# Study of anomalous coupling sensitivity at the Compact Linear Collider

###### Abstract

The Compact Linear Collider (CLIC) is one of the most popular linear colliders, planned to realize collisions in three energy stages of , , and TeV. It has an energy scale never reached by any existing lepton collider. In this study, we present the sensitivity studies of the anomalous quartic gauge boson coupling (aQGC) in the three different processes , , and at the CLIC. We obtained confidence level limits on the aQGC parameters and at various integrated luminosities and center-of-mass energies. The best limits obtained on the aQGCs and in these processes are at the order of TeV and TeV, respectively.

## I Introduction

In the Standard Model (SM) of particle physics, triple and quartic interactions of the gauge bosons have not been determined with great accuracy. The possible deviation of gauge boson self-interactions from the SM predictions will be an important sign of new physics beyond the SM. Assuming that the energy scale associated with the new physics is sufficiently high compared to the masses of the gauge bosons, we are motivated to use Effective Lagrangian method effek , which is model independent and can converge to the SM at lower energy scale. The aQGCs are parameterized by effective operators which do not induce new trilinear gauge couplings. Therefore, we consider that aQGCs can be examined independently from the effects induced by trilinear gauge couplings.

In the literature, the aQGC operators have been described by either linear or non-linear effective Lagrangians. While non-linear effective Lagrangians have dimension 6, linear effective Lagrangians have dimension 8. However, the dimension 6 Lagrangian only contains vertices with two photons. Assuming that there is no Higgs boson in the low energy spectrum we employ a nonlinear representation of the spontaneously broken gauge symmetry. C and P conserving non-linear effective Lagrangian for aQGCs are given by lag2 ; lag3

(1) |

(2) |

(3) |

where is triplet, is the tensor for electromagnetic field strength, and and are the dimensionless. is a mass-dimension parameter associated with the energy scale of the new degrees of freedom which have been integrated out.

In addition, the dimension 8 Lagrangian induces quartic couplings among the neutral gauge bosons and it is obtained by using a linear representation of gauge symmetry that is broken by the conventional SM Higgs mechanism. However, after the recent discovery of a new particle which is consistent with the SM Higgs boson, it becomes important to examine aQGCs based on linear effective Lagrangian. On the other hand, since many aQGC-related studies have been generally investigated using non-linear effective Lagrangian, which we also choose in our aQGC study. In addition, our limits can compare with the results of the linear parameterizations which is published by CMS collaboration ilk . For this reason, the non-linear effective Lagrangian can be expressed by the linear effective Lagrangian ww1 ; ww2 ; ww3 . Therefore, aQGC parameters obtained from non-linear operators can be easily translated into parameters of linear operator.

In effective Lagrangian approach, the cross sections of aQGC processes increase with the coupling strength, and hence unitarity is violated for sufficiently high energy collisions. As stated by Ref. cin , a plausible choice of form factor can really provide the unitarity requirement. Form factor formalism is commonly described by the formula ebol ; D0

(4) |

where represents new physics scale. However, because of selection of form factor is arbitrary in the literature, we will examine aQGC parameters without using any form factor in this study.

In the presence of the effective Lagrangian in Eqs.(2-3), the vertex functions for are obtained respectively by

(5) |

(6) |

aQGCs were examined in OPAL, D0 and CMS experiments. The C. L. limits on aQGCs and through the processes and by the OPAL collaboration at LEP2 collider are given by TeV, TeV OPAL . Also, the current limits on aQGCs and have been provided through the processes by D0 collaboration at the Tevatron D1 and CMS collaboration at the Large Hadron Collider (LHC) CMS . The aQGC limits obtained by D0 collaboration are

(7) |

(8) |

The most restrictive limits on the parameters of aQGCs were determined by CMS collaboration

(9) |

(10) |

In the literature, aQGCs at the linear colliders were studied through the processes ee1 ; ee2 ; stir ; ee3 , stt , sato ; sato1 , and mem ; mem1 , where or and or . In addition, these couplings were examined at the LHC via the processes had ; had1 , had2 , mur ; ebol ; cin and had3 .

In this study, we probe the aQGCs by analyzing three different processes , , and at the CLIC.

