Search for the RS model with a small curvature through photon-induced process at the LHC

In this paper, potential of the LHC to explore the phenomenology of the Randall-Sundrum-like scenario with the small curvature for the process $pp \to p \gamma \gamma p \to p \mu^- \mu^+p $ through the subprocess $\gamma \gamma \to \mu^- \mu^+$ is examined for two forward detector acceptances, $0.0015<\xi<0.5$ and $0.1<\xi<0.5$. This process is known to be one of the most clean channels. The sensitivity bounds on the anomalous model parameters have been found at the $95\%$ confidence level for various LHC integrated luminosity values.


I. INTRODUCTION
The Standard Model (SM) gives very satisfactory description of high energy physics at an energy scale of electroweak interactions. It is in perfect agreement with current experimental results. However, there are many open equations at the SM. One of this is the hierarchy problem. The extra dimensional models in high energy physics provide possible candidates for this problem. In this respect, these models have drawn attention over the recent years.
There are many phenomenology paper of new physics models at the LHC. These searches generally include the usual proton-proton inelastic processes in which the interacted protons dissociate into jets. Due to these jets, such interactions give a very crowded environment.
These formed jets create some uncertainties and make it difficult to detect the signals from the new physics which is beyond the SM. However, exclusive production pp → pXp provides a very clean environment. These type of processes have been much less examined in the literature. Both interacted protons remain intact, hence they do not dissociate into partons in the exclusive productions. ATLAS and CMS collaborations prepared a physics program of forward physics with extra detectors symmetrically located in a distance from the interaction point. These new detectors are equipped with charged particle trackers and they provides to tag intact scattered protons after the collision. Additionally, forward detectors can detect intact outgoing protons in the interval ξ min < ξ < ξ max where ξ is the momentum fraction loss of the intact protons ξ = (|E| − |E ′ |)/|E| where E and E ′ are the energies of the incoming and intact scattered proton, respectively. These intervals are known as the acceptance of the forward detectors. If these machines are located closer to main detectors, a higher ξ can be created. With applying this new detectors it is possible to obtain high energy photonphoton process with exclusive two particle final states such as leptons or photons. The programs about these detectors were prepared by ATLAS Forward Physics Collaboration (AFP) and CMS-TOTEM Precision Proton Spectrometer (CT-PPS) [1,2]. AFP cover 0.0015 < ξ < 0.15, 0.015 < ξ < 0.15 detector acceptance ranges. Similarly, CT-PPS has the acceptance ranges of 0.0015 < ξ < 0.5, 0.1 < ξ < 0.5. Two types of measurements are planed by AFP with high precision: 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.) [3,4].
The CT-PPS experiment main motivations are the investigation of the proton-proton total cross-section, elastic proton-proton interactions, and all of the diffractive processes. Roman Pots detector is used in this experiment. In the forward location, almost all inelastic physical interactions can be detected by the charged particle detectors. A large solid angle unable to covered with the support of the CMS detector. In this way, the detectors can be used to precise studies [5][6][7]. Due to high energy and high luminosity, this kind of interactions may cause a number of pile-up events. However, these backgrounds enable to rejected by applied exclusivity conditions, kinematics and timing constraints with use of forward detectors in conjunction with central detectors. Moreover, two lepton final states have very small backgrounds in the presence of pile-up events. Because, there are no other charged particles on the two lepton interaction vertex. Therefore, final state leptons are highly back-to-back with almost equivalent p t [8,9].
Photon induced reactions were studied in by the CDF collaboration [10,11]. Obtained results in these experiments are consistent in theoretical expectations with pp → pℓ − ℓ +p through the subprocess (γγ → ℓ − ℓ + ). At the LHC, the CMS collaboration have recently examined to measurement of the photon-induced reactions through the processes [12,13]. In this experiments show that the both the number of candidates and the kinematic distributions are in agreement with the expectation for exclusive and semi-exclusive e − e + production γγ → e − e + process. Similarly, ATLAS Collaboration reported a measurement of the exclusive γγ → l − l + (l = e, µ ) cross-section in proton-proton collisions at a center-of-mass energy of 7 TeV [14]. In this measurement, it is obtained that when proton absorptive effects due to the finite size of the proton are taken into account in the theory calculation, the obtained cross-sections are found to be consistent with the theoretical calculations. Another similar measurement was made by the ATLAS Collaboration at a center-of-mass energy of 8 TeV [15]. In this experiment, it was obtained that the single-differential cross section as a function of m ℓ − ℓ + (ℓ = e, µ). The leptons channel measurements are combined and a total experimental precision of better than 1% is achieved at low m ll . In the literature, other phenomenological papers based on photon-induced reactions at the LHC for new physics beyond the SM can be found in .
