Feasibility at the LHC, FCC-he and CLIC for sensitivity estimates on anomalous \tau-lepton couplings

# Feasibility at the LHC, FCC-he and CLIC for sensitivity estimates on anomalous τ-lepton couplings

Apartado Postal C-580, 98060 Zacatecas, México.
M. Köksal222mkoksal@cumhuriyet.edu.tr Deparment of Optical Engineering, Sivas Cumhuriyet University, 58140, Sivas, Turkey.
A. A. Billur333abillur@cumhuriyet.edu.tr Deparment of Physics, Sivas Cumhuriyet University, 58140, Sivas, Turkey.
Apartado Postal C-585, 98060 Zacatecas, México.
July 29, 2019
###### Abstract

In this paper, we present detailed studies on the feasibility at , and colliders for model-independent sensitivity estimates on the total cross-section and on the anomalous interaction through the tau pair production channels , and at the mode. Measurements of the anomalous couplings of the -lepton and provide an excellent opportunity to probing extensions of the Standard Model. We estimate the sensitivity at the Confidence Level, and we consider that the -lepton decays leptonically or semi-leptonically. We found that of the three considered colliders, the future CLIC at high energy and high luminosity should provide the best sensitivity on the dipole moments of the -lepton and , which show a potential advantage compared to those from LHC and FCC-he.

###### pacs:
13.40.Em, 14.60.Fg
Keywords: Electric and Magnetic Moments, Taus, LHC, FCC-he, CLIC.

## I Introduction

The study of the -lepton by the ATLAS and CMS Collaborations ATLAS-tau (); ATLAS1-tau () at the Large Hadron Collider (LHC) has developed significantly in recent years to the point where they have a very active physical program. Furthermore, with the existence of a Data2018 () scalar boson Englert (); Higgs (); Higgs1 (); Guralnik () established by the ATLAS Atlas () and CMS CMS () experiments, making it possible to complete the Standard Model (SM) of particle physics, that is the theory that describes the particles of matter we know, and their interactions. However, there are fundamental problems in the SM like: dark matter, dark energy, hierarchy problem, neutrino masses, the asymmetry between matter and antimatter, etc.. These problems demand the construction of new machines that operate at a much higher energy than the LHC, with cleaner environments and that allow exploring other components of the universe. For these and other reasons, the scientific community of High Energy Physics has the challenge of discovering that the universe is made in its entirety.

There are several proposals to build new, powerful high-energy and high-luminosity hadron-hadron (), lepton-hadron () and lepton-lepton ) colliders in the future at CERN for the post LHC era that will open up new horizons in the field of fundamental physics.

The future colliders, such as the Large Hadron Electron Collider (LHeC) Klein (); Fernandez (); Bruening () and the Future Circular Collider Hadron Electron (FCC-he) FCChe (); Fernandez (); Fernandez1 (); Fernandez2 (); Huan (); Acar (); Bruening (), are a hybrid between the and colliders, and they will complement the physical program of the LHC. These colliders have the peculiarity that can be installed at a much lower cost than that of the collider. Furthermore, they provide invaluable information on the Higgs and top sectors, as well as of others heavy particles as the -lepton. The FCC-he study puts great emphasis in the scenarios of high-intensity and high energy frontier colliders. These colliders, with its high precision and high-energy, could extend the search of new particles and interactions well beyond the LHC. In addition, in comparison with the LHC, the FCC-he has the advantage of providing a clean environment with small background contributions from QCD strong interactions. In the case of the future collider as the Compact Linear Collider (CLIC) Abramowicz (), although with a much lower center-of-mass energy than the colliders, is ideal for precision measurements due to very low backgrounds.

