# Search for the electromagnetic moments of the lepton in photon-photon collisions at the LHeC and the FCC-he

###### Abstract

We examine the potential of the process at the Large Hadron Electron Collider (LHeC) and the Future Circular Hadron Electron Collider (FCC-he) to examine non-standard coupling in a model independent way by means of the effective Lagrangian approach. We perform pure leptonic and semileptonic decays for production in the final state. Furthermore, we use fb with at TeV and we consider systematic uncertainties of . The best sensitivity bounds obtained from the process on the anomalous couplings are and , respectively. Therefore, our results show that the process at the LHeC and FCC-he are a very good prospect for probing the anomalous magnetic and electric dipole moments of the lepton at mode of the future collider.

## I Introduction

The magnetic dipole moment of the electron which is responsible for the interaction with the magnetic field in the Born approximation is given as follows

(1) |

Here, is the Lande g-factor or gyromagnetic factor, is the Bohr magneton and represents the spin of the electron. For the electron, the value of in the Dirac equation is . It is traditional to point out the deviation of from 2 in terms of the value of the so-called anomalous magnetic moment. The anomalous magnetic moment of the electron is a dimensionless quantity and is described by

(2) |

The without anomalous and radiative corrections is equal to . Besides, it was firstly found from Quantum Electrodynamics (QED) using radiative corrections by Schwinger as 1 ().

The accuracy of the has been studied so far in many works. These works have provided the most precise determination of fine-structure constant , since is quite senseless to the strong and weak interactions. However, the anomalous magnetic moment of the muon enables testing the Standart Model (SM) and investigating alternative theories to the SM. Especially, the is more sensitive to new physics beyond the SM by a factor of than to the case of the . Similarly, due to the large mass of the lepton, a precise measurement of the anomalous magnetic moment provides an excellent opportunity to reveal the effects of the new physics beyond the SM.

The and have been examined with high sensitivity through spin precession experiment. On the other hand, spin precession experiment is not appropriate to investigate the anomalous magnetic moment because of the relatively short lifetime s of the lepton tau (). Instead of those experiments, we focus on highly precise measurements by comparing the measured cross section with the SM cross section in colliders with high center-of-mass energies in production processes.

The SM contribution for the anomalous magnetic moment is obtained by the sum of QED, electroweak and hadronic terms. The theoretical contribution from the QED to the anomalous magnetic moment up to three loops is calculated as 2 (); 3 ()

(3) |

In addition, the sum of the one and two loop electroweak effects is given by

(4) |

The hadronic contribution to the anomalous magnetic moment arising from QED diagrams including hadrons is

(5) |

By collecting all these additives, we obtain the anomalous magnetic moment

(6) |

The SM involves three sources of CP violation. One of them appears by complex couplings in the Cabibbo-Kobayashi-Maskawa (CKM) matrix of the quark sector 8 (). In the SM, neutrinos are massless. With correction of the SM to contain neutrino masses, CP violation can occur in the mixing of leptons. The last source of this phenomenology is possible in flavor conserving strong interaction processes. Besides, the experimental upper limit on the neutron electric dipole moment indicates that this ( term in the SM Lagrangian is at best tiny, . This is generally known as the strong CP problem. As mentioned above, although there is CP violation in the SM, it is not enough to explain for the observed baryon asymmetry of the universe given the limits on baryon number violation. It is clear that there must be CP violation beyond the SM.

The electric dipole moment of the lepton allows a direct investigation of CP violation 9 (); 10 (), a property of the SM and new physics beyond the SM. However, CP violation in the quark sector induces a small electric dipole moment of the lepton. One has to go at least to three loop level to create a non-zero contribution. Itâs crude estimate gives as follows 111 ()

(7) |

Thus, the electric dipole moment of the lepton is undetectably small with the contributions arising from the SM. Besides, the electric dipole moment of this lepton may cause detectable size due to interactions arising from the new physics beyond the SM such as leptoquarks 11 (); 12 (), supersymmetry 13 (), left-right symmetric models 14 (); bok () and more Higgs multiplets 15 (); 16 ().