The forward detectors allow high energy photon-photon interaction as mention above.
The photons are generated by the high energetic protons can be considered as an intense photon beam. These almost-real photons have very low virtuality so that it can be assumed that they are on-mass-shell. They radiate off the incoming protons with small angles and low transverse momentum. Intact protons thus deviate slightly from their trajectory along the beam path without being detected by central detectors. Intact protons and ξ are measured by the forward detectors with a very large pseudorapidity. As a result, the final state X is obtained through the process pp → pγγp → pXp and measured at the main detector. The schematic diagram for this process is shown in Fig.1. Since, energy loses protons can be detected by the forward detectors, the invariant mass of the central system W = 2E √ ξ 1 ξ 2 can be known. The photon-induced reactions can be described by the equivalent photon approximation (EPA) [37,38]. According to this method, quasi-real photons with low vitalities (Q 2 = −q 2 ) interact to create final state X through the subprocess γγ → X. In the EPA framework, emitted quasi-real photons bring a spectrum that is a function of virtuality Q 2 and the Here m p proton mass. In the dipole approximations the other terms given as follows, where, µ 2 p = 7.78 is the square of the magnetic moment of the proton, F E and F M are the relative the electric and magnetic form factors of the proton, respectively. In the photonphoton collisions, luminosity spectrum dL γγ dW can be found with using EPA as follows, where, Q 2 max = 2 GeV 2 is taken in the above equation since the contribution of higher virtualities more than this value is negligible. The cross section for the pp → pγγp → pXp can be derived by integrating selected subprocess γγ → X cross section over the photon spectrum, In this paper, we have investigated the RS-like scenario with the small curvature (the details is given in the next section) for the process pp → pγγp → pµ − µ + p through the subprocess γγ → µ − µ + . Because of the main contribution comes from the high energy region, we have made this calculation for two acceptance ranges 0.0015 < ξ < 0.5 and 0.1 < ξ < 0.5. Fig. 2 shows the behavior of the γγ luminosity spectrum as a function of the W for these two forward acceptance ranges.

II. RSSC MODEL OF WARPED EXTRA DIMENSION WITH SMALL CURVA-TURE
The Randall-Sundrum model with two branes (RS1 model [39]) was proposed as an alternative to the scenario with large flat extra dimensions (ADD model [40][41][42]). The RS1 model is described by the following background warped metric where η µν is the Minkowski tensor with the signature (+, −, −, −), and y is an extra coordinate. The periodicity condition y = y + 2πr c is imposed, and the points (x µ , y) and  is assumed that there are two branes located at these points (called Planck and TeV brane, respectively). All the SM fields are confined to the TeV brane.
The function σ(y) in (6) was obtained in [39] to be where κ is a parameter with a dimension of mass. It defines the curvature of the 5dimensional space-time. Recently, a generalization of the warp factor σ(y) war derived in [43] σ where C is y-independent quantity, with the fine tuning relations Here Arccos(z) is a principal value of the multivalued inverse trigonometric function arccos(z). This generalized solution (i) obeys the orbifold symmetry y → −y; (ii) makes the jumps of σ ′ (y) on both branes; (iii) has the explicit symmetry with respect to the branes [43].
By taking C = 0 in (10), we get the RS1 model (9), while putting C = πκr c , we come to the RS-like scenario with the small curvature of the space-time (RSSC model [44]- [46]).
What are main features of the RSSC model in comparison with those of the RS1 model?
µν with the SM fields on the TeV brane are given by the effective Lagrangian density wereM Pl = M Pl / √ 8π is the reduced Planck mass, T µν (x) is the energy-momentum tensor of the SM fields. The coupling constant is equal to The hierarchy relation looks like (κπr c ≫ 1) The masses of the KK gravitons are proportional to the curvature parameter κ [45] m n = x n κ , n = 1, 2, . . . , where x n are zeros of the Bessel function J 1 (x). If we put κ ≪M 5 ∼ 1 TeV, the mass splitting will be small, ∆m ≃ πκ, and we come to an almost continuous mass spectrum, similar to the mass spectrum of the ADD model [40]. This is in contrast to the RS1 model, in which the gravitons are heavy resonances with masses above one-few TeV.
