Graviton Signals in Central Production at the LHC

Abstract

We study central production, in the inclusive case, to look for graviton signals in a large extra dimensional model using dilepton and diphoton channels. We carefully analyze signal and possible standard model background processes and study the feasibility of such new physics searches in a relatively clean environment as in central production where the proton fragments are mostly emitted in the forward direction, and there is a clear rapidity gap between them and the centrally produced system. Our analysis shows that the LHC with 14 TeV center of mass energy and 100 fb⁻¹ (300 fb⁻¹) of integrated luminosity can probe the effective gravity scale up to 3.6 TeV (4.4 TeV) in both the dilepton and diphoton channels.

1. Introduction

The standard model (SM) has been extremely compatible with all the experimental tests so far. However, there are many theoretical attempts to go beyond the SM in order to answer some fundamental issues in the SM, namely, the hierarchy problems, number of fermion generations. Theories with extra spatial dimensions have been proposed to answer the hierarchy problems of the SM [1, 2]. In what follows, in this paper, we look for the graviton signals as predicted in the large extra dimensional model (proposed by Arkani-Hamed et al. [1]) in the central diffractive processes at the LHC.

Forward physics has been used with great success in studying the (SM) physics at colliders. In particular, hadron-electron ring accelerator (HERA) has used forward detectors to greatly enhance our knowledge of perturbative quantum chromodynamics (QCD) and its effects in deep inelastic scattering (DIS) experiments. Moreover, the observation of diffractive events, those with large rapidity gaps between the final state particles, have been one of the most interesting aspects of HERA physics.

In the last decade, the importance of forward physics has been recognized for Large Hadron Collider (LHC) physics too. This has resulted in a proposal to install forward particle detectors at ATLAS and CMS in order to detect protons emitted in the forward direction, implying the existence of a centrally produced system with a rapidity gap between it and the protons [3, 4]. More specifically, we are talking of a process of the kind 𝑝+𝑝→𝑝+𝑀+𝑝, also known as Central Exclusive Production (CEP). In this case, the outgoing protons are emitted with barely any loss of energy (typically less than 2% of their longitudinal momenta) along with a centrally produced system 𝑀 with a rapidity gap between it and the outgoing protons on either side. This is a clean production of the central system since it corresponds to just two forward protons in the final state along with the decay products used to identify the central particle 𝑀 and no hadronic activity between them. This has therefore been described as being like a vacuum fluctuation as the two protons pass each other, producing the central particle. Some good review on the phenomenology of exclusive processes can be found in [5–7]. Recently, in [8] a general framework of exclusive double diffractive processes and its prospects are discussed.

One of the main aim of the forward physics at the LHC is to look for the exclusive Higgs production events. First in [9], it was pointed out that Higgs can be produced in diffractive processes with rapidity gaps on either side at hadron-hadron colliders. Later, this interesting scenario has been discussed in a series of papers [10–18] by several authors. Recently, to compute the background for Higgs searches, in [19], authors have estimated the exclusive 𝑏𝑏 pair production cross-section at the LHC. In exclusive processes, Higgs would be produced almost at rest through gluon-gluon fusion. The final state contains two protons and a Higgs and, since the process is exclusive, the invariant mass of the Higgs is directly related to the energy loss of the outgoing protons [20]. In exclusive production, measuring the energy loss of the protons, one can determine the mass of the central system without looking at its decay products.

It was discussed in [21], if the outgoing protons scatter through small angles, the two-gluon system is in a 𝐽_𝑧=0, C-even, P-even state, where the 𝑧-axis is the proton beam axis. This means that any new resonance must carry the 0⁺⁺ quantum number in CEP. In other words, this enables a clean determination of the quantum numbers of any new resonance [22].

With the possibility of tagging the forward outgoing protons, the LHC can be turned into an effective gluon-gluon, photon-proton, and photon-photon collider, giving rise to a major QCD and electroweak physics program [23, 24]. In addition, the possibility of producing quarkonium states like 𝜒, 𝐽/𝜓, is something being envisaged at the ALICE detector. As discussed in [25, 26], the measurement of the two-body decay channels of 𝜒_𝑐0 to light mesons in an exclusive process can be used to study the dynamics of heavy quarkonia and to test the QCD framework of CEP.

However, all these studies require a QCD based model that allows us to couple the digluon system to the protons. The preferred model to describe that this is the socalled Durham model which we briefly describe in the next section. Before we do that however, we should point out that various other processes in the CEP scenario have recently been observed in the Tevatron at CDF which have given physicists the confidence that the Durham model is on the right track. Even though the Durham model was proposed almost a decade ago, it was not until 2006 that CDF saw diphoton production in the central rapidity region which was consistent with the gluon-gluon fusion Durham model [27]. Subsequently, CDF also found another exclusive process—𝑒⁺𝑒⁻ and 𝜇⁺𝜇⁻ production—produced by two photons [28]. Here, the Tevatron (and in future the LHC) acts as a photon-photon collider. These processes also allow the calibration of the forward proton detectors proposed by CMS and ATLAS. Quarkonium production of 𝐽/𝜓 and 𝜓 (2S) as well as 𝜒⁰_𝑐 have also been observed through their decay products [29]. Exclusive dijet production has been observed, at CDF [30] and D0 [31] in line with the predictions of the Durham model. These processes already seen at the Tevatron have given confidence that we can use digluon fusion in accordance with the Durham model to observe the signatures of new physics.

It was indicated in [32] that new physics like SUSY, extra dimensions, and so forth can be searched for in exclusive processes. In the exclusive configuration, various BSM signatures, namely, MSSM Higgs [33], charged Higgs [34], triplet Higgs [35], radions [36, 37], massive gravitons [37], long lived gluinos [38], effects of quantum gravity [39], and so forth, have been discussed in the literature. In this work, we study inclusive double diffractive process as a probe of large extra dimensions (the ADD model) using dilepton and diphoton channels via the exchange of KK graviton.

In Section 2, we briefly introduce the ADD model. In Section 3, we discuss diffractive processes and summarize the Durham model. In Section 4, we consider dilepton and diphoton production via KK graviton. We discuss important SM backgrounds in Section 5. In Section 6, we summarize our numerical results and present LHC discovery potential of extra dimensions. Finally, in Section 7, we offer our conclusions.

2. The ADD Model

In this section, we give a very brief introduction of the ADD model [1]. This model assumes that space is 4+𝑛 dimensional where 𝑛 is the number of extra spatial dimensions. All the SM particles are confined to the usual (3+1)𝐷 spacetime which is called “brane,” whereas gravity can propagate in the extra dimensions. In this model, the 4D Planck scale (𝑀_Pl∼10¹⁹ GeV) is a derived scale which is related to the fundamental Planck scale (𝑀_𝑆∼TeV) by 𝑀²_Pl=𝑉_𝑛𝑀_𝑆^2+𝑛,𝑉_𝑛=(2𝜋𝑅)^𝑛, (1) where 𝑉_𝑛 is the extra dimensional volume and 𝑅 is the compactification radius. An important consequence of the ADD model is the appearance of a tower of Kaluza-Klein (KK) modes as a solution to the linearized Einstein equation in 4+𝑛 dimensions. These KK modes are almost degenerate and separated in mass by 𝒪(1/𝑅) (1/𝑅∼10⁻⁴ eV to ~100 MeV for 𝑛= 2–7) terms. After KK decomposition, we have massive spin-2 KK gravitons (ℎ^𝑘_𝜇𝜈) which have interactions with the brane localized SM states via the energy momentum tensor 𝑇^𝜇𝜈 of the SM 𝜅ℒ=−2_𝑘𝑇^𝜇𝜈(𝑥)ℎ^𝑘_𝜇𝜈(𝑥), (2) where 𝜅=√16𝜋/𝑀_Pl and the summation runs over all KK modes. The ADD model has been studied extensively both theoretically as well as experimentally. Recently, the extra dimensions are being searched for using dilepton and diphoton channels at the LHC, both by ATLAS [40, 41] and CMS [42, 43]. These searches have already put the lower bound on 𝑀_𝑆 quite high.

