and CMS to measure the Higgs decays to a µ + µ − pair [13,14] and by CMS to an e + e − pair [14], only upper limits have been obtained so far. Indirect limits on these decays could be potentially obtained from a very precise measurement of the total Higgs decay width. While future linear colliders can play an important role in this [15,16], at present, the Higgs width is only fairly loosely bounded to Γ H < 22 MeV (22.7 MeV) at 95% C.L from CMS (ATLAS) [17,18] using highly model-dependent off-shell coupling measurements [19,20]. Further, a global coupling fit to Higgs data indicates that the Higgs boson can have a sizable non-standard branching fraction, i.e. BR non-std < 0.26 at 95% C.L [21]. Hence, within the present experimental findings exotic Higgs decays into first and second-generation fermions, gluons or missing energy can be significantly enhanced compared to SM predictions.
Among the many different plausible non-standard decay modes of the Higgs boson, one of the most intriguing are flavour violating Higgs decays. In the SM, these decays are highly suppressed, thus any experimental confirmation of such a process will be conclusive evidence of physics beyond the standard model (BSM). During run-I of the LHC, a number of searches have been carried out both by CMS [22,23] and ATLAS [24,25]. The result reported by CMS constrains the branching ratio BR(h → µτ ) < 1.51 % at 95% C.L, while the upper limit on the branching ratio reported by ATLAS is 1.43%. Remarkably, both CMS and ATLAS reported a mild excess in the h → µτ e channel with a local significances of 2.4σ and 1.3σ respective 1 . These results can be explained with the best-fit branching ratio of BR(h → τ µ) = 0.84%(0.77%) for CMS (ATLAS). In addition, CMS also looked for flavour violation in the eµ and eτ channels [23]. Recently, ATLAS updated their results from the h → eτ analysis at 8 TeV [24].
Thus, motivated by the recent searches carried out by CMS and ATLAS, we present a detailed analysis of flavour violation in direct and indirect experimental searches and compare their sensitivities. We first review the different experimental constraints on Higgs lepton-flavour violating and non-violating couplings in section 2. Here we also discuss some of the future limits of low energy experiments, e.g. MEG-II, Belle-II and super KEKB. After establishing the reach of low energy constraints using an EFT framework, we study the sensitivity of future high luminosity LHC runs on lepton flavour violating decays in section 3. We provide a case study for h → eτ at the ILC in section 4. In sections 3 and 4, we consider the respective LFV branching ratios as free parameters and evaluate how well they can be constrained in collider measurements. Finally we discuss our findings and summarise them in section 5.

Higgs couplings in the lepton sector
In the SM, the Yukawa couplings are proportional to the masses of the fermions. Confirmation of this hypothesis requires independent measurements of the fermion masses and their coupling strength to the Higgs boson. That is a strenuous task, particularly for the Higgs boson couplings to the first and second generation fermions. A precise statement about the relation between Yukawa couplings and fermion masses is still at stake. Because of the observation of the recent excess reported by CMS in the h → µτ channel, we are restricting ourselves to a study of LFV interactions 2 . In this section we review and update the existing searches on such LFV decays and summarise their bounds.
To give an interpretation of the measurements in terms of LFV interactions we consider an effective theory where the interaction between Higgs and fermions are given by the Yukawa interactions where we use Y ii = m i v and f L and f R are the charged leptons. The Yukawa coupling matrix is parametrised [27] by where F L is the fermion doublet, F R is the singlet and H is the SM scalar doublet. For Λ → ∞, we recover the SM Yukawa structure. There are also some gauge invariant dimension-6 operators involving derivatives, induced by [27] However, these operators do not contribute to the Hff couplings as shown in Eq. 2.1 after EWSB and hence we disregard further discussions of these operators in the present study 3 .