Let us calculate the scattering amplitude for the subprocess γγ → l − l + by adding schannel KK graviton exchange to the SM electromagnetic contribution, which is defined as where k 1 , k 2 , p 1 , p 2 and e ρ (k i ) are incoming photon, outgoing lepton momenta and polarization vectors of photons. The coherent sum is over KK modes. Vertex functions Γ αβρσ 1 for KK-γγ and Γ µν 2 for KK-ℓℓ are given below Explicit forms of the tensors C αβρσ and D αβρσ are given by eqs.
while B µναβ is a tensor part of the graviton propagator The total amplitude squared consists of electromagnetic, KK and interference parts [47] where Hereŝ,t are Mandelstam variables of the subprocess γγ → l − l + , and g 2 e = 4πα em . The s-channel contribution of the KK gravitons in (22) and (23) is given by the expression were Γ n denotes the total width of the KK graviton with the mass m n . The sum (24) has been calculated in ref. [46] S where The virtual graviton exchange should lead to deviations from the SM predictions both in a magnitude of the cross sections and in an angular distribution of the final leptons because of the spin-2 nature of the gravitons.

III. NUMERICAL ANALYSIS
The KK graviton exchange studied in the previous Section should lead to deviations from the SM in magnitudes both of differential cross sections and of total cross sections for the photon-induced process pp → pµ + µ − p at the LHC which goes via subprocess γγ → µ + µ − .
Our goal is to calculate the dependence of these deviations on the parameters of the RSSC model.
As it was mentioned in Introduction, the process can be detected by using forward de-
Before describing our numerical results, it is worth to make a few remarks about using the cut imposed on the muon rapidities during our calculations. In the c.m.s. of the colliding protons, two-muon system X moves with the rapidity where Correspondingly, the muon rapidities in the c.m.s. of the colliding protons are equal to It is clear that the rapidity cuts |η (1,2) pp | < η max are equivalent to the inequality In our case η max = 2.4. Taking into account the cut imposed on the transverse momenta of the muons, |p t | > p t,min , one can transform inequality (30) into the bound on the scattering angle of the muons in the c.m.s of two photons where It is convenient to use the rapidity cut in such a form (32). The reason is that the electromagnetic part of the matrix element |M| 2 (21) depends only on cos θ γγ , while other two terms in |M| 2 , (22) and (23), are functions of the variable cos θ γγ multiplied by functions of s. Thus, the corresponding integrals (a = em, KK, int) can be calculated analytically.
The inequality |η X | < η max means that allowable values of the muon transverse momenta are bounded from below where W min = 2ξ min E. Thus, for fixed p t > p t,0 , the following kinematical bounds take place where W max = 2ξ max E.  The results of our numerical calculations of the differential cross sections dσ/dp t as a function of the transverse momenta of the muons are presented in figs. 3-5. As one can see, dσ/dp t exceeds the SM contribution dσ SM /dp t for p t > 500 GeV, and the difference between dσ/dp t and dσ SM /dp t increases as p t grows. The effect is more pronounced for small values ofM 5 , for which dσ/dp t becomes dominant already for p t 500 ÷ 600 GeV.
Some comments should be made on a possibleM 5 -dependence of dσ/dp t . One may naively expect from eq. (25) that the KK contribution to the differential cross section should be Nevertheless, the results of our numerical calculations shows that at small and moderate values of p t . As a result, the differential cross section dσ/dp t follows thisM 5 -dependence. At large p t theM 5 -dependence tends to the formM −6 5 . It can be explained as follows [48]- [49]. In the RSSC model the invariant part of the scattering amplitude (25) is a sum of rather sharp resonances whose widths are proportional to κ 4 /M 3 5 . The contribution of one resonance can be estimated as [48]- [49] dσ KK dp t one res Taking into account that the total number of the graviton resonances which contribute to the differential cross section ∼ W/κ, we come to eq. (37). Figure 5 demonstrates us that the differential cross section is almost independent of the curvature parameter κ, with the exception of its weak dependence on κ around the point p t = 750 GeV. Thus, we can put limits on the fundamental gravity scaleM 5 regardless of the parameter κ (provided κ ≪M 5 is satisfied). This is in contrast to the RS1 model [39] in which all cross sections depend substantially on the ratio β = κ/M Pl .
