Production of \gamma\gamma + 2jets from double parton scattering in proton-proton collisions at the LHC

Cross sections for the production of pairs of photons plus two additional jets produced from double parton scattering in high-energy proton-proton collisions at the LHC are calculated for the first time. The estimates are based on the theoretical perturbative QCD predictions for the productions of \gamma\gamma at next-to-next-to-leading-order, jet + jet and \gamma + jet at next-to-leading-order, for their corresponding single-scattering cross sections. The cross sections and expected event rates for \gamma\gamma + 2 jets from double parton scattering, after typical acceptance and selections, are given for proton-proton collisions with the center-of-mass energies of 13 TeV and an integrated luminosity of 100 fb-1 planned for the following two years, and also center-of-mass energies of 14 TeV with 3000 fb-1 of integrated luminosity as LHC designed.


Introduction
In proton-proton (pp) collisions with higher energies at the Large Hadron Collider (LHC), particle production is dominated by multiple interactions of their constituent partons, with most particles from the hardest protonproton scattering and the radiation and fragmentation of secondary partonic actions. The higher centre-of-mass energy leads to enhanced parton densities which cause a sizable probability for two or more parton-parton scatterings within the same pp interaction [1,2]. At LHC, various measurements of the differential distributions in W + jets [3,4] and J/ψ + W [5] show that the excesses above the expectations from single parton scattering (SPS) are consistent with double parton scattering (DPS). Various measurements in other pp and pp collisions at √ s = 63 GeV [6], 630 GeV [7], and 1.8 TeV [8] are consistent with DPS contributions to multijet final states, as well as to γ + 3-jet events at √ s = 1.8 TeV [9] and 1.96 TeV [10]. The measurements of DPS processes can provide valuable information on the transverse distribution of partons in the proton [11] and on the multi-parton correlations in the hadronic wave function [12]. DPS also constitutes the background for new physics searches at the LHC [13][14][15]. Additional searches for DPS have been proposed via double Drell-Yan, four jets and same-sign WW production [16][17][18].
In this paper, the cross section for the production of pairs of photons plus two additional jets produced from double parton scattering (DPS) in high-energy protonproton collisions at the LHC are calculated for the first time. γγ final states have played a crucial role in the recent discovery of a new boson at the LHC [19,20] and are also important in many New Physics searches [21][22][23][24], in particular the search for extra spatial dimensions or cascade decays of heavy new particles. In particular, diphotons in combination with jets and missing energy occur in gauge mediated SUSY scenarios. γγ or plus two additional jets offers also an important test of both perturbative and non-perturbative quantum chromodynamics (QCD) [25][26][27]. γγ + 2 jets is also the main irreducible background for other physics analyses with γγ and jets in the final state at the LHC, such as Higgs produced in vector boson fusion. For the production of γγ + 2 jets, a sizeable contribution from DPS with γγ produce in one scattering while the second scattering yielding two jets can be expected.
The structure of this paper is organized as follows. In section 2, a generic way of the DPS cross section as the product of the SPS cross sections and its parameter are briefly introduced. The details of the cross section of γγ + 2 jets calculation and the cross section of different SPS processes estimated from higher order theoretical predictions are described in section 3. The results including the cross section of γγ + 2 jets with √ s = 13 TeV and 14 TeV at LHC and the expected event rates, with typical selections,a re summarized in section 4. The summary and outlook are given in section 5.
2 Generic formule of DPS For a composite system (A+B) in hadronic collisions, its production cross section from DPS, σ DP S pp→AB , can be written model-independently as the product of the cross sections of A and B originated from single parton scattering, σ SP S pp→A and σ SP S pp→B , normalized by an effective cross section σ ef f [28] where m ia a symmetry factor accounting for distinguishable (m=2) and indistinguishable (m=1) final-states. The effective cross section σ ef f is a measure of the transverse distribution of partons inside the colliding hadrons and their overlap in a collision. It is independent of the process and of the phase-space under consideration.  Fig. 1 shows a comparison of the effective cross section σ ef f measured by different experiments using different processes at various centre-of-mass energies. CMS (W + 2 jets) ATLAS (W + 2 jets) CDF (4 jets) + 3 jets) γ CDF ( + 3 jets) γ Corrected CDF ( + 3 jets) γ D0 ( UA2 (4 jets -lower limit) AFS (4 jets -no errors given) Fig. 1. σ ef f measured by different experiments using different processes [3, 4, 6-10, 29]. The "Corrected CDF" data point indicate the σ ef f value corrected for the exclusive event selection [29].