Present status of Higgs couplings in the lepton sector
We list the current constraints on the Higgs Yukawa couplings in the lepton sector. First, we briefly summarise the status of the flavour diagonal and off-diagonal leptonic decays from direct searches during run-I at the LHC. Next we discuss the low energy constraints on the LFV Yukawas in an EFT framework.

Limits from direct searches
• h → e + e − : An upper limit on its branching ratio has been obtained by CMS [14] of BR(h → e + e − ) < 0.19%, which is about 3.7 × 10 5 times that of the SM expectation.
• h → τ + τ − : Both ATLAS [6] and CMS [7]  • h → µτ : The search was conducted in the channel pp → h → µτ , followed by the leptonic as well as hadronic decays of τ . CMS reported a slight excess of events around m h = 125 GeV in the h → µτ e channel with a local significance of 2.4σ [22]. From this they obtained BR(h → µτ ) < 1.51% at 95% CL with a best-fit of (0.84 +0.39 −0.37 )%. ATLAS has set an upper limit of 1.43% on this branching ratio at 95% CL [24]. From the upper limit on BR(h → µτ ) from CMS one obtains (2.7) In our calculation, we have used that the partial decay width of 125 GeV Higgs into two fermions is, where α = β = e, µ, τ and α = β. The branching ratio for this decay mode is where Γ SM = 4.1 MeV.

Limits from low-energy measurements
We motivate below the different bounds on these Yukawa couplings that emerge from the low energy flavour violating processes and summarise these limits in Table 1.
• Y µτ is constrained by the non-observation of processes like τ → µγ and τ → 3µ. The branching ratio of the process τ → µγ is bounded by BR(τ → µγ) < 4.4 × 10 −8 at 90% C.L. [70,71]. Assuming the low energy dynamics to be governed by the two flavour violating couplings Y µτ and Y τ µ , the decay width of this process reads as [27] Γ where the Wilson coefficients, c L and c R at one loop are given by (2.11) In the above expression, for simplicity, we assume that Y τ µ = Y µτ . Higher-order corrections to c L/R receive contributions from the top Yukawa coupling and hence can be large [27]. For example, including two-loop contributions, c L/R increases by a factor O(4). For a full discussion on the dependence of Y tt we refer to Ref. [27]. Hence by assuming Y τ µ = Y µτ and also a SM-like Y τ τ , we obtain Y µτ 0.011.
In presence of flavour violating Yukawa interactions, the process τ → 3µ has a decay width of Taking into account the one-loop contribution, the constraint on the Yukawa coupling is, Y µτ < 0.177. Similar to the previous limit, this also depends on the couplings Y τ τ , Y µµ and Y tt and their SM values have been assumed in deriving the limit on Y µτ . This limit is weaker than τ → µγ due to an additional factor of α (where α is the fine-structure constant).
• The coupling Y τ e is also constrained from similar low energy flavour violating processes, such as, τ → eγ and τ → 3e. The decay width for τ → eγ has a similar expression as given in Eq. 2.10, where µ has to be replaced by e [27]. Under similar assumptions, one obtains Y τ e 0.01 from τ → eγ, while from τ → 3e, this is relaxed owing to the extra α factor. Assuming Y τ e = Y eτ , we show the bounds on the Yukawas in Table. 1.
• The Yukawa coupling Y µe is severely constrained by the µ → eγ limit from MEG [72].
• The stringent constraint on µ → eγ branching ratio can also be used to set the limit on the product of the flavour violating couplings Y τ µ and Y τ e . The limit from MEG, Note that, in the limit Y µe = 0, the two loop Barr-Zee diagram does not occur in this case.

Future limits
In this subsection we briefly discuss the expected future limits on the flavour diagonal and flavour violating couplings of the Higgs boson in the lepton sector.