In this paper we have based our study on three phenomenological analyses for finding physics beyond the Standard Model (BSM) at present and future colliders to be able to compare the electromagnetic properties of the -lepton. We consider collisions at the LHC with 13, 14 TeV and luminosities 10, 30, 50, 100, 200 . Another scenario is the FCC-he with 7.07, 10 TeV and . The CLIC at CERN is another option with 1.5, 3 TeV and luminosities 100, 300, 500, 1000, 1500, 2000, 3000 have been assumed. With a large amount of data and collisions at the TeV scale, LHC, FCC-he and CLIC provide excellent opportunities to model-independent sensitivity estimates on the total cross-section of the production channels , and , as well as of the Magnetic Dipole Moment (MDM) and Electric Dipole Moment (EDM) of the -lepton and .

The theoretical prediction on the MDM of the -lepton in the SM is well known with several digits Passera1 ():

 SM:aτ=0.00117721(5), (1)

while the DELPHI DELPHI (), L3 L3 (), OPAL OPAL (), BELLE BELLE () and ARGUS ARGUS () Collaborations report the current experimental bounds on the MDM and the EDM in Table I.

The best experimental results on the MDM and the EDM are reported by the DELPHI and BELLE collaborations using the following processes and , respectively. The EDM of the -lepton, is a very sensitive probe for CP violation induced by new CP phases BSM Yamanaka1 (); Yamanaka2 (); Engel (). It is worth mentioning that the current Particle Data Group limit was obtained by DELPHI Collaboration DELPHI () using data from the total cross-section at LEP2.

The MDM and EDM of the -lepton allow a stringent test for new physics and have been deeply investigated by many authors, see Refs. Bernreuther0 (); Iltan1 (); Dutta (); Iltan2 (); Iltan3 (); Iltan4 (); Gutierrez1 (); Gutierrez2 (); Gutierrez3 (); Gutierrez4 (); Ozguven (); Billur (); Sampayo (); Passera1 (); Eidelman1 (); Koksal3 (); Arroyo1 (); Arroyo2 (); Xin (); Pich (); Atag1 (); Lucas (); Passera2 (); Passera3 (); Bernabeu (); Bernreuther (); Koksal1 (); Aranda (); Fomin () for a summary on sensitivities achievable on the anomalous dipole moments of the -lepton in different context.

A direct comparison between Eq. (1) and the results given in Table I clearly shows that the experiment is far from determining the anomaly for the MDM of the -lepton in the SM. It is therefore of great interest to investigate and propose mechanisms model-independent to probe the dipole moments of the -lepton with the parameters of the present and future colliders, i.e. the LHC, FCC-he and CLIC, rendering such an investigation both very interesting and timely.

The outline of the paper is organized as follows: In Section II, we introduce the -lepton effective electromagnetic interactions. In Section III, we show sensitivity estimates on the total cross-section and the -lepton MDM and EDM through at the LHC, at the FCC-he and at the CLIC. Finally, we present our conclusions in Section IV.

## Ii The effective Lagrangian for τ-lepton electromagnetic dipole moments

We following Refs. Escribano1 (); Escribano2 (); Eidelman1 (), in order to analyze in a model-independent manner the total cross-section and the electromagnetic dipole moments of the -lepton through the channels at the LHC, at the FCC-he and at the CLIC and using the effective Lagrangian description. This approach is appropriate for describing possible new physics effects. In this context, all the heavy degrees of freedom are integrated out leading to obtain the effective interactions with the SM particles spectrum. Furthermore, this is justified due to the fact that the related observables have not shown any significant deviation from the SM predictions so far. Thus, below we describe the effective Lagrangian we use with potential deviations from the SM for the anomalous coupling and fix the notation:

 Leff=LSM+∑nαnΛ2O(6)n+h.c., (2)

where, is the effective Lagrangian which contains a series of higher-dimensional operators built with the SM fields, is the renormalizable SM Lagrangian, is the mass scale at which new physics expected to be observed, are dimensionless coefficients and represents the dimension-six gauge-invariant operator.

