Search for noncommutative interactions in $\gamma\gamma\rightarrow\gamma\gamma$ process at the LHC

The noncommutative QED (NCQED) has a non-Abelian nature due to the presence of 3- and 4-photon vertices in the lagrangian. Thus, NCQED predicts a new physics contribution to the $\gamma\gamma\rightarrow\gamma\gamma$ scattering already at the tree level. We have examined NCQED by studying the light-by-light process at the 14 TeV LHC with intact protons. Our results show that the NC scale up to $\Lambda_{\mathrm{NC}} = 1.65(1.35)$ TeV can be probed in the $pp\rightarrow p(\gamma\gamma)p \rightarrow p'(\gamma\gamma)p'$ collision for the time-space (space-space) NC parameters. These bounds are stronger than the limits that can be obtained in the light-by-light scattering at high energy linear colliders.


Introduction
The quantization of the electromagnetic field in a noncommutative (NC) space-time has a long story [1,2]. To a large extent, an interest in NC quantum field theories [3]- [5] was mainly motivated by string theory [6]- [9].
In NC field theories the conventional coordinates are represented by noncommutative operators, where θ µν is the NC constant of dimension (mass) −2 . In what follows, where the dimensionless elements of the antisymmetric matrix c µν are assumed to be of order unity. Let the fieldΦ(X) be an element of the algebra (1). The noncommutivity of the space-time can be implemented by the Weyl-Moyal correspondence [10]- [14]Φ where k, x are real variables. Thus, we associateΦ(X) with a function of classical variable x. As it follows from (3), Φ 1 (X)Φ 2 (X) = 1 (2π) 4 d 4 k d 4 p e ikX φ(k) e ipX φ(p) , = 1 (2π) 4 d 4 k d 4 p e i(k+p)X−k µ p ν [Xµ,Xν ]/2 φ(k)φ(p) , where we used the Backer-Campbell-Hausdorff formula. Thus, the NC version of a field theory is given by replacing field products by the star product defined aŝ It obeys the associative law. To the leading order in θ, the star product is given by It is useful to define a generalized commutator known as the Moyal bracket (MB) by the relation As one can see, the MB of coordinates, in agreement with the commutator relation on the NC space-time (1). There is a relation between the matrix c µν (2) and the Maxwell field strength, since in string theory the quantization of NC quantum field theory is described by the excitations of D-branes in the presence of the background EM field [6]- [9]. The c 0i coefficients are defined by the direction of a background electric field, E = (c 01 , c 02 , c 03 )/Λ 2 NC . The c ij elements are related to a background magnetic field, B = (c 23 , c 02 , −c 12 )/Λ 2 NC . Note that theories with nonzero c 0i in (1) do not generally obey unitarity [15]- [18]. However, theories with only space-space noncommutativity, c ij = 0, c 0i = 0, are unitary.

Noncommutative QED
Noncommutative QED (NCQED), based on the group U(1), has been studied in a number of papers [19]- [28]. It was shown that unbroken U(N) gauge theory is both gauge invariant and renormalizable at the one-loop level [19], [23]. The pure noncommutative U(1) Yang-Mills action is defined as with We see that even in the U(1) case the potential A µ couples to itself. One can easily check that the action (9) is invariant under U(1) transformation defined as where The covariant derivative transforms covariantly. The * product admits only the fields ϕ with charge 0 or ±1 [21]. The field strength transforms as Using relations U * U −1 = U −1 * U = I and the cyclic property of the star product under the integral [22], we find that The 2-point photon function is identical in NC and commutative spaces, since the quadratic term in (15) remains the same, Due to the presence of the * product and MB bracket, the theory reveals non-Abelian nature. Namely, both 3-point and 4-point photon vertices are generated. The Feynman rules of the pure NCQED [29]- [32] are shown in Fig. 1 They are given by the following expressions where the wedge product is defined as Note that p ∧ k = −k ∧ p and p ∧ p = 0. As one can see, the Feynman rules are very similar to those in non-abelian gauge theory, with the structures constants replaced by factors 2 sin[(p i ∧ p j )/2]. These factors arise as a consequence of the MB. Indeed, we have Thus, NCQED predicts new tree-level contributions to the γγ → γγ collision. Note that c µν (2) in not a tensor. It means that the Lorentz symmetry is explicitly violated. However, it is quite different from Lorentz breaking models discussed often in the literature, since it can hold only at energies of order Λ NC . Moreover, since the NCQED is CPT invariant, available experimental bounds on observables which are simultaneously CPT and Lorentz violating cannot be used to constrain NCQED.