• h → e + e − : Ref. [80] discusses the present and future bounds on the Y ee Yukawa coupling. The ACME bound on the EDM puts a strong constraint on the imaginary part of this Yukawa, i.e. < 1.7 × 10 −2 times the SM electron Yukawa, Y e . The deviations of the real part are far less constrained. The authors find that the constraint coming from 8 TeV LHC data is the strongest. A limit κ e < 611 can be derived, where κ e is a multiplicative complex parameter to the SM Yukawa, which parametrises the deviation from the SM Yukawa coupling. Noting that for the 14 TeV run, the production cross-section of the Higgs will increase by a factor of ∼ 2.5, they project that κ e ∼ 260(150) with L = 300(3000) fb −1 . For a 100 TeV pp collider with L = 3000 fb −1 , κ e ∼ 75.

Searches
Experimental limit on Limits on Yukawas branching ratios • h → µ + µ − : In Ref. [81], it is mentioned that by combining the gluon fusion and weak boson fusion channels, it is possible to obtain a 3σ significance for h → µ + µ − at an integrated luminosity of 300 fb −1 . It is also projected by CMS and ATLAS [82, 83] that for the 14 TeV run with an integrated luminosity of around 1200 fb −1 , one can observe the h → µ + µ − mode with a 5σ significance.
• h → τ + τ − : Future runs of the LHC and the ILC are expected to improve the sensitivity of this coupling. From Ref. [16] one finds that the uncertainty on this coupling measurement decreases to about 12.5% and about 1.5% respectively at future runs of LHC and ILC.
• Finally, we briefly mention the novel proposal given in Ref. [86][87][88] which outlines an experimental technique to put bounds on the the flavour diagonal Higgs couplings to mostly the first generation fermions. The authors propose to achieve this by measuring isotope shifts in atomic clock transitions. This method can potentially bound the Higgs-light fermion couplings better than the present and future runs of the LHC. By studying the isotope shift of the Y b ion they show that one can bound the couplings to where ∆ is the isotope shift measurement uncertainty.
We summarise the constraints from the direct searches, low energy experiments and sensitivities from the future experiments in Table 1. For simplicity, we assume that Y βα = Y αβ with α, β = e, µ, τ . We find that the strongest individual constraints on Y µτ and Y eτ come from the run-I searches at the LHC. These constraints however have been obtained assuming no correlation between the two flavour violating couplings. Assuming the validity of an EFT and a non-zero Y µτ explaining the excess seen by CMS, the limits on Y eτ are strongest from a measurement of BR(µ → eγ) which sets an upper limit on |Y µτ Y eτ | at 3.98×10 −8 . Thus, combining the excess in h → µτ with the best-fit branching ratio of BR(h → µτ ) = 0.84% and the MEG limit [72] on µ → eγ, we get Y eτ ∼ 2 × 10 −5 , which is stronger than present LHC limit (Y eτ < 1.7 × 10 −3 at 95% CL). For generic complex Yukawas, we show the constraints on Re(Y eτ ) and Im(Y eτ ) in Fig. 1. In the left panel, we show the existing constraints from the direct searches and the low energy experiments. The gray region is in agreement with the flavour violating low energy processes, i.e. τ → eγ, τ → 3e, electron EDM and electron g − 2 as summarised in Table 1. Finally we show the combined constraint from h → µτ (CMS) and µ → eγ (MEG). The blue (solid) and red (dashed) lines represent BR(h → µτ ) = 1.51% and 0.84% respectively. In this figure, for the charged lepton decays τ → eγ and τ → 3e, we have considered the one loop contributions. The two loop contribution for µ → eγ depends on the Y µe coupling and vanishes in the limit Y µe = 0. In the right panel, we show the individual limits from τ → eγ (pink region) and the future sensitivity from Belle-II [85] (red line) which will constrain the flavour violating Yukawa couplings by a further factor of O(0.1). The gray region in the right panel satisfies the constraint from h → µτ and µ → eγ decays, where Figure 1. the branching ratio of h → µτ ranges from 0.84 − 1.51%. From Fig. 1, it is evident that the limit on Y τ e is more stringent than the current LHC limit, provided that BR(h → µτ ) lies between 0.84 − 1.51% which is required to explain the excess seen by CMS. In case Y µτ is negligibly small, then BR(h → eτ ) can be as large as 0.69% from the direct LHC searches. The future sensitivity of MEG-II [84] will constrain this coupling even further. In addition, the future constraints coming from the Mu2e experiment can become even more severe [89]. However, these constraints depend strongly on the validity of the effective field theory. If the underlying degrees of freedoms are light enough, then the EFT description will not be valid [32,40]. In the following section 3, we analyse the collider reach to probe lepton-flavour violating Higgs interactions in the eµ, µτ and eτ decay modes for the 14 TeV LHC and its future upgrades with 3000 fb −1 . Subsequently, we analyse the reach of h → eτ at the ILC.