### ii.1 τ+τ−γ vertex form factors

The most general structure consistent with Lorentz and electromagnetic gauge invariant for the vertex describing the interaction of an on-shell photon with two on-shell fermions can be written in terms of four form factors Passera1 (); Grifols (); Escribano (); Giunti (); Giunti1 ():

 Γατ=eF1(q2)γα+ie2mτF2(q2)σαμqμ+e2mτF3(q2)σαμqμγ5+eF4(q2)γ5(γα−2qαmτq2). (3)

In this expression, is the four-momentum of the photon, and are the charge of the electron and the mass of the -lepton. Since the two leptons are on-shell the form factors appearing in Eq. (3) are functions of and only, and have the following interpretations for .

parameterize the vector part of the electromagnetic current and it is identified with the electric charge:

 Qτ=F1(0). (4)

defines the anomalous MDM:

 aτ=F2(0). (5)

describes the EDM:

 dτ=e2mτF3(0). (6)

is the Anapole form factor:

 FA=−F4q2. (7)

It is worth mentioning that in the SM at tree level, and . In addition, should be noted that the term behaves under C and P like the SM one, while the term violates CP.

### ii.2 Gauge-invariant operators of dimension six for τ-lepton dipole moments

Theoretically, experimentally and phenomenologically most of the -lepton anomalous electromagnetic vertices involve off-shell -leptons. In our study, one of the -leptons is off-shell and measured quantity is not directly and . For this reason deviations of the -lepton dipole moments from the SM values are examined in a model-independent way using the effective Lagrangian formalism. This formalism is defined by high-dimensional operators which lead to anomalous coupling. For our study, we apply the dimension-six effective operators that contribute to the MDM and EDM Buchmuller (); 1 (); eff1 (); eff3 () of the -lepton:

 Leff=1Λ2[C33LWQ33LW+C33LBQ33LB+h.c.], (8)

where

 Q33LW = (¯ℓτσμντR)σIφWIμν, (9) Q33LB = (¯ℓτσμντR)φBμν. (10)

Here is the tau leptonic doublet and is the Higgs doublet, while and are the and gauge field strength tensors.

After electroweak symmetry breaking from the effective Lagrangian given by Eq. (8), the Higgs gets a vacuum expectation value GeV and the corresponding CP even and CP odd observables are obtained:

 κ = 2mτe√2υΛ2Re[cosθWC33LB−sinθWC33LW], (11) ~κ = 2mτe√2υΛ2Im[cosθWC33LB−sinθWC33LW], (12)

where, as usual is the sine (cosine) of the weak mixing angle.

The effective Lagrangian given by Eq. (8) gives additional contributions to the electromagnetic moments of the -lepton, which usually are expressed in terms of the parameters and . They can be described in terms of and as follows:

 ~aτ = κ, (13) ~dτ = e2mτ~κ. (14)

## Iii The total cross-sections in pp, e−p and e+e− colliders

As we mentioned above, the -lepton anomalous couplings offer an interesting window to physics BSM. Furthermore, usually the current and future colliders probing the feasibility of measured the anomalous couplings that are enhanced for higher values of the particle mass, making the -lepton the ideal candidate among the leptons to observe these new couplings.

We point out that the total cross-section for the channels at the LHC, at the FCC-he and at the CLIC are large enough to allow for a study of the anomalous electromagnetic couplings of the -lepton. The schematic diagram corresponding to these processes is given in Fig. 1, and the subprocess can be produced via the set of Feynman diagrams depicted in Fig. 2.

