# Analysis of the anomalous quartic couplings at the LHeC and the FCC-he

###### Abstract

The quartic gauge boson couplings that identify the strengths of the gauge boson self-interactions are exactly determined by the non-Abelian gauge nature of the Standard Model. The examination of these couplings at collisions with high center-of-mass energy and high integrated luminosity provides an important opportunity to test the validity of the Standard Model and the existence of new physics beyond the Standard Model. The quartic gauge boson couplings can contribute directly to multi-boson production at colliders. Therefore, we examine the potential of the process at the Large Hadron Electron Collider and the Future Circular Collider-hadron electron to study non-standard couplings in a model independent way by means of the effective Lagrangian approach. We present an investigation on measuring production in pure leptonic and semileptonic decay channels. In addition, we calculate the sensitivity limits at Confidence Level on the anomalous , , , , , , and couplings obtained by dimension-8 operators through the process for the Large Hadron Electron Collider and the Future Circular Hadron Electron Collider’s different center-of-mass energies and integrated luminosities. Our results show that with the process at the Large Hadron Electron Collider and the Future Circular Collider-hadron electron the sensitivity estimated on the anomalous couplings can be importantly strengthened.

## I Introduction

Since the Large Hadron Collider (LHC) began to receive data, there is no significant deviation from the Standard Model (SM). Instead, with the ultimate discovery of the approximately 125 GeV Higgs boson at the LHC in 2012, the SM has achieved an important success higgs1 (); higgs2 (). Even so, new physics beyond the SM is needed to clarify the deficiencies of the SM such as neutrino oscillations, the strong CP problem, and matter-antimatter asymmetry. The examination of the gauge boson self-interactions is important for the precise testing of SM and new physics research beyond the SM. Thus, it is possible to carry out different measurements in processes involving multi-boson production in colliders with high center-of-mass energy and high luminosity. However, contributions to the quartic gauge boson couplings in the SM that can arise from new physics can be explained in a model independent way through the effective Lagrangian approach that is parameterized by high dimensional operators. These operators that define the anomalous quartic gauge boson couplings are based on the non-linear or linear representation realization of the gauge symmetry bel1 (); bel2 ().

It is assumed that there is no Higgs boson at low energy spectrum in the non-linear representation. Besides, with the discovery of the Higgs particle, a linear representation of gauge symmetry that broken by the conventional Higgs mechanism is probable. Therefore, in linear representation, the lowest order operators that define the possible deviations of the quartic gauge boson couplings from the SM are dimension-8 baa ():

(1) |

where the operators possess couplings and is a characteristic scale. There are 17 different operators that define the anomalous quartic gauge boson couplings. The indices S, T, and M of the operators depict three different class: operators including only , operators containing and field strength and operators exhibiting four field strength tensors.

The first class has two independent operators as follows

(2) | |||||

(3) |

where is Higgs doublet field.

The eight operators of second class are given by

(4) | |||||

(5) | |||||

(6) | |||||

(7) | |||||

(8) | |||||

(9) | |||||

(10) | |||||

(11) |

Here, and that are the electroweak field strength tensors are given as follows

(12) |

where stands for the unit of electric charge and is the Weinberg angle, represents the generators, and are coupling constant of and , respectively.

The final class is expressed as follows

(13) | |||||

(14) | |||||

(15) | |||||

(16) | |||||

(17) | |||||

(18) | |||||

(19) | |||||

(20) |

Table I shows the changes in the quartic gauge boson couplings in the SM caused by the above operators.

Many studies have been carried out for processes involving the anomalous quartic gauge boson couplings via dimension-6 and dimension-8 operators x1 (); x2 (); x3 (); x4 (); x5 (); x6 (); x7 (); x8 (); x9 (); x91 (); x92 (); x10 (); x11 (); x12 (); x13 (); x14 (); x141 (); x15 (); x16 (); x17 (); x18 (); x19 (); x20 (); x21 (); x22 (); x33 (); yy (); zz (); pp (). However, in this study, we consider , , , , , , and effective Lagrangians which contribute to the anomalous quartic couplings. As can be seen in Table I, , , , , , , and couplings can be examined via , , , , , , and effective Lagrangians, respectively.

In the literature, the ATLAS Collaboration at the LHC has been investigated and couplings for triboson production in two decay channels atlas (). In Ref. ebo (), the sensitivity limits on and couplings have been examined at Confidence Level for TeV through the reaction . As can be seen from the results of Snowmass, the sensitivity limits on coupling with Confidence Level in pure leptonic channel through the triboson production have been studied deg (). In Ref. yu (), and couplings have been investigated through the process at the LHC with TeV and at the future hadron colliders with TeV. All of the obtained limits on the anomalous and couplings in these studies are given in Table II.