Direct detection of LFV at the HL-LHC
From the previous section, we see that the strongest constraint on Y µτ arises from a direct search at CMS. However, the constraints from low energy measurements on Y eτ and Y eµ are still considerably stronger. However, it is important to note that all constraints derived from the low energy experiments are subject to correlations among various Yukawa couplings, which are innate in any EFT approach. In this section, we evaluate the highluminosity LHC's potential to set limits on the three LFV Higgs decays, while being com-pletely agnostic about low-energy constraints. If direct searches find evidence for both h → µτ and h → eτ in the near future -and if then as a result the bound on |Y µτ Y eτ | is found to be weaker than obtained from the low energy experiments an interpretation in terms of an effective field theory approach will be at stake and has to be augmented with a less constraining theory assumption.
Taking a cue from Sec. 2.1, here we evaluate the possible reach of a 14 TeV highluminosity LHC with integrated luminosities up to 3000 fb −1 in measuring lepton-flavour violation in the Higgs sector. Hence, we implement the relevant flavour violating interactions of the Higgs and the charged leptons in FeynRules [90]. The generated Universal FeynRules Output (UFO) [91] model is then used as input to the Monte-Carlo (MC) event generator MadGraph5 aMC@NLO [92]. Parton-showering and hadronisation is carried out using Pythia 6 [93]. Thus, the τ decays are simulated using TAUOLA [94]. Finally we perform a detector simulations using Delphes 3 [95] where the jets are reconstructed using the anti-k t algorithm [96] with a jet parameter of R = 0.5, as implemented in FastJet [97]. In the following three subsections we show the prospects of each of the lepton flavour violating Higgs decays.

Prospect of µτ channel
In this subsection we focus on the channel where an excess of events have already been seen, i.e. in h → µτ . Among all the possible final states, e.g. eµ + / E T , µµ + / E T and µ + τ h + / E T , the channel where τ → µ + / E T is the cleanest one. In order to reduce the backgrounds, we implement the CMS-like selection cuts as listed in Ref. [22]. The different backgrounds for this final state are listed in Table. 2. The order to which these backgrounds are computed is discussed in section 3.2. For the signal, we first consider the branching ratio of h → µτ as 0.84%, that corresponds to the central value for the excess reported by CMS. Hence, for the 14 TeV LHC, the signal and background events for pp → h → eµ + / E T with L ∼ 37 fb −1 are 251 and 1135 respectively. This results in a 2.1σ statistical significance. We further analyse the sensitivity reach of the HL-LHC for this channel. In Table 3, we list the number of signal and background events for an integrated luminosity of 3000 fb −1 . We find that this branching ratio h → µτ can be constrained down to 0.76% (1.90%) with a 2σ (5σ) statistical significance. However, this can be further optimised by adding more cuts or using a multi-variate analysis. We discuss such an optimisation for the h → eτ channel in the section 3.2.
To derive the above significances and reach, we define the statistical significance, S 1 by assuming a flat 10% systematic uncertainty on the backgrounds where κ = 10% in our case. We further note that for a very low value of systematic uncertainty (κ 0), the significance is given by and one achieves 2σ significance for a branching ratio of ∼ 0.025%.