It must be noticed that, unlike direct processes ARGUS (); Sher (), L3 (), OPAL (); Grifols () and Galon (), the two-photon processes offers several advantages to study the electromagnetic tau couplings at the LHC, FCC-he and CLIC. The characteristics that distinguish them from the direct processes are mainly: 1) High sensitivity on and . 2) Increase of the cross-section for high energies and high luminosity. 3) They are extremely clean reactions because there is no interference with weak interactions as they are purely quantum electrodynamics (QED) reactions. 4) The photon-photon fusion processes are free from the uncertainties originated by possible anomalous couplings. 5) Since the photons in the initial state are almost real and the invariant mass of the tau-pairs is very small, we expect the effects of unknown form-factors to be negligible. 6) Furthermore, a very important feature is that the present and future colliders such as LHC, FCC-he and CLIC can produce very hard photons at high luminosity in the Equivalent Photon Approximation (EPA) of high energy , and beams, with which the final state photon identification has the advantage to determine the tau pair identification.

The main theoretical tool of our study for sensitivity estimates on the total cross-section of the processes , and and on the anomalous -lepton couplings, is the EPA. In the literature this approach is commonly referred to as the Weizsacker-Williams Approximation (WWA) Weizsacker (); Williams (). In general, EPA is a standard semi-classical alternative to the Feynman rules for calculation of the electromagnetic interaction cross sections. This approximation has many advantages. It helps to obtain crude numerical estimates through simple formulas. Furthermore, this approach may principally ease the experimental analysis because it gives an opportunity one to directly achieve a rough cross-section for subprocess through the research of the reaction , where symbolizes objects generated in the final state. The essence of the EPA is as follows, photons emitted from incoming charged particles which have very low virtuality are scattered at very small angles from the beam pipe and because the emitted quasi-real photons have a low virtuality, these are almost real.

It is worth mentioning that the exclusive two-photon processes can be distinguished from fully inelastic processes by the following experimental signatures: after of the elastic emission of a photon, incoming charged particles (electron or proton) are scattered with a small angle and escapes detection from the central detectors. This generate a missing energy signature called forward large-rapidity gap, in the corresponding forward region of the central detector Albrow (). This method have been observed experimentally at the LEP, Tevatron and LHC Abulencia (); Aaltonen1 (); Aaltonen2 (); Chatrchyan1 (); Chatrchyan2 (); Abazov (); Chatrchyan3 ().

Also, another experimental signature can be implemented by forward particle tagging. These detectors are to tag the electrons and protons with some energy fraction loss. One of the well known applications of the forward detectors is the high energy photon induced interaction with exclusive two lepton final states. Two almost real photons emitted by charged particles beams interact each other to produce two leptons . Deflected particles and their energy loss will be detected by the forward detectors mentioned above but leptons will go to central detector. Produced lepton pairs have very small backgrounds Albrow2 (). Use of very forward detectors in conjunction with central detectors with a precise synchronization, can efficiently reduce backgrounds from pile-up events Albrow1 (); Albrow2 (); Tasevsky (); Tasevsky1 ().

CMS and TOTEM Collaborations at the LHC began these measurements using forward detectors between the CMS interaction point and detectors in the TOTEM area about away on both sides of interaction point Sirunyam (). However, LHeC and CLIC have a program of forward physics with extra detectors located in a region between a few tens up to several hundreds of metres from the interaction point Fernandez (); CLIC2018 ().

### iii.1 Benchmark parameters, selected cuts and χ2 fitting

In this work, to evaluate the total cross-section , and and to probe the dipole moments and , we examine the potential of LHC, FCC-he and CLIC based colliders with the main parameters given in Table II. Furthermore, in order to suppress the backgrounds and optimize the signal sensitivity, we impose for our study the following kinematic basic acceptance cuts for events at the LHC, FCC-he and CLIC:

 (15)

Here the cuts given by Eq. (15) are applied to the photon transverse momentum , to the photon pseudorapidity , which reduces the contamination from other particles misidentified as photons, to the tau transverse momentum for the final state particles, to the tau pseudorapidity which reduces the contamination from other particles misidentified as tau and to , and which give the separation of the final state particles. It is fundamental that we apply these cuts to reduce the background and to optimize the signal sensitivity to the particles of the final state.