The measured values of σ ef f from TeV experiments at Tevetron (CDF and D0) and LHC (ATLAS and CMS) are consistent with each other within their uncertainties. In the following calculations, a numerical value σ ef f ≈ 15 mb was used to estimate the production cross section of γγ + 2 jets from DPS with √ s = 13 TeV and 14 TeV at LHC. A number 5 mb was assigned as its uncertainty to estimate its effects on the final results. The uncertainty on σ ef f is the dominant uncertainty for the calculation of the production cross section of γγ + 2 jets from DPS in the following sections 3 σ DP S pp→γγ+2jets calculation According to the descripition in above section, the production cross section of γγ + 2 jets from DPS in pp collisions can be written as γγ production has been calculated at next-to-leadingorder (NLO) some time ago [30], supplemented also by gluon initiated subprocesses beyond the leading order [31] and soft gluon resummation [32,33]. Recently, next-to-next-to-leading-order (NNLO) corrections to direct diphoton production also have become available [34]. The measurements from LHC [25][26][27] show that the NNLO can give much better agreement with measured data than the lower order predictions. For the integrated cross section, the predicted values by NNLO are almost exactly the same [27] or consistent within the uncertainties [25, 26] with the measured ones. So for the production cross sections of γγ final-state from SPS, σ SP S pp→γγ , with different √ s will be obtained from the NNLO calculation with the package 2γNNLO.
For the dijet cross section, the measured data at LHC [35,36] can be well described by NLO perturbative QCD (pQCD) calculations from NLOJet++ program [37] corrected to account for non-perturbative and electroweak effects. From [35], the non-perturbative correction is within 3% for jet reconstructed with the anti-k t clustering algorithm [38] and distance parameter or cone size R=0.4. The corrections for the electroweak effect can be negligible if the dijet mass less than about 1 TeV. In this analysis, the NLO calculations of σ SP S pp→2jets are performed using NLOJet++ (version 4.1.3) within the framework of the fastNLO package (version 2.3.1) [39].
NLO pQCD prediction from the program JETPHOX (version 1.3.1) [40] is used for the calculation of γ + jet cross sections from SPS in this paper. This program includes a full NLO QCD calculation of both the directphoton and fragmentation contributions to the cross section. The number of flavours was set to five. Compared with the measurements of γ + jet cross sections at the LHC, the predictions from JETPHOX multiplied by a factor close to unity for the corrections of hadronisation and underlying-event effects give a good description of the E γ T and p jet T measured cross sections [41, 42].
Different PDF sets are used for the calculations of these three SPS processes. MSTW2008NNLO [43] is used for σ SP S pp→γγ calculations with 2γNNLO. CT10NLO [44] is used for both σ SP S pp→2jets with NLOJet++ in fastNLO package and σ SP S pp→γ+jet with JETPHOX. The calculations of σ SP S pp→γγ are performed with the factorization and renormalization scales equal to the invariant mass of two photons, µ F = µ R = m γγ . The scale uncertainty and PDF uncertainty are also considered. A simplified and less computationally intensive estimate of the renormalization (µ R ) and factorization (µ F ) scale uncertainties is performed by varying these scales simultaneously by a factor of two up and down around m γγ , µ F = µ R = 2m γγ and µ F = µ R = 0.5m γγ . 41 eigenvector sets of MSTW2008NNLO are used to build the PDF uncertainty envelope.
Calculations of σ SP S pp→2jets are derived using NLO-Jet++ within the framework of the fastNLO package at a factorization and renormalization scale equal to the average transverse momentum (p ave T ) of the two jets (µ F = µ R = p ave T ). The uncertainty due to the choice of factorization and renormalization scales is estimated as the maximum deviation at the six points (µ F /µ, µ R /µ)= (0.5, 0.5), (2, 2), (1, 0.5), (1, 2), (0.5, 1), (2, 1) with µ = p ave T . 52 eigenvector sets of CT10NLO are used to build the PDF uncertainty envelope.
