Search for the anomalous $WW\gamma$ couplings through the process $e^-e^+\,\rightarrow\,\nu_e\bar{\nu}_e\gamma$ at ILC with unpolarized and polarize beams

We investigate the anomalous $W^+W^-\gamma$ couplings through the process $e^-e^+\,\rightarrow\,\nu_e\bar{\nu}_e\gamma$ for unpolarized and polarized electron (positron) beams at the International Linear Collider. We give the 95$\%$ Confidence Level limits on the anomalous couplings with and without the systematic uncertainties for various values of center-of-mass energies and the integrated luminosities. We show that the obtained limits on the anomalous couplings through the process $e^-e^+\,\rightarrow\,\nu_e\bar{\nu}_e\gamma$ can highly improve the current experimental limits.


I. INTRODUCTION
The Standard Model (SM) of particle physics has proven a remarkably successful field theory at the electroweak scale and below. The gauge boson self-interactions are determined by the non-Abelian SU(2) L × U(1) Y gauge symmetry of the electroweak sector of the SM and also described by the triple gauge couplings (TGCs) such as W + W − V , ZγV and ZZV (V = γ, Z) [1,2]. W + W − V vertex involves charged couplings whereas ZγV and ZZV involve neutral TGCs. Neutral TGCs at the tree level is forbidden due to lack of the electric charge of the Z boson. Neutral gauge boson self-couplings are permitted with loop diagrams in the SM. Therefore, studying the TGCs are of crucial importance to test the validity of the SM. Any deviation from the SM predictions would be a sign of the presence of new physics beyond the SM.
The effective Lagrangian method is based upon the assumption that at higher energy regions beyond the SM, there is a more fundamental physics which reduces to the SM at lower energy regions. The model-independent approach via this effective Lagrangian method is used to investigate the new physics effect on W + W − γ interactions. In this approach, in order to achieve effective interactions with SM particles, all heavy degrees of freedom are incorporated.
We examine the effects of anomalous W + W − γ couplings described with the effective Lagrangian method between W and γ for the process e − e + → ν e ν e γ at the International Linear Collider (ILC). New physics beyond the SM occurs with new interactions among the known particles. These new interactions contribute to the effective Lagrangian as higher dimensional operators, which are invariant under the SM symmetries and suppressed by the new physics scale Λ [3]: where d is the dimension of the operators. This effective Lagrangian reduces to the SM one in the limit Λ → ∞. Since the coefficients of the higher dimensional operators, C i , are fixed by the complete high energy theory, any extension of the SM can be parameterized by this effective Lagrangian, where C i are free parameters. Now, we will identify the effective Lagrangian of new physics including dimension-six operators that modify the interactions between electroweak gauge bosons: Only operators with even dimension can be constructed when baryon and lepton numbers are conserved. As a result, the largest contribution for new physics beyond the SM comes from dimension-six operators. Three CP-conserving dimension-six operators: and two CP-violating dimension-six operators: where Φ is the Higgs doublet field. The D µ covariant derivative, W µν and B µν field strength tensors of the SU(2) I and U(1) Y gauge fields are respectively as follow: where τ i are the SU(2) I generators with Tr[τ i τ j ] = 2δ ij (i, j = 1, 2, 3). g and g ′ are SU(2) I and U(1) Y couplings, respectively. The effective Lagrangian for W + W − γ interaction can be then parameterized by [4]: is the field strength tensor for photon. In Eq. (11), g γ 1 , κ γ and λ γ anomalous parameters are both C and P conserving while g γ 4 , g γ 5 ,κ γ andλ γ anomalous parameters are C and/or P violating. Electromagnetic gauge invariance requires that g γ 1 = 1. In the SM, the anomalous coupling parameters are given by κ γ = 1 (∆κ γ = 0) and λ γ = 0 at the tree level. However, CP-violating interactions can be confined individually to specially designed CP-odd observables that are insensitive to CP-even effects. Thus, the CP-conserving and violating interactions can be separated from each other. Here, the anomalous κ γ and λ γ coupling parameters can be reframed in terms of the couplings of the operators in Eq. (2) and transformed into c W W W /Λ 2 , c W /Λ 2 and c B /Λ 2 [5]. Thus, the effective field theory approach allows the following the relations between parameters: Similarly, the values of above c W W W /Λ 2 , c W /Λ 2 and c B /Λ 2 parameters lead to deviations from the SM for W + W − γ couplings and determine new physics contributions. In the SM, the anomalous coupling parameters are given by In theoretical side, the aTGC such as the anomalous W + W − V (V = γ, Z) couplings have been discussed previously in the literature [4,. The anomalous W + W − γ couplings have been studied experimentally on the parameters of κ γ and λ γ at the LEP [34][35][36], the Tevatron [37][38][39][40] and the LHC [41][42][43][44]. The limits of the anomalous coupling parameters on the aTGC obtained in some experimental and the phenomenological studies are given in Table I.  The SM is a successful theory that answers many important questions in particle physics, such as describing electromagnetic, weak and strong interactions in the universe and predicting all known elementary particles. Although the SM has passed all experimental tests, some significant arguments such as the hierarchy problem, the non-unification of fundamental forces, the baryon-antibaryon asymmetry, non-explained dark matter demonstrate that the SM has some shortcomings to be final theory of everything. For this reason, there is a great desire to search for the new physics beyond the SM.