Tau identification efficiency depends of a specific process, background processes, some kinematic parameters and luminosity. For the processes examined, investigations of tau identification have not been examined yet for LHC, FCC-he and CLIC detectors. In this case, identification efficiency can be detected as a function of transverse momentum and rapidity of the -lepton. We have considered the following cuts for the selection of the -lepton as used in many studies Galon (); Atag2 () , .

The above cuts on the -leptons ensure that their decay products are collimated which allows their momenta to be reconstructed reasonably accurately, despite the unmeasured energy going into neutrinos Howard ().

Another important element in our study is the level or degree of sensitivity of our results. In this sense, to estimate the Confidence Level (C.L.) sensitivity on the parameters and , a fitting is performed. The distribution murat (); Billur () is defined by

 χ2(~aτ,~dτ)=(σSM−σBSM(√s,~aτ,~dτ)σSM√(δst)2+(δsys)2)2, (16)

with is the total cross-section incorporating contributions from the SM and new physics, is the statistical error and is the systematic error. The number of events is given by , where is the integrated luminosity of the , and colliders. The -lepton decays almost of the time into an electron and into two neutrinos, of the time, it decays in a muon and in two neutrinos. While, in the remaining of the occasions, it decays in the form of hadrons and a neutrino. Thus, we assume that the branching ratio of the -lepton pair in the final state to be or Data2018 ().

On the other hand, it should be noted that in all the processes considered in this article, the total cross-section of the , and signals are computed using the CalcHEP package Belyaev (), which can computate the Feynman diagrams, integrate over multiparticle phase space, and simulate events.

### iii.2 The total cross-section of the pp→pγ∗γ∗p→pτ+τ−γp signal at LHC

In the EPA, the quasireal photons emitted from both proton beams collide with each other and produce the subprocess . The spectrum of photon emitted by proton can be written as follows Belyaev (); Budnev ():

 fγ∗p(x)=απEp{[1−x][φ(Q2maxQ20)−φ(Q2minQ20)], (17)

where and is maximum virtuality of the photon. The minimum value of the is given by

 Q2min=m2px21−x. (18)

The function is given by

 φ(θ)= (1+ay)[−In(1+1θ)+3∑k=11k(1+θ)k]+y(1−b)4θ(1+θ)3 (19) +c(1+y4)[In(1−b+θ1+θ)+3∑k=1bkk(1+θ)k].

with

 y=x2(1−x), (20)
 a=1+μ2p4+4m2pQ20≈7.16, (21)
 b=1−4m2pQ20≈−3.96, (22)
 c=μ2p−1b4≈0.028. (23)

Therefore, in the EPA the total cross-section of the signal is given by

 σpp→pγ∗γ∗p→pτ+τ−γp=∫fγ∗p(x)fγ∗p(x)d^σγ∗γ∗→τ+τ−γdE1dE2. (24)

With all the elements considered in subsection A, that is to say the CalcHEP package, selected cuts, fitting and with 13 and 14 TeV at the LHC, the determination of the total cross-section in terms of the anomalous parameters and , translate in the following results:

For :

 σ(κ) = [1.28×107κ6+2.61×103κ5+4.71×103κ4+1.99κ3 (25) + σ(~κ) = [1.28×107~κ6+4.71×103~κ4+1.39~κ2+1.03×10−4](pb). (26)

For :

 σ(κ) = [1.78×107κ6+3.52×103κ5+5.18×103κ4+2.21κ3 (27) + σ(~κ) = [1.78×107~κ6+5.18×103~κ4+1.50~κ2+1.05×10−4](pb). (28)

In these expressions the independent terms of and correspond to the cross-section of the SM, that is . In the next section, the calculated cross-sections in Eqs. (25)-(28) are used to sensitivity estimates on the anomalous MDM and EDM of the -lepton.

