Anomalous $WW\gamma$ couplings with beam polarization at the Compact Linear Collider

We study the anomalous $WW\gamma$ couplings at the Compact Linear Collider through the processes $e^{+}e^{-}\to W^+W^-$, $e^{-}e^{+} \to e^{-} \gamma^{*} e^{+} \to e^{+} \nu_{e} W^-$ and $e^{-}e^{+}\to e^{-} \gamma^{*} \gamma^{*} e^{+} \to e^{-} W^+ W^- e^{+} $ $ (\gamma^{*}$ is the Weizsacker-Williams photon). We give the 95\% confidence level limits for unpolarized and polarized electron (positron) beam on the anomalous couplings for various values of the integrated luminosities and center-of-mass energies. We show that the obtained limits on the anomalous couplings through these processes can highly improve the current experimental limits. In addition, our limits with beam polarization are approximately two times better than the unpolarized case.


I. INTRODUCTION
The Standard Model (SM) has been so far successful in describing the below the electroweak scale with high precision. Therefore, electroweak interactions are known very well in this model. Self-interactions of the gauge bosons are outcomes of the SU L (2) × U Y (1) gauge symmetry of the SM. Determination of these type of interactions plays an important role to test the non-Abelian gauge symmetries of the electroweak sector. Searching that kind of interactions can generate extra confirmation of the SM with a higher sensitivity or reveal some information for new physics beyond the SM. Any measurement which conflicts with the SM expectations would lead to the existence of new physics.
The traditional approach to investigate new physics effect to W W γ interactions introduces in a model independent way by means of the effective Lagrangian method. The theoretical motivations of such a method would be based on the guess that at higher energy regions beyond the SM, there is a main physics which reduces to the SM at lower energy regions. Such a procedure is quite general and independent of the details of the model. Hence, this method is generally known model independent analysis. The effective Lagrangian for W W γ interaction which conserves charge and parity can be given as follows [1,2], Here g W W γ = e , V µν = ∂ µ V ν −∂ ν V µ (V µ = W µ , A µ ), g γ 1 , κ, λ are the dimensionless anomalous parameters. They are related to magnetic dipole and electric quadrupole moments of W boson. In the SM, the couplings are obtained g γ 1 = 1, κ = 1 and λ = 0 at the tree level. The g γ 1 = 1 value is fixed for on-shell photons at tree level by electromagnetic gauge invariance to its SM value. Then, the Feynman rule for the anomalous vertex can be found from Eq.(1), where all of momentums are incoming the vertex.
However, after the recent discovery of a new particle which is consistent with the SM Replacing the Higgs doublet field by its vacuum expectation value in the above equation, nonvanishing anomalous W W γ gauge couplings in Eq. 1 can be expressed as There are a lot of phenomenological studies for W W γ interactions at the linear and hadron colliders [4][5][6][7][8][9][10][11]. The experimental sensitivity limits on anomalous W W γ couplings ∆κ = κ − 1 and λ are obtained by the LEP [12], Tevatron [13] and LHC [14][15][16]. The obtained results have been shown in Table I  The linear e + e − collider with high energy and high luminosity can give opportunity to higher precision than the hadron collider. One of the possibilities of new type linear collider is the Compact Linear Collider (CLIC). CLIC energies span from 0.5 to 3 TeV and luminosity up to 590 f b −1 and we have taken these parameters to conform with [17][18][19].The linear colliders may have an another option that polarized beam collisions. These type of collisions give new perspectives such as on the hadronic structure and high precision measurements on the electroweak mixing angle [20]. Beam polarization could be important role in the next linear colliders as well as RHIC and HERA. It is expected that 80% polarization of lepton beam can be achievable at the future linear colliders [21]. In this work, we take into account one beam can be ±80% polarization (+80% means that eighty of percent are right polarized) and one beam can be -60% (this means that sixty of percent are left polarized).