The colliders in experimental particle physics are often classified according to their shape (linear/circular) and the type of colliding particles (hadron/lepton). All major differences between the hadron and lepton colliders depend on the nature of the colliding particles. The discovery potential of the LHC would be complemented by the ILC, which is a linear electron-positron collider in the design phase [45]. The ILC is planned to reach tunable center-of-mass energy up to 500 GeV (upgradeable up to 1 TeV) with published the Technical Design Report (TDR) for the ILC accelerator [46,47]. The electron and the positron beams are longitudinally polarized to 80% and 30%, respectively, which have different signs of polarization from each other. The positron beams are foreseen to polarize about 60% in the upgraded option of the ILC. While polarized electron beams are produced by photoproduction with a polarized laser, polarized positron beams are produced in pair conversion γ → e + e − , where the photon is produced by a high energy electron beam passing through a superconducting undulator [48]. A Compton polarimeter will perform the primary polarimeter measurement at the ILC. Polarimeters can be placed upstream or downstream of the Interference Point (IP). Because of distortion and radiation in the beam-beam collision process luminosity-weighted beam polarization will differ from measured polarization. The difference between luminosity-weighted beam polarization and polarimeter measurement is expressed as dP = P lum-wt The beam direction at the IP of the polarimeter (Compton IP) needs to be aligned within 50 µrad with the collision axis at e + e − IP for achieve the minimization of dP [49]. Using both upstream and downstream polarimetry will provide to obtain the desired minimum dP and predict the systematic error.
Thanks to the clean event environment, the tunable collision energy and the potential to polarize beams, it is possible for the ILC to observe the smallest deviation from SM predictions indicating new physics as well as to discover new particles and to make their precise measurements [49]. The possibility of both electron and positron beam polarization in the ILC is of great importance in reaching the major goals of particle physics. The two polarized beams in the ILC are very powerful tools to reveal the structure of the underlying physics, determine new physics parameters in model-independent analysis and also test basic The cross section of any process is determined from the four possible pure chiral cross sections with electron beam polarizations P e − and positron beam polarization P e + by [50] σ (P e − , P e + ) = where σ LR represents the cross section if the electron beam is left-handed polarized (P e − = −1) and the positron beam is right-handed polarized (P e + = +1). Other cross sections are similarly defined. The unpolarized cross section σ 0 is expressed as Also, the other significant definitions are the effective polarization and the effective luminosity [51][52][53][54][55]. The formulas for these definitions are given below: The effective polarization and the ratio L eff /L according to some electron and positron beam polarizations are obtained from Eqs. (16)-(17) and given in Table II. Since beam polarization plays a significant role by effectively enhancing the signal and suppressing the background rates, a scaling factor comparing the cross-sections with two different polarization configurations is parametrized by: where for two different cross sections, the one with the index (a) indicates that only the electron beam is polarized and the one with the index (b) indicates that both beams are polarized. A scaling factor can range from 0 to maximum of 2. For example, using the results at Table IV in Section III of this paper, it is seen that the cross sections for c W W W /Λ 2 coupling and Cut-3d are 29.27 pb at (P e − , P e + ) = (+80%, 0%) polarization, 20.48 pb at (P e − , P e + ) = (+80%, −30%) polarization and 11.78 pb at (P e − , P e + ) = (+80%, −60%) polarization. As a result, the scaling factors are calculated from Eq. (18) to about 0.7 between (P e − , P e + ) = (+80%, −30%) and (P e − , P e + ) = (+80%, 0%) polarizations and about 0.4 between (P e − , P e + ) = (+80%, −60%) and (P e − , P e + ) = (+80%, 0%) polarizations.

