Diphoton signals in theories with large extra dimensions to NLO QCD at hadron colliders

We present a full next-to-leading order (NLO) QCD corrections to diphoton production at the hadron colliders in both standard model and ADD model. The invariant mass and rapidity distributions of the diphotons are obtained using a semi-analytical two cut-off phase space slicing method which allows for a successful numerical implementation of various kinematical cuts used in the experiments. The fragmentation photons are systematically removed using smooth-cone-isolation cuts on the photons. The NLO QCD corrections not only stabilise the perturbative predictions but also enhance the production cross section significantly.


Introduction
The gauge hierarchy problem has been one of the main motivations to go beyond the standard model (SM). A novel idea that addresses this problem was put forward by Arkani-Hamed, Dimopoulos and Dvali (ADD) wherein they introduced extra spatial dimensions and allowed only gravity to propagate in the extra dimensions, keeping the SM fields confined to a 3-brane [1]. As the inverse square law behavior of gravity has so far been tested down to sub-millimeter length scales, the size of the extra dimensions, in this model, should be much smaller than sub-millimeter. The apparent weakness of gravity as compared to the other forces seen in nature, can now be accounted for through the volume of the extra dimensions. The relation between the fundamental scale M s at which the new physics sets in (above which the extra dimensions are dynamically accessible) and the Planck scale M P is given by where d is the number of extra spatial dimensions and R, the size of the extra dimensions.
Since R is of order of a milli-meter, the scale M s can be as low as a few TeV, which circumvents the hierarchy problem. The propagation of a massless graviton in 4 + d dimensions, after compactifying the extra dimensions on a d-dimensional torus, manifests itself as an infinite tower of massive Kaluza-Klein (KK) modes on the 3-brane. Each KK mode couples with SM field through energy momentum tensor with a coupling proportional to κ ∼ 1/M P . However, the effective coupling after summing over all the KK modes is enhanced significantly due to large multiplicity of KK modes. In any typical scattering process at colliders, the gravity can enter through their KK propagator as well as through the real emission of KK states. These KK states are large in number. Hence the suppression resulting from coupling κ is compensated by the large multiplicity factor resulting either from the sum of KK propagator D(Q 2 ) or from the phase space of large number of real KK states . For example, if the KK states enter through a propagator, we find that any typical amplitude will be proportional to where Λ = M s is the explicit cut-off on the KK sum and the function I can be found in [2]. Thus, for M s ∼ O(TeV), the gravity effects can become significant and hence the collider phenomenology associated with this model is very interesting [2]. To exemplify, the virtual effects of the KK modes could lead to the enhancement of the cross sections of pair productions in the processes like Drell-Yan, diphoton and dijet while the real emissions could lead to large missing E T signals giving some new observable like monojet, mono-photon in an experiment. Owing to a very high centre-of-mass energy of √ S = 14 TeV and a large gluon flux at the large hadron collider (LHC), rich collider signals resulting from this model have been reported in the literature [2][3][4][5][6][7]. However, these results are based on leading order (LO) calculations. At the hadron colliders like LHC, the QCD effects are often considerably large and hence the quantum corrections can influence the predictions significantly. In the ADD model, QCD effects [8] have been shown to increase the di-lepton productions and also to stabilise the perturbative predictions. Hence in this paper we study the impact of the QCD corrections for the diphoton signal in the ADD model.
In QCD, the infra-red safe observable exhibit a feature called factorisation, according to which collinear singularities can be factored out from the partonic cross sections in a process independent way and then they are either absorbed into the bare parton distribution functions (PDF) if they originate from initial state partons or into fragmentation functions if they are from final state partons. This procedure introduces a scale called factorization scale µ F , which is arbitrary. In addition, ultra-violate renormalisation introduces renormalisation scale µ R which is again arbitrary. The truncated perturbative expansion leaves our theoretical predictions µ F and µ R dependent, these scale dependence will go down as we include more and more terms in the expansion. In addition, the fitted PDFs are usually not fully constrained due to insufficient experimental data.
Hence, predictions beyond LO are often more reliable than LO ones.
Diphoton production process is an important probe for the Higgs boson search at the LHC. NLO QCD corrections to this process in the SM are available in the literature [9][10][11][12] and hence the diphoton signal has been a useful tool for precision studies. This process has also been used to search for the physics beyond the standard model, such as extra dimensional models, super symmetry and the unparticle physics. Di-photon production [5] at Tevatron has set stringent constraints on the parameters of the ADD model [13]. It will also play an important role at LHC. The DØcollaboration [13] assumed a K-factor for their analysis but a full NLO QCD calculation for the ADD model does not exist for the diphoton production. In this paper, we have systematically computed all the QCD effects to NLO in perturbation theory to various important observable in diphoton production that are sensitive to the ADD model. Quantitative estimates of QCD corrections to these observable are presented and our predictions are expected to be less sensitive to the factorisation scale.