Let’s examine structure of the interaction of the lepton to a photon. The most general anomalous vertex function describing interaction for two on-shell ’s and a photon can be parameterized below hu (); fa (),

(8) |

where , is the momentum transfer to the photon and GeV is the mass of the lepton. The -dependent form factors and have familiar interpretations in limit :

(9) |

In many studies investigating the anomalous magnetic and electric dipole moments of the lepton, photon or leptons in couplings in the examined processes are off-shell. Then, the quantity investigated in these studies is not actually the anomalous and couplings due to the lepton is off-shell. For this reason, instead of and we can call the anomalous magnetic and electric dipole moments of lepton examined as and . Therefore, the possible deviation from the SM predictions of couplings could be investigated in a model independent way by means of the effective Lagrangian approach. In this approach, the anomalous couplings are described by means of high-dimensional effective operators. In our analysis, we assume the dimension-six effective operators that contribute to the electromagnetic dipole moments of the lepton.

The experimental bounds on the at Confidence Level (C.L.) are provided by L3 and OPAL Collaborations through the reaction at LEP at 4 (); 5 ()

(10) |

(11) |

The present most restrictive bounds on the are obtained by the DELPHI Collaboration from the process total cross section measurements at GeV 6 ()

(12) |

The present experimental limits on the anomalous coupling of the lepton at the LEP by L3, OPAL and DELPHI Collaborations are

(13) |

(14) |

(15) |

Besides, the most restrictive experimental bounds are given by BELLE Collaboration

(16) |

(17) |

In the literature, there have been many studies for the anomalous and couplings at linear and hadron colliders. The linear colliders and their operating modes of and have analyzed via the processes 4 (), ee (); ee1 (), al (); sat (), al2 (), al3 (), al3 () and al4 (). Also, these couplings at the LHC have examined through the processes al5 (), al6 (), al7 (). Finally, there is a lot of work related to the anomalous magnetic moment of the lepton al8 (); al9 (); al10 (); al11 ().

In the investigated processes for examining the electromagnetic dipole moments of the lepton in particle accelerators, it is not possible that all the particles are on-shell. For this reason, we can use the effective Lagrangian approach to study the anomalous magnetic and electric dipole moments of this particle. In our analysis, we consider dimension-six operators mentioned in Ref. al12 () related to the electromagnetic dipole moments of the lepton. These operators are given as follows

(18) |

(19) |

where and are the Higgs and the left-handed doublets, are the Pauli matrices and and are the gauge field strength tensors. Therefore, the effective Lagrangian is parameterized as follows,

(20) |

After the electroweak symmetry breaking, contributions to the electromagnetic dipole moments of the lepton can be written as

(21) |

(22) |

where is the vacuum expectation value and is the weak mixing angle. The relations of CP even parameter and CP odd parameter to the anomalous magnetic and electric dipole moments of the lepton are given as follows

(23) |

The LHC may not provide highly precision measurements due to strong interactions of collisions. An collider may be a good idea to complement the LHC physics program. Since colliders have high center-of-mass energy and high luminosity, new physics effects beyond the SM may appear by examining the interaction of the lepton with photon which requires to measure couplings precisely. The Large Hadron Electron Collider (LHeC) and the Future Circular Hadron Electron Collider (FCC-he) are planned to generate collisions at energies from TeV to TeV al13 (); al14 (). The LHeC is a suggested deep inelastic electron-nucleon scattering machine which has been planned to collide electrons with an energy from GeV to possibly GeV, with protons with an energy of TeV. In addition, FCC-he is designed electrons with an energy from GeV to GeV, with protons with an energy of TeV.

The remainder of the study is structured as follows: In Section II, we observe the total cross sections and the anomalous magnetic and electric dipole moments of the lepton via the process . Finally, we discuss the conclusions in Section III.

## Ii The Total Cross Sections

The well-known applications of colliders are , and collisions where the emitted quasireal photon is scattered with small angles from the beam pipe of electron or proton beams. Since has a low virtuality, it is almost on the mass shell. , and collisions are defined by the Weizsacker-Williams Approximation (WWA). 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. Nevertheless, in many studies, new physics investigations are examined by using the WWA s1 (); s2 (); s3 (); s4 (); s5 (); s6 (); s7 (); s8 (); s9 (); s10 (); s11 (); s12 (); s13 (); s14 (); s15 (); s16 (); s17 (); s18 (); s19 (); s20 (); s21 (); s22 (); sa1 (); sa2 (); sa3 (); sa4 (); sa5 (); sa6 (); sa7 (); sa8 (); sa9 (); sa10 (); sa11 ().

