Model-independent bounds probing on the electromagnetic dipole moments of the \tau-lepton at the CLIC

# Model-independent bounds probing on the electromagnetic dipole moments of the τ-lepton at the CLIC

M. Köksal111mkoksal@cumhuriyet.edu.tr Deparment of Optical Engineering, Cumhuriyet University, 58140, Sivas, Turkey.
A. A. Billur222abillur@cumhuriyet.edu.tr Deparment of Physics, Cumhuriyet University, 58140, Sivas, Turkey.
A. Gutiérrez-Rodríguez333alexgu@fisica.uaz.edu.mx Facultad de Física, Universidad Autónoma de Zacatecas
Apartado Postal C-580, 98060 Zacatecas, México.
M. A. Hernández-Ruíz444mahernan@uaz.edu.mx Unidad Académica de Ciencias Químicas, Universidad Autónoma de Zacatecas
Apartado Postal C-585, 98060 Zacatecas, México.
July 18, 2019
###### Abstract

We establish model independent bounds on the anomalous magnetic and electric dipole moments of the tau-lepton using the two-photon processes and . We use of data collected with the future linear collider such as the CLIC at and we consider systematic uncertainties of . Precise bounds at C. L. on the anomalous dipole moments to the tau-lepton and are set from our study. Our results show that the processes under consideration are a very good prospect for probing the dipole moments of the tau-lepton at the future linear collider at the mode.

###### pacs:
13.40.Em, 14.60.Fg, 12.15.Mm
Keywords: Electric and Magnetic Moments, Taus, Neutral Currents.

## I Introduction

One of the greatest achievement of the Standard Model (SM) Glashow (); Weinberg (); Salam () is the measurement of the electric (EDM) and magnetic (MM) dipole moments of the electron and muon , which have been measured with a excellent precision of

 aExpe=1159652180.73(28)×10−12[0.24 ppb]% \@@cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{% Hanneke}{\@@citephrase{(}}{\@@citephrase{)}}}, (1)
 aExpμ=11659209.1(5.4)(3.3)×10−10[0.54%  ppm]\@@cite[cite]{\@@bibref{Authors Phrase1YearPhras% e2}{Bennett}{\@@citephrase{(}}{\@@citephrase{)}}}, (2)

respectively, and the theoretical prediction of the SM Data2016 () is given by

 aSMμ=116591803(1)(42)(26)×10−11. (3)

On the other hand, in comparison with the electron or muon mass, the tau-lepton has a large mass of Data2016 (), allowing one to expected an essential enhancement in the sensitivity to the effects of new physics beyond the Standard Model (BSM), such as its dipole moments Passera1 (). However the very short lifetime of this unstable particle makes it impossible to directly measure their electromagnetic properties. Therefore, indirect information must be obtained by precisely measuring cross sections and decay rates in processes involving the emission of a real photon by the tau-lepton.

With respect to the anomalous magnetic moment of the -lepton, the SM prediction is Samuel (); Hamzeh () and the respective EDM is generated by the GIM mechanism only at very high order in the coupling constant Barr (). The error with an order of magnitude of is an indication that SM extensions predicting values for above this level they are worth studying, as well as it is worthwhile to study new mechanisms and new modes of production of tau pairs in association with a photon at the future linear collider at the mode.

The SM predicts CP violation, which is necessary for the existence of the EDM of a variety physical systems. The EDM provides a direct experimental probe of CP violation Christenson (); Abe (); Aaij (), a feature of the SM and BSM physics. Precise measurement of the EDM of fundamental charged particles provides a significant probe of physics BSM.

The sensitivity to the MM and EDM of the tau-lepton has been studied in different context, both theoretical and experimental and some of which are summarized in Table I. Furthermore, there is an extensive theoretical work done in models BSM that contribute to dipole moments of charged leptons: Extra dimensions Iltan1 (), Seesaw model Dutta (), version III of the 2HDM Iltan2 (), Non-commutative geometry Iltan3 (), Non-universal extra dimensions Iltan4 (), Left-Right symmetric model Gutierrez1 (), Superstring models Gutierrez2 (), Simplest little Higgs model Gutierrez3 (), 331 model Gutierrez4 (). There are also bounds independent of the model such as collisions Koksal1 (), scattering Ozguven () and collisions Billur (); Sampayo (). Other limits on the MM and EDM of the -lepton are reported in Refs. Passera1 (); Eidelman1 (); Galon (); Arroyo1 (); Arroyo2 (); Xin (); Pich (); Atag1 (); Lucas (); Gutierrez5 (); Passera2 (); Passera3 (); Bernabeu (); Gutierrez6 (); Bernreuther ().