A number of noncommutative extensions of the Standard Model is constructed [33]- [41]. Since we are interested in the light-by-light collision, we will work in the framework of the NCQED.

Light-by-light scattering at the LHC
The observable signatures in a number of NCQED processes in e + e − collisions has been considered in [42]- [47]. The light-by-light (LBL) scattering in ultraperipheral Pb+Pb collisions in the NCQED context have been recently studied in [48,49]. Our goal is to examine the LBL scattering in pp collisions at the 14 TeV LHC through the process pp → p(γγ)p → p ′ (γγ)p ′ . Here the final state photons are detected in the central detector and the scattered intact protons are measured with forward detectors.
To detect the protons scattered at small angles, so-called forward detectors are needed. The ATLAS is equipped with the Absolute Luminosity For ATLAS (ALFA) [50,51] and ATLAS Forward Physics (AFP) [52,53]. The CMS collaboration uses the Precision Proton Spectrometer (PPS) as a subdetector which was born from a collaboration between the CMS and TOTEM [54] (previously named CT-PPS). The ALFA system is made of four Roman Pot stations located in a distance of about 240 m at both sides of the AT-LAS interaction point. The AFP detector consists of four detectors placed symmetrically with respect to the ATLAS interaction point at 205 m (NEAR stations) and 217 m (FAR stations). The PPS detector has four Roman Pots on each side placed symmetrically in the primary vacuum of the LHC beam pipe, at a distance between 210 m and 220 m from the CMS interaction point. These forward detectors are installed as close as a few mm to the beamline to tag the intact protons after elastic photon emission. It allows detecting the fractional proton momentum loss in the interval ξ min < ξ < ξ max . The larger value of ξ can be achieved when a detector is installed closer to the beam pipe.
Two types of examinations included by the AFP are i) exploratory physics (anomalous couplings between γ and Z or W bosons, exclusive production, etc.); ii) standard QCD physics (double Pomeron exchange, exclusive production in the jet channel, single diffraction, γγ physics, etc.). PPS experiments aim at a study of the elastic proton-proton interactions, the proton-proton total cross-section and other diffractive processes. Moreover, precise search can be done with the forward detectors [55]- [57]. In such interactions involving high energy and high luminosity, the pile-up background may be formed. This background can be extremely reduced by using kinematics, timing constraints, and exclusivity conditions [58]- [60]. There are many phenomenological papers that use photon-induced reactions for searching new physics at the LHC [61]- [82].
We examine the process pp → pγγp → p ′ (γγ)p ′ . Emitted photons have very small virtualities, hence they are almost-real photons and can be considered as on-mass-shell particles. The main detector (ATLAS or CMS) registers the final state γγ, while the proton momentum loss ξ is measured by the forward detector (AFP or CT-PPS). This makes it possible to determine the invariant energy of the γγ collision, W = 2E √ ξ 1 ξ 2 , where E is the energy of the incoming protons. These type of collision can be studied using equivalent photon approximation (EPA) [83]- [85]. In the EPA, a photon emitted with small angles by the protons shows the following spectrum in photon virtuality Q 2 and energy fraction where Here, m p is the mass of the proton, µ p is its magnetic moment, F E and F M are electric and magnetic form factors of the proton. To obtain the cross section of the process pp → p(γγ)p → p ′ (γγ)p ′ , the cross section dσ γγ→γγ of the subprocess γγ → γγ should be integrated over the photon spectrum, The effective photon luminosity in (24) is given by We put Q 2 max = 2 GeV 2 , since the contribution of more than this value is very small.