### iii.3 The total cross-section of the e−p→e−γ∗γ∗p→e−τ+τ−γp signal at FCC-he

To determine the total cross-section of the signal at FCC-he, we must take into account that in the EPA approach, the spectrum of first photon emitted by electron is given by Belyaev (); Budnev ():

 fγ∗e(x1)=απEe{[1−x1+x21/2x1]log(Q2maxQ2min)−m2ex1Q2min (1−Q2minQ2max)−1x1[1−x12]2log(x21E2e+Q2maxx21E2e+Q2min)},

where and is maximum virtuality of the photon. The minimum value of the is given by

 Q2min=m2ex211−x1. (30)

For the spectrum of the second photon emitted by proton we consider the expression given by Eq. (17). Therefore, the total cross-section of the reaction is obtained from

 σe−p→e−γ∗γ∗p→e−τ+τ−γp=∫fγ∗e(x1)fγ∗p(x2)d^σγ∗γ∗→τ+τ−γdx1dx2. (31)

We have performed a global fit (and apply the cuts given in Eq. (15)), as a function of the two independent anomalous couplings and , with 7.07 and 10 TeV at FCC-he to the following studied observables:

For :

 σ(κ) = [2.85×107κ6+2.33×103κ5+1.87×104κ4+11.14κ3 (32) + σ(~κ) = [2.85×107~κ6+1.87×104~κ4+7.30~κ2+6.09×10−4](pb). (33)

For :

 σ(κ) = [2.10×108κ6+3.26×104κ5+6.40×104κ4+13.85κ3 (34) + σ(~κ) = [2.10×108~κ6+6.40×104~κ4+12.11~κ2+8.50×10−4](pb). (35)

### iii.4 The total cross-section of the e+e−→e+γ∗γ∗e−→e+τ+τ−γe− signal at CLIC

The total cross-section for the elementary processes at CLIC is determined in the context of EPA, where the quasi-real photons emitted from both lepton beams collide with each other and produce the subprocess .

The form of the spectrum in two-photon collision energy is a very important ingredient in the EPA. In this approach, the photon energy spectrum is given by Eqs. (29) and (30).

The elementary process participates as a subprocess in the main process , and the total cross-section is given by

 σe+e−→e+γ∗γ∗→e+τ+τ−γe−=∫fγ∗e−(x)fγ∗e+(x)d^σγ∗γ∗→τ+τ−γdE1dE2. (36)

We presented results for the dependence of the total cross-section of the process on and . We consider the following cases at CLIC:

For :

 σ(κ) = [2.09×107κ6+5.09×104κ5+5.86×104κ4+63.89κ3+60κ2 (37) + 2.10×10−2κ+6.9×10−3](pb) σ(~κ) = [2.09×107~κ6+5.86×104~κ4+60~κ2+6.9×10−3](pb). (38)

For :

 σ(κ) = [4.22×108κ6+2.49×105κ5+2.89×105κ4+1.25×102κ3 (39) + 1.22×102κ2+2.79×10−2κ+9.74×10−3](pb) σ(~κ) = (40)

In the next section, the calculated cross-sections in Eqs. (25)-(28), (32)-(35) and (37)-(40) are used to sensibility estimates on the anomalous MDM and EDM of the -lepton.

## Iv Sensitivity estimates on the dipole moments of the τ-lepton at the LHC, FCC-he and CLIC

### iv.1 Sensibility on the dipole moments of the τ-lepton from pp→pτ+τ−γp at LHC

In this subsection phenomenological projections on the total cross-section and on the dipolar moments and of the -lepton though the signal at LHC are presented.

For our numerical analysis we starting from the expressions given by Eqs. (25)-(28) and we obtained the total cross-sections plots of Figs. 3-6. These four figures represent the same observable, but just expressed in terms of different anomalous parameters, that is , and , respectively. From these figures, a strong dependence of the total cross-section with respect to the anomalous parameters , , as well as with the center-of-mass energies of the LHC is clearly observed. Furthermore, a direct comparison between the results for the SM, that is to say with (see Eqs. (25)-(28)) and the corresponding ones obtained in Figs. 3-6, show a great difference of the order of on the total cross-section.