In this paper, using and reactions, we establish model independent limits on the dipole moments of the tau-lepton, and we improve the existing bounds on and . An interesting feature of these reactions is that they are extremely clean process because it has not interference with weak interactions, being a purely process of Quantum Electrodynamics (QED). Furthermore, the high center-of-mass energies proposed for the Compact Linear Collider (CLIC) make of it an appropriate machine to probing the MM and EDM which are more sensitive with the high energy and high luminosity of the collider.

The CLIC Linssen (); Accomando (); Abramowicz (); Dannheim () is a proposal for a future linear collider at CERN in the High Luminosity-Large Hadron Collider (HL-LHC) era. The machine is designed to make full use of the physics potential of CLIC with initial operation at center-of-mass energy of and luminosity of . The following stages at center-of-mass energies of ) and ) focus on exploring physics BSM. In summary, the CLIC project offers a rich physics program for about 20 years, with discovery potential to new physics, that can reach scales of up to several tens of TeV, through indirect searches with precision measurements.

For our study, we consider the following parameters of the CLIC: , and we consider systematic uncertainties of , with these parameters as input, we established model independent bounds on the electromagnetic dipole moments of the -lepton at C.L.. We get stringent limits in comparison with the bounds obtained by the DELPHI, L3, OPAL, BELLE and ARGUS Collaborations Abdallah (); Acciarri (); Ackerstaff (); Inami (); Albrecht () (see Table I).

The remainder of the paper is organized as follows: In Section II, we study the total cross section and the electromagnetic dipole moments of the tau-lepton through the and reactions. Section III provides the conclusions.

## Ii Two-photon processes γγ→τ+τ− and γγ→τ+τ−γ

For our study, we will take advantage of our previous works on the collision modes , and Billur (); Gutierrez7 (); Koksal2 (); Koksal3 () for calculate the total cross section for the and reactions. The corresponding Feynman diagrams for these processes are given in Figs. 1 and 2, respectively.

In our study we deduce bounds on the electromagnetic dipole moments of the -lepton and via the two-photon processes and . These processes are of interest for a number of reasons: First, are sensitive to the and . Additionally, increased cross sections for high energies and the absence or stong suppression of weak contributions are further complementary aspects of the two-photon processes in the contrast with the direct processes DelAguila (); Albrecht (), Acciarri () and Ackerstaff (); Grifols (). Furthermore, an important point is the availability of high luminosity photon beams due to Bremsstrahlung as a byproduct in planned high energy colliders. Also, very hard photons at high luminosity may be produced in Compton backscattering of laser light off high energy beams, as is the case of the future CLIC.

In order to determine the bounds on the MM and EDM of the -lepton, we calculate the total cross section of the reactions and . The most general parametrization for the electromagnetic current between on-shell tau-lepton and the photon is given by Passera1 (); Grifols (); Escribano (); Giunti ()

 Γατ=eF1(q2)γα+ie2mτF2(q2)σαμqμ+e2mτF3(q2)σαμqμγ5+eF4(q2)γ5(γα−2qαmτq2), (4)

where is the charge of the electron, is the mass of the tau-lepton, represents the spin 1/2 angular momentum tensor and is the momentum transfer. In the static (classical) limit the -dependent form factors have familiar interpretations for : is the electric charge; its anomalous MM and with its EDM. is the Anapole form factor.

In phenomenological and experimental searches most of the tau-lepton electromagnetic vertices search involve off-shell tau-leptons. Indeed in these studies one of the tau-lepton is off-shell and measured quantity is not directly and . For this reason deviations of the tau-lepton dipole moments from the SM values are examined in an effective lagrangian approach. It is usually common to study new physics in a model independent way through the effective lagrangian approach. This approach is defined by high-dimensional operators which lead to anomalous coupling. In this study, we will apply the dimension-six effective operators that contribute to the electric and magnetic dipole moments of the tau-lepton at the tree level given by Ref. 1 (); eff1 (); eff3 ():

 Leff=1Λ2[C33LWQ33LW+C33LBQ33LB], (5)

where

 Q33LW=(¯ℓτσμντR)σIφWIμν, (6)
 Q33LB=(¯ℓτσμντR)φBμν. (7)

Here, and are the Higgs and the left-handed doublets, are the Pauli matrices and and are the gauge field strength tensors.