The diagrams describing the γγ → γγ scattering in the order e 2 are presented in Fig. 2. The nonzero independent helicity amplitudes of NCQED have been derived in [44,49] M NC ++++ (p 1 , p 2 ; k 1 , k 2 ) = −32πα where α = e 2 /(4π), and s = ( Mandelstam variables of the γγ → γγ process (s + t + u = 0). To derive expressions (27) form those presented in [44], we have used the Moyal-Weyl star product Jacobi identity in momentum space [49] sin The amplitude M NC +−+− is defined by the crossing relation The other nonzero NC helicity amplitudes are related to the amplitudes (27) and (29) by the parity relations After simple arithmetic we find from (27), (29), and (30) As it is shown in Appendix A, this equation can be written as in a full agreement with eq. (93) in [49], after changing momenta notations, (p 1 , p 2 , k 1 , k 2 ) → (k 1 , k 2 , k 4 , k 3 ). Note that the second term in the brackets in eq. (32) is obtained from the first term if one uses the replacements k 1 ⇄ k 2 , t ⇄ u. Analogously, the third term comes from the first one after the replacements k 1 ⇄ p 2 , t ⇄ s. There is a one-to one correspondence between the color ordering in QCD and the star product in pure NCQED [86]. As it was mentioned in the end of section 2, all vertices of NCQED are similar to gluon vertices in QCD in which the structure constants f a i a j c are replaced by 2 sin[(l i ∧ l j )/2]. A correspondence between NCQED and QCD amplitudes can be achieved, if one makes in (32) the following replacements [49] 2 sin where (l i , l j , l k , l r ) is a combination of the photon momenta p 1 , p 2 , k 1 , k 2 , and the sum over all color indices is assumed. The gluon-gluon amplitude square is a sum of helicity amplitudes, and the differential cross section is given by After summation over Mandelstam variables in (32), and replacement α → α s , we come from (32)- (35) to the well-known expression for the differential cross section of the gluon-gluon scattering [87] dσ dΩ Thus, our NC amplitude square and its counterpart in QCD are closely connected.

Numerical analysis
We work in the c.m.s. of the colliding protons. Let x 1 and x 2 be momentum fractions of the protons carried by the scattered photons which can be considered to be on-shell particles. Then the photon momenta look like Thus, the γ(p 1 ) + γ(p 2 ) → γ(k 1 ) + γ(k 2 ) collision goes in the non-center-of mass system. For the massless case the momenta of the outgoing photons are given by where Here c θ = cos θ , s θ = sin θ , c φ = cos φ , s φ = sin φ, with θ and φ being the scattering angles of the outgoing photon with momentum k 1 .
Then the wage products of the photon momenta in the formulas for the NC helicity amplitudes take the form [44,49] The Mandelstam variables of the γγ → γγ process are given by The differential cross section of the γγ → γγ process in the frame (38)-(39) is given by [49] dσ dΩ (γγ → γγ) = E 1 4πs where dΩ = sin θdθdφ. Note that the first factor in the right-hand side of this equation coincides with the conventional factor 1/(64π 2 s) only if x 1 = x 2 .
To do our calculations, we choose the region of 0.015 < ξ < 0.15 for both protons which is a standard acceptance in the central detectors at the nominal accelerator and beam conditions. It is in accordance with the intervals for ξ used by the ATLAS [88] and CMS-TOTEM [89] collaborations, as well as in several papers that examine processes with the proton tagging at the LHC [90]- [96]. We also have applied the cut on the rapidity of the outgoing photons, |η| < 2.5 in all calculations.
The SM contribution to the γγ → γγ scattering is described by diagrams with charged fermion loops, W boson loops, and gluon loops [97]- [101]. As it is shown in [73], when the center of mass energy is greater than 200 GeV, QCD contributions are negligible, since they are very small relative to W and fermion loops contributions. In our study, minimum diphoton mass energy could be 2 × 7000 GeV×0.015 = 210 GeV due to ξ min = 0.015 value. Therefore, we could safely ignore the QCD loop contributions. Explicit analytical expressions for the SM helicity amplitudes, both for the fermion and W boson terms are too long. That is why we do not present them here. They can be found in [102]. In such high-energy and luminosity collisions, pile-up backgrounds can occur. With the use of kinematics, exclusivity conditions, and timing constraints such backgrounds can be extremely reduced [58,59].