III. CROSS SECTIONS AND SENSITIVITY ANALYSIS OF THE PROCESS
e − e + → ν e ν e γ AT THE ILC The Feynman diagrams for the process e − e + → ν e ν e γ are given in Fig. 1. The first of the five Feynman diagrams includes the anomalous W + W − γ coupling and it contributes to the new physics. We can see from Fig. 1 that one of the advantages of the process e − e + → ν e ν e γ is that they can isolate W + W − γ couplings from W + W − Z couplings. Also, the neutrinos are of major importance to the elementary particle theory, astrophysics and cosmology [56].
In our calculations, the total cross sections of the process e − e + → ν e ν e γ with the configurations of electron-positron beam polarization are simulated using MadGraph5 aMC@NLO [57]. At this program, we use the EWdim6 model file for the operators that examine interactions between the electroweak gauge boson we have described above in dimension-six. We examine the potential of the process e − e + → ν e ν e γ at the ILC with √ s = 500 GeV. The configurations of electron-positron beam polarization and their corresponding integrated luminosities are studied at the ILC: (P e − , P e + ) = (∓80%, ±60%) and L = 1350 fb −1 , (P e − , P e + ) = (∓80%, ±30%) and L = 1600 fb −1 , (P e − , P e + ) = (±80%, 0%) and L = 2000 fb −1 , unpolarized electron-positron beam and L = 4000 fb −1 .
Here, we have used − sign for left polarization and + sign for right polarization.
We apply the kinematic selection cuts to suppress the backgrounds and to optimize the signal sensitivity. p ν T is transverse momentum of the final state neutrinos, |η γ | is the pseudorapidity of the photon and p γ T is the transverse momentum of the photon. The outgoing particles are required to satisfy these kinematic cuts for ν e ν e γ events at the ILC.
We consider p ν T > 25 GeV with tagged Cut-1, |η γ | < 2.5 with tagged Cut-2 and four different values of the transverse momentum of the photon, p γ T > 10, 15, 20, 25 GeV with tagged Cut-3a, Cut-3b, Cut-3c and Cut-3d, respectively. A summary of the kinematic cuts is given in Table III.

Cuts
Definitions In this analysis, we focus on CP-conserving c W W W /Λ 2 , c W /Λ 2 and c B /Λ 2 couplings via the process e − e + → ν e ν e γ at the ILC. The total cross sections of the process e − e + → ν e ν e γ as a function of anomalous c W W W /Λ 2 , c W /Λ 2 and c B /Λ 2 couplings parameters for kinematic cuts defined in Table III are presented in Fig. 2 with unpolarized electron-positron beam, in Fig. 3 with polarized electron-positron beam (P e − , P e + ) = (+80%, 0%), in Fig. 4 with polarized electron-positron beam (P e − , P e + ) = (−80%, 0%), in Fig      We present the total and SM cross sections of the process e − e + → ν e ν e γ for the kinematic cuts of seven polarization scenarios with respect to c W W W /Λ 2 , c W /Λ 2 and c B /Λ 2 couplings in  Table IV correspond to c W W W /Λ 2 , c W /Λ 2 and c B /Λ 2 = 5 TeV −2 , respectively. The ratios arise from the total cross sections divided by the SM cross sections and increase after each applied kinematic cut. As the applied kinematic cuts increase, the SM cross section is suppressed, thus the signal becomes more apparent.   The total cross sections of the process e − e + → ν e ν e γ as a function of c W W W /Λ 2 , c W /Λ 2 and c B /Λ 2 at the largest cut (Cut-3d) according to the polarization configurations are compared in Fig. 9 and thus the effect of polarizations on the total cross sections is observed.