The LHC may not be the best platform to investigate genuine quartic gauge couplings due to remnants of usual deep inelastic processes. On the other hand, since and are fundamental particles, lepton colliders can determine aQGC parameters with much higher precision measurements than hadron colliders. The CLIC, which has high energy and luminosity, is one of the foreseen linear colliders. It is designed to be running at three center-of-mass energy stages: , , and TeV 17 . The fundamental parameters of the three energy stages are given in Table I. In the CLIC’s first energy stage, it provides good opportunities for the precise measurements of different observables and characteristics of the SM Higgs boson, top quark and gauge bosons, etc.. The second phase will further open the window to the discovery of new physics beyond the SM. In addition, Higgs boson properties such as the Higgs self-coupling and rare Higgs decay modes will be examined. The third phase, which has a maximum energy of TeV, is considered to be able to make the most precise measurements of the SM, and to directly determine the pair-production of new heavy particles with masses up to TeV 18 .

The linear collider also has and run modes to examine the new physics search. In and run modes, high energy real photons can be obtained by converting the incoming leptons beam into a photon beam via the Compton backscattering mechanism las1 ; las2 ; las3 ; yeni ; yeni1 . In addition, the linear collider allows us to study photon-induced and reactions arising from almost real photons. Here, is emitted by any of the incoming lepton immediately after it collides with the other lepton. Hence, it is possible to investigate and collisions at the CLIC. The photons in these processes have been described as a suitable framework by the Weizsaecker-Williams approximation (WWA) WWA ; WWA1 ; WWA2 ; WWA3 ; WWA4 ; WWA5 . WWA is used to define processes of electro-production in the case of very small angle of or scattering. In this instance, since the virtuality of photon emitted from incoming leptons in WWA is very low, it is assumed to be on the mass shell. It gives a possibility to reduce the process of electro-production to the photo-production one with an appropriate photon spectrum. As a result, and processes are produced in a natural way from the process itself. The diagrams defining these processes are presented in Fig. and . In the literature, photon-induced reactions through the WWA have been widely examined at the LEP, Tevatron, and LHC 399 ; 400 ; 401 ; 402 ; 43 ; 44 ; 45 ; q1 ; q2 ; q3 ; q4 ; q5 ; q6 ; q7 ; kok ; kok2 ; 4777 ; 46 ; 47 ; kok22 ; kok1 ; q8 ; q9 ; q10 ; q11 ; q12 ; q13 .

## Ii Cross Sections

The processes and the subprocesses , and at the CLIC are described by tree-level Feynman diagrams in Figs. -. We can see from these figures that only one of these diagrams involves anomalous vertex while the others show the contributions arising from the SM. We have used the COMPHEP-4.5.1 program for numerical calculations in this study comp . In our study, only one aQGC coefficient is set to 0 at one time while the other one is set to non-zero. The total cross sections as functions of and aQGCs for these processes at the CLIC with and TeV are given in Figs. -. We observe from Figs. - that the cross section values induced by non-zero are larger than the values of the coupling. Hence, the obtained limits on the coupling are anticipated to be more restrictive than the limits on coupling.

## Iii Limits on the aQGCs

During statistical analysis, we determined C.L. limits on the aQGC parameters and using a simple one-parameter analysis. The analysis is defined by the following formula

(11) |

where is the SM cross section, is the total cross section with aQGC, is the statistical error: is the number of events. First, the number of events for the process is given by

(12) |

where is the integrated luminosity and is the SM cross section. The boson is heavy enough to decay both hadronically and leptonically. It decays approximately of the time leptonically (for electron or muon) and of the time to hadrons WB . So we consider one of the bosons decays leptonically and the other hadronically for the signal. Therefore, we consider that the branching ratio of the bosons pairs in the final state to be . In addition, we impose the acceptance cuts on the pseudorapidity and the transverse momentum GeV for the final state photon. After applying these cuts, the SM background cross sections in the process , we obtain as pb at TeV, pb at TeV, and pb at TeV.

In Table II, we calculate limits on the aQGC parameters and for various integrated luminosities and center-of-mass energies of the process . We observe from Table II that the obtained limits on the aQGCs via the process with TeV are less restrictive than the best limits attained by the current experimental limits. On the other hand, our limits at and TeV of center-of-mass energies of the same process are, even with low integrated luminosities, more restrictive than the experimental limits. Especially, we have found the limits of the aQGC as TeV while the limits on aQGC as TeV at the integrated luminosity of fb and center-of-mass energy of TeV.

In the second analysis, the number of events for the process is obtained as

(13) |

Here, the boson can decay leptonically, but in this case, it will introduce a great uncertainty due to the production of two neutrinos in the final state of our process. For this reason, we consider the hadronic decay of the boson. Also, we apply the cuts GeV and for the photon in the final state. Therefore, we calculate the SM cross sections as pb at TeV, pb at TeV, and pb at TeV. We show the limits on the and aQGCs for the process at various integrated luminosities and center-of-mass energies in Table III. As we can see from this table, the best limits obtained on and aQGCs through process are found to be TeV and TeV, respectively for fb and TeV at the CLIC.