As it follows from eqs. (27), (29), (30), and (41), two options, c 01 = 1, with all other c µν vanishing, and c 13 = 1, with all other c µν vanishing, result in the same NC helicity amplitudes. The same is true for the amplitudes with c 02 = 1 and c 23 = 1. Given only one of the parameters c µν is taken to be nonzero, the NC amplitudes are insensitive to its sign, since they are invariant under the c µν → −c µν replacement. Note also that four-photon NC helicity amplitudes do not depend on c 12 . Thus, only three possibilities should be addressed: (i) c 03 = 1, with all other c µν vanishing; (ii) c 13 = 1, all other NC parameters are zero; (iii) c 23 = 1, with all other c µν vanishing.
In Fig. 3 we show the results of our calculations of the total and SM differential cross section for the diphoton production at the 14 TeV LHC with intact protons. The cross section is presented as a function of the invariant mass of outgoing photons, m γγ , for two values of the NC scale Λ NC . We see that in the region m γγ > 400(600) GeV the NC contribution strongly dominate the SM one for Λ NC = 0.5(1.0) TeV. For both values of the NC scale, the time-space NC (blue curves in Fig. 3) is larger that the spacespace NC (red curves). Only for Λ NC = 0.5 TeV and m γγ > 1800 GeV the time-space and space-space curves merge. From eqs. (27), (29), and (41) one makes sure that after integrations over angular variables, the NC amplitudes should give the same contribution to the differential cross section both for c 13 = 1 and for c 23 = 1. Our numerical calculations confirm this statement. That is why, we do not present curves for case (iii) in Fig. 3.
In Fig. 4 the total and SM cross sections σ(m γγ > m γγ, min ) versus m γγ, min , the minimal invariant mass of the outgoing photons, is presented. As one can see, for Λ NC = 0.5 TeV this cross section is approximately two order of magnitude larger than the SM cross section in all mass region. For Λ NC = 1 TeV the deviation of the cross section from the SM one is also very large and becomes more and more prominent as m γγ, min grows. Thus, the bigger is the value of m γγ, min , the larger is the difference between the new physics and SM. Note that the time-space NC exceeds the space-space NC for all m γγ, min . Only for Λ NC = 0.5 TeV it becomes comparable with the space-space NC in the large m γγ, min region.
Knowing cross sections and SM backgrounds, we have calculated upper bounds on the NC scale Λ NC which can be obtained from the pp → pγγp → p ′ γγp ′ scattering at the LHC. To obtain the exclusion region, we applied the where S is the number of signal events and B is the number of background events. We define the regions SS 1.645 as the regions that can be excluded at the 95% C.L. To reduce the SM background, we used the cut m γγ > 1000 GeV . The results are shown in Fig. 5

Conclusions
We have examined the light-by-light scattering in the noncommutative (NC) quantum electrodynamics (NCQED) in the pp → pγγp → p ′ γγp ′ collision at the 14 TeV LHC. The NC geometry of space-time appears within a framework of string theory. NCQED is based on the U(1) group, but it exhibits the non-Abelian nature in having both 3-photon and 4-photon vertices in the lagrangian. That is why the γγ → γγ scattering must be sensitive to the NC contributions at the tree level, see Fig. 2.
We have calculated the differential cross sections as functions of the invariant mass of the outgoing photons (Fig. 3), as well as the cross sections as functions of the minimum invariant mass of the outgoing photons (Fig. 4). Both time-space and space-space NC parameters have been considered. The SM background is defined by the contributions from the W and charged fermion loops. It allowed us to estimate the 95% C.L. bounds for the NC scale Λ NC . We have shown that the scales up to Λ NC = 1.64(1.35) TeV can be probed at the LHC, for the time-space (space-space) NC parameters, see Fig. 5. Our bounds are stronger than the limits that can be obtained in the 4-photon scattering at high energy linear colliders [44]. Note that γγ → γγ is the more appropriate channel to probe Λ NC also in Pb-Pb collisions at the LHC [49].