The photons emitted from both electron and proton beams collide with each other, and collisions are generated. The process participates as a subprocess in the process . The Feynman diagrams for the subprocess are shown in Fig. 1. In addition, the diagram of the process is given in Fig.2.

For the subprocess that has two Feynman diagrams, the polarization summed amplitude square are obtained as follows

Here, shows the lepton charge, is the fine-structure constant and are the Mandelstam invariants.

In the WWA, two photons are used in the subprocess . The spectrum of first photon emitted by electron is given as al15 ()

where and is maximum virtuality of the photon. The minimum value of is shown as follows

(28) |

Second, the spectrum of second photon emitted by proton can be written as follows al15 ()

(29) |

where the function is given by

Here,

(31) |

(32) |

(33) |

(34) |

Therefore, we find the total cross section of the main process by integrating the cross section for the subprocess . The total cross section of this process is obtained as follows,

(35) |

In this work, all analyzes have been calculated in the CalcHEP package program, including non-standard couplings al15 (). We apply the photon spectrums in the WWA [23] embedded in the CalcHEP. Also, for the process , we consider the following basic acceptance cuts to reduce the background and to maximize the signal sensitivity:

(36) |

(37) |

(38) |

Here, is the pseudorapidity which reduces the contamination from other particles misidentified as tau, is the transverse momentum cut of the final state particles and is the separation of the final state particles.

Using the cuts given above, we give numerical fit functions for the total cross sections of the process as a function of the anomalous couplings for center-of-mass energies of TeV. The numerical fit functions for this process are

For TeV

(39) |

(40) |

For TeV

(41) |

(42) |

For TeV

(43) |

(44) |

For TeV

(45) |

(46) |

In the above equations, the total cross sections at give the SM cross section. Also, as can be understood from Eqs. (39)-(46), the linear, quadratic and cubic terms of the anomalous couplings arise from the interference between the SM and anomalous amplitudes, whereas the quartic terms are purely anomalous.

## Iii Bounds on the anomalous magnetic and electric dipole moments of the lepton at the LHeC and the FCC-he

We represent the total cross sections of the process as a function of the anomalous and couplings in Figs. - for center-of-mass energies of TeV. In this analysis, we consider that only one of the anomalous couplings deviate from the SM at any given time. We can easily understand from these figures that the total cross sections of the examined process increase when the center-of-mass energy increases. In addition, as can be seen from Eqs. 39-46, while the total cross sections are symmetric for the anomalous coupling, it is nonsymmetric for . For this reason, we expect that while the bounds on the anomalous magnetic dipole moment are asymmetric, the bounds on the electric dipole moment are symmetric. It is easily understood from Figs. - that the deviation from the SM of the anomalous cross sections containing and couplings at TeV is larger than those of including and at TeV. Therefore, the obtained bounds on the anomalous and couplings at TeV are anticipated to be more restrictive than the bounds at TeV.

In addition, to visualize the effects of the anomalous and couplings on the total cross section of the process , we give Figs. 5-8. Figs. 5-8 show that the surfaces of these curves strongly depend on the anomalous and couplings.

We need statistical analysis to probe the sensitivity to the anomalous and dipole moments of the lepton. For this reason, we use the usual the test with a systematic error

(47) |

where represents only the SM cross section, is the total cross section containing contributions from the SM and new physics, is the statistical error, . The is the only lepton that has the mass necessary to disintegrate, most of the time in hadrons. In of the time, the 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 our analysis, we assume pure leptonic and semileptonic decays for production in the final state of the examined process. Therefore, we use that branching ratios of the tau pairs are for pure leptonic decays and for semileptonic decays.

Systematic uncertainties may occur in colliders when tau lepton is identified. Due to these uncertainties, 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. On the other hand, in the literature, there are a lot of experimental and theoretical investigations to study the anomalous magnetic and electric dipole moments of the lepton with systematic errors. For example, as seen in Table I, DELPHI Collaboration at the LEP was examined these couplings via the process with systematic errors between and . In Refs. al5 (); al6 (), the processes and at the LHC have studied from to with systematic uncertainties. The sensitivity bounds on the anomalous magnetic and electric dipole moments of the lepton through the processes , , and at the Compact Linear Collider (CLIC) have calculated by considering of systematic errors: and al (); al12 (); al13 (). On the other hand, we could not have any information on the systematic uncertainties of the process we examined in the LHeC and FCC-he studies. Taking into consideration the previous studies, we consider the total systematic uncertainties of and .

