# Search for the anomalous electromagnetic moments of tau lepton through electron-photon scattering at CLIC

###### Abstract

We have examined the anomalous electromagnetic moments of the tau lepton in the processes ( is the Compton backscattering photon) and ( is the Weizsacker-Williams photon) with unpolarized and polarized electron beams at the CLIC. We have obtained 95 confidence level bounds on the anomalous magnetic and electric dipole moments for various values of the integrated luminosity and center-of-mass energy. Improved constraints of the anomalous magnetic and electric dipole moments have been obtained compared to the LEP sensitivity.

###### pacs:

14.60.Fg,13.40.Gp## I Introduction

The magnetic dipole moment of a particle is given by ger (); uhlen (). Here, represents the strength of the magnetic dipole moment in units of Bohr magneton, and defined as g-factor or gyromagnetic factor. The value of for a point-like particle is obtained 2 as a result of the Dirac equation. However, in quantum electrodynamics, the interactions of particle are much more complex and so there is a deviation from schwinger (). This deviation is known as anomalous magnetic moment. For any spin- particle with mass, the anomalous magnetic moment is represented as . The anomalous magnetic moments of electron and muon can be constrained with high accuracy at low energy spin precession experiments. The latest experimental data for the anomalous magnetic moment of the electron has been found as hanna (); data (); berin (). Anomalous magnetic moment prediction can also be performed for the muon which has a mass about times the electron mass. However, a disagreement has been observed for different Standard Model (SM) predictions and experimentally performed measurements for hagi (); bennet (). The latest experimental data has been determined as through the experiment hoec (). The experimental measurement of the anomalous magnetic moment of any particle contains its estimated value and some new physics effects that are just unpredictable in the SM. To find out the new physics contributions, it is an advantage that the mass of the tau lepton is enormous compared to the mass of the muon. However, the lifetime of the tau lepton is very short so measuring electric and magnetic dipole moments of the tau lepton is quite difficult with spin precession experiments. As a result, using colliders to study the anomalous magnetic moment of the tau lepton is highly preferable.

For the SM predictions, following numerical values can be found by summing all contributions discussed above 11 (); 12 (); hamzeh (); samu ():

(1) |

(2) |

(3) |

(4) |

However, experimental restrictions on have been obtained through by measuring the total cross-section at the C. L. in LEP in the following ranges L3 (); opal (); del ():

L3: ,

OPAL: ,

DELPHI:

The SM does not provide enough information to adequately understand the origin of CP violation chris (). Another interesting contribution in the interaction of photon with the tau lepton is CP violation which is generated by electric dipole moment. This phenomenon has been identified within the SM by the complex couplings in the CKM matrix of the quark sector koba (). In fact, there is no CP violation in the leptonic couplings (an exception is the neutrino mixing with different masses which is another source of CP violating barr ()). Additional sources beyond the SM for the CP violation in the lepton sector are leptoquark 18 (); 19 (), SUSY 20 (), left-right symmetric 21 (); 22 () and Higgs models 23 (); 24 (); 25 (); 26 (). CP violation in the quark sector induces electric dipole moment of the leptons in the three loop level. Due to this contribution of the SM, it is very difficult to determine the electric dipole moment of the tau lepton. However, the electric dipole moment of this particle may cause detectable size due to interactions arising from the new physics beyond the SM.

The SM value for is obtained as hoo (). However, the most restrictive experimental bounds on the electric dipole moment of the tau lepton have been obtained through by BELLE in the following ranges belle ():

,

.

The main motivation of the present study is to investigate vertex contributions with anomalous electromagnetic form factors to the SM. In the SM, these form factors arise from radiative corrections. In this manner, to characterize the interaction of the tau lepton with the photon, the electromagnetic vertex factor can be parametrized by

(5) |

with , where and are the photon momentum and the mass of the tau lepton, respectively. are the electric charge, the anomalous magnetic dipole and the electric dipole form factors of the tau lepton. The electromagnetic form factor parametrizes electric charge coupling in the SM and electromagnetic coupling of the tau lepton for vertex is in compact form J (); P (); M (). There are a lot of phenomenological studies about this subject phe1 (); phe2 (); phe3 (); phe4 (); phe5 (); phe6 (); phe7 ().

