Anomalous Triple Gauge Boson Couplings in $e^{-}e^{+} \to \gamma \gamma$ for Non Commutative Standard Model

We investigate $e^{+}e^{-}\to \gamma\gamma$ process within the Seiberg-Witten expanded noncommutative standard model(NCSM) scenario in the presence of anomalous triple gauge boson couplings. This study is done with and without initial beam polarization and we restrict ourselves to leading order effects of non commutativity i.e. $O(\Theta)$. The non commutative(NC) corrections are sensitive to the electric component($\vec{\Theta}_E$) of NC parameter. We include the effects of earth rotation in our analysis. This study is done by investigating the effects of non commutativity on different time averaged cross section observables. We have also defined forward backward asymmetries which will be exclusively sensitive to anomalous couplings. We have looked into the sensitivity of these couplings at future experiments at the International Linear Collider(ILC). This analysis is done under realistic ILC conditions with the Center of mass energy(c.m.) $\sqrt{s}=800$GeV and integrated luminosity L=500fb${}^{-1}$. The scale of non commutativity is assumed to be $\Lambda = 1$TeV. The limits on anomalous couplings of the order $10^{-1}$ from forward backward asymmetries while much stringent limits of the order $10^{-2}$ from total cross section are obtained if no signal beyond SM is seen.


I. INTRODUCTION
Triple gauge boson couplings arise in the Standard Model(SM) due to the non Abelian nature of the theory and thus gives the possibility of exploring the bosonic sector of the SM.
Many of such couplings do not appear in SM at the tree level and even at higher orders thus they are expected to be very small. Hence the precise measurement of these couplings could indicate a signal for new physics beyond the SM even if there is no direct production of particles beyond SM spectrum. A linear collider at a center of mass energy of 800 GeV or more in high luminosity regime provide a unique opportunity to measure such couplings with an unprecedented accuracy where we can distinguish such effects from the SM predictions.
Moreover the availability of initial beam polarization option, can significantly enhance the sensitivity to such effects.
In this study we investigated the expected sensitivity of triple gauge boson couplings Zγγ and γγγ, to leading order that will contribute in e − e + → γγ process at proposed International Linear Collider(ILC) [1,2]. This investigation is done within the frame work of non commutative SM(NCSM).
Quantum field theories constructed on non commutative(NC) space time have been extensively explored in the past few years. This field has received much attention due to its possible connection with quantum gravity and because of its natural origin in string theories. Infact Seiberg and Witten [3] described how NC gauge theory can emerge as a low energy manifestation of open string theory. This work has stimulated many paper on non commutative models [4][5][6][7][8][9].
Hence keeping in mind the above considerations it is reasonable to investigate field theories, and in particular the standard model of particle physics on non commutative space time. Here we adopt an approach based on Seiberg-Witten Map(SWM) popularized by the Munich group [10][11][12][13][14][15][16][17][18].
In this approach, to construct the NC extension of the standard model (SM) [13,14,17,18], which uses the same gauge group and particle content one expands the NC gauge fields in non linear power series of Θ [3,11,12]. At face value it can be seen from the above map that SW approach leads to a field theory with an infinite number of vertices and Feynman graphs thereby leading to an uncontrolled degree of divergence inturn giving an impression of complete failure of perturbative renormalization. But over the years a number of studies have shown that it is possible to construct anomaly free, renormalizable, and effective theories at one loop and first order in Θ [19][20][21][22][23][24][25][26][27]. The above mentioned studies provide confidence in using the using NC SW expanded SM for phenomenological purposes.
However it should be mentioned here that the celebrated IR/UV mixing does not exist in the above Θ expanded approach. Though this is not a drawback in the scales of our interest there do exist certain phenomena that require all orders of the NC parameter be retained.
This led to the so called Θ-exact approach, that is from the exact solutions of the SW equations. The phenomenological consequences of this have been explored in [28][29][30][31].
In this work we have calculated (O(Θ)) corrections for pair annihilation with and without initial beam polarization. Here, unlike NCQED case, non commutative effects at leading order also appear in unpolarized cross section due to the presence of axial vector coupling of the Z boson. We have also taken into account the effect of earth's rotation [57][58][59][60] on observable signals of non commutativity. The effects of Non commutativity is studied on various time averaged observables to check the sensitivity of anomalous couplings. Here we note that this process has been studied previously [61] with unpolarized initial beams and polarized final states. Note that observation of final state polarization of photons is not possible at the high energies of ILC.
We have looked at the sensitivity of the anomalous couplings at the International Linear Collider(ILC) [1,2] with realistic beam luminosity of 500 fb −1 . The availability of longitudinal beam polarization of one or both of the e − e + beams, can give the opportunity to test the couplings which otherwise are absent in the observables with unpolarized initial beam. If these beams become available at future Linear Colliders then it will serve as crucial test for these anomalous couplings of NCSM in the process we discuss.
The rest of the paper is organized as follows. In section II, we give the calculational details of our work for the mentioned process. In section III, we will present our numerical results. Finally we conclude with a section on summary of our results.