Prospect of eτ channel
Here we consider the flavour violating Higgs decay to eτ , followed by the hadronic as well as leptonic decays of τ . For the 14 TeV eτ analysis we follow the proposal of Ref.
[98] and adopt a τ tagging and mistagging efficiencies of (40%, 0.2%) 4 . Here we consider the following final states: The major SM backgrounds for the processes mentioned above are τ τ + jet (computed at the next-to-next-to leading order (NNLO) [99]), V V (with V = W ± , Z) (at the nextto leading order (NLO) [100]), W + jets (at NLO [101]), with W decaying leptonically and one of the jets mistagged as a τ -hadron, ee+jets (computed at NNLO [102]), tt (at next-to-next-to-next-to leading order (N 3 LO) [103]), single-top (at NLO [104]) and the SM-Higgs backgrounds (also computed at NNLO), i.e. h → τ + τ − . For the single-top background, the W t mode has the dominant contribution for our scenario, whereas the s-and t-channel contributions are negligible. So, for simplicity, we multiplied our leading order (LO) cross-section by the NLO k-factor for the W t mode. Besides these, there are some fake backgrounds like QCD multi-jets, where the jets might fake leptons, and W γ, where the photon might convert to an electron-positron pair. However, it is very difficult to get a proper estimate for these backgrounds without doing a full detector simulation or without using data driven methods. For all practical purposes, these backgrounds will not significantly alter our quoted results. Hence we neglect these in the present analysis.
For the τ τ + jet background, viz., the major background for the eµ final state, we perform an ME-PS MLM matching in the MadGraph framework. For the W + jets background, which is the single most important background for the eτ had final state, we also simulate a matched sample with up to two partonic jets. We select only those events where at least one of the jets fake a τ -hadron. We perform an inclusive study and demand no jet veto. Similarly a matched sample was generated for the ee+jet background, which is the dominant background for the ee final state. To optimise the signal sensitivity, we adopt similar cuts as done by the CMS 8 TeV analysis for h → eτ final state [23]. In addition, we also optimise over the p T cuts in order to gain maximum sensitivity.
We use different selection cuts for the three different final states ee + / E T , µe + / E T and eτ had + / E T . For the eµ + / E T final state we use the following selection cuts: • / E T > 30 GeV • The collinear mass variable: 105 GeV < M µe collinear < 145 GeV, where the collinear mass is the following, with the visible momentum fraction of the τ decay products being, • In addition, we use 10 sets of cuts for p e T and p µ T optimised around the CMS-like cut, viz. p e T > 50 GeV and p µ T > 10 GeV. For the ee + / E T final state, the cuts are exactly same as in the previous case with the following transformations µ → e 2 , e → e 1 , where e 1 is the electron coming from the Higgs decay and e 2 comes from the τ decay. For the remaining final state eτ had + / E T , we use the following selection cuts: In addition to the above CMS-like selection cuts, we implement the cut on / E T and the optimised p T cut giving the maximum sensitivity (shown in Table 4). We show the number of events after all the trigger and selection cuts in Table 5 and show the exclusion limit of the BR(h → eτ ) in Table 6.
Channel optimal p T cut eµ + / E T p e T > 50 GeV and p µ T > 10 GeV ee + / E T p e 1 T > 50 GeV and p e 2 T > 10 GeV eτ had + / E T p e T > 55 GeV and p τ had T > 50 GeV Table 4. Optimized p T cuts for the three final states.
From Table 6, one can observe a BR(h → eτ ) of around 1.5% at 5σ for the eµ final state. For the ee final state we require a somewhat larger branching ratio to have a 5σ statistical significance. Whereas for the hadronic final state one can not go below ∼ 5.1% in order to obtain a 5σ significance with such a cut-based analysis. If we statistically combine these three significances in quadrature, then we attain a 2σ significance for a branching ratio of 0.50%. Combining them additively, one achieves 2σ for as low a branching ratio as ∼ 0.32%. To see if one can probe lower branching ratios in the eµ channel, we exploit the kinematics of both the signal and the background in more details. We perform a multivariate analysis with the Boosted Decision Tree (BDT) algorithm using the root based TMVA [105] framework. We choose 11 kinematic variables for this purpose, viz.