3. Diffractive Processes

We consider two types of double diffractive (DD) mechanisms at the LHC as follows: (1) central exclusive process (CEP): 𝑝𝑝→𝑝+𝑀+𝑝 as shown in Figure 1(a); (2) central inclusive process (CIP): 𝑝𝑝→𝑋+𝑀+𝑌 as shown in Figure 1(b). Here, 𝑀 denotes the centrally produced system (we also denote the invariant mass of the central system with the same notation “𝑀”) and the “+” signs denote the presence of rapidity gaps between the system 𝑀 and the outgoing protons (in exclusive process) or proton dissociated products 𝑋, 𝑌 (in inclusive process). The differential cross-sections for exclusive and inclusive DD processes shown in Figure 1 can be expressed in the folllowing factorized form [32]: 𝑀²𝑑²𝜎𝑑𝑦𝑑𝑀²=ℒ_𝑔𝑔𝑀²,𝑦𝜎𝑀², (3) where 𝜎 is the cross-section of the hard subprocess which produces the color singlet system of invariant mass 𝑀 (𝑔𝑔→𝑀) and ℒ_𝑔𝑔 is the effective luminosity for the production of a system of invariant mass 𝑀 at rapidity 𝑦. The total cross-section 𝜎(𝑝𝑝→𝑋+𝑀+𝑌) can be calculated using 𝜎=𝑑𝑦𝑑𝑀²1𝑀²ℒ_𝑔𝑔𝑀²,𝑦𝜎𝑀². (4)

(a)

(b)

Figure 1 

Double diffractive production of a system of mass 𝑀 with rapidity gap Δ𝜂 on either side via (a) the central exclusive process and (b) the central inclusive process at the LHC.

In our calculation, we keep ℒ_𝑔𝑔 and 𝜎 in a factorized form as the effective luminosity is independent of the subprocess, and for different processes, we need to compute just the subprocess cross-sections. Although we keep ℒ_𝑔𝑔 and 𝜎 in a factorized form, we must remember that there might exist specific selection rules for different configurations. In the exclusive configuration, the fusing 𝑔𝑔 state obeys a special selection rule in the limit that the protons scatter through zero angle. The 𝑧-component of the angular momentum of a centrally produced system vanishes as the incoming state which consists of the fusing hard gluons has 𝐽_𝑧=0 with positive 𝐶 and 𝑃 parity [32]. For inclusive production on the other hand, there is no such selection rule, and therefore, the production rate is larger than that for the exclusive production. Although the signals are very clear for CEP, the luminosity and the event rates are very small due to the restricted kinematics. A much larger phase space is available due to the dissociation of the incoming protons in the inclusive DD case. The luminosities for exclusive and inclusive processes can be computed using the known parton distribution functions (PDF) of the protons.

3.1. The Durham Model

To estimate ℒ_𝑔𝑔, we will use what is known as the “Durham model,” described in [14, 15, 18, 32]. The ℒ_𝑔𝑔 depends on off-diagonal or skewed gluon distribution functions (off-diagonal, since we need to consider the coupling of two gluons with different momentum fractions, 𝑥_𝑖 and 𝑥^′_𝑖, with a proton). However, as we see, for 𝑥^′_𝑖≪𝑥_𝑖 and 𝑘_𝑖𝑡≈𝑄_𝑡, the relevant kinematic region for CEP, it is possible to approximate the off-diagonal distributions using the integrated gluon distribution function [44]. For CIP, the dominant (leading logarithm) contribution comes from the asymmetric configuration, 𝑄²_𝑡≪𝑘²_𝑖𝑡. At the partonic level, the inclusive production is equivalent to the exclusive production, and the unintegrated gluon distributions in partons may be calculated perturbatively in terms of nonforward BFKL amplitudes. The effective luminosity depends also on two survival probabilities of the rapidity gaps—one to the QCD radiation and another to the nonperturbative soft rescattering of the protons. The first is expressed as a Sudakov factor which ensures that the fusing gluons remain intact up to the hard scale 𝑀/2. The second factor is included as an explicit multiplicative factor 𝒮², known as the “gap survival factor.” It accounts for the probability that apart from the perturbative processes, no other particle is produced via soft interactions between the protons [45–48]. In general, 𝒮² is expected to depend on the kinematics of the process and can be calculated using various QCD based models [47–52]. Typically, for LHC energies, 𝒮² for CEP is estimated to be very small [47, 48] (e.g., in [32] authors found 𝒮²=0.020(0.026) for 14 TeV (8 TeV) LHC). For CIP, relatively smaller absorption cross-section leads to a larger 𝒮². For simplicity, in our analysis, we keep 𝒮²=0.1 for CIP, as a constant multiplicative factor.

We use the MSTW2008 LO parton density [53] in our computation. Since MSTW parametrization for the gluon distribution function 𝐺(𝑥,𝑄)=𝑥𝑔(𝑥,𝑄) does not go below 𝑄=1 GeV, for 𝑄≤1 GeV, we take the following extrapolation function: 𝐺(𝑥,𝑄)=𝛼(𝑥)𝑄^{[2+{𝛽(𝑥)−2}𝑄]}, (5) so that for 𝑄²→0, 𝐺(𝑥,𝑄)∼𝑄² [18]. Here, 𝛼 and 𝛽 are the two functions of 𝑥 and can be computed using the following relations: 𝛽(𝑥)=11+ln𝑄₀𝐺^′𝑥,𝑄₀𝐺𝑥,𝑄₀−2𝑄₀𝛼(𝑥)=𝐺+2,𝑥,𝑄₀𝑄^{[2+{𝛽(𝑥)−2}𝑄0]}₀, (6) where 𝐺^′=𝜕𝐺/𝜕𝑄 and 𝐺, 𝐺^′ are evaluated at 𝑄₀=1 GeV. To compute the Sudakov factor, we use the LO 𝛼_𝑆 (strong coupling) with Λ_QCD=220 MeV and 𝑁_𝑓=5. The full formalism to calculate effective luminosity for exclusive and inclusive configuration is given in [32]. In Appendix A, we briefly present the computation of ℒ_𝑔𝑔 for CIP.

4. Diffractive Production

With diffractive processes being dominated by two-gluon exchange as shown in Figure 1, it is obvious that these processes would be observed only for the final states with a substantial coupling to a gluon pair. In this work, we consider the central production of KK gravitons (𝐺) via DD processes. The spin-2 gravitons can have three possible spin projections 𝐽_𝑧=0, 1, and 2. Due to the presence of special 𝐽_𝑧=0 selection rule in the exclusive configuration, graviton cannot be produced via leading order (LO) exclusive processes (although there is no 𝐽_𝑧=0 point-like 𝑔𝑔 coupling to a 2⁺ graviton, it can still be produced at next to LO (NLO) in the exclusive configuration.) [32]. In the unitary gauge, the coupling of the spin-0 projection of 𝐺 with two gluons is zero as the LO coupling of spin-0 component with any two vector bosons is proportional to the square of the mass of the vector boson, and gluons are massless in this case [54]. By the Landau-Yang theorem [55, 56], it is impossible to produce a spin-1 particle via the fusion of two on-shell vector bosons, but the production is not forbidden if those vector bosons are off-shell. Though the gluons coming from protons are not purely on-shell, the degree of virtuality of the gluons is very small. Thus, the contribution in any process from spin-1 projection of graviton is subdominant [57]. Thus, the 𝐽_𝑧=2 is the only possible spin projection that can couple to a gluon pair at LO. Therefore, in this paper, we discuss the LHC signatures of 𝐺 in inclusive configuration. In what follows, we analyze the diphoton and the dilepton channels via the exchange of 𝐺 arising in ADD type models [1] as a signal of extra dimension. We carefully consider these signal processes and the relevant background processes for them and look for regions of parameter space where the signal would dominate. Since these processes can only occur in inclusive configuration, the event rates could be substantial to search for at the LHC. Moreover, the nature of the intermediate particle and the coupling with SM particles reflect in the angular distribution of the final state particles.