To estimate the sensitivity of the LHC to the anomalous couplings and we consider and 14 TeV and integrated luminosities . To this effect, in Figs. 7 and 8, we use Eq. (25)-(28) to illustrate the region of parameter space allowed at C.L.. The best sensitivity estimated from Figs. 7 and 8, taken one coupling at a time are given by:

 κ=(−0.007,0.007),95% C.L.,~κ=(−0.008,0.008),95% C.L., (41)

at and . These results are consistent with those reported in Table III for and as in Eq. (42):

 ~aτ=(−0.0067,0.0065),95% C.L.,|~dτ|=3.692×10−17ecm,95%% C.L.. (42)

Our results are an order of magnitude better than the best existing limit for the -lepton anomalous MDM and EDM comes from the process as measured by DELPHI Collaboration DELPHI () at LEP2 (see Table I), as well as of the study of by BELLE Collaboration BELLE () (see Table I).

We next consider the sensibility estimated for the anomalous observables and , considering different values of and at C.L.. We consider both cases: pure-leptonic and semi-leptonic. Our results for these cases are shown in Table III, where the semi-leptonic case provides more sensitive results on and .

### iv.2 Sensitivity on the dipole moments of the τ-lepton from e−p→e−τ¯τγp at FCC-he

We now turn our attention to the associated production of a photon with a -lepton pair, via the signal, as is show in Figs. 9-12. The motivation to study this process is simple and already mentioned above, the gauge invariance of the effective Lagrangian relates the dipole couplings of the -lepton to couplings involving the photon. At the same time a similar study of the total cross-section as a function of -lepton dipole couplings and are realized. Our results show that the total cross-section depends significantly on and , in addition to . We find that the difference with respect to the SM is of the order of , which is several orders of magnitude best than the result of the SM.

Figs. 13 and 14 show the sensitivity contour bands in the plane of vs for the FCC-he with center-of-mass energies and luminosities . The sensitivity estimates at C.L. on the anomalous parameters are found to be:

 κ=(−0.0035,0.0025),95% C.L.,~κ=(−0.0025,0.0030),95% C.L.. (43)

Here, it was studied using data collected by the DELPHI experiment at LEP2 during the years 1997-2000. The corresponding integrated luminosity is . However, the corresponding integrated luminosity related to BELLE is .

The comparison with the limits of the present DELPHI and BELLE Collaborations with the corresponding ones obtained by the FCC-he on the anomalous couplings searches, indicates that the sensitivity estimates of the FCC-he at C.L are still stronger that for both experiments.

In Table IV, we list the C.L. sensitivity estimates on the observables and , based on di-tau production cross-section via the process at FCC-he. At present, DELPHI and BELLE experimental measurements on tau pair production and give the most stringent bounds on and DELPHI (); BELLE (). However, note that our sensitivity estimates on and are about ten times better than those for DELPHI and BELLE Collaborations, corroborating the impact of the signal, in addition of the parameters of the FCC-he:

 ~aτ=(−0.00265,0.00246),95% C.L.,|~dτ|=1.437×10−17ecm,95%% C.L.. (44)

with and .

### iv.3 Sensitivity on the dipole moments of the τ-lepton from e+e−→e+τ¯τγe− at CLIC

Before beginning with the study of the sensitivity on the dipole moments of the -lepton through the process at CLIC, it should be noted that experimentally, the processes that involving single-photon in the final state can potentially distinguish from background associated with the process under consideration. Besides, the anomalous coupling can be analyzed through the process at the linear colliders. This process receives contributions from both anomalous and couplings. But, the subprocess isolate coupling which provides the possibility to analyze the coupling separately from the coupling. Generally, anomalous parameters and tend to increase the cross-section for the subprocess , especially for photons with high energy which are well isolated from the decay products of the taus L3 (). Furthermore, the single-photon in the final state has the advantage of being identified with high efficiency and purity.