For NLO calculations of σ SP S pp→γ+jet using JETPHOX, the renormalization, factorization and fragmentation (µ f ) scales are chosen to be photon's transverse momentum, µ F = µ R = µ f = E γ T . Same as above, 52 eigenvector sets of CT10NLO are used to build the PDF uncertainty envelope.
Above calculations were performed with the strong coupling constant at two-loop order with α s (m Z ) = 0.118 in CT10NLO and 0.117 in MSTW2008NNLO. The uncertainty on α s (m Z ) was not considered in this study. The uncertainty from scales, pdf and α s (m Z ) is around 10%, 5% and 1% respectively [25-27, 35,36,41,42]. Compared to the larger uncertainty on the σ ef f with more than 30% used in this study as told in the end of section 2, the effect on the final results from the uncertainty on α s (m Z ) can be negligible.
4 Results of σ DP S pp→γγ+2jets and expected event rates at LHC In this paper, several sets of typical selections at LHC were tried to calculate the production cross section of γγ + 2 jets from DPS in pp collisions, σ DP S pp→γγ+2jets . Due to the high level trigger requirements for γγ events at LHC for the higher energy and higher luminosity collisions, five sets of requirements on the photons' transverse momentum were considered,  ) GeV with γ 1 representing the maximum E T photon and γ 2 the minimum E T one of two photons. So for single photon requirement in the γ +jet, 3 cases with E γ T > (20, 30, 40) GeV were considered. The photon should be also constrained in the pseudorapidity region |η| <2.5. An isolation requirement is applied on the photon to fulfill the isolation requirement from experimental measurements [25-27, 41, 42]. The standard isolation, the E T sum of partons in a cone of size ∆R=0.4 around the photon required to less than 5 GeV, is applied in JET-PHOX for the calculation of σ SP S pp→γ+jet . For 2γNNLO, the smooth Frixione isolation [45] on the photons is applied where E iso T (∆R) is the E T sum of partons in a cone of size ∆R, ∆R 0 = 0.4, ǫ = 5GeV, and n = 0.1. This criterion is found to have the same efficiency as the standard isolation used for the other generators within a few percent [25,26]. Additional the angular separation between two photons is required to be at least larger than 0.4 (∆R γγ >0.4) to ensures one photon will not enter the isolation cone of the other photon, which is similar to the requirement applied in the data analyses at LHC experiments ATLAS and CMS [25][26][27].
In this study, jet is reconstructed with the anti-k t clustering algorithm and cone size R=0.5. Jets are in the acceptance region with |η jet | <4.5. Two tries on jet p jet T were performed, p jet T > 20 or 25 GeV. For the dijet events, two jets should be separated by requiring their angular distance ∆R jj greater than 1.0 to avoid the overlapping of two jet cones. For the γ + jet production, the angular distance between γ and jet should be greater than 0.5 (∆R γj > 0.5) to ensure that the partons belong to the jet will not enter to the isolation cone of γ. Fig. 2 shows the cross sections of σ SP S pp→γγ computed from the 2γNNLO at √ s = 13 TeV and 14 TeV with scales and pdf uncertainties considered, for different sets of E T requirements on diphotons. The scale uncertainty is around 10% and the pdf uncertainty is about 4%. The selection sets in x-axis are the 5 sets of requirements on   (40,40) GeV. Scale and pdf uncertainties are included. Table 1.