4.1. Dilepton Production

We consider the process where dilepton pair is produced from gluon fusion via the exchange of 𝐺 as shown in Figure 2. The Feynman rules for this process are taken from [54] where one can find definitions of all the terms which appear here. We also show these Feynman rules in Appendix B. The matrix element for this process is given by 𝑖ℳ^𝑎𝑏=−𝑖𝜅2𝛿^𝑎𝑏𝜖^𝛼₁𝑘₁𝜖^𝛽₂𝑘₂𝑉_{𝜇𝜈;𝛼𝛽}×(𝑖/2)𝐵^{𝜇𝜈;𝜇′𝜈′}𝑃²−𝑀²_𝐺−+𝑖𝜀𝑖𝜅8𝑢𝑘₃Γ_{𝜇′𝜈′}𝑣𝑘₄, (7) where the coupling 𝜅 is related to the Newton constant 𝐺_𝑁 as 𝜅=16𝜋𝐺_𝑁. Since the reduced Planck mass 𝑀^∗_Pl=1/8𝜋𝐺_𝑁 (𝑀^∗_Pl≈2.4×10¹⁸ GeV), each KK mode coupling to the SM particles is Planck mass suppressed. A summation over high multiplicity of KK modes lying below the UV cutoff scale 𝑀_S compensates the suppression and gives rise to a substantial effective coupling strength. Therefore, we replace the graviton propagator in (7) by the following effective propagator: 𝒟_eff(𝑠)̂=_𝑘𝑖𝑠̂−𝑀²_𝐺𝑘+𝑖Γ_𝐺𝑘𝑀_𝐺𝑘, (8) where 𝑘 sums over all KK towers below 𝑀_𝑆 and Γ_𝐺𝑀_𝐺=𝜀 and “hat” notation is used for subprocess quantities. Here, √𝑠̂=𝑀_𝑙𝑙 is the invariant mass of the lepton pair. Since, KK modes are quasicontinuous, this summation can be done by defining KK state density as shown in Appendix B [54]. For 𝑛 number of extra dimensions, considering the contributions from resonant and nonresonant KK states, the 𝒟_eff is given by 𝒟_eff(𝑠)̂=16𝜋𝑠̂^𝑛/2−1𝜅²Γ(𝑛/2)𝑀_𝑆^𝑛+2𝑀𝜋+2𝑖𝐼^𝑆√𝑠̂, (9) where the real part comes from the summation over all resonant contributions below 𝑀_𝑆 and the imaginary part is the summed contribution coming from all the nonresonant states. The definition of the 𝐼 function can be found in Appendix B. After summing over the spins of the final state leptons, the effective squared matrix element takes the following form: |ℳ|²=𝜅⁴𝒟1024^eff2𝜖^𝛼1₁𝑘₁𝜖^∗𝛼2₁𝑘₁×𝜖^𝛽1₂𝑘₂𝜖^∗𝛽2₂𝑘₂𝑉_{𝜇1𝜈1;𝛼1𝛽1}𝑉_{𝜇2𝜈2;𝛼2𝛽2}×𝐵^{𝜇1𝜈1;𝜇′1𝜈′1}𝐵^{𝜇2𝜈2;𝜇′2𝜈′2}×tr𝑢𝑘₃Γ_{𝜇′1𝜈′1}𝑣𝑘₄𝑣𝑘₄Γ^†_{𝜇′1𝜈′1}𝑢𝑘₃𝛿_𝑎𝑏𝛿^𝑎𝑏. (10)

Figure 2 

The subprocess Feynman diagram for dilepton production.

In the zero mass limit of the produced lepton, the angular dependence of |ℳ|² takes a very simple form as follows: |ℳ|²=𝛿_𝑎𝑏𝛿^𝑎𝑏𝜅⁴𝒟64^eff2𝑠⁴̂1−cos⁴𝜃, (11) where 𝜃 is the scattering angle of the leptons in the center of mass (CM) frame of the 𝐺. Averaging |ℳ|² over eight colors and two polarizations of the initial gluons, one can evaluate the subprocess differential cross-section after including the phase space factor as 𝑑𝜎𝑔𝑔→𝑙⁺𝑙⁻𝑑||=1cos𝜃𝑠̂⋅132𝜋8²⋅12²𝜅⋅8⋅⁴𝒟64^eff2×𝑠⁴̂1−cos⁴𝜃. (12)

If we consider that two types of leptons (i.e., electron and muon) are contributing in the dilepton final state, we should multiply the previous differential cross-section by an extra factor of two.

4.2. Diphoton Production

We consider the process where a photon pair is produced from gluon fusion via the exchange of 𝐺 as shown in Figure 3. As mentioned earlier, the Feynman rules for this process are taken from [54] where one can find definitions of all the terms which appear here and also can be found in Appendix B. The matrix element for this process is given by 𝑖ℳ^𝑎𝑏=𝜅−𝑖2𝛿^𝑎𝑏𝜖^𝛼₁𝑘₁𝜖^𝛽₂𝑘₂𝑉_{𝜇𝜈;𝛼𝛽}(𝑖/2)𝐵^{𝜇𝜈;𝜇′𝜈′}𝑃²−𝑀²_𝐺×𝜅+𝑖𝜀−𝑖2𝑉_{𝜇′𝜈′;𝛼′𝛽′}𝜖^∗𝛼′₃𝑘₃𝜖^∗𝛽′₄𝑘₄. (13) After summing over all KK states up to 𝑀_𝑆 contributing to a physical process and summing over two polarizations of the final state photons, the effective squared matrix element takes the following form: |ℳ|²=𝜅⁴𝒟64^eff2𝜖^𝛼1₁𝑘₁𝜖^∗𝛼2₁𝑘₁𝜖^𝛽1₂𝑘₂𝜖^∗𝛽2₂𝑘₂×𝜖^∗𝛼′1₃𝑘₃𝜖^𝛼′2₃𝑘₃𝜖^∗𝛽′1₄𝑘₄𝜖^𝛽′2₄𝑘₄×𝑉_{𝜇1𝜈1;𝛼1𝛽1}𝑉_{𝜇′1𝜈′1;𝛼′1𝛽′1}𝑉_{𝜇2𝜈2;𝛼2𝛽2}×𝑉_{𝜇′2𝜈′2;𝛼′2𝛽′2}𝐵^{𝜇1𝜈1;𝜇′1𝜈′1}𝐵^{𝜇2𝜈2;𝜇′2𝜈′2}𝛿_𝑎𝑏𝛿^𝑎𝑏. (14) After doing some tedious algebra, the angular dependence of |ℳ|² takes the following form: |ℳ|²=𝛿_𝑎𝑏𝛿^𝑎𝑏𝜅⁴𝒟64^eff2𝑠⁴̂1+6cos²𝜃+cos⁴𝜃, (15) where 𝜃 is the scattering angle of the photons in the CM frame of the 𝐺. Averaging |ℳ|² over eight colors and two polarizations of the initial gluons, one can evaluate the subprocess differential cross-section after including the phase space factor as 𝑑𝜎𝑔𝑔→𝛾𝛾𝑑||=1cos𝜃2⋅1𝑠̂⋅132𝜋8²⋅12²𝜅⋅8⋅⁴𝒟64^eff2×𝑠⁴̂1+6cos²𝜃+cos⁴𝜃. (16) Here, we have included an extra 1/2 factor because of the presence of two identical particles in the final state.

Figure 3 

The subprocess Feynman diagram for diphoton production.

5. SM Backgrounds

In the diffractive configuration, there are possibilities of producing a central system from 𝑔𝑔, 𝛾𝛾, 𝑞𝑞, or 𝑊𝑊 fusion processes. In a same kinematic region, the effective luminosity for the 𝑔𝑔 fusion (ℒ_𝑔𝑔) is the largest at the LHC among all the alternatives, whereas the effective luminosities for 𝛾𝛾, 𝑞𝑞, or 𝑊𝑊 fusion processes are much smaller compared to ℒ_𝑔𝑔. Estimations of effective luminosities for the 𝑔𝑔, 𝛾𝛾, and 𝑊𝑊 can be found in [32] and for the 𝑞𝑞 in [58]. While computing the SM backgrounds for the dilepton and the diphoton signal channels, we can sometimes neglect 𝛾𝛾, 𝑞𝑞, and 𝑊𝑊 initiated background processes due to the small luminosities of the fusing particles.