Finally, for the process , in order to get promising results the number of events is obtained as

(14) |

Here, we find the SM cross sections to be pb for TeV, pb for TeV, and pb for TeV. In Table IV, we give the limits of the and aQGCs for at CLIC at and TeV. As shown in all these tables, we realize that the sensitivities of and aQGCs are rapidly enhanced when the center-of-mass energy of the CLIC increases.

In addition, the combined limits obtained on and aQGCs through three different processes with the integrated luminosity of fb at the CLIC with TeV are calculated as TeV and TeV, respectively. The combined limits are worse than the limits derived in the process since the other two processes have worse limits.

## Iv Conclusions

The CLIC is a future linear collider at the planning stage. Having it running with high energy and high luminosity is quite important new physics research beyond the SM. aQGCs are defined by dimension 6 effective Lagrangians, which have very strong energy dependences. Hence, cross section with anomalous couplings interactions has a higher momentum dependence than the SM cross section. We can easily understand that the contribution to the cross section of aQGCs quickly increases with center-of-mass energy. The analyzed processes that have very high energy and clean experimental environment such as CLIC, are anticipated to be more sensitive to aQGCs. Therefore, we are motivated to study the the aQGC parameters and through the processes , , and at the CLIC. We have shown that the best limits obtained on aQGCs and in these processes are at the order of and TeV, respectively. Since our limits are derived based on the total cross section with only statistical uncertainty, they is significantly better than the limits derived in experimental studies.