The left-polarized electron (right-polarized positron) beam enhances the cross sections due to the structure of the e − ν e W − (e + ν e W − ) vertex in the first Feynmann diagram of Fig. 1, which contains the largest contribution with the anomalous W W γ coupling [32,58]. As seen in Fig. 9, (P e − , P e + ) = (−80%, +60%) polarization has larger cross sections compared to other polarization configurations. We have estimated using χ 2 analysis with a systematic error to obtain the constraints on the anomalous coupling parameters at the 95% C.L.. χ 2 function is defined by [59][60][61]: Here, σ SM is the cross section in the SM and σ N P is the cross section containing both the SM and new physics contributions. δ st = 1 √ N SM and δ sys are the statistical error and the systematic error. The number of SM events is presented by N SM = L × σ SM , where L is the integrated luminosity.
The systematic uncertainty value has been taken into account in previous electron positron collider studies. The systematic uncertainty in the total cross section analysis for the process e − e + → tt in electron positron collider is considered to be 3% and also the systematic uncertainty in determining the cross section has been reduced from 3% to 1% in the LEP [62]. Since the ILC will be built in the coming years, it can be assumed that systematic uncertainties will be lower with the development of future detector technology.
Taking into consideration the previous studies, we consider the systematic uncertainties of 0%, 1% and 3% in this paper. The measurements with small enough systematic uncertainties are needed to provide the required sensitivity for the new physics research.
Also, in our study, we consider the effect of the uncertainty on the electron and positron beam polarization. There are very important to know the exact polarization because deviation from the expected polarization can fake contributions from the anomalous couplings.
The measurement of the polarization is performed by a polarimeter. The systematic uncertainties affecting the polarization measurements are (a) laser polarization, (b) detector linearity, (c) analyzing power calibration and (d) electric noise which led to up to an uncertainty of ∆P e /P e ∽ 0.5%. For this reason, our analyses include polarization uncertainty of both the electron and positron beam polarization of 0.5%, 0.25% or 0.1%. Also, the crucial point is that the systematic uncertainties can be significantly reduced when both beams are polarized [49]. Here, we model the systematic uncertainty by adding uncertainties of the values of δP e + and δP e − to the systematic uncertainty introduced in χ 2 analysis.
Polarization scenarios Cuts δ sys = 0% δ sys = 1% δ sys = 3%   the process e − e + → ν e ν e γ at the ILC. The total cross section and the limit analysis were performed according to the anomalous c W W W /Λ 2 , c W /Λ 2 and c B /Λ 2 coupling parameters that lead the deviations from SM. The cutflow has created by p ν T , |η γ | and p γ T cuts. According to this cutflow and polarization scenarios, the total cross sections were calculated against the anomalous coupling parameters. The polarization scenarios affecting the size of the total cross section in the largest cut (Cut-3d) have compared with each other. The ratios of total cross section to SM cross section on the anomalous c W W W /Λ 2 , c W /Λ 2 and c B /Λ 2 coupling parameters for polarization scenarios have determined and the contributions of the kinematic cuts to the signal have investigated. Using the χ 2 analysis, the limits have obtained at 95% C.L. for the anomalous coupling parameters. If we look at the limits for each kinematic cuts corresponding to unpolarized and polarized beams in Tables V-VII, we can notice that the proper polarization of the leptons improve the limits on the anomalous couplings. We find that the polarization (P e − , P e + ) = (−80%, +60%) provide the best limits on the all anomalous coupling parameters at the ILC.
These limits can be easily compare with the limits of the phenomenological studies in Table I. The λ γ coupling in Eq. (23) appears to be more sensitive than the limits obtained for ILC, CLIC and CEPC in the phenomenological studies in Table I. But, we cannot say the same for the ∆κ γ coupling in Eq. (23). Because, although the positive limit of ∆κ γ coupling in Eq. (23) is more sensitive than the positive limits obtained for ILC, CLIC and CEPC in phenomenological studies in Table I, the sensitivity of the negative limit of ∆κ γ coupling is lower than the others. In addition, considering the systematic uncertainties of δ sys = 1%, 3%, although the sensitivities of limits decrease, it is seen that they are better compared to the sensitivity of the limits in Ref. [44].
As a result, we highlight that the sensitivities of the limits in this study are better than the sensitivity of the experimental limits reported for the LHC. Using polarized beams at the ILC to examine the anomalous W + W − γ coupling through the process e − e + → ν e ν e γ provides great advantages for sensitivity studies by guaranteeing precise measurements.