The next two figures 6-7 shows us the total cross sections with and without KK graviton exchange versus the minimal transverse momentum of the final muons p t,min . The comparison with the pure SM predictions is also given. From all said above it is not surprising that the quantity σ(p t > p t,min ) does not depend on κ (see fig. 6). For both acceptance regions, its deviation from the SM gets higher as p t,min grows (see fig. 7). When the two figures are compared, we can see that the 0.1 < ξ < 0.5 case has almost the same behaviour as the case 0.0015 < ξ < 0.5 with p t,min ∼ 500 GeV. Moreover, for the 0.1 < ξ < 0.5 acceptance region, as the p t,min changes, the cross sections almost do not change if 0 < p t,min < 500. Therefore, it can be said that a high value of ξ min mimics a high value of p t,min .
In this motivation, we have obtained the limits on theM 5 for two cases: 0.0015 < ξ < 0.5 for p t > 500 GeV and 0.1 < ξ < 0.5 for p t > 30 GeV. In sensitivity analysis, the likelihood method are used. In this method, it is assumed that the entire number of background events in every signal location respects the normal distribution with a fractional uncertainty. The statistical significance is obtained as follows [50], where S, B are the signal and background events number, respectively. Here, we have assumed that the uncertainty of the background is negligible. Using our results on the cross sections, we have calculated 95% C.L. bounds onM 5 for two acceptance regions, 0.0015 < ξ < 0.5 with p t > 500 GeV and 0.1 < ξ < 0.5 with p t > 30 GeV with using eq. ( 39). The bounds are presented in fig. 8 as a function of the integrated LHC luminosity.
Here, we assumed that the κ = 1 GeV. Note that the cut |η| < 2.4 was imposed on the rapidities on the final muons. The results are presented in fig. 8 which have nearly the same behavior for the two acceptances regions. It can be see that 95% C.L. sensitivity ofM 5 is about 1430 GeV for the 0.0015 < ξ < 0.5 and 1265 GeV for the 0.1 < ξ < 0.5 when the LHC integrated luminosity values are equal to 3000 fb −1 .

IV. CONCLUSIONS
Photon-induced exclusive processes pp → pγγp → pXp are of great importance for high-energy physics. They provide one with unique precision measurements of the electroweak sector of the SM. They also allow us to study physics beyond the SM. For instance, (semi)exclusive W W production by photon-photon interactions is very sensitive to quartic gauge anomalous couplings.
The dilepton production at the electroweak scale have been studied both at the Tevatron and LHC colliders. However, all previous experiments was done without a proton tag. Recently, the dimuon production in the process pp → pγγp * → pµ + µ − p * have been studied with In the present paper we have studied the photon-induced production of the muon pair at the LHC for 14 TeV in the RSSC model with the warped extra dimension and small curvature [44]- [46]. For two acceptance regions, 0.0015 < ξ < 0.5 and 0.1 < ξ < 0.5, where ξ is the proton energy fraction loss, the distributions in the muon transverse momenta p t are calculated as a function of the reduced fundamental gravity scaleM 5 and curvature parameter κ. It is shown that the deviation from the SM gets higher as p t grows. The obtained cross sections σ(p t > p t,min ) almost do not change in the region 0 < p t,min < 500 GeV for the case 0.1 < ξ < 0.5. It means that the high value of ξ min mimics the high value of p t,min . The 95% L.C. discovery limits onM 5 are obtained for the acceptance region 0.0015 < ξ < 0.5 with p t > 500 GeV and for 0.1 < ξ < 0.5 with p t > 30 GeV. Note that the cut |η| < 2.4 was imposed on the rapidities of the final muons.
Let us stress that our limits onM 5 do not depend on the curvature parameter κ. In the RSSC model this fact also takes place for other processes, provided the inequality κ ≪M 5 is satisfied [48]. It is in contrast to the RS1 model, in which κ ∼M 5 ∼ M Pl . As a result, the RS1 discovery limits on m 1 for all processes, including photon-induced collisions at the LHC [47], depend on a chosen value of κ.