After the electroweak symmetry breaking from the effective lagrangian in Eq. (5), the CP even and CP odd parameters are obtained

 κ=2mτe√2υΛ2Re[cosθWC33LB−sinθWC33LW], (8)
 ~κ=2mτe√2υΛ2Im[cosθWC33LB−sinθWC33LW], (9)

where GeV and is the weak mixing angle.

These parameters are related to contribution of the anomalous magnetic and electric dipole moments of the tau-lepton through the following relations:

 κ=~aτ, (10)
 ~κ=2mτe~dτ. (11)

### ii.1 γγ→τ+τ− cross section

All signal cross sections in this paper are computed using the package CALCHEP 3.6.30 Calhep (), which can computate the Feynman diagrams, integrate over multiparticle phase space and event simulation. In addition, for our study we consider the following basic acceptance cuts for events at the CLIC:

 pτ,¯τt>20GeV,|ητ,¯τ|<2.5,ΔR(τ,¯τ)>0.4, (12)

we apply these cuts to reduce the background and to optimize the signal sensitivity. In Eq. (12), is the transverse momentum of the final state particles, is the pseudorapidity which reduces the contamination from other particles misidentified as tau and is the separation of the final state particles.

The tau-lepton was discovered by Martin Lewis Perl in 1975 Perl (); Perl1 (). It was discovered in the Stanford Positron and Electron Accelerator Ring (SPEAR) of SLAC with the MARK I detector. With these tools, Perl and his team managed to distinguish leptons, hadrons and photons fairly accurately. The tau-lepton was discovered from certain anomalies detected in the disintegration of the particles. The observed event was as follows

 e+e−→τ+τ−→e±+μ±+νe+¯νμ+ντ+¯ντ.

When making the energy balance between the initial and final states, it was observed that the final energy was lower. At no time did the muons, hadrons and photons sum the energy necessary to equal the initial state. Then it was thought that the energy that made the electron and the positron collider created a pair of new particles very massive, which soon decay into the other observed particles. This theory was difficult to verify because the energy needed to produce the tau-antitau pair was similar to that required to create a pair of mesons. Subsequent experiments carried out in DESY and in SLAC confirmed the existence of the tau lepton and provided more precise values for its mass and spin.

The tau is the only lepton that has the mass necessary to disintegrate, most of the time in hadrons. of the time the tau decays into an electron and into two neutrinos; in another of the time, it decays in a muon and in two neutrinos. In the remaining of the occasions, it decays in the form of hadrons and a neutrino. In Table II the main -decay branching ratios are shown.

Since its discovery in 1975, the lepton-tau has been important to check different aspects of the SM. In particular, since the tau is the charged lepton of the third generation, the verification of its properties can give light on the problems related to the replication of generations, which is one of the open problems of the SM. Furthermore, it is the heavier lepton which makes it especially more sensitive to new physics, since its coupling to the dynamics responsible for the generation of masses, whatever it is, is more intense. In addition, it is the only lepton heavy enough to disintegrate into hadrons, which makes this particle a particularly suitable system for studying Quantum Chromodynamics (QCD) at low energies.

The square matrix elements for the process as a function of the Mandelstam invariants , and are given by:

 |M1|2 = 16π2Q2τα2e2m4τ(^t−m2τ)2[48κ(m2τ−^t)(m2τ+^s−^t)m4τ−16(3m4τ−m2τ^s+^t(^s+^t))m4τ (13) + 2(m2τ−^t)(κ2(17m4τ+(22^s−26^t)m2τ+^t(9^t−4^s)) + ~κ2(17m2τ+4^s−9^t)(m2τ−^t))m2τ+12κ(κ2+~κ2)^s(m3τ−mτ^t)2 − (κ2+~κ2)2(m2τ−^t)3(m2τ−^s−^t)],
 |M2|2 = −16π2Q2τα2e2m4τ(^u−m2τ)2[48κ(m4τ+(^s−2^t)m2τ+^t(^s+^t))m4τ (14) + 16(7m4τ−(3^s+4^t)m2τ+^t(^s+^t))m4τ + 2(m2τ−^t)(κ2(m4τ+(17^s−10^t)m2τ+9^t(^s+^t)) + ~κ2(m2τ−9^t)(m2τ−^t−^s))m2τ + (κ2+~κ2)2(m2τ−^t)3(m2τ−^s−^t)],
 M†1M2+M†2M1 = 16π2Q2τα2em2τ(^t−m2τ)(^u−m2τ) (15) × [−16(4m6τ−m4τ^s)+8κm2τ(6m4τ−6m2τ(^s+2^t)−^s)2 + 6^t)2+6^s^t)+(κ2(16m6τ−m4τ(15^s+32^t)+m2τ(15^s)2 + 14^t^s+16^t)2)+^s^t(^s+^t))+~κ2(16m6τ−m4τ(15^s+32^t) + m2τ(5^s)2+14^t^s+16^t)2)+^s^t(^s+^t)))−4κ^s(κ2+~κ2) × (m4τ+m2τ(^s−2^t)+^t(^s+^t))−4~κ(κ2+~κ2)(2m2τ−^s−2^t) × ϵαβγδpα1pβ2pγ3pδ4−2^s(κ2+~κ2)2(m4τ−2^tm2τ+^t(^s+^t))],