The Diphoton Production
In the SM, at leading order (LO), diphoton production proceeds via quark anti-quark annihilation subprocess q + q → γ + γ 5 . In the ADD model, the SM fields couple to KK modes through the energy-momentum tensor of the SM fields with a strength denoted by κ. Hence, diphotons are produced in (i) quark antiquark annihilation (q +q → γ +γ) and (ii) gluon fusion process (g +g → γ +γ) via the exchange of KK modes. A comprehensive phenomenology taking into account all the above LO processes has been done in [5]. It was observed that unitarity restricts the maximum value of the invariant mass Q of the diphotons. Following [5], we restrict the invariant mass Q to Q < 0.9 M s .
At NLO, the SM as well as ADD leading order quark antiquark annihilation processes get O(α s ) QCD radiative corrections through virtual gluons in q + q → γ + γ + one loop and real gluon emissions in q+q → γ+γ+g processes. To this order, q(q)+g → q(q)+γ+γ process also shows up in both SM and ADD. The LO gluon fusion process in the ADD model gets NLO QCD corrections to order α s through g + g → γ + γ + one loop and g + g → γ + γ + g processes. Since KK modes appear at the propagator level, the LO SM (ADD) processes interfere with the corresponding NLO ADD (SM) processes giving order α s NLO QCD corrections. We have incorporated all these NLO QCD corrections in this article for the study that follows.
The NLO partonic cross sections are often ill-defined due to soft and collinear singularities that result from the presence of zero momentum gluons and mass-less partons. In addition to these singularities, we encounter collinear (QED) singularities that originate when the photon in the final state becomes collinear to the quark or the anti-quark emitting it. These (QED) singularities go away if we also include the diphoton production channels resulting from the fragmentation of partons. This involves introduction of nonperturbative fragmentation functions. These functions are poorly constrained. Hence, in our study we do not include fragmentation photons but consider only direct photons. Alternatively, we can suppress QED collinear singularities using the smooth-cone-isolation prescription proposed by Frixione [14]. In the rapidity-azimuthal angle (y, φ) plane the amount of transverse hadronic energy E T in any cone of radius r = (∆y) 2 + (∆φ) 2 with r < r 0 centered around the photon must satisfy The above prescription safely removes all the photons from the fragmentation processes without disturbing soft and collinear partons.
An analytical computation incorporating smooth-cone-isolation and other kinematical constraints at NLO level is hard to achieve. Hence, we resort to a semi-analytical approach called two cutoff phase space slicing method [15]. In this method, two small slicing parameters δ s and δ c are introduced to isolate the cross sections that are sensitive to soft and collinear singularities. The remaining part of the cross section denoted by

Numerical Results
In this section, we present our results for invariant mass (Q) and rapidity (Y) distributions of the photon pair at LHC. We have employed the kinematical cuts given by ATLAS collaboration [16]: the transverse momentum p γ T > 40 GeV for the harder photons, p γ T > 25 GeV for the softer photon, and the rapidity |y γ | < 2.5 for each photon. In addition, the photons are isolated from hadronic activity according to Eq. with the limits from [13].
We have first checked our numerical code by studying the dependence of observable on the slicing parameters, δ s and δ c . In the left (right) panel of Fig. 1  in [11] with their choice of parameters.
In Fig. 2 The cross sections do depend on the isolation criterion. The E iso T at the partonic level need not be the same as that of the hadrons at the detector level, which gives rise to the dependency of the cross sections on E iso T . In the smooth cone isolation prescription discussed above, large logarithms of E iso T often spoil the reliability of fixed order computation. We can study the effect the these logarithms by varying the function that appear in the isolation criterion. We present in Fig. 4, the dependency of our cross sections on the choices of E iso T (varied between 5 GeV and 30 GeV), and n, (varied between 1 and 2). We find that the dependency is unnoticeable making our predictions reliable for experimental study.
Finally we consider the invariant mass distribution at the Tevatron for both the SM and ADD model to NLO QCD. We have used M s value which is consistent with the experimental bounds [13] for the di-electromagnetic signal which is the combined e + e − and γγ final state. In this analysis we are hence interested only in gauging the impact of the QCD corrections to these studies. In Fig. 5 we plot the invariant mass distribution of the diphoton system in the range 100 < Q < 1000 GeV at the Tevatron ( √ S = 1.96 GeV) for both the SM and including the ADD contribution at LO and NLO in QCD.

Conclusions
In this article, we have systematically computed NLO QCD corrections to the diphoton production process at the hadron colliders in SM as well as in ADD model. We use a semianalytical two cut-off phase space slicing method to compute invariant mass as well as rapidity distributions of the diphotons system. We have applied the kinematical cuts used by the ATLAS detector collaboration for our study. A smooth-cone-isolation prescription on the diphotons has been used to reject poorly known fragmentation photons. Our method takes care of all the soft and collinear singularities that appear at NLO level in QCD. We have explicitly shown that our NLO results are least sensitive to the slicing parameters δ s and δ c . Our SM results are in good agreement with those given in the literature. Predictions for invariant mass distribution of diphotons in ADD model with M s = 2 TeV are found to be large compared to those in SM for invariant mass Q > 600 GeV. This is due to large gluon flux at the LHC which enhances the gluon initiated production channels over the rest. In addition, the QCD corrections are significantly large both in the SM and in the ADD over the entire range of Q considered. For the rapidity distribution, we have integrated Q in the region 600 ≤ Q ≤ 1100 GeV where the gravity (through KK modes) contributes significantly. We find that the QCD corrections are important throughout the region |Y | ≤ 2.0. In addition, our results are expected to be less sensitive to the uncertainties coming from the choice of factorisation scale.
In summary, we have accomplished an important task of computing all the partonic contributions at NLO level in QCD to diphoton production at hadron colliders both in SM and ADD model. These QCD corrections for the ADD model and its interference with the SM are being presented for the first time, while to this order the SM results already exist in the literature. The NLO QCD effects are found to be large and they are expected to reduce theoretical uncertainties, thus providing an excellent opportunity to put stringent bounds on the parameters of the ADD model when the experimental results are available. Quantitative impact of the NLO QCD corrections to both the ADD and RS model would be addressed in a future publication [18].