On the other hand, CLIC, aims to accelerate and collide electrons and positrons at TeV nominal energy. It is a linear collider with high energy and high luminosity that is planned to be constructed at future date braun (); clc (). In addition, CLIC can be constructed with and collider modes with real photons. This real photon beam is obtained by the Compton backscattering of laser photons off linear electron beam. Moreover, most of these photon beams can be in the high-energy region.

Linear colliders make it possible to use and interactions possible to examine the new physics beyond the SM. The emitted photons from the incoming electrons scatter at very small angels from the beam pipe. Therefore, these photons have very low virtuality and we say that these photons are âalmost-realâ. The Weizsäcker-Williams approximation is a facility in phenomenological studies because it permits to obtain cross sections for the process approximately through the study of the main process. Here, X represents particles obtained in the final state. Also, these interactions have very clean experimental conditions.

With these motivations, we have obtained the sensitivity bounds on new physics parameters through ( is the Compton backscattering photon) and ( is Weizsäcker-Williams photon) in next subsections. In the next section, we briefly outline details of our numerical calculation and results. The final section is devoted to our conclusions.

## Ii Numerical Analysis

In the SM, electromagnetic form factors are reduced to . However, due to the vertex, in other words, contributions from loop effects or arising from the new physics, and could not be taken as zero J (); P (); M (). While considering the limit of , the form factors become

(6) |

which relates to the static properties of fermions pich ().

In this study, the validity of the approach can be easily understood. As shown in the Feynman diagrams in Fig.1, the anomalous electromagnetic moments contribution of the tau lepton only comes from the diagram (b). As seen from this diagram, the photon in the vertex is either a Compton backscattering photon for the process or a Weizsacker-Williams photon for the process. There is no other intermediate photon and Compton backscattering photon is on the mass-shell (). So, the limit we use () is appropriate for this process.

In Weizsäcker-Williams approximation beam particles (electrons) scatter at very small angels. So, electrons may not be observed in the central detector. If scattered electrons of the beams are detected, maximum and minimum values of incoming photon energies can be detected. In the other case, final energy or momentum cuts of produced final state particles can be used to specify minimum photon energy. In this approximation, the photon virtuality is given by,

(7) |

Here is the ratio of the energy of the photon and the energy of the incoming electron, and is the transverse momentum of the photon. is given by

(8) |

is very small due to the electron mass (See Eq.8). In addition, since the electrons are scattered at very small angles, their transverse momentum are very small. For this reason, transverse momentum of the emitted photons must be very small due to momentum conservation. When all these arguments are taken into account, it can be understood that the virtuality of the photons in Weizsäcker-Williams approximation should be small. In other words, the photon must be almost-real. The moment of the tau lepton was also investigated by the DELPHI collaboration using multiperipheral collisions through the process del (). In this study, the virtuality of of the photons was obtained as GeV using the appropriate experimental techniques. In this motivation, we have taken the maximum photon virtuality GeV as in other phenomenological studies.

In our calculations in this article, we have used the following kinematic cuts,

(9) |

In sensitivity analysis, we take into account method,

(10) |

where is the total cross section which includes SM and new physics, ; is the statistical error and is the systematic error.

Systematic errors can arise for the following reasons. First, we take into account experimental uncertainties. However, we do not know exactly what the value of systematic errors of the two processes are since they are not examined in any of the CLIC reports 6 (); 7 (); 8 (). 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 del (). Relative systematic errors on cross-section of the process are given in Table 1. Also, at center-of-mass energies GeV GeV, the process was studied with the L3 detector at LEP L3 (). The total systematic uncertainty in this work was estimated between and . Even though the process at the LHC has not been examined experimentally, the process at TeV has been reported using data corresponding to an integrated luminosity of . An overall relative systematic uncertainty on the signal have obtained by summing quadratically all uncorrelated contributions 10 (). In addition, the anomalous magnetic and electric dipole moments of the tau lepton via the process with of the total systematic errors at the LHC was investigated phenomenologically in Ref 3 (). As a result, we think that the systematic error in CLIC will be much smaller than these experimental studies because it will be a new generation accelerator with innovative technologies.

Secondly, there may be uncertainties arising from the identification of tau lepton. The tau lepton has several different decay channels. These channels are classified according to the number of charged particles in the last state: one prong or three prong. Since the particles in the tau decays are always greater than one, these are called tau jets. The determination of hadronic decay channels are more problematic than the leptonic modes due to QCD backgrounds. For hadronic decays tau jets can be separated from other jets due to the its topology. Work in this regard is done by ATLAS and CMS groups atlastau1 (); atlastau2 (); cmstau (). Tau tagging efficiencies also studied for ILD ild (). 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.