After high energy linear colliders have been constructed, its operating modes of eγ and γγ [22,23] are expected to be made. Here real photons are obtained by Compton backscattering mechanism. However, γ * γ * and eγ * interactions can appear spontaneously with respect to γγ and eγ interactions [24][25][26][27]. Therefore, γ * γ * and eγ * collisions are more realistic than the Compton backscattering procedure search for the new physics beyond the SM. These reactions occur with quasi-real photons are emitted from one (or two) of the lepton beams.
In this approximation, the virtuality of the photons are very small. Therefore, scattered angels of the emitting photons from the leptons trajectory along the actual beam path should be very small. The use of the WWA provides a lot of benefits. With simple formulas, it let to obtain simple numerical estimations [30]. Also, this method provides a facility in the experimental studies since it allows to give events number for γ * γ * → X process approximately through the examination of the e − e + → e − Xe + scattering [30]. Moreover, these processes have a very clean experimental environment, since they have no interference with weak and strong interactions.
There are many phenomenological and experimental analysis about the WWA at the LEP, Tevatron and LHC colliders . Furthermore, many studies on new physics beyond the SM using the WWA at the CLIC in the literature have been phenomenologically investigated.
In this study, we search for e − e + → W + W − , e − e + → e − γ * e + → W − v e e + , e − e + → e − γ * γ * e + → e − W − W + e + processes to investigate W W γ anomalous couplings. One of the advantages of γ * γ * and γ * e processes is that they can isolate W W γ couplings from W W Z There are several terms in tree-level cross section. These are proportional to ∆κ 2 , λ 2 , ∆κ, λ and ∆κλ additional to the SM cross section. In the effective Lagrangian, the energy dependence of ∆κ and λ terms are different as seen from (Eq.1). Especially, limits on λ are stronger than ∆κ (see Table I). Furthermore, it can be seen from Table I [74,75]. Subsequently, ∆κ, and λ couplings in this effective Lagrangian are defined in Variables file. Other files are not be any change. Finally, routine of the rules given in Refs. [74,75] are performed numerical calculations for the three processes including new physics beyond the SM.
To obtain limits, one-parameter sensitivity analysis we take into account χ 2 test, where σ(∆κ, λ) is the total cross section including SM and new physics, We have used that only one of the anomalous coupling is non zero at any given time, while the other one anomalous coupling is taken to zero. For the total cross section of the e + e − → W + W − and γ * γ * → W + W − processes, we assume that one of the bosons decays is hadronic and the other is leptonic. For these processes, we take into account BR = 0.145. For the process e − γ * → W − ν e , we assume hadronic decay channel to a violation of unitarity at some energy. At CLIC energies these deflection are expected to be small. Therefore, it will be difficult to detect this signal. However, these effects can raise up using polarized electrons. This process has been investigated for the Compton backscattering photons in [78,79].
We show the total cross section as a function of ∆κ and λ anomalous couplings for three different center-of-mass energies at Figs.16 and 17. It can be seen from these figures that total cross section increases with center-of-mass energy. For ∆κ coupling, this increment is much less than λ due to the energy dependence of the coupling. The cross section is sensitive to the sign of the ∆κ as seen from the Fig.16. In the case of 500 GeV center of mass energy, the cross-section is less sensitive to the sign of ∆κ. Thus, we see from the in Fig.16 that deviation of the the cross section of ∆κ from linearity increases at higher energies. In Fig.17 the cross section almost symmetric under the sign of the λ. Therefore, main contribution comes from the λ 2 term of the cross section due to energy dependence of the λ.
We also examine these couplings for polarized case. C. Anomalous couplings via the process γ * γ * → W − W + There is another contribution to W − W + production via γ * γ * → W − W + → (qq ′ lν l ) process with W W γ couplings. The e + e − → W + W − and e − γ * → W − ν e processes includes only 3-boson interactions. Also specific W W γγ vertex is predicted in SM.