5.1. Dilepton Backgrounds

In Figure 4, we show the Feynman diagrams of some important background subprocesses for the dilepton channel proceed through 𝑔𝑔, 𝛾𝛾, 𝑞𝑞, and 𝑊𝑊 initial states. When one considers lepton pair production within the SM in the diffractive configuration, one would not expect a significant rate for a lepton pair unaccompanied by other particles (like radiation jets/photons or other SM particles). Although there is always a chance to miss those extra particles other than the lepton pair or to misidentify other particles as leptons, in our background computation, we do not include those reducible backgrounds as these are expected to be very small. Using the Landau-Yang theorem, dual considerations of angular momentum conservation and Bose symmetry prevent a pair of gluons couple to an on-shell 𝛾/Z (via a quark loop, see Figures 4(a) and 4(b)) or a 𝑊𝑊 pair couple to an on-shell 𝛾/Z (see Figures 4(g) and 4(h)). While such a coupling, although very small, is indeed permitted for an off-shell 𝛾/Z. The pieces in the corresponding vertex function are proportional to the degree of virtuality of the initial particles or to the momentum of the 𝛾/Z itself. Moreover, 𝑔𝑔→𝑙𝑙 processes involve a quark loop, and 𝑊𝑊→𝑙𝑙 processes are suppressed due to very low 𝑊𝑊 luminosity. Because of these reasons, we are not including 𝑔𝑔 and 𝑊𝑊 initiated processes in our computation. We also neglect 𝑞𝑞 initiated processes in Figures 4(d) and 4(e) due to very low luminosity for 𝑞𝑞. Consequently, these processes lead only to subdominant contributions.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 4 

The subprocess Feynman diagrams for dilepton channel.

The dominant SM background for the dilepton channel comes from the inelastic photoproduction of the lepton pair as shown in Figure 4(c) where photons are emitted from the quarks inside protons. This contribution can be very large in the kinematic region where the invariant mass of the lepton pair is small. This is because the collinear emission of equivalent photons from quark lines can give large contribution. The inelastic 𝑝𝑝→𝑙𝑙 cross-section has been computed in [59] for the Weizsäcker-Williams photon pairs, and an outline of the computation is presented in Appendix C. The contributions from the interference terms among various background processes might be important at low energies. Since we are only interested in the high energy regions, we do not include them for simplicity.

5.2. Diphoton Backgrounds

In Figure 5, we show possible background subprocesses for the diphoton channel initiated from 𝑔𝑔, 𝛾𝛾, 𝑞𝑞, and 𝑊𝑊 initial states. The dominant background for the diphoton channel comes from the 𝑔𝑔→𝛾𝛾 box diagram (via quark loop) shown in Figure 5(a), and cross-section is proportional to 𝛼²_𝑆𝛼²_𝑒𝑚 (𝛼_𝑒𝑚 is the QED coupling). There are other higher order diagrams that contribute to 𝑔𝑔→𝛾𝛾 process as shown in Figures 5(b) and 5(c) where Higgs is involved. The Higgs is produced from 𝑔𝑔 fusion via a top quark loop and then decayed to 𝛾𝛾 via another top quark or a 𝑊 loop. There are also Delbruck type (light-by-light scattering) backgrounds possible shown in Figures 5(d) and 5(e). In this case, quarks from the dissociated protons radiate two photons which then give two final state photons via a fermion or 𝑊 in the loop. This contribution is much suppressed as cross-section is proportional to 𝛼⁴_𝑒𝑚. The other backgrounds which are initiated from 𝑞𝑞 (see Figure 5(f)) and 𝑊𝑊 (see Figures 5(g) and 5(h)) initial states are negligible as the corresponding effective luminosities are very low. We note that all the 𝑔𝑔 and 𝛾𝛾 initiated background processes occur only via loop diagrams, and they are usually very small. If the signal rate is substantial in the diphoton channel, searching the signatures of extra dimensions in the diphoton channel might be very promising.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 5 

The subprocess Feynman diagrams for diphoton channel.

We estimate the dominant 𝑔𝑔→𝛾𝛾 box diagram background cross-section using 𝛾𝛾→𝛾𝛾 helicity amplitudes taken from [60]. Relevant formulae are given in Appendix D. At low energy, the fermion loop contribution in the box diagram dominates, but above a few hundred GeV, this contribution starts falling rapidly, whereas the 𝑊 loop contribution dominates in the total cross-section of the 𝛾𝛾→𝛾𝛾 process at high energies. Other higher order 𝛾𝛾→ℎ→𝛾𝛾 processes similar to Figures 5(b) and 5(c) are not considered here. We do not include 𝑞𝑞 initiated process in our background computation as the contribution originates from 𝑞𝑞  𝑡-channel exchange in Figure 5(f) is roughly two order of magnitude lower than the 𝑔𝑔 contributions [58]. As mentioned earlier, we neglect 𝑊𝑊 initiated processes for low 𝑊𝑊 luminosity and all background interference terms for simplicity.

6. Numerical Results

In all our numerical computations, we use the MSTW2008 LO parton densities to obtain the unintegrated parton distributions. Here, we once again give the numerical values of some quantities which are used. To compute the Sudakov form factors, we use NLO 𝛼_𝑆 with Λ_QCD=220 MeV and number of “active” flavor 𝑁_𝑓=5. We take a fixed value of the “gap survival factor,” 𝒮²=0.1 in our analysis. Our results depend on two model parameters, the number of extra dimensions 𝑛 and the ultraviolet cutoff 𝑀_𝑆 of the theory, and four kinematical variables, invariant mass 𝑀 and rapidity 𝑦 of the centrally produced system, rapidity gap Δ𝜂 between the central system and the outgoing protons or proton dissociated products, and the CM energy √𝑆 of two colliding protons at the LHC. The cutoff scale 𝑀_𝑆 is of the order of few TeV for low scale gravity theories. The lower limit of 𝑀_𝑆 is bounded from experiments, and the bounds are already quite high as stated in Section 2. If we increase 𝑀_𝑆, the signal cross-sections decrease as usual. In our case, the signal falls off very rapidly with increasing 𝑀_𝑆 because of the presence of 𝑀⁸_𝑆 in the denominator of the signal cross-sections (see (12) and (16)). This makes the ADD model very difficult to search for experimentally.

In Figure 6(a), we show the dependence of the effective luminosity ℒ_𝑔𝑔 on the invariant mass 𝑀 of the central system produced at rapidity region 𝑦=0 at the 14 TeV LHC. The ℒ_𝑔𝑔 shows a decreasing nature with increasing 𝑀 since it is harder to produce more massive objects while keeping other parameters fixed. In Figure 6(b), we plot the dependence of ℒ_𝑔𝑔 on 𝑦 with 𝑀=500 GeV at the 14 TeV LHC. In the diffractive processes, it is more likely to produce an object centrally, that is, near 𝑦=0 region. Thus, a system which is produced away from the central region is less probable, and we see a decreasing behavior of ℒ_𝑔𝑔 with increasing  𝑦.

(a)

(b)

Figure 6 

(a) The effective luminosity ℒ𝑔𝑔 as a function of the invariant mass 𝑀 of the central system produced at 𝑦=0 at the 14 TeV LHC and (b) ℒ𝑔𝑔 as a function of 𝑦 for the production of a system with mass 𝑀=500 GeV at the 14 TeV LHC. For both plots we choose four values of the rapidity gap, that is, Δ𝜂= 1.5, 2.0, 2.5, and 3.0.

In Figure 7, we show the total signal cross-section in the dilepton (𝜎_𝑙𝑙 where 𝑙={𝑒,𝜇}) channel as a function of √𝑆. In Figure 7(a), we show 𝜎_𝑙𝑙 for different Δ𝜂 taking 𝑛=2 and 𝑀_𝑆=2 TeV. The cross-section increases as we increase √𝑆 as expected. If we increase Δ𝜂, the 𝜎_𝑙𝑙 decreases as the effective luminosity decreases with increasing Δ𝜂 (see Figures 6(a) and 6(b)). In Figure 7(b), we show the 𝜎_𝑙𝑙 for different 𝑛 varied from two to seven. We see that as we increase 𝑛, the 𝜎_𝑙𝑙 decreases. This is because 𝜅²|𝒟_eff|∼2/(𝑛−2) for 𝑛>2, in the limit 𝑀²_𝑆≫𝑠̂ (see (B.9)). Here, we compute the total cross-section after applying the following selection cuts: 𝑦_𝜓𝜓≤2,cos𝜃_𝜓𝜓≤1,100GeV≤𝑀_𝜓𝜓<𝑀_𝑆−10GeV, (17) where 𝜃 is the scattering angle of 𝜓 (where 𝜓=𝑙, 𝛾) in the rest frame of the centrally produced system and 𝑀_𝜓𝜓 is the invariant mass of the 𝜓𝜓 pair. We have collected events with the rapidity of the central system |𝑦_𝜓𝜓|≤2. The upper limit of 𝑀_𝜓𝜓 can go to 𝑀_𝑆 up to which the theory is valid. Here, we have taken the upper limit on 𝑀_𝜓𝜓 slightly smaller than 𝑀_𝑆 as the 𝐼 function in (9) diverges at 𝑀_𝑆. The cross-sections are only few 𝑓𝑏, and we need higher CM energy at the LHC to observe these events.