Thirdly, there may be theoretical uncertainties. One of these uncertainties may arise from photon spectra. Another theoretical uncertainty comes from loop calculations in the SM, at tree level, , and . Besides, in the loop effects arising from the SM and the new physics, and may not be equal to zero. For example, the anomalous coupling is given by

(11) |

where is the contribution of the SM and is the contribution of the new physics eidel (); c1 (); c2 (); L3 (); c4 (). As mentioned above, is the SM prediction comes from three parts (which occur the SM loop effects). Higher order corrections to the anomalous magnetic moment for tau lepton are searched for several authors in the literature. The total error on tau lepton anomalous magnetic moment which comes from QED, electroweak and hadronic loop contributions is approximately eidel (); Roberts:2010zz (); Passera:2004bj (); Passera:2006gc (); Fael:2014vyp (). This value is negligible compared to standard model value due to this reason not included uncertainty calculations. As a result, we take into account the SM loop effects by using the electromagnetic vertex factor of the tau lepton.

We assume that the tau lepton decays into hadrons hence we take in all calculations. While getting the bound in this article, when fitting for we set to the SM value (zero), and vice versa.

### ii.1 Analysis with Compton Backscattering Photons

In this subsection, we show the numerical results for the . We have used the CalcHEP package calchep (); 48 () for all numerical analysis. This program allows automatic calculations of the distributions and cross sections in the SM as well as their extensions at the tree level. We have considered TeV and TeV CLIC center-of-mass energies in our calculations.

The photon distribution function for the Compton backscattering photons is given by,

(12) |

where

(13) |

with

(14) |

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

Using the above function, the cross section can be obtained as

(15) |

with . Here is related to , the square of the center of mass energy of collision, by .

In Table 2, we present the C.L. sensitivity bounds on the anomalous couplings for Compton backscattered photon and unpolarized electron beam (), TeV and TeV center-of-mass energies and integrated CLIC luminosities. The bounds are found with no systematic error () and with systematic errors of , . Similarly, the limits on are shown in Table 3. It can be understood that the bounds on the anomalous couplings are sensitive to the values of the center-of-mass energy and luminosity. Also, we can see from these tables that our bounds on the are better than the current experimental limits even for fb and TeV. In Figs. 2 and 3, we show the contour bounds in the plane for TeV with fb and TeV with fb, respectively. The region outside the resulting ellipsoid are the regions of exclusion. From these figures, the best bounds on anomalous couplings are obtained for the TeV and fb.

For the numerical analysis, we have used polarized electron beams. For a process with electron and positron beam polarizations, the cross section can be defined as pol (),

(16) |

where represents the obtained cross section with fixed helicities for positron and for the electron. and are the polarization degree of the electron and positron, respectively. The process which is examined in this paper, has three Feynman diagrams each of them have weak charged boson vertex. Due to the weak bosons couple to left handed fermions, negative helicity polarization can increase cross section and as a consequence of the increment in the cross section, the stronger bounds on the anomalous electromagnetic moments can be achieved. Hence, we have applied electron polarization. We give C.L. sensitivity bounds on the anomalous and couplings in Tables 4 and 5 respectively. Here, we have considered for and TeV with different luminosity values. As seen from the tables obtained sensitivity bounds on the anomalous couplings are better than unpolarized beams. In Figs. 4 and 5 for polarized electron beam, we show the contour bounds in the plane for TeV with fb and TeV with fb, respectively. Comparison of Figs. 2 (3) and 4 (5) shows that the excluded area of the model parameters which we have obtained from the polarized beams () expands to wider regions than the cases of the unpolarized electron beams.