For the MVA, we take care of the issue of overtraining of the signal/background. The Kolmogorov Smirnov (KS) test is used to check for the overtraining of a sample. The test sample is not overtrained if the KS probability lies between 0.1 to 0.9. In most cases, a critical KS probability value more than 0.01 [106] ensures that the signal and background samples are not overtrained.
We find after a proper training of the sample, that an optimised cut on the BDT yields a better reach on the branching ratio. In table 7, we tabulate the signal and dominant background events at an integrated luminosity of 3000 fb −1 after an optimised cut on the BDT variable.  Table 5. Signal events for BR(h → eτ ) = 0.1% after all selection cuts. The superscript "optimal" signifies the number of events for the optimal p T cuts. We also show the corresponding background events for the same set of cuts. The number of events are computed for L = 3000 fb −1 .
Channel BR % (S optimal ) We find that one can go down to as low as ∼ 0.48% in order to achieve a 2σ significance. To achieve a 5σ discovery one can not go below a branching ratio of 1.20%. We see that  Table 7. Same as in Table 5 for the h → eµ + / E T channel after an optimal cut on the BDT variable.
the MVA analysis improves the reach by a factor of ∼ 1.28 for the eµ + / E T final state. We also note that in 3.2 we obtained the reach on BR(h → µτ ) with the CMS-like cuts. With an MVA, however, we expect a similar improvement in this channel as in the eτ sector.
We also note that one can attain a 2σ significance in the eµ + / E T channel with the cut-based analysis for a branching ratio of 0.028% by using Eq. 3.2.

Prospect of eµ
Inspired by CMS's direct search for the flavour violating decay h → eµ [23], we study the prospect of observing this channel at the HL-LHC. For this analysis, we apply the following simple set of cuts : • p T (e) > 40 GeV and p T (µ) > 40 GeV • |η e | < 1.479 and |η µ | < 0.8 (in the barrel) • / E T < 20 GeV • 123 GeV < m h < 127 GeV.
Here also we compute the backgrounds at the orders specified in section 3.2. The major backgrounds are eµ + / E T (dominantly from W W production), Drell-Yan production of τ τ , tt, e(µ) + τ + / E T (also dominantly from W W production), τ τ + / E T (mainly coming from W W and ZZ) and single top (dominantly from the W t production). In Table 8 we list the number of signal and background events after all the selection cuts for the HL-LHC at 3000 fb −1 . Finally in Table 9, we show the prospective reach.
Here also we note that one can attain a 2σ significance in the eµ channel with the this analysis for as low a branching ratio of 1.65 × 10 −3 % by using Eq. 3.2.

h → eτ at ILC
In the previous section, we discussed the prospects of observing a lepton flavour violating Higgs in all the three possible channels at a 14 TeV LHC with an integrated luminosity going up to 3000 fb −1 . We saw that one can definitely expect improvements compared to the 8 TeV results but due to the large backgrounds and huge uncertainties, these are  not so dramatic as one would like. We know that LHC is plagued with huge backgrounds and hence we can expect better precision at lepton colliders. In this section, we repeat the analysis for h → eτ for centre of mass energies of √ s = 250 GeV and 1 TeV at an ILC machine. Here we just want to point out the improvement over the LHC. A similar improvement can be expected for the h → µτ as well. For eµ, we do not expect a significant improvement at ILC because in section 3.3, we already saw that the reach for the branching ratio can be as low as 1.95 × 10 −2 %. The number of signal events are expected to be very low at the ILC for such small branching ratios.