(a)

(b)

Figure 7 

The total signal cross-section (𝑝𝑝→𝑋+𝑙𝑙+𝑌) in dilepton channel (𝜎𝑙𝑙) as a function of LHC CM energy. In (a), we plot 𝜎𝑙𝑙 for Δ𝜂= 1.5, 2.0, 2.5, and 3.0 for 𝑛=2 and 𝑀𝑆=2 TeV. In (b), we plot 𝜎𝑙𝑙 for 𝑛= 2–7 for 𝑀𝑆=2 TeV and Δ𝜂=2. In both plots, 𝜎𝑙𝑙's are computed after applying the cut defined in (17).

In Figure 8, we show the total signal cross-section in the dilepton (𝜎_𝑙𝑙) channel as a function of 𝑀_𝑆 at the 14 TeV LHC. In Figure 8(a), we show 𝜎_𝑙𝑙 for different Δ𝜂 taking 𝑛=2, and in Figure 8(b), we show 𝜎_𝑙𝑙 for different 𝑛 taking Δ𝜂=2. Here, we compute the total cross-section after applying the kinematical cuts defined in (17). The cross-section decreases rapidly as we increase 𝑀_𝑆 since 𝑀⁸_𝑆 is present in the denominator of the total cross-section.

(a)

(b)

Figure 8 

The total signal cross-sections (𝑝𝑝→𝑋+𝑙𝑙+𝑌) in dilepton channel (𝜎𝑙𝑙) as a function of 𝑀𝑆 at the 14 TeV LHC. In (a), we plot 𝜎𝑙𝑙 for Δ𝜂= 1.5, 2.0, 2.5, and 3.0 for 𝑛=2. In (b), we plot 𝜎𝑙𝑙 for 𝑛= 2–7 for Δ𝜂=2. In both plots, 𝜎𝑙𝑙's are computed after applying the cut defined in (17).

In Figure 9(a), we show signal and background invariant mass distributions (𝑑𝜎_𝑙𝑙/𝑑𝑀_𝑙𝑙) of the lepton pair after applying the selection cuts defined in (18) at the 14 TeV LHC. To compute 𝑑𝜎_𝑙𝑙/𝑑𝑀_𝑙𝑙, we have not integrated cos𝜃_𝜓𝜓 in the full range from −1 to 1 because the background cross-section diverges as |cos𝜃_𝜓𝜓|→1 (see (C.5)). We plot signal 𝑑𝜎_𝑙𝑙/𝑑𝑀_𝑙𝑙 for 𝑀_𝑆= 1.5, 2.0, and 2.5 TeV taking 𝑛=2 and Δ𝜂=2 for both signal and background. The signal 𝑑𝜎_𝑙𝑙/𝑑𝑀_𝑙𝑙 is a monotonically increasing function of 𝑀_𝑙𝑙 (𝑀²_𝑙𝑙=𝑠̂) because the subprocess cross-section increases roughly as  𝑠³̂ (see (12)). The background cross-section is computed only considering the dominant 𝛾𝛾→𝑙𝑙 channel. For small values of 𝑀_𝑙𝑙, the background is quite large but falls off rapidly as we increase 𝑀_𝑙𝑙. Thus, to observe an excess over the background one should collect lepton pairs with sufficiently high 𝑀_𝑙𝑙: 𝑦_𝑙𝑙≤2,cos𝜃_𝑙𝑙≤0.8. (18)

(a)

(b)

Figure 9 

(a) The invariant mass distributions (𝑑𝜎𝑙𝑙/𝑑𝑀𝑙𝑙) for signal and background (𝑝𝑝→𝑋+𝑙𝑙+𝑌) of the lepton pair after applying the cuts defined in (18) at the 14 TeV LHC. (b) The angular distributions (𝑑𝜎𝑙𝑙/𝑑cos𝜃𝑙𝑙) for signal and background of the lepton pair after applying the cuts defined in (19) at the 14 TeV LHC. For both plots, we take 𝑀𝑆= 1.5, 2.0, and 2.5 TeV and 𝑛=2 for signal and Δ𝜂=2 for both signal and background.

In Figure 9(b), we show signal and background angular distributions (𝑑𝜎_𝑙𝑙/𝑑cos𝜃_𝑙𝑙) of the lepton pair after applying the selection cuts defined in (19) at the 14 TeV LHC. To compute 𝑑𝜎_𝑙𝑙/𝑑cos𝜃_𝑙𝑙, we apply a very high invariant mass cut on the lepton pair to reduce large background in the small 𝑀_𝑙𝑙 region. We show signal 𝑑𝜎_𝑙𝑙/𝑑cos𝜃_𝑙𝑙 for 𝑀_𝑆= 1.5, 2.0, and 2.5 TeV taking 𝑛=2 and Δ𝜂=2 for both signal and background. The angular distribution carries spin information of the intermediate particle. For dilepton production, the angular distributions for signal and background show a very contrasting behavior. For signal 𝑑𝜎_𝑙𝑙/𝑑cos𝜃_𝑙𝑙∼(1−cos⁴𝜃_𝑙𝑙) (see (12)) which attains its maximum at cos𝜃_𝑙𝑙=0. On the other hand, background attains its minimum at cos𝜃_𝑙𝑙=0 since it goes as (1+cos²𝜃_𝑙𝑙)/sin²𝜃_𝑙𝑙 and diverges when |cos𝜃_𝑙𝑙|→1 (see (C.5)). This contrasting feature can be used as a unique signature of the extra dimensions 𝑦_𝑙𝑙≤2,750GeV≤𝑀_𝑙𝑙<1250GeV. (19)

In Figure 10, we show the total signal cross-section in the diphoton channel (𝜎_𝛾𝛾) as a function of √𝑆 after applying the cut defined in (17). In Figure 10(a), we show 𝜎_𝛾𝛾 for different Δ𝜂 taking 𝑛=2 and 𝑀_𝑆=2 TeV. In Figure 10(b), we show 𝜎_𝛾𝛾 for different 𝑛 varied from two to seven. In Figure 11, we show the 𝜎_𝛾𝛾 as a function of 𝑀_𝑆 at the 14 TeV LHC. In Figure 11(a), we show 𝜎_𝛾𝛾 for different Δ𝜂 taking 𝑛=2, and in Figure 11(b), we show 𝜎_𝛾𝛾 for different 𝑛 taking Δ𝜂=2. Here, we compute the 𝜎_𝛾𝛾 after applying the kinematical cuts defined in (17). The behavior of these plots is very similar to the corresponding plots for dilepton channel, and we are not repeating those features here. An interesting point to notice is that 𝜎_𝑙𝑙 and 𝜎_𝛾𝛾 are similar even quantitatively. This is because surprisingly 𝜎_𝑙𝑙 and 𝜎_𝛾𝛾 are equal after cos𝜃 integration from −1 to 1 as follows: _{𝑙=𝑒,𝜇}¹₋₁𝑑()𝑑𝜎cos𝜃𝑔𝑔→𝑙⁺𝑙⁻𝑑|cos𝜃|=¹₋₁𝑑()𝑑𝜎cos𝜃𝑔𝑔→𝛾𝛾𝑑||.cos𝜃 (20)

(a)

(b)

Figure 10 

The total signal cross-sections (𝑝𝑝→𝑋+𝛾𝛾+𝑌) in diphoton channel (𝜎𝑙𝑙) as a function of LHC CM energy. In (a), we plot 𝜎𝑙𝑙 for Δ𝜂= 1.5, 2.0, 2.5, and 3.0 for 𝑛=2 and 𝑀𝑆=2 TeV. In (b), we plot 𝜎𝑙𝑙 for 𝑛= 2–7 for 𝑀𝑆=2 TeV and Δ𝜂=2. In both plots, 𝜎𝑙𝑙's are computed after applying the cut defined in (17).

(a)

(b)

Figure 11 

The total signal cross-sections (𝑝𝑝→𝑋+𝛾𝛾+𝑌) in diphoton channel (𝜎𝛾𝛾) as a function of 𝑀𝑆 at the 14 TeV LHC. In (a), we plot 𝜎𝛾𝛾 for Δ𝜂= 1.5, 2.0, 2.5, and 3.0 for 𝑛=2. In (b), we plot 𝜎𝛾𝛾 for 𝑛= 2–7 for Δ𝜂=2. In both plots, 𝜎𝛾𝛾's are computed after applying the cut defined in (17).