### ii.2 Analysis with Weizsacker-Williams Photons

We have analyzed the anomalous dipole moments of the tau lepton via the main process in this subsection. In Weizsacker-Williams approximation, the photon spectrum used in the CalcHep program is

(17) |

where is the mass of the electron, , is the ratio of the energy of the photon and energy of the incoming electron, is the fine structure constant. Using Eq.(17), the cross section can be obtained by using Eq.15

In Tables 6 and 7 we present the C.L. sensitivity bounds on the anomalous and parameters for the unpolarized electron beams and different systematic error values, respectively. We can understand from the tables that the sensitivity bounds of the anomalous couplings enhance with the increasing center-of-mass energy and luminosity. The obtained bounds for the are also better than the current experimental limits. On the other hand, bounds with a Compton backscattering photon (Tables 2 and 3 ) are more sensitive than the bounds for the Weizsäcker-Williams approximation (Table 6 and 7). Main reason of this situation is that the Compton backscattering photon spectrum gives higher effective than the Weizsäcker-Williams photon spectrum in high energy regions com1 (); com2 (); ep1 (); ep2 (); ep3 (); ep4 (); ep5 (); wwa (). However, the application of the Weizsäcker-Williams approximation gives a lot of benefits in experimental and phenomenological studies ph1 (); ph2 (); ph3 (); ph4 (); ph5 (); ph6 (); ph7 (); qm1 (); qm2 (); qm3 (); qm4 () as mentioned in Section I. We present the C.L. sensitivity bounds on the anomalous and parameters for the polarized electron beams in Table 8 and 9 for TeV with fb and TeV with fb, respectively. Best bounds on the anomalous couplings have been obtained in this situation as we expected due to above discussions.

In Figs. 6 and 7 for unpolarized electron beam we present the contour bounds in the plane for TeV with fb and TeV with fb, respectively . Fig.8 (Fig.9) is the same as Fig. 6 (Fig.7) but for polarized electron beams. Limits for the polarized case are strong compared to the unpolarized case. However, these bounds are weaker compared to the Compton backscattered case.

## Iii Conclusion

We analyzed the tau lepton anomalous dipole moments through the processes and . These processes have a very clean environment. The deviation of the anomalous couplings from the expected values of the SM would evidence the existence of new physics. In this study, we compared the electromagnetic dipole moments of the tau lepton using the Weizsäcker approximation and Compton back-scattering photons. We have found the process gives better bounds than the other. However, processes that have and initial states require a special collider setup. On the other hand, and occur spontaneously during collisions.

Additionally, we used polarized and unpolarized electron beam in our study. We understood the polarization enhances the sensitivity bounds as mentioned in Section II. Our predictions for the expected limits on are better than the current experimental limits. Based on the finding of this paper, we can conclude that CLIC provides new opportunities for examination of tau physics beyond the SM using and modes.

## Iv Acknowledgements

This work has been supported by the Scientific and Technological Research Council of Turkey (TUBITAK) in the framework of Project No. 115F136.

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Trigger efficiency | ||||
---|---|---|---|---|

Selection efficiency | ||||

Background | ||||

Luminosity | ||||

Total |

TeV | Luminosity() | ||||
---|---|---|---|---|---|

10 | (-0.0105, 0.0105) | (-0.0227, 0.0227) | (-0.0291, 0.0291) | (-0.0343, 0.0343) | |

100 | (-0.0059, 0.0059) | (-0.0224, 0.0225) | -0.0290, 0.0290) | (-0.0342, 0.0343) | |

1.4 | 500 | (-0.0039, 0.0039) | (-0.0224, 0.0224 | (-0.0289, 0.0290) | (-0.0342, 0.0342) |

1500 | (-0.0030, 0.0030) | (-0.0224, 0.0224) | (-0.0289, 0.0290) | (-0.0342, 0.0342) | |

10 | (-0.0046, 0.0046) | (-0.0092, 0.0092) | (-0.0118, 0.0118) | (-0.0139, 0.0139) | |

500 | (-0.0017, 0.0017) | (-0.0091, 0.0091) | (-0.0117, 0.0117) | (-0.0139, 0.0139) | |

3 | 1000 | (-0.0014, 0.0014) | (-0.0091, 0.0091) | (-0.0117, 0.0117) | (-0.0139, 0.0139) |

2000 | (-0.0012, 0.0012) | (-0.0091, 0.0091) | (-0.0117, 0.0117) | (-0.0139, 0.0139) |

TeV | Luminosity() | ||||
---|---|---|---|---|---|

10 | |||||

100 | |||||

1.4 | 500 | ||||

1500 | |||||

10 | |||||

500 | |||||

3 | 1000 | ||||

2000 |

TeV | Luminosity() | ||||
---|---|---|---|---|---|

10 | (-0.0091, 0.0091) | (-0.0226, 0.0225) | (-0.0290, 0.0290) | (-0.0343, 0.0342) | |

100 | (-0.0051, 0.0051) | (-0.0225, 0.0224) | (-0.0290, 0.0289) | (-0.0343, 0.0342) | |