Because of high background arising from QCD interactions, colliders such as LHC may not be convenient for the precise measurements. A cleaner environment compared to the LHC can provide with colliders. Therefore, new physics beyond the SM can be examined with colliders with high center-of-mass energy and high luminosity. The Large Hadron electron Collider (LHeC) lhec () and Future Circular Collider-hadron electron (FCC-he) fcc () are planned for the new generation colliders in the near future projects. The future colliders are considered to produce electron-proton collisions at center-of-mass energies from 1.30 TeV to 10 TeV.

This study presents a research on di-boson production through discovering the potential of measuring final state with full leptonic decay and semileptonic decay channels at 1.30,1.98 TeV LHeC and 7.07,10 TeV FCC-he and examining the anomalous quartic coupling.

The remaining part of the study is planned as follows: in the following Section, the cross sections of the process at the LHeC and the FCC-he are obtained. We focus model independent sensitivity estimates on the anomalous quartic couplings in Section III. Conclusions are discussed in Section 4.

## Ii The cross section of the process

In the production of signal process, the effective Lagrangians with the anomalous quartic couplings are implemented to FeynRules package rul () and embedded into MadGraphaMCNLO mad () as a Universal FeynRules Output ufo (). For parton distribution functions, the CTEQ6L1 is adopted cteq (). In order to investigate the possibilities of colliders as an option to sensitivity estimates on the anomalous quartic couplings, we concentrate signal processes. Here, we apply the following set of cuts in order to suppress the backgrounds and enhance the signal in the process including the anomalous quartic vertex. For pure leptonic decay channel, these cuts are given by

(21) |

(22) |

(23) |

For semileptonic decay channel, applied cuts are

(24) |

(25) |

(26) |

where is the pseudorapidity, and are the transverse momentum and the separation of the final state particles, respectively.

In the LHeC and the FCC-he, production can be generated via the process , the schematic diagram of this process is represented in Fig. 1. For pure leptonic decay channel, the total Feynman diagram of the process is 308, while the process with semileptonic decay channel has 310 Feynman diagrams. For pure leptonic and semileptonic decay channels, as shown in Figs. 2-9, the total cross sections of the process in terms of the anomalous , , , , , , and couplings for the LHeC and the FCC-he are displayed. The results demonstrate a clear dependence of the cross section of the process with respect to the anomalous quartic couplings and the center-of-mass energies of the LHeC and the FCC-he.

The cross section of signals for a certain value of the anomalous , , , , , , and parameters after cuts given in Eqs. 21-26 of the processes and at the LHeC and the FCC-he with and TeV has been shown in Tables III-VI. By examining Tables III-VI, a comparison can be made about the cross sections as a function of the anomalous quartic couplings. As seen from these Tables, the cross section of the process with TeV is up to five orders of magnitude larger than that of the examined process with TeV. Similarly, the cross section of the process has changed up to three orders of magnitude. Especially, the largest deviation from the SM cross section between the cross sections including the anomalous couplings is seen in the cross section containing coupling. In this case, we expect that the obtained limits on parameter should be more restrictive than the other parameters.

## Iii Model independent sensitivity estimates on the anomalous quartic couplings

We examine the potential of production at the future colliders, namely at the LHeC and the FCC-he, on the anomalous , , , , , , and parameters of the anomalous quartic couplings. To realize our work, we focus on the processes with pure leptonic and semileptonic decay channels.

To study the sensitivity on the anomalous , , , , , , and parameters we apply test and investigate Confidence Level sensitivities. The statistical method test is given by

(27) |

where is the statistical error, .

In Tables VII-XVI, constraints on the anomalous , , , , , , and parameters at 1.30, 1.98, 7.07 and 10 TeV future hadron electron colliders via production pure leptonic and semileptonic decay channels with different integrated luminosities are shown. Table VII shows that the obtained limits on and couplings through the process with center-of-mass energy of 1.30 TeV and integrated luminosity of 10 fb are given as [-2.16; 2.00] 10 TeV and [-1.34; 1.33] 10 TeV which are almost the same with the current limits obtained by the LHC, respectively. We see from Table VIII that best sensitivities on and couplings are one order of magnitude better than the experimental limits. However, the our limits for TeV with fb from Table X are far beyond the sensitivities of the LHC.

Furthermore, we compare our calculations with the phenomenological work at the LHC. In Ref. ebo (), the limits on the anomalous and couplings arising from dimension-8 effective Lagrangians are calculated as [-2.20; 2.40] 10 TeV and [-2.50; 2.50] 10 TeV, respectively. As can be seen in Table XIV, our best limits on these couplings by probing the process are more restrictive with respect to the limits obtained in Ref. ebo (). Also, the best limits on coupling for pure leptonic decay channels of the production in the final state of the process at 10 TeV FCC-he with an integrated luminosity of 100 fb are up to four times better than the limits of Ref. ebo ().

In Snowmass paper deg (), coupling at Confidence Level in pure leptonic channel via the triboson production is examined. We observe that our best limit on this coupling at TeV FCC-he with fb is almost twenty times worse than the best limits derived in Snowmass paper for TeV with fb.