In Figure 12(a), we show signal and background invariant mass distributions (𝑑𝜎_𝛾𝛾/𝑑𝑀_𝛾𝛾) of the diphoton pair after applying the selection cuts defined in (21) at the 14 TeV LHC: 𝑦_𝛾𝛾≤2,cos𝜃_𝛾𝛾≤1. (21) We plot signal 𝑑𝜎_𝛾𝛾/𝑑𝑀_𝛾𝛾 for 𝑀_𝑆= 1.5, 2.0, and 2.5 TeV taking 𝑛=2 and Δ𝜂=2 both for signal and background. The signal 𝑑𝜎_𝛾𝛾/𝑑𝑀_𝛾𝛾 is a monotonically increasing function of 𝑀_𝛾𝛾 (𝑀²_𝛾𝛾=𝑠̂) because the subprocess cross-section increases roughly as  𝑠³̂ (see (16)). The background cross-section is computed including 𝑔𝑔 and 𝛾𝛾 initiated processes as shown in Figure 5. Although the background in Figure 12(a) can beat signal in the small 𝑀_𝛾𝛾 region, it dies out very rapidly in the high 𝑀_𝛾𝛾 region. This is because the diphoton background goes roughly as 1/𝑀²_𝛾𝛾 and becomes almost negligible at large 𝑀_𝛾𝛾. In Figure 12(b), we show signal and background angular distributions (𝑑𝜎_𝛾𝛾/𝑑cos𝜃_𝛾𝛾) of the diphoton pair after applying the selection cuts defined in (22) at the 14 TeV LHC: 𝑦_𝛾𝛾≤2,250GeV≤𝑀_𝛾𝛾<1250GeV. (22)

(a)

(b)

Figure 12 

(a) The invariant mass distributions (𝑑𝜎𝛾𝛾/𝑑𝑀𝛾𝛾) for signal and background (𝑝𝑝→𝑋+𝛾𝛾+𝑌) of the photon pair after applying the cuts defined in (18) at the 14 TeV LHC. (b) The angular distributions (𝑑𝜎𝛾𝛾/𝑑cos𝜃𝛾𝛾) for signal and background after applying the cuts defined in (19) at the 14 TeV LHC. For both plots, we take 𝑀𝑆= 1.5, 2.0, and 2.5 TeV and 𝑛=2 for signal and Δ𝜂=2 for both signal and background. The background in (b) is shown after multiplying by a factor of 50.

We see that diphoton signal and background angular distributions behave just in the opposite way to the dilepton signal and background angular distributions, respectively. The signal 𝑑𝜎_𝛾𝛾/𝑑cos𝜃_𝛾𝛾∼(1+6cos²𝜃+cos⁴𝜃) has a minimum at cos𝜃=0 and attains its maximum value at |cos𝜃|=1, whereas the 𝑑𝜎_𝛾𝛾/𝑑cos𝜃_𝛾𝛾 for the dominant 𝑔𝑔→𝛾𝛾 background attains its maximum value at cos𝜃=0 as shown in Figure 12(b). This is because of the nature of angular dependent part in (D.1). The similar angular behavior is true for subdominant  𝛾𝛾→𝛾𝛾 background also. Whereas, there is no angular dependency of other subdominant 𝑔𝑔→ℎ→𝛾𝛾 backgrounds because of the spin-0 nature of the intermediate Higgs.

6.1. LHC Discovery Potential

We define the luminosity requirement for the discovery of KK gravitons at the LHC as follows: 𝐿_𝐷=Max𝐿₅,𝐿₁₀, (23) where 𝐿₅ denotes the luminosity required to attain 5𝜎 statistical significance and 𝐿₁₀ is the luminosity required to observe 10 signal events. We compute 𝐿_𝐷 after applying some kinematical cuts which we call the “discovery cuts” as defined in the following.

6.1.1. Discovery Cuts

Motivated by the invariant mass and angular distributions in Figures 9 and 12, we construct some kinematical cuts to separate the signal from the background in the dilepton and diphoton channels as follows: (1) dilepton channel: |𝑦_𝑙𝑙|≤2; |cos𝜃_𝑙𝑙|≤0.8; 𝑀_𝑆/2≤𝑀_𝑙𝑙<𝑀_𝑆−10GeV;(2) diphoton channel: |𝑦_𝛾𝛾|≤2; |cos𝜃_𝛾𝛾|≤1; 250GeV≤𝑀_𝛾𝛾<𝑀_𝑆−10GeV. We call these cuts as the “discovery cuts” for the discovery of KK graviton at the LHC. In Figure 13, 𝐿_𝐷 goes as 𝐿₁₀ for both dilepton and diphoton channels in the whole parameter space we have displayed. In case of the diphoton channel, the background is already quite small compared to the signal, whereas for the dilepton channel, background becomes under control after the “discovery cuts.” With the “discovery cuts,” one can probe 𝑀_𝑆 roughly up to 3.6 TeV (4.4 TeV) with 100 fb⁻¹ (300 fb⁻¹) integrated luminosity at the 14 TeV LHC.

(a)

(b)

Figure 13 

The required luminosity (𝐿𝐷) for the discovery of KK graviton in (a) dilepton and (b) diphoton channels as a function of 𝑀𝑆 at the 14 TeV LHC with “discovery cuts” (see text for the definitions of the cuts). The 𝐿𝐷𝑙𝑙 and 𝐿𝐷𝛾𝛾 are computed for 𝑛= 2–7 with Δ𝜂=2 at 𝑝𝑝→𝑋+𝜓𝜓+𝑌 level where 𝜓={𝑙,𝛾}.

7. Conclusions

In this paper, we have discussed the possibility of looking for new physics, in particular, the signatures of large extra dimensions, in diffractive processes at the LHC. We have concentrated on the central inclusive production where the outgoing protons breakup, but the fragments are mostly emitted in the forward direction. We have studied in detail the signal of KK gravitons via the dilepton and the diphoton channels. From angular momentum conservation, the spin-2 graviton cannot be produced at LO in exclusive processes as discussed in the text, but it can be produced in an inclusive configuration. Although the backgrounds are larger for inclusive production, the larger cross-section and event rate for signals make the collider search for extra dimensional theories very promising. Since the intermediate particle is spin-2 in nature, we also get a unique angular distribution which can be used to distinguish signals from backgrounds. We have shown that it is very unlikely to produce a lepton or a photon pair in the SM with very high invariant mass in the inclusive configuration. Thus, experimentally, if we find a reasonable number of lepton or photon pairs with high invariant mass, it would clearly indicate the signature of new physics possibly the signature of large extra dimensions.

In this work, we have indicated the possibility of using dilepton and diphoton in the final state to signal the presence of extra dimensions. We have carefully estimated the sizes of the signal and the background and found that the signal rates are not too big. Nevertheless, it is possible within reasonable parameter ranges as suggested by present day studies to use central production, with forward detectors for the proton fragments, to look for KK gravitons as a signal for ADD-like theories. The great advantage of using such channels for the study of new physics is that they are clean, and the signals are easily distinguished from the backgrounds. On the flip side, the event rates are considerably smaller than other standard new physics searches. This can be compensated by runs at higher energies and greater luminosities at some future date. There is another general issue with high luminosity run at the LHC—the presence of huge “pile-up” background. This background can overwhelm the rapidity gap signatures. If we consider inclusive processes and sacrifice the forward proton tagging (applicable only for exclusive processes), we can detect rapidity gap events at the LHC [8].

All higher dimensional models are valid up to a cutoff scale. In our case, the signal cross-sections depend on the eighth power of 1/𝑀_𝑆 (𝑀_𝑆 is the cutoff scale of the theory). Thus, numerical results are very sensitive to the UV cutoff. The experimental lower bound of 𝑀_𝑆≈3 TeV for 𝑛=2. Future experiments can push up the lower bound and make the signatures less visible. We show that LHC with √𝑆=14 TeV and 100 fb⁻¹ (300 fb⁻¹) of integrated luminosity can probe 𝑀_𝑆 up to 3.6 TeV (4.4 TeV) in both the dilepton and the diphoton channels.