1.4 | 500 | (-0.0034, 0.0034) | (-0.0224, 0.0224) | (-0.0290, 0.0289) | (-0.0343, 0.0342) |

1500 | (-0.0026, 0.0026) | (-0.0224, 0.0224) | (-0.0290, 0.0289) | (-0.0343, 0.0342) | |

10 | (-0.0039, 0.0039) | (-0.0091, 0.0091) | (-0.0117, 0.0117) | (-0.0138, 0.0138) | |

500 | (-0.0015, 0.0015) | (-0.0090, 0.0091) | (-0.0117, 0.0117) | (-0.0138, 0.0138) | |

3 | 1000 | (-0.0012, 0.0012) | (-0.0090, 0.0091) | (-0.0117, 0.0117) | (-0.0138, 0.0138) |

2000 | (-0.0010, 0.0010) | (-0.0090, 0.0091) | (-0.0117, 0.0117) | (-0.0138, 0.0138) |

TeV | Luminosity() | ||||
---|---|---|---|---|---|

10 | |||||

100 | |||||

1.4 | 500 | ||||

1500 | |||||

10 | |||||

500 | |||||

3 | 1000 | ||||

2000 |

TeV | Luminosity() | ||||
---|---|---|---|---|---|

10 | (-0.0287, 0.0285) | (-0.0404, 0.0402) | (-0.0499, 0.0496) | (-0.0582, 0.0579) | |

100 | (-0.0162, 0.0160) | (-0.0379, 0.0376) | (-0.0486, 0.0483) | (-0.0574, 0.0571) | |

1.4 | 500 | (-0.0109, 0.0106) | (-0.0376, 0.0374) | (-0.0485, 0.0482) | (-0.0573, 0.0571) |

1500 | (-0.0083, 0.0081) | (-0.0376, 0.0373) | (-0.0484, 0.0482) | (-0.0573, 0.0571) | |

10 | (-0.0144, 0.0144) | (-0.0221, 0.0220) | (-0.0276, 0.0275) | (-0.0323, 0.0323) | |

500 | (-0.0054, 0.0054) | (-0.0210, 0.0209) | (-0.0271, 0.0270) | (-0.0320, 0.0320) | |

3 | 1000 | (-0.0046, 0.0045) | (-0.0210, 0.0209) | (-0.0271, 0.0270) | (-0.0320, 0.0320) |

2000 | (-0.0039, 0.0038) | (-0.0210, 0.0209) | (-0.0271, 0.0270) | (-0.0320, 0.0320) |

TeV | Luminosity() | ||||
---|---|---|---|---|---|

10 | |||||

100 | |||||

1.4 | 500 | ||||

1500 | |||||

10 | |||||

500 | |||||

3 | 1000 | ||||

2000 |

TeV | Luminosity() | ||||
---|---|---|---|---|---|

10 | (-0.0248, 0.0246) | (-0.0392, 0.0390) | (-0.0493, 0.0490) | (-0.0578, 0.0575) | |

100 | (-0.0140, 0.0138) | (-0.0377, 0.0375) | (-0.0485, 0.0483) | (-0.0574, 0.0571) | |

1.4 | 500 | (-0.0094, 0.0092) | (-0.0376, 0.0373) | (-0.0485, 0.0482) | (-0.0573, 0.0571) |

1500 | (-0.0072, 0.0069) | (-0.0376, 0.0373) | (-0.0485, 0.0482) | (-0.0573, 0.0571) | |

10 | (-0.0124, 0.0124) | (-0.0216, 0.0215) | (-0.0274, 0.0273) | (-0.0322, 0.0322) | |

500 | (-0.0046, 0.0046) | (-0.0210, 0.0209) | (-0.0271, 0.0270) | (-0.0320, 0.0320) | |

3 | 1000 | (-0.0039, 0.0039) | (-0.0210, 0.0209) | (-0.0271, 0.0270) | (-0.0320, 0.0320) |

2000 | (-0.0033, 0.0033) | (-0.0210, 0.0209) | (-0.0271, 0.0270) | (-0.0320, 0.0320) |

TeV | Luminosity() | ||||
---|---|---|---|---|---|

10 | |||||

100 | |||||

1.4 | 500 | ||||

1500 | |||||

10 | |||||

500 | |||||

3 | 1000 | ||||

2000 |