Ref. yu () investigated the anomalous quartic couplings parameters , and at 100 TeV future collider via production pure leptonic decay channel with integrated luminosity of 3000 fb. They found TeV, TeV and TeV. As understood from Table X, the limits on and couplings derived by Ref. yu () are very close to our best limits, but the best limits we find for are one order of magnitude worse.

In the literature, in addition to the , and parameters examined for the anomalous couplings, we also investigate the anomalous , , , and parameters. The best limits on , , , and parameters at the FCC-he with TeV and fb are obtained as [-2.47; 2.42] 10 TeV, [-9.51; 9.41] 10 TeV, [-1.76; 1.79] TeV, [-6.17; 5.46] 10 TeV and [-10.27; 9.98] 10 TeV, respectively.

Particularly, the sensitivity of the process to the anomalous , , ,, , , and parameters rapidly increases with the center-of-mass energy and the luminosity. As we can see from our results, we find that the process is the most sensitive to coupling that has stronger energy dependence than other couplings.

To make our study more effective, for TeV FCC-he with an integrated luminosity of fb, we find sensitivities on the anomalous , , , , , , and couplings by using statistical significance

(28) |

For pure leptonic and semileptonic decay channels, observation sensitivities on the anomalous couplings are given Tables XV-XVIII. For both decay channels, the obtained limits using analysis at Confidence Level are better than the best limits derived from signal significance at and . Our best limits obtained on the anomalous couplings by using statistical significance at are up to two times better than the best limits derived for .

There are other studies on the anomalous quartic couplings through the nonlinear parametrization for the electroweak symmetry breaking sector in the literature 11 (); 12 (); baa (). We can compare the limits obtained from these studies with our own limits via the following relations baa ()

(29) |

(30) |

However, in this study, we will only focus on the researches that examine the anomalous quartic couplings through the linear parametrization for the electroweak symmetry breaking sector.

## Iv Conclusions

The new physics effects beyond the SM may appear in the electroweak sector that requires to measure the gauge boson self-interactions. Possible deviation of the quartic gauge boson couplings from the predictions of the SM would be a sign to new physics. The LHeC and the FCC-he with a rich physics program can provide many important information to better understand the SM and to probe new physics. In this study, we offer an analysis to constrain the anomalous quartic gauge boson couplings in the electroweak sector by considering an effective Lagrangian approach with the content of several dimension-8 operators.

The best sensitivities obtained from the process on the anomalous , , , , , , and couplings vary from the order of to the order of . In addition, the best sensitivities derived on the anomalous , , , , , , and couplings from the process change from the order of to the order of .

We observe that the best sensitivities obtained from pure leptonic decay channel can set more stringent sensitive to the best sensitivity derived from the semileptonic channel. The significance gain in pure leptonic channel is bigger than semileptonic channel, which may be due to that the QCD backgrounds increase much faster than the pure leptonic background.

A prominent advantage of the process at the LHC is that it isolates coupling from the other quartic couplings. As it can be understood by examining the process diagrams, the anomalous quartic coupling by examining the process can be distinguished the other anomalous quartic couplings. Therefore, the examination of coupling alone is only specific to the and colliders.

Also, in this study, we give the first limits obtained on the anomalous , , , and couplings derived by dimension-8 operators in the literature. For these reasons, colliders such as the LHeC and FCC-he provide an ideal platform to probe the anomalous quartic couplings.

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

atlas () | 8 | ||||

ebo () | 14 | ||||

yu () | 14 | ||||

deg () | 14 | ||||

deg () | 14 | ||||

deg () | 33 | ||||

deg () | 100 | ||||

yu () | 100 | ||||

deg () | 100 |

Couplings | Signal at 1.30 TeV LHeC (pb) | Signal at 1.98 TeV LHeC (pb) |
---|---|---|

1.66 | 0.06 | |

2.40 | 5.39 | |

1.41 | 1.99 | |

1.60 | 1.63 | |

2.20 | 3.42 | |

5.66 | 2.31 | |

2.30 | 9.93 | |

7.52 | 3.12 |

Couplings | Signal at 7.07 TeV FCC-he (pb) | Signal at 10 TeV FCC-he (pb) |
---|---|---|

1.46 | 28.01 | |

0.10 | 1.94 | |

0.03 | 0.52 | |

6.86 | 5.77 | |

1.55 | 1.62 | |

57.65 | 1.20 | |

25.91 | 5.29 | |

8.45 | 1.70 |

Couplings | Signal at 1.30 TeV LHeC (pb) | Signal at 1.98 TeV LHeC (pb) |
---|---|---|

6.77 | 0.14 | |

7.32 | 1.21 | |

3.80 | 4.10 | |

3.65 | 3.87 | |

7.83 | 1.04 | |

2.28 | 5.70 | |

8.15 | 2.34 | |

2.76 | 7.25 |