Appendices

A. Inclusive Luminosity

An outline of the derivation of effective luminosity for the inclusive configuration is presented in [32]. Here, we briefly give some useful formulae for computing inclusive luminosity. At the partonic level, the inclusive process in Figure 14 can be expressed as the exclusive production of the system 𝑀 with rapidity gaps on either side, that is, 𝑎₁𝑎₂⟶𝑎₁+𝑀+𝑎₂. (A.1)

Figure 14 

The inclusive double diffractive amplitude multiplied by its complex conjugate. A central system 𝑀 is produced with rapidity gaps Δ𝜂1 and Δ𝜂2 between proton dissociated products 𝑋 and 𝑌, respectively.

The probabilities to find partons 𝑎_𝑖 (𝑖={1, 2}) in the protons is given by the effective parton densities 𝒢(𝑥_𝑖, 𝑘²_𝑖𝑡) evaluated at scale 𝑘²_𝑖𝑡 with momentum fraction 𝑥_𝑖 as given by 𝒢𝑥_𝑖,𝑘²_𝑖𝑡=𝑥_𝑖𝑔𝑥_𝑖,𝑘²_𝑖𝑡+1681_𝑞𝑥_𝑖𝑞𝑥_𝑖,𝑘²_𝑖𝑡+𝑞𝑥_𝑖,𝑘²_𝑖𝑡, (A.2) where 𝑔, 𝑞, and 𝑞 are the usual PDFs. The rapidity gaps can be filled up due to gluon bremsstrahlung effect. The mean number of gluons emitted in transverse momentum interval, 𝑄_𝑡<𝑃_𝑡<𝑘_𝑖𝑡 (also 𝑄²_𝑡≪𝑘²_𝑖𝑡, asymmetric 𝑡-channel gluon exchange) with rapidity gap Δ𝜂_𝑖 is given by 𝑛_𝑖=3𝛼_𝑆𝜋Δ𝜂_𝑖𝑘ln^2𝑖𝑡𝑄²_𝑡. (A.3) Therefore, the amplitude for no emission in the rapidity interval Δ𝜂_𝑖 can be expressed as 𝐴_𝑖=exp(−𝑛_𝑖/2)Φ(𝑌_𝑖). The amplitude 𝐴_𝑖 is called the nonforward BFKL amplitude which is computed resumming the double logarithms in (A.3). The factor Φ(𝑌_𝑖) accounts for the longitudinal BFKL logarithm where 𝑌_𝑖=(3𝛼_𝑆/2𝜋)Δ𝜂_𝑖. For rapidity gap Δ𝜂_𝑖≲4, we have 𝑌_𝑖≲0.4. In the asymmetric region, 𝑄²_𝑡≪𝑘²_𝑖𝑡, it is sufficient to keep only the 𝒪(𝑌_𝑖) term, that is, Φ(𝑌_𝑖)≈1+𝑌_𝑖𝑄²_𝑡/𝑘²_𝑖𝑡≈1.1±0.1 [61]. There is another suppression that comes in the form of the Sudakov factor 𝑇(𝑘_𝑖𝑡,𝜇) which accounts for the survival probability of a gluon in the interval 𝑘_𝑖𝑡<𝑃_𝑡<𝜇 where 𝜇 is related to the hard scale as 𝜇=𝑀/2. The Sudakov factor reads as 𝑇𝑘_𝑖𝑡,𝜇=exp−^𝜇2_{𝑘2𝑖𝑡}𝛼_𝑆𝑘²_𝑡2𝜋𝑑𝑘²_𝑡𝑘²_𝑡×₀^1−Δ𝑧𝑃_𝑔𝑔(𝑧)+_𝑞𝑃_𝑞𝑔(𝑧)𝑑𝑧, (A.4) where 𝑃_𝑔𝑔(𝑧) and 𝑃_𝑞𝑔(𝑧) are the LO DGLAP splitting functions and Δ=𝑘_𝑡/(𝜇+𝑘_𝑡). We obtain the effective luminosity (ℒ_𝑔𝑔) by multiplying the inclusive amplitude by its complex conjugate as shown in Figure 14. To compute ℒ_𝑔𝑔, we carefully account for all the factors, that is, nonforward BFKL amplitudes, the Sudakov factors, gap survival factor, and appropriate color factors, and finally arrive at the expressionℒ_𝑔𝑔=𝒮²¹_𝑥1min𝒢𝑥₁,𝑘²_1𝑡𝑑𝑥₁𝑥₁×¹_𝑥2min𝒢𝑥₂,𝑘²_2𝑡𝑑𝑥₂𝑥₂𝛼⁴_𝑆𝜋²𝑁²_𝑐𝑁²_𝑐−1²ℐ_𝑔𝑔, (A.5) where ℐ_𝑔𝑔=𝑑𝑄²_𝑡𝑄²_𝑡𝑑𝑄^′2_𝑡𝑄^′2_𝑡𝑑𝑘²_1𝑡𝑘²_1𝑡𝑑𝑘²_2𝑡𝑘²_2𝑡𝐴₁𝐴₂𝐴^′₁𝐴^′₂𝑇₁𝑇₂𝑇^′₁𝑇^′₂. (A.6) Here, all the primed quantities are arising from the complex conjugate of the amplitude. The minimum of the momentum fraction 𝑥_𝑖^min maintains the rapidity gap Δ𝜂_𝑖 and is given by 𝑥_𝑖^min=𝑀√𝑆𝑦+𝑘exp^𝑖𝑡√𝑆exp𝑦+Δ𝜂_𝑖. (A.7)

The momentum of the screening gluon 𝑄 is very small. Therefore, in the limit 𝑄²≪𝑘²_𝑖, we have 𝑡_𝑖=(𝑄−𝑘_𝑖)²≈−𝑘²_𝑖𝑡≈−𝑘^′2_𝑖𝑡. After performing 𝑄²_𝑡 and 𝑄^′_𝑡² integrations, ℐ_𝑔𝑔 takes the simplified form as follows: ℐ_𝑔𝑔=1𝑌₁+𝑌₂²𝑑𝑡₁𝑡₁𝑑𝑡₂𝑡₂exp−3𝛼_𝑆𝜋𝑡Δ𝜂ln¹𝑡₂×𝑇𝑡₁,𝜇𝑇𝑡₂,𝜇, (A.8) where Δ𝜂=Δ𝜂₁ if |𝑡₁|>|𝑡₂|, but Δ𝜂=Δ𝜂₂ if |𝑡₁|<|𝑡₂|. In our computation of ℒ_𝑔𝑔, we have used this simplified form of ℐ_𝑔𝑔 taking Δ𝜂₁=Δ𝜂₂≡Δ𝜂.

B. Feynman Rules and Graviton Propagator

In this appendix, we give the Feynman rules for 𝑉𝑉𝐺 (where 𝑉 is a vector boson with mass 𝑀_𝑉) and 𝑓𝑓𝐺 (where 𝑓 is a fermion with mass 𝑀_𝑓) vertices [54]: (B.1) where 𝜅=16𝜋𝐺_𝑁. The 𝐺_𝑁=(4𝜋)^𝑛/2/𝑅^𝑛𝑀_𝑆^2+𝑛 is the (4+𝑛) dimensional Newton constant. All the tensors appear in the Feynman rules are defined as 𝐶_{𝜇𝜈,𝜌𝜎}=𝜂_𝜇𝜌𝜂_𝜈𝜎+𝜂_𝜇𝜎𝜂_𝜈𝜌−𝜂_𝜇𝜈𝜂_𝜌𝜎,where𝜂_𝜇𝜈{},𝐷=diag+1,−1,−1,−1_{𝜇𝜈,𝜌𝜎}𝑘₁⋅𝑘₂=𝜂_𝜇𝜈𝑘_1𝜎𝑘_2𝜌−𝜂_𝜇𝜎𝑘_1𝜈𝑘_2𝜌+𝜂_𝜇𝜌𝑘_1𝜎𝑘_2𝜈−𝜂_𝜌𝜎𝑘_1𝜇𝑘_2𝜈+,𝐸𝜇⟷𝜈_{𝜇𝜈,𝜌𝜎}𝑘₁⋅𝑘₂=𝜂_𝜇𝜈𝑘_1𝜌𝑘_1𝜎+𝑘_2𝜌𝑘_2𝜎+𝑘_1𝜌𝑘_2𝜎−𝜂_𝜈𝜎𝑘_1𝜇𝑘_1𝜌+𝜂_𝜈𝜌𝑘_2𝜇𝑘_2𝜎+𝜇⟷𝜈. (B.2)