II. CROSS SECTIONS IN THE LABORATORY FRAME
This process in NCQED proceed at the tree level by the following diagrams (Fig.1). The first two diagrams also appears in pure QED while 3rd one arises just because of non commutative nature of space time and is a contact interaction. The squared amplitude for the above process is given by the expression Here as mentioned in introduction we are restricting ourselves only to O(Θ) thus the interference between SM and NC term can provide required corrections to cross section.
Since non commutative parameter is considered as elementary constant in nature so its direction is fixed in some non rotating coordinate system(can be taken to celestial sphere).
FIG. 2: Non Minimal NCSM: s-channel Feynman Diagrams with anomalous couplings. Here However the experiment is done in laboratory coordinate system which is rotating with earth's rotation. So one should take into account these rotation effects on Θ µν in this frame before moving towards the phenomenological investigations.
These effects were considered in many previous studies [57][58][59][60] but we are here following the lines of [59]. In the laboratory coordinate system, the orthonormal basis of the non rotating(primary) coordinate system can be written as (1) Here we have used the abberivations c α = cos α, s α = sin α etc. (δ, a) defines the location of experiment with −π/2 ≤ δ ≤ π/2 and 0 ≤ a ≤ 2π. More details can be found in Ref. [59].
Thus the NC parameter in the Laboratory frame is given by electric and magnetic com- Here (η, ξ) specifies the direction of NC parameter(Θ µν ) w.r.t primary coordinates system. Θ E is the absolute values of its electric components with corresponding scale Λ E .
Our results are based on Feynman rules for NCSM given in Ref. [17,18] where θ is the polar angle and φ is the azimuthal angle, with initial beam direction chosen as the z-axis.
Due to the breaking of Lorentz invariance for fixed Θ background, non commutativity of space time lead to dependence of cross section on azimuthal angle which is absent in Standard Model. The final cross section formulae for different cases are given by: A. Unpolarized Case The differential cross section for e − e + unpolarized case is given by: B. Polarized Case The differential cross section for e − in Right polarized state is given by: The differential cross section for e − in Left polarized state is given by: C V and C A are the vector and axial vector coupling of Z boson with electron and are given by (-1+ 4 sin 2 θ W )/2 and -1/2 respectively. Since it is difficult to get time dependent data so one have to take average over full day to be compared with the experiment. So we will consider here following cross section observables to reveal the effects of non commutativity where σ ≡ In addition to these observables, since the terms containing anomalous couplings in cross section flip sign under transformation (θ → (π − θ), φ → (π + φ)), so we can define following forward backward asymmetry(with appropriate cuts on azimuthal angle(φ)) which will only be sensitive to these couplings.
= C 1 (4θ 0 − π)s E sin a cos δ cos η π(cos θ 0 + 2 log[tan θ 0 2 ]) where Similarly one can also define polarized forward backward asymmetry where where L and R refer to the helicity of the incident electron beam; F and B stand for forward and backward. For unpolarized case, the coefficient of K Zγγ in forward backward asymmetry is smaller by a factor of 2C V /C A then to its polarized counterpart so the corresponding limit for K Zγγ will be stringent in this case.