The two main topologies that we study here are the associated production of the Higgs with a Z-boson and the Higgs produced in association with neutrinos through the t-channel fusion diagram. At 250 GeV, the associated production with a Z-boson offers the largest cross-section with the Z decaying hadronically, whereas for the leptonic modes of Z, the total cross-section is suppressed. Hence, we study in detail the Zh, Z → qq production for √ s = 250 GeV. For the 1 TeV study, we consider the / Eh channel which includes both the t-channel contribution mediated by W -boson and an s-channel contribution from the Zh topology. However, here the contribution coming from the latter is nominal. The different final states that we study in this section can be summarised as : • e + e − → Zh, h → τ e, with Z →≥ 2j and τ → eν, µν or τ tagged as τ had • e + e − → / Eh, h → τ e, with τ → eν, µν or τ tagged as τ had .

ILC at √ s = 250 GeV
In this subsection, we simulate a prospective analysis at the ILC with √ s = 250 GeV. The dominant backgrounds for the various final states are: • Z νν, Z and W ν , followed by the hadronic decays of Z and W , viz. Z → qq and W → qq . In short, we denote these backgrounds with the tag, 3V -like, • eeZ and τ τ Z, followed by Z → qq. In short, we denote these backgrounds as 2V -like, where q = light jets + b-jets and = e, µ, τ .
Here we perform a cut-based analysis with the following set of general selection cuts: • Demand that at least two jets reconstruct the Z-boson mass with a window of 25 GeV, i.e., M Z − 25 GeV < M jj < M Z + 25 GeV.
• For the associated production Zh, we apply the cuts on the invariant mass of the visible products: The visible invariant mass lies in 110 GeV < M eµ < 140 GeV for eµ.
The visible invariant mass lies in 120 GeV < M ee < 130 GeV for ee.
The visible invariant mass lies in 110 GeV < M eτ had < 140 GeV for eτ had .
• In addition to the above cuts, we demand a cut on | cos(θ jj )| < 0.8 for the eejj channel in order to reduce the 2V -like background.
In Table 10, we list the number of signal and background events after the selection cuts for the dominant modes. Finally in Table 11, we show the reach of the ILC for BR(h → eτ ) in these dominant final states with √ s = 250 GeV. For this purpose, we use a different formula for the significance as compared to LHC, owing to the fact that the systematic uncertainties are expected to be significantly less for the ILC. We use the standard formula as quoted in the ILC Snowmass report [107], viz.
On statistically combining these three significances in quadrature, we attain a 2σ significance for a branching ratio of 0.38%. Combining them additively, one achieves 2σ for as low a branching ratio as ∼ 0.25%. So we see that the ILC at √ s = 250 GeV performs comparably with the high luminosity LHC for these sets of simple cuts. To see if we have better prospects at the 1 TeV ILC, we study the vector boson fusion topology in details in section 4.2.

ILC at √ s = 1 TeV
The ILC at 1 TeV motivates us to study the Higgs in association with missing energy because of much cleaner backgrounds. Another reason for studying the prospects at the 1 TeV ILC is the 1000 fb −1 integrated luminosity. Here, we implement the following selection cuts:   • For the eµ + / E channel, we apply : The invariant mass of the visible particles, 50 GeV < M eµ < 120 GeV • For the ee + / E channel, we apply : 2.0 < ∆φ e 1 / E < 3.0, where e 1 is the electron with the hardest p T | p vis | < 200 GeV 2.0 < ∆R e 1 / E < 4.5 ∆R e 1 e 2 < 2.8, where e 2 is the second hardest electron.
| cos(θ e 1 e 2 ) < 0.8| For the eτ had + / E channel, we apply : 1.5 < ∆φ τ had / E < 3.0, where e 1 is the electron with the hardest p T | p vis | < 200 GeV where the notations of the variables are self-explanatory.
The dominant backgrounds for these channels can be summarised as: • 2 + 2ν and where = e, µ, τ .