## References

- (1) W. Buchmuller and D. Wyler, Nucl. Phys. 268, 621 (1986).
- (2) G. Belanger and F. Boudjema Phys. Lett. B 288, 201 (1992).
- (3) G. Belanger and F. Boudjema Phys. Lett. B 288, 210 (1992).
- (4) CMS Collaboration, Phys. Rev. D 90 032008 (2014).
- (5) M. Baak , arXiv:1310.6708 [hep-ph].
- (6) C. Degrande , arXiv:1309.7890 [hep-ph].
- (7) O.J.P. Eboli, M.C. Gonzalez-Garcia and J.K. Mizukoshi, Phys. Rev. D 74 073005 (2006).
- (8) D. Yang, Y. Mao, Q. Li, S. Liu, Z. Xu, and K. Ye, JHEP 1304 108 (2013).
- (9) O. J. P. Eboli, M. C. Gonzalez-Garcia, and S. M. Lietti, S. F. Novaes Phys. Rev. D 63 075008 (2001).
- (10) V. M. Abazov , D0 Collaboration, Phys. Lett. B 695 67 (2011).
- (11) G. Abbiendi , OPAL Collaboration, Phys. Rev. D 70 032005 (2004).
- (12) V. M. Abazov , D0 Collaboration, Phys. Rev. D 88 012005 (2013).
- (13) S. Chatrchyan CMS Collaboration, JHEP 1307 116 (2013).
- (14) G. Belanger, F. Boudjema, Y. Kurihara, D. Perret-Gallix, and A. Semenov, Eur. Phys. J. C 13 283-293 (2000).
- (15) W. J. Stirling and A. Werthenbach, Eur. Phys. J. C 14 103-110 (2000).
- (16) G. A. Leil and W. J. Stirling, J. Phys. G 21 517-524 (1995).
- (17) P.J. Dervan, A. Signer, W.J. Stirling, and A. Werthenbach, J. Phys. G 26 607-615 (2000).
- (18) W.J. Stirling and A. Werthenbach, Phys. Lett. B 466 369-374 (1999).
- (19) S. Atag and ˙İ. Sahin, Phys. Rev. D 75 073003 (2007).
- (20) O. J. P. Eboli, M. C. Gonzalez-Garcia, and S. F. Novaes, Nucl. Phys. B 411 381-396 (1994).
- (21) O. J. P. Eboli, M. B. Magro, P. G. Mercadante, and S. F. Novaes, Phys. Rev. D 52 15-21 (1995).
- (22) İ. Sahin, J. Phys. G 36 075007 (2009).
- (23) T. Pierzchala and K. Piotrzkowski, Nucl. Phys. Proc. Suppl. 179-180 257 (2008).
- (24) E. Chapon, C. Royon and O. Kepka, Phys. Rev. D 81 074003 (2010).
- (25) A. Senol, arXiv:1311.1370.
- (26) P. J. Bell, Eur. Phys. J. C 64 25-33 (2009).
- (27) O. J. P. Eboli, M. C. Gonzalez-Garcia, and S. M. Lietti, Phys. Rev. D 69 095005 (2004).
- (28) D. Dannheim , CLIC e+e- Linear Collider Studies, arXiv:1305.5766v1.
- (29) D. Dannheim , CLIC Linear Collider Studies, arXiv:1208.1402.
- (30) I. F. Ginzburg, G. L. Kotkin, V. G. Serbo and V. I. Telnov, Nucl. Instr. and Meth. 205 47 (1983).
- (31) I. F. Ginzburg, G. L. Kotkin, S. L. Panfil, V. G. Serbo and V. I. Telnov, Nucl. Instr. and Meth. 219 5 (1984).
- (32) V. I. Telnov, Nucl. Instr. and Meth. A 294 72 (1990).
- (33) D. V. Soa , Mod. Phys. Lett. A 27 1250126 (2012).
- (34) J.P. Delahaye, Mod. Phys. Lett. A 26 2997-3023 (2011).
- (35) V.M. Budnev, I.F. Ginzburg, G.V. Meledin and V.G. Serbo, Phys. Rept. 15 181 (1974).
- (36) G. Baur , Phys. Rep. 364 359 (2002).
- (37) E. Yehudai, Phys. Rev. D 41 33 (1990).
- (38) C. Carimalo, P. Kessler and J. Parisi, Phys. Rev. D 20 1057 (1979).
- (39) J.E. Cieza Montalvo, N. V. Cortez, M. D. Tonasse, Phys.Rev. D 77 095015 (2008).
- (40) R. Nisius, Phys. Rept. 332 165-317 (2000).
- (41) J. Abdallah , DELPHI Collaboration, Eur. Phys. J. C 35 159 (2004).
- (42) A. Abulencia , CDF Collaboration, Phys. Rev. Lett. 98 112001 (2007).
- (43) T. Aaltonen , CDF Collaboration, Phys. Rev. Lett. 102 222002 (2009).
- (44) T. Aaltonen , CDF Collaboration, Phys. Rev. Lett. 102 242001 (2009).
- (45) S. Chatrchyan , CMS Collaboration, JHEP 1201 052 (2012).
- (46) S. Chatrchyan , CMS Collaboration, JHEP 1211 080 (2012).
- (47) CMS Collaboration, JHEP 07 116 (2013).
- (48) S. Atag and A. Billur, JHEP 11 060 (2010).
- (49) K. Piotrzkowski, Phys. Rev. D 63 071502 (2001).
- (50) H. Murayama and M. E. Peskin, Ann. Rev. Nucl. Part. Sci. 46 533-608 (1996).
- (51) S. Atag, S. C. İnan and İ. Şahin, Phys. Rev. D 80 075009 (2009).
- (52) İ. Şahin and S. C. İnan, JHEP 09 069 (2009).
- (53) S. C. İnan, Phys. Rev. D 81 115002 (2010).
- (54) İ. Şahin and M. Köksal, JHEP 11 100 (2011).
- (55) M. Köksal and S. C. İnan, arXiv:1305.7096.
- (56) M. Köksal and S. C. İnan, AHEP Volume 2014, Article ID 315826, 8 pages (2014).
- (57) M. Köksal, arXiv:1402.2915.
- (58) A. A. Billur and M. Köksal, Phys. Rev. D 89 037301 (2014).
- (59) A. A. Billur and M. Köksal, arXiv:1311.5326.
- (60) A. Senol, Phys. Rev. D 85 113015 (2012).
- (61) İ. Şahin , Phys.Rev. D 88 095016 (2013).
- (62) S. C. İnan and A. Billur, Phys. Rev. D 84 095002 (2011).
- (63) R. S. Gupta, Phys. Rev. D 85014006 (2012).
- (64) İ. Şahin, Phys. Rev. D 85 033002 (2012).
- (65) L. N. Epele , Eur. Phys. J. Plus 127 60 (2012).
- (66) İ. Şahin and B. Şahin, Phys. Rev. D 86 115001 (2012).
- (67) A. A. Billur, Europhys. Lett. 101 21001 (2013).
- (68) K. Nakamura , J. Phys. G 37, 075021 (2010).
- (69) A. Pukhov , Report No. INP MSU 98-41/542; arXiv:hep-ph/9908288; arXiv:hep-ph/0412191.

Parameter | Unit | Stage | Stage | Stage |
---|---|---|---|---|

TeV | ||||

fb | ||||

nm | ||||

nm | ||||

m |

(TeV) | (fb) | (TeV) | (TeV) |
---|---|---|---|

(TeV) | (fb) | (TeV) | (TeV) |
---|---|---|---|

(TeV) | (fb) | (GeV) | (GeV) |
---|---|---|---|