Cross sections in unit of pb of γγ predicted by 2γ NNLO at √ s = 13 TeV and 14 TeV. The uncertainties include the scale and pdf uncertainties.   Combined the photon E γ T requirements for the γγ productions and the jet p jet T requirements for the jet+jet productions, cross section of σ SP S pp→γ+jet with six sets of selections on the transverse momentums of photon and jet with (E γ T , p jet T ) > (20, 20) GeV, (30,20) GeV, (40,20) GeV, (20, 25) GeV, (30,25) GeV and (40, 25) GeV were calculated. Fig. 4 shows the differential cross sections of σ SP S pp→γ+jet as a function of photon E γ T with (E γ T , p jet T ) > (40, 20) GeV, |η γ | < 2.5, |η jet | < 4.5 and separation ∆R γj > 0.5, computed from JETPHOX with √ s = 14 TeV. The contributions from the direct photon production and photon from fragmentation are shown in the same plot. The scales and pdf uncertainties are also plotted in the same figure. The integrated cross sections are listed in Table 2 for different sets of selections and collision energies. The scale and pdf uncertainties are also listed in this table, with about 10% uncertainty from scales and around 4% from pdf.  According to Eq.2 and the above cross sections of the SPS processes, the cross sections for the production of pairs of photons plus two additional jets produced from double parton scattering (DPS) in high-energy protonproton collisions at the LHC are calculated for the first time. The results are summarized in Table 3. Two jets in the same pp → γγ +2jets event from DPS have the same p T cut thresholds, both p jet T > 20 GeV or 25 GeV simultaneously. The calculated cross section can be around 0.1 pb to ≈1 pb with the selections considered in this paper. The uncertainty on the cross section ia around 50%, with the dominant contribution from the uncertainty of σ ef f . Table 3.
Cross sections in unit of pb of σ DP S pp→γγ+2jets calculated for √ s = 13 TeV and 14 TeV with the selections described in the paper. The total uncertainties including scale uncertainty, pdf uncertainty and also the σ ef f uncertainty are also list in this table. With an integrated luminosity of 100 f b −1 at √ s = 13 TeV accumulated in the following years, about 85k pp → γγ+2jets events from DPS can be obtained with the loosest selections, diphoton (E γ 1 T , E γ 2 T )> (30, 20) GeV and both jets p jet T > 20 GeV. These events can be triggered by the diphoton paths proposed at the LHC for √ s = 13 TeV. When the integrated luminosity increasing at √ s = 14 TeV, tighter E T thresholds on diphoton for the trigger will be used. With the tighter selections, diphoton (E γ 1 T , E γ 2 T )> (40, 30) GeV and both jets p jet T > 20 GeV, about 940k pp → γγ+2jets events from DPS can be obtained with an integrated luminosity of 3000 f b −1 . Even with the tightest selections studied in this paper, diphoton (E γ 1 T , E γ 2 T )> (40, 40) GeV and both jets p jet T > 25 GeV, we can also get about 260k pp → γγ + 2jets events from DPS with 3000 f b −1 as designed by LHC.

Summary and outlook
In this paper, the cross sections for the production of pairs of photons plus two additional jets produced from double parton scattering in high-energy protonproton collisions at the LHC with √ s = 13 TeV and 14 TeV (LHC Run2) are calculated for the first time.
With the generic formula, the cross sections have been computed based on the theoretical perturbative QCD predictions for the productions of γγ at next-to-next-toleading-order , jet + jet and γ + jet at next-to-leadingorder, with their corresponding single-scattering cross sections. From the LHC measurements with the collision data obtained in years 2011 and 2012 (LHC Run1), these theoretical predictions for these three SPS processes can give the best agreements with the measured data. With the typical acceptance and selections used at LHC, the cross sections σ DP S pp→γγ+2jets can be estimated to be around 0.1 pb to 1 pb with the collision energy √ s = 13 TeV or 14 TeV. The expected event rates for γγ + 2jets from DPS, with some sets of selections, are given for protonproton collisions with the collision energy √ s = 13 TeV and an integrated luminosity of 100 f b −1 planned for the following two years, and also √ s = 14 TeV with 3000 f b −1 of integrated luminosity as LHC designed. The uncertainties on the cross section and the events rates are mainly dominated by the σ ef f uncertainty. The scale and pdf uncertainties for the productions of these three SPS processes are also considered.
With the incoming LHC Run2 data, there are enough pp → γγ + 2jets events from DPS for investigations. It needs further studies on the variables, such as the angles between two photons and two jets, to be chosen for the discrimination of pp → γγ + 2jets events from DPS and pp → γγ + 2jets events from SPS when performing the data analysis. Also the contributions from the DPS to the whole pp → γγ+2jets event rates on the distributions of some typical variables are need detailed investigations in the LHC Run2 data analysis.