where , , , and and are the four-momenta of the incoming photons, and are the momenta of the outgoing tau-lepton, is the tau-lepton charge, is the fine-structure constant and is the mass of the tau.

The most promising mechanism to generate energetic photon beams in a linear collider is Compton backscattering. Compton backscattered photons interact with each other and generate the process . The spectrum of Compton backscattered photons is given by

 fγ(y)=1g(ζ)[1−y+11−y−4yζ(1−y)+4y2ζ2(1−y)2], (16)

where

 g(ζ)=(1−4ζ−8ζ2)log(ζ+1)+12+8ζ−12(ζ+1)2, (17)

with

 y=EγEe,ζ=4E0EeM2e,ymax=ζ1+ζ. (18)

Here, and are energy of the incoming laser photon and initial energy of the electron beam before Compton backscattering and is the energy of the backscattered photon. The maximum value of reaches 0.83 when .

The total cross section is given by,

 σ=∫fγ(x)fγ(x)d^σdE1dE2. (19)

Next, we present the total cross section as a polynomial in powers of . This provides more precise and convenient information for the study of the process . We consider the following cases:

For .

 σ(κ) = [(9.73×106)κ4+(8.18×104)κ3+(8.13×104)κ2+(1.11×102)κ+38.75](pb), σ(~κ) = [(9.73×106)~κ4+(8.26×104)~κ2+38.75](pb). (20)

For .

 σ(κ) = [(1.54×108)κ4+(8.42×108)κ3+(8.80×104)κ2+(17.5)κ+6](pb), σ(~κ) = [(1.54×108)~κ4+(8.81×104)~κ2+6](pb). (21)

For .

 σ(κ) = [(6.17×108)κ4+(9.13×104)κ3+(8.72×104)κ2−(1.21)κ+1.97](pb), σ(~κ) = [(6.17×108)~κ4+(8.83×104)~κ2+1.97](pb). (22)

From Eqs. (20)-(22), the linear, quadratic and cubic terms in arise from the interference between SM and anomalous amplitudes, whereas the quartic terms are purely anomalous. The independent term of correspond to the cross section at and represents the contribution of the cross section of the SM.

### ii.2 Bounds on the ~aτ and ~dτ through γγ→τ+τ− at the CLIC

For our numerical analysis of the total cross section , as well as of the electromagnetic dipole moments of the tau-lepton, where the free parameters are the center-of-mass energy , the integrated luminosity of the CLIC and the factors and , we also consider the acceptance cuts given in Eq. (12). In addition, we take into account the systematic uncertainties for the collider. For this purpose, we use the usual formula for the function Koksal1 (); Ozguven (); Billur (); Sahin1 ():

 χ2=(σSM−σNP(√s,κ,~κ)σSMδ)2, (23)

where is the total cross section including contributions from the SM and new physics, , is the statistical error, is the systematic error and is the number of signal expected events where is the integrated CLIC luminosity. Furthermore, as the tau-lepton decays roughly of the time leptonically and of the time to one or more hadrons (see Table II), then for the signal we consider one of the tau-leptons decays leptonically and the other hadronically. Therefore, for our study we assume that the Branching ratio of the two-tau in the final state to be BR=0.46.