The propagator for spin-2 KK graviton with mass 𝑀_𝐺 and decay width Γ_𝐺 is given by Δ_{𝜇𝜈,𝜌𝜎}(𝑘)=(𝑖/2)𝐵_{𝜇𝜈,𝜌𝜎}(𝑘)𝑘²−𝑀²_𝐺+𝑖Γ_𝐺𝑀_𝐺, (B.3) where 𝐵_{𝜇𝜈,𝜌𝜎}(𝑘)=𝜂_𝜇𝜌−𝑘_𝜇𝑘_𝜌𝑀²_𝐺𝜂_𝜈𝜎−𝑘_𝜈𝑘_𝜎𝑀²_𝐺+𝜂_𝜇𝜎−𝑘_𝜇𝑘_𝜎𝑀²_𝐺𝜂_𝜈𝜌−𝑘_𝜈𝑘_𝜌𝑀²_𝐺−23𝜂_𝜇𝜈−𝑘_𝜇𝑘_𝜈𝑀²_𝐺𝜂_𝜌𝜎−𝑘_𝜌𝑘_𝜎𝑀²_𝐺. (B.4)

Mass of the 𝑘th KK mode is 𝑀_𝑘=4𝜋²𝑘²/𝑅², and mass separation between two adjacent KK modes is a 𝒪(1/𝑅) term. Therefore, KK modes become quasicontinuous, and it is convenient to define KK state density. The number of KK states in the mass interval 𝑀²_𝑘 and 𝑀²_𝑘+𝑑𝑀²_𝑘 for 4+𝑛 dimensional space can be expressed as Δ𝑘²=𝜌(𝑀_𝑘)𝑑𝑀²_𝑘, where 𝜌𝑀_𝑘=𝑅^𝑛𝑀_𝑘^𝑛−2(4𝜋)^𝑛/2Γ().𝑛/2 (B.5)

The effective interaction due to all KK states contributing to a physical process can be obtained after summing over all the propagators as follows: 𝒟(𝑠)̂=_𝑘𝑖𝑠̂−𝑀²_𝑘+𝑖𝜀=^∞₀𝑑𝑀²_𝑘𝜌𝑀_𝑘𝑖𝑠̂−𝑀²_𝑘,+𝑖𝜀 (B.6) where 𝜀=𝑀_𝐺Γ_𝐺 and 𝑠̂=𝑘². The effective propagator 𝒟(𝑠)̂ valid up to 𝑀_𝑆 looks as 𝒟(𝑠)̂=𝑠̂^𝑛/2−1𝑅^𝑛Γ(𝑛/2)(4𝜋)^𝑛/2𝑀𝜋+2𝑖𝐼^𝑆√𝑠̂, (B.7) where 𝐼𝑀_𝑆√𝑠̂=−∑^𝑛/2−1_𝑘=11𝑀2𝑘√𝑠^𝑆̂^2𝑘−12𝑀ln^2𝑆𝑠̂−−1for𝑛=even_{(𝑛−1)/2𝑘}∑₌₁1𝑀2𝑘−1√𝑠^𝑆̂^2𝑘−1+12𝑀ln^𝑆+√𝑠̂𝑀_𝑆−√𝑠̂for𝑛=odd. (B.8) In the limit 𝑀²_𝑆≫𝑠, the effective propagator 𝒟(𝑠)̂ becomes |𝒟(𝑠)̂|=16𝜋𝜅²𝑀⁴_𝑆𝑀𝒦where𝒦=ln^2𝑆𝑠̂2for𝑛=2𝑛−2for𝑛>2. (B.9) In our analysis, we have used the exact form of 𝒟(𝑠)̂ given in (B.7).

C. Dilepton Background

We consider the inelastic photoproduction of a lepton pair at the LHC shown in Figure 15 as follows: 𝑝𝑝⟶𝑋+𝛾𝛾+𝑌⟶𝑋+𝑙⁺𝑙⁻+𝑌. (C.1)

Figure 15 

Inelastic photoproduction of a lepton pair at the LHC. The photons are coming from the quarks inside protons.

A photon with momentum fraction 𝑧₁ (𝑧₂) with respect to the quark momentum is emitted from the quark 𝑞₁ (𝑞₂) with momentum fraction 𝑥₁ (𝑥₂) of a proton. The total inelastic 𝑝𝑝→𝑙⁺𝑙⁻ cross-section can be computed (this has been thoroughly computed in [59]) using the Weizsäcker-Williams equivalent photon approximation [62, 63] as given by 𝜎(𝑆)=_𝑞1,𝑞2¹_{4𝑚2𝑙/𝑆}𝑑𝑥₁¹_{4𝑚2𝑙/𝑆𝑥1}𝑑𝑥₂¹_{4𝑚2𝑙/𝑆𝑥1𝑥2}𝑑𝑧₁×¹_{4𝑚2𝑙/𝑆𝑥1𝑥2𝑧1}𝑑𝑧₂𝑒²_𝑞1𝑒²_𝑞2×𝑓_𝑞1/𝑝𝑥₁,𝑄²𝑓_𝑞2/𝑝𝑥₂,𝑄²×𝑓_𝛾/𝑞1𝑧₁𝑓_𝛾/𝑞2𝑧₂𝜎(𝑠)̂, (C.2) where 𝑞₁, 𝑞₂ are the quarks (we denote 𝑞≡{𝑞₁, 𝑞₂}), 𝑚_𝑙 is the mass of the lepton, 𝑒_𝑞 is the EM charge of 𝑞 in the unit of an electron's charge, and 𝜎 is the subprocess cross-section with CM energy √𝑠̂ (𝑠̂=𝑥₁𝑥₂𝑧₁𝑧₂𝑆,𝑆 is the LHC CM energy). The quark density inside the proton is denoted as 𝑓_𝑞/𝑝. The 𝑓_𝛾/𝑞 is the photon spectrum inside a quark and is given by 𝑓_𝛾/𝑞(𝑧)=𝛼_𝑒𝑚()2𝜋1+1−𝑧²𝑧𝑄In²¹𝑄²₂. (C.3) We choose the scale 𝑄₁=√𝑠/̂2 and 𝑄₂=1 GeV. The subprocess cross-section for 𝛾𝛾→𝑙𝑙 is given by 𝜎=4𝜋𝛼²_𝑒𝑚𝑀²_𝑊𝑠̂3−𝛽⁴2ln1+𝛽1−𝛽−2𝛽+𝛽³, (C.4) where 𝛽=1−4𝑚²_𝑙/𝑠̂. We have used 𝛼_𝑒𝑚=1/137 and 𝛼²_𝑒𝑚(𝑀²_𝑊)=1/128 in our analysis.

The differential cross-section of the subprocess in the high energy limit is 𝑑𝜎𝑑||=cos𝜃2𝜋𝛼²_𝑒𝑚𝑀²_𝑊𝑠̂1+cos²𝜃sin²𝜃, (C.5) where 𝜃 is scattering angle of the lepton in the CM frame. We see that 𝑑𝜎/𝑑|cos𝜃|→∞ if |cos𝜃|→1.

D. Diphoton Background

We compute the dominant background cross-section for the diphoton channe 𝑔𝑔→𝛾𝛾 via a quark loop using the 𝛾𝛾→𝛾𝛾 helicity amplitudes given in [60]. The differential cross-section of 𝑔𝑔→𝛾𝛾 background in the high energy limit and for small 𝜃 (scattering angle of 𝛾 in the CM frame) is given by 𝑑𝜎𝑑||=1cos𝜃2⋅1𝑠̂⋅132𝜋8²⋅12²⋅8⋅𝑁_𝑓𝑁_𝑐⋅𝛼²_𝑆𝛼²_𝑒𝑚|ℳ|². (D.1) Origin of some factors in the previous equation is once explained in Section 4.2, and we are not explaining them again. The factor 𝑁_𝑓𝑁_𝑐 is the number of possible quarks that can contribute in the loop. The |ℳ|² is the sum of two helicity amplitudes, which are the only two surviving amplitudes in the high energy limit, |ℳ|²=ℳ₊₊₊₊²+ℳ₊₋₊₋²=32In⁴2||.1+cos𝜃 (D.2)

Acknowledgment

The authors are grateful to Subhadip Mitra for many fruitful discussions and assistance in certain numerical aspects of this work.