To assess future CLIC sensitivity to the dipole moments, as well as for the total cross-section from searches for the signal, we perform several figures, as well as a table that illustrates the sensitivity on the dipole moments.

In Figs. 15-18 we show the expected vs , vs and vs cross-sections for the signal with . All analysis cuts given in Eq. (15) are applied. Obviously of the plots we observed that the cross-section depends strongly on , and , throughout the range defined for these observables, as well as of . An improvement of the order of with respect to the SM is obtained.

In Figs. 19 and 20, we show the exclusion contours on the two-parameter and . For comparison, we also include several energies and luminosities. The results of the CLIC improve the sensitivity of the existing limits for the MDM and the EDM of the -lepton given in Table I. What is more, there is also a significant improvement in the cross-section constraining and parameters because our observables are sensitive to the parameters of the collider. Thus, from Fig. 20, it is straightforward to obtain that the sensitivity estimates on the anomalous dipole moment are:

 κ=(−0.00125,0.00115),95% C.L.,~κ=(−0.00125,0.00125),95% C.L., (45)

where the results obtained in Eq. (45) are for and .

Our final results are summarised in Table V below and agree with the experimental determinations of the -lepton dipole moments which are given in Table I for the DELPHI, L3, OPAL, BELLE and ARGUS Collaborations. From Table V, our best sensitivity projected correspond to:

 ~aτ=(−0.00128,0.00105),95% C.L.,|~dτ|=6.439×10−18ecm,95%% C.L.., (46)

and the results obtained in Eq. (46) are for and at CLIC.

It is worth mentioning that, the above sensitivity estimates are completely model-independent and no assumption has been made on the anomalous couplings in the effective Lagrangian given by (8). For the sake of comparison with published data for the DELPHI, L3, OPAL, BELLE and ARGUS Collaborations DELPHI (); L3 (); OPAL (); BELLE (); ARGUS (), we have presented the limits that can be found by considering separately only operator or only operator in Eqs. (8).

## V Conclusions

The sensitivity estimates of the vertex at the LHC, FCC-he and CLIC at CERN are discussed in this paper. We propose to measure this vertex in the , and channels at the mode. Furthermore, to the total cross-section measurement with the EPA, method provides powerful tools to probe the anomalous structure of the coupling. Additionally, in order to select the events we implementing the standard isolation cuts, compatibly with the detector resolution expected at LHC, FCC-he and CLIC to reduce the background and to optimize the signal sensitivity.

A very important aspect in our study and worth mentioning is the following, in most of the above mentioned experiments some of the particles in the anomalous coupling are off-shell. The off-shell form factors are problematic since they can hardly be isolated from other contributions and gauge invariance can be a difficulty. However, in the effective Lagrangian approach which we use in this paper, all those difficulties are solved because form factors are directly related to couplings, which are gauge invariant. Therefore, stringent and clean sensitivity estimates on the anomalous MDM and EDM of the -lepton are obtained.

In conclusion, new mechanism are proposed in this paper obtain the anomalous MDM and EDM of the -lepton produced in the high energy , and colliders. With the information from the effective Lagrangian formalism, a significant improvement can be achieved as shown in Figs. 3-20 and Tables III-V. Under this framework, it is predicted that with of data that will be collected by LHC, a sensitivity of , C.L., can be achieved for the -lepton, and , C.L. can be achieved for the EDM. In the case of the FCC-he, it is feasibility that with it is possible to obtain a sensitivity of and , at C.L.. While for the CLIC, the projections with of data that will be collected by CLIC are and , at C.L.. The precision of the -lepton is about of the SM prediction, therefore in this framework and with the large amount of data collected at current and future colliders can constrain BSM much better than before. In summary, the future CLIC at high energy and high luminosity should provide the best sensitivity on the MDM and EDM of the -lepton, and shows a potential advantage compared to those from LHC and FCC-he.