III. NUMERICAL ANALYSIS
In this section we will provide the numerical results of our investigation. In order to check the sensitivity of anomalous couplings(K Zγγ , K γγγ ), we studied previously defined forward backward asymmetries(A F B ) and total cross section( σ T ). We fixed the initial beam energy at √ s(= E com ) = 800 GeV. The position of Lab system is fixed by taking δ = π/4 and Here for studying total cross section we applied a cut of (0−π) on azimuthal angle φ since non commutative effects disappear once we integrate over the full azimuthal angle(0 − 2π).
Our results are useful for case s/Λ 2 < 1 since in this domain one can safely ignore higher order corrections to cross section.
For deriving limits on anomalous couplings we will make use of expressions for forward backward asymmetry along with total cross section. Fig. 4 shows variation of previously defined forward backward asymmetries plotted against polar cutoff angle θ 0 for different values of anomalous couplings. As evident from the plots, the magnitude of asymmetries become larger for higher values of polar cutoff angle(θ 0 ). Thus using higher θ 0 will give better limits on couplings.
The asymmetries are then used to calculate 90% CL limits with realistic integrated luminosities in the absence of any signal at ILC. The limit on the coupling at a polar cut off angle(θ 0 ) is related to the value of A(θ 0 ) of the asymmetry by [64,65]: where |A(θ 0 )| is the absolute value of the asymmetry for unit value of the coupling.
From Eq. 18 we see that A U F B solely depends on K Zγγ , therefore an independent limit can be placed on it. However A P F B depends on K Zγγ as well as on K γγγ . Thus for evaluating limit on K γγγ we have assumed other anomalous coupling to be zero. From Figs. 6 it is clear that the best limit for |K Zγγ | and |K γγγ | is achieved for an cutoff angle θ 0 = 75 • .
Following the same procedure one can obtain limits from total time averaged cross section.
In Fig. 5 we have plotted the total time averaged cross section section for different values of anomalous couplings. Here unlike asymmetries significant deviation from SM case is obtained for smaller values of cut-off angle(θ 0 ). We have also derived the limits on these couplings in case of no excess in signal events is observed at ILC.
However here the limits are obtained by using the condition that the excess number of events beyond expected from SM should be smaller than the statistical error in the number of SM events. This translates to L|σ N P (θ 0 )| < 1.64 σ tot (θ 0 ) · L where σ N P is the NC contribution to the total cross section and L is the integrated luminosity which we assumed to be 500 fb −1 for current study.
From Eqs.4, 7 we see that σ T for unpolarized case solely depends on K Zγγ , therefore an independent limit can be placed on it. However σ T for unpolarized case depends on K Zγγ as well as on K γγγ . Thus for evaluating limit on K γγγ we have assumed other anomalous coupling to be zero. From Figure 7 it is clear that the best limit for |K Zγγ | and |K γγγ | is obtained for an cutoff angle θ 0 = 30 • .
Coupling Limits-Unpolarized case Limits-Polarized case  In Table I we have quoted the derived limits from asymmetries as well as from time averaged total cross section. The asymmetries give a limit of order 10 −1 while from total cross section limits are much more stringent of the order of 10 −2 . Thus total cross section is proven to be much useful observable then forward backward asymmetries.

IV. SUMMARY
The extension of SM to NC space time with motivations coming from string theory and quantum gravity provides interesting phenomenological implications since scale of non commutativity could be as low as a few TeV, which can be explored at the present or future colliders. In the present work we focused on exploring the sensitivity of anomalous couplings(K Zγγ , K γγγ ) that will contribute to the process e + e − → γγ process at ILC.
We have done our study with unpolarized as well as taking into account the initial beam polarization effects. We restricted ourselves to the leading order effects of non commutativity to be occurred at leading order in Θ(i.e. O(Θ)) at cross section level. Unlike NCQED case non commutative effects at O(Θ) also appear in unpolarized cross section due to the presence of axial vector coupling of Z boson.
In this analysis we have also taken into account the apparent time variation of non commutative parameter(Θ µν ) in Laboratory frame. We have used time averaged observables for this study.
The NC corrections to the considered process are sensitive to the electric component( Θ E ) of NC parameter ( Θ). However for checking the sensitivity of anomalous couplings at ILC we used time averaged total cross section and forward backward asymmetries as observables.
This analysis is done under realistic ILC conditions with the Center of mass energy(c.m.) √ s = 800GeV and integrated luminosity L=500fb −1 . The scale of non commutativity is assumed to be Λ = 1TeV and Lab coordinates are fixed to be (δ, a) = (π/4, π/4).
The observables for unpolarized case are only sensitive to K Zγγ while for polarized case they are sensitive to both couplings. For putting limits from polarized case we have assumed one coupling to be zero at a time. The asymmetries give a limit of order 10 −1 while limits from total cross section are much more stringent, of the order of 10 −2 on absolute value of the anomalous couplings.