For pure and semileptonic decay channels, the estimated sensitivities at C.L. on the anomalous and dipole moments of the lepton through the process at the LHeC and FCC-he, as well as for center-of-mass energies of TeV and systematic errors of are given in Tables II-IX. As can be seen in Table III, the process at TeV for semileptonic decay channel with fb improves approximately the sensitivity of dipole moment by up to a factor of 8 compared to the LEP. Our bounds on the anomalous couplings at TeV are competitive with those of LEP. As can be seen in Table IX, the best sensitivities obtained on and are and , respectively. Therefore, the collision of FCC-he with TeV and fb without systematic error probes the anomalous and dipole moments with a far better than the experiments bounds. Tables VI-IX represent that the bounds with increasing values at the FCC-he are almost unchanged with respect to the luminosity values and for the center-of-mass energy values. The reason of this situation is which is much smaller than .

Figs. - show bounds values obtained the anomalous magnetic and electric dipole moments of the lepton at C.L. through the process at the LHeC and FCC-he for center-of-mass energies of TeV without systematic uncertainty. From these figures we can compare the bounds obtained from four different center-of-mass energies more easily.

We compare our process with photon-photon collisions that are the cleanest process to examine the anomalous and dipole moments of the lepton. First, in Ref. al (), collisions at the TeV CLIC with an integrated luminosity of fb show that the bounds on the anomalous and couplings are calculated as and , respectively. We observe that the bounds obtained from collisions at the TeV FCC-he are at the same order with those reported in Ref. al ().

Another collision studied for investigation the anomalous magnetic and electric dipole moments of the lepton is the process at the CLIC al13 (). We understand that the sensitivities on and dipole moments expected to be obtained for the future collisions that generate Compton backscattering photons are roughly 10 times better than our limits.

In addition, we probe the bounds of Ref. al15 (), in which the best bounds on anomalous couplings by examining the the process at the LHC with center-of-mass energy of TeV and the integrated luminosity of fb are obtained. We see that the anomalous and dipole moments we found from our process for semileptonic decay channel with TeV and fb are at the same order with those reported in Ref. al15 (). Our best bounds on the anomalous couplings can set more stringent sensitive by one order of magnitude with respect to the best sensitivity derived from production at the LHC.

Finally, in Figs. 9-12, we show contours for the anomalous and couplings for the process at the LHeC and FCC-he for various integrated luminosities and center-of-mass energies. As we can see from these figures, the improvement in the sensitivity on the anomalous couplings is achieved by increasing to higher center-of-mass energies and luminosities.

## Iv Conclusions

The new physics effects beyond the SM may appear by examining the interaction of the lepton with photon which requires to measure coupling precisely. Nevertheless, the possible deviation from the SM predictions of coupling would be a sign for the presence of new physics beyond the SM. colliders with high center-of-mass energy and high luminosity such as the LHeC and FCC-he may be able to provide a lot of information on new physics beyond the SM as well as providing precise measurements of the SM. For this purpose, possible non-standard coupling at the LHeC and FCC-he is examined in an effective Lagrangian approach. It is usually common to investigate new physics in a model independent way via effective Lagrangian approach. This approach is defined by high-dimensional operators which lead to anomalous coupling.

In this study, we examine the potential of the process at colliders with TeV to study the anomalous dipole moments of the lepton. This process is the cleanest production mechanism for colliders. Furthermore, the subprocess isolates coupling which provides the possibility to analyze coupling separately from coupling. Also, colliders with high center-of-mass energy and luminosity are important for new physics research. Since non-standard couplings defined via effective Lagrangian have dimension-six, they have very strong energy dependences. So, the anomalous cross sections including vertex have a higher energy than the SM cross section. Finally, the subprocess may be effective efficient identification due to clean final state when compared to collisons of the LHC.

In our analysis, we find that the process at colliders lead to a remarkable improvement in the existing experimental bounds on the anomalous and couplings. Therefore, we show that collision at the LHeC and FCC-he are quite suitable for studying the anomalous and dipole moments of the lepton.

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