Systematic uncertainties may occur when tau-lepton is identified due to some of the reasons described below: Although, we do not have any CLIC reports 6 (); 7 (); 8 () to know exactly what the systematic uncertainties are for our processes which are investigated, we will make some approaches about the systematic uncertainties. The DELPHI Collaboration examined the anomalous magnetic and electric dipole moments of the tau-lepton through the process in the years at collision energy between and GeV Abdallah (). Relative systematic errors on cross section of the process are given in Table 3. Also, the process was studied with the L3 detector for center-of-mass energies GeV GeV at LEP Acciarri (). The anomalous magnetic and electric dipole moments of the tau-lepton via the process with of the total systematic uncertainties at the LHC was investigated phenomenologically in Ref Atag (). Work in this regard is done by ATLAS and CMS groups atlastau1 (); atlastau2 (); Bagliesi (). Tau tagging efficiencies also studied for the International Large Detector (ILD) ild (), a proposed detector concept for the International Linear Collider (ILC). Due to these difficulties, tau identification efficiencies are always calculated for specific process, luminosity, and kinematic parameters. These studies are currently being carried out by various groups for selected productions. For a realistic efficiency, we need a detailed study for our specific process and kinematic parameters. For all these reasons, in this work, kinematic cuts contain some general values chosen by detectors for lepton identification. Hence, in this paper, tau-lepton identification efficiency is considered within systematic errors. Taking into consideration the previous studies, and of total systematic uncertainties were taken in this study. It can be assumed that this accelerator will be built in the coming years and the systematic uncertainties will be lower when considering the development of future detector technology.

With all these elements that we taken into consideration, we made and presented a set of figures, as well as tables which illustrate our results.

The total cross sections are presented as a function of the anomalous couplings in Fig. 3 and in Fig. 4 for the center-of-mass energies of , respectively. The total cross section clearly shows a strong dependence with respect to the anomalous parameters , and with the center-of-mass energy of the collider . Additionally, the as function of and are shown in Figs. 5-7. In these figures, the surfaces are increased for the lower and upper limits of the parameters and , showing a strong dependence with respect to these parameters.

C. L. allowed regions in the plane for the process for the first, second and third stage of operation of the CLIC, where a fixed center-of-mass energies of is assumed with luminosities of , and , respectively, and considering systematic uncertainties of Abdallah (); Achard (), they are displayed in Figs. 8-10. For a complementary description on the uncertainties we suggest the reader consult Refs. Chatrchyan (); Atag ().

The achievable precision in the determination of the anomalous magnetic moment and the electric dipole moment are summarized in Figs. 11-14 and Tables IV-VI and are compared with experimental results of earlier studios for a linear collider as published by the BELLE, DELPHI, L3 and OPAL Collaboration Inami (); Abdallah (); Acciarri (); Ackerstaff (); Albrecht (). Our results show that the two-photon process at the CLIC improve the sensitivity bounds on anomalous electromagnetic dipole moments of tau-lepton with respect to the existing experimental bounds (see Table I) by two orders of magnitude. Our best bounds obtained on and are and , respectively, as shown in Tables IV-VI.

### ii.3 γγ→τ+τ−γ cross section

Experimentally, the processes that involving single-photon in the final state can potentially distinguish from background associated with the process under consideration. Furthermore, the anomalous coupling can be analyzed through the process at the linear colliders. This process receives contributions from both anomalous and couplings. However, the processes and isolate coupling which provides the possibility to analyze the coupling separately from the coupling. In general, anomalous values of and tend to increase the cross section for the process , especially for photons with high energy which are well isolated from the decay products of the taus Acciarri (). Additionally, the single-photon in the final state has the advantage of being identifiable with high efficiency and purity.

They may also provide a clear signal in the detector for new physics, for new phenomena such as the dipole moments of fermions. Also, the selection criteria used for the analysis allow the search for events having the characteristics of single-photon.

We now turn attention to the process at future collider. On the technical side, for the calculation of the total cross section of , the analytical expression for the amplitude square is quite lengthy so we do not present it here. Instead, we present numerical fit functions for the total cross sections with respect to center-of-mass energy and in terms of the parameters and . Furthermore, in the case of the process , we apply the following kinematic cuts to reduce the background and to maximize the signal sensitivity:

 (24)

for the photon transverse momentum , the photon pseudorapidity which reduces the contamination from other particles misidentified as photon, the tau transverse momentum for the final state particles, the tau pseudorapidity which reduces the contamination from other particles misidentified as tau and , and are the separation of the final state particles. In conclusion, by using these cuts given in Eq. (24) in our manuscript we have taken into account isolation criteria to optimize the signal to the particles of the final state. The cases considered are:

For .

 σ(κ) = [(3.22×107)κ6+(4.08×104)κ5+(3.36×105)κ4+(1.87×103)κ3 (25) + (1.24×103)κ2+(0.515)κ+0.21](pb), σ(~κ) = [(3.22×107)~κ6+(3.43×105)~κ4+(1.24×103)~κ2+0.21](pb). (26)

For .

 σ(κ) = [(8.29×109)κ6+(8.65×106)κ5+(1.15×107)κ4+(3.90×103)κ3 (27) +