These optimised cuts are applied to compute the significance in these three channels. We must note here that the ∆φ distributions for the τ + τ − background peak at 0 and π. Hence our ∆φ cuts almost completely eradicate this background. The signal and background events after imposing the above sets of selection cuts are found in Table 12  Here also, on statistically combining the three significances in quadrature, we obtain a 2σ significance for a branching ratio of 0.22%. Combining them additively, one achieves 2σ for as low branching a ratio as ∼ 0.16%. We see that the ILC at 1 TeV with L = 1000 fb −1 performs better. But, we must note that with neither scenario we can measure a branching ratio of less than 0.1%.
In doing the computations for the ILC, we consider unpolarised beams for the 250 GeV analysis because there is no significant enhancement in the signal. But for the 1 TeV analysis we consider the e − and e + polarisations as (-80,20). Here we see a significant increase in the signal cross-section by ∼ 2.15 times even though the dominant background, i.e. 2 2ν also increases by a factor of 2. So we gain in sensitivity for the polarised beams in this scenario. In contrast to the LHC studies, here we have performed a leading order analysis. However, the next-to-leading order effects are not expected to change our conclusions appreciably.
Here, we must comment on the fact that the ILC capabilities [15] on measuring an invisible branching ratio (which can very well be read as non-standard branching ratio) can be as low as 0.54% for a 250 GeV ILC machine with an integrated luminosity of 250 fb −1 and this decreases to around 0.22% for √ s = 1 TeV with L = 1000 fb −1 . As we can see that the numbers that we have obtained are in the ballpark of these quoted limits. We also mention that our analysis in the ILC sector can be further improved by optimising the cuts to a greater degree or by using a multivariate technique after identifying interesting variables.

Summary
The run-I results from CMS and ATLAS gave us the first hint at flavour violation in the Higgs sector in the channel h → µτ with a best-fit branching ratio of less than a percent. The 8 TeV collider searches in the other two LFV channels, i.e. h → eτ and h → eµ did not have any significant excess over the respective SM backgrounds. Confirmation of any of these flavour violating processes will necessarily indicate the existence of new physics.
In this work, we performed rigorous collider analyses to get an estimates of the discovery prospect of all the three lepton flavour violating Higgs decays, at the high luminosity run of the 14 TeV LHC. We also performed a case study for h → eτ at the ILC with √ s = 250 GeV and 1 TeV.
Below we summarise our findings from the collider analyses.
• We analysed the prospect of h → µτ and h → eτ at the 14 TeV LHC with L = 3000 fb −1 and found that a BR(h → µτ /eτ ) of ∼ 0.5 % can be probed with a 2σ significance.
• We obtained the prospects of observing the cleaner channel h → eµ at the HL-LHC with an integrated luminosity of 3000 fb −1 . We found that to achieve a 2σ significance, one can go as low as 1.95 × 10 −2 % in the branching ratio.
• In addition to the LHC, we also showed the prospects of studying Higgs flavour violation at the ILC with √ s = 250 GeV and 1 TeV. A branching ratio, BR(h → eτ ) as low as 0.24% can be probed for the the h + / E channel with a 2σ statistical significance and with an integrated luminosity of 1000 fb −1 at the 1 TeV ILC. We also expect similar improvement in the µτ sector compared to HL-LHC.
Hence, we see that the discovery of lepton flavour violation in h → µτ and/or the h → eτ at collider experiments will imply large branching ratios of few times O(0.1%). The simultaneous discovery of both these channels at the 14 TeV HL-LHC or at the ILC will no doubt question the validity of an EFT approach. This will prompt us in building models with relatively light massive states. However, if there is only evidence for one of the processes, we can not make strong statements about the existence of light degrees of freedom. Our present study thus encourages the experimental groups to look for these unique signatures and gives an estimate of the smallest branching ratios that can be probed at the near-future colliders.