Next-to-Leading-Order Monte Carlo Simulation of Diphoton Production in Hadronic Collisions
aa r X i v : . [ h e p - ph ] J un Next-to-Leading-Order Monte CarloSimulation of Diphoton Production inHadronic Collisions
Luca D’Errico
Institut f¨ur Theoretische Physik,University of Karlsruhe, KIT, 76128, Germany;Institute of Particle Physics Phenomenology, Department of Physics,University of Durham, DH1 3LE, UK;Email: [email protected]
Peter Richardson
Institute of Particle Physics Phenomenology, Department of Physics,University of Durham, DH1 3LE, UK;Email: [email protected]
KA-TP-11-2011SFB/CPP-11-30MCNET-11-15DCPT/11/68IPPP/11/34
Abstract
We present a method, based on the positive weight next-to-leading-order matching formal-ism (POWHEG), to simulate photon production processes at next-to-leading-order (NLO).This technique is applied to the simulation of diphoton production in hadron-hadron col-lisions. The algorithm consistently combines the parton shower and NLO calculation,producing only positive weight events. The simulation includes both the photon fragmen-tation contribution and a full implementation of the truncated shower required to correctlydescribe soft emissions in an angular-ordered parton shower. ——————————————————————————————————
Contents ¯ B (Φ B ) . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1 Real emission contribution . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Virtual contribution and collinear remainders . . . . . . . . . . . . . 123.3 Generation of the hard process . . . . . . . . . . . . . . . . . . . . . . 134 The generation of the hardest emission . . . . . . . . . . . . . . . . . 134.1 The hardest QED emission . . . . . . . . . . . . . . . . . . . . . . . . 134.2 The hardest QCD emission . . . . . . . . . . . . . . . . . . . . . . . . 145 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19——————————————————————————————————— The production of photons via perturbative processes is very important for both thesearch for the Higgs boson and other new physics, via photon pair production, andfor the study of QCD and experimental effects, in particular the jet-energy scale,in the production of a photon in association with a jet. To study these processesin detail in hadron-hadron collisions we need an accurate Monte Carlo simulation.In this paper we present a new approach for the simulation of these processes andillustrate it with the simulation of photon pair production.Monte Carlo event generators simulate events by combining fixed-order matrix el-ements, parton showers and hadronization models. The first programs used leading-order (LO) matrix elements, together with the parton shower approximation whichdescribes soft and collinear emission. Recently different approaches correcting theemission of high transverse momentum, p T , partons have been introduced .Different algorithms have been developed to provide a better description of thehardest emission, including the full next-to-leading-order cross section. In the ap-proach of Frixione and Webber (MC@NLO) [31, 32], the parton shower approxima-tion is subtracted from the real emission contribution to the next-to-leading-ordercross section and combined with the virtual correction. This method was success-fully applied to many different processes [33–40]. However, this approach has twodrawbacks: it generates weights that are not positive definite and its implementationdepends on the parton shower algorithm.These drawbacks have been addressed with a new method introduced by Na-son [41,42], the POsitive Weight Hardest Emission Generator (POWHEG) approach. See Ref. [1] for a recent review of the older techniques [2–19] and techniques for improving thesimulation of multiple hard QCD radiation [20–30]. This method generates positive weights and is implemented in a way that does notdepend on the details of the parton shower algorithm. Nevertheless, the partonshower algorithm must have a well defined structure: a truncated shower simulatingwide angle soft emission; followed by the emission with highest transverse momen-tum ( p T ); followed by a vetoed shower simulating softer radiation. The hardestemission is generated by a Sudakov form factor that includes the full matrix ele-ment for real emission. The truncated shower generates emission at a higher scale (inthe evolution variable of the parton shower), while the vetoed shower simulates ra-diation at a lower evolution scale than the one at which the hardest emission isgenerated. The POWHEG method has been successfully applied to a wide range ofprocesses [43–62] .These approaches have yet to be applied to processes involving the productionof photons due to the complications which arise in the experimental measurement,simulation and calculation at higher orders in perturbation theory of these pro-cesses. Collider experiments do not measure inclusive photons because of the highbackground due to the production of photons in meson decays. Indeed, the in-clusive production rate of high p T π , η , ω mesons is orders of magnitude biggerthan for direct photon production. For this reason the experimental selection ofdirect photons requires the use of an isolation cut. Different criteria for the isolationof photons include: the cone approach [66, 67], the democratic approach [68] andthe smooth isolation procedure [69]. In fixed-order calculations this contributionis included using the measured photon fragmentation function, the probability of aparton fragmenting to produce a photon with a given fraction of the parent parton’smomentum, whereas Monte Carlo simulations instead rely on the parton shower andhadronization models to simulate this contribution. This presents a problem in sim-ulating these processes at NLO where some of the singularities in the real emissionprocesses are absorbed into photon fragmentation function in fixed-order calcula-tions. In this paper we will present a method for simulating these processes usingthe POWHEG approach which still relies on the parton shower and hadronizationmodels to simulate the photon fragmentation contribution. This approach is simi-lar in its philosophy to the method of Ref. [70] for combining leading-order matrixelements and the parton shower.We illustrate this approach with the simulation of diphoton ( γγ ) production.Diphoton production is important as it provides a large background for the discov-ery of the Higgs boson decaying into a pair of photons, for both the Tevatron [71]and LHC experiments [72, 73]. It is also an important background in new physicsmodels, for example in heavy resonance models [74], models with extra spatial di-mensions [75] and cascade decays of heavy new particles [76]. Experimental mea-surements of γγ production have a long history in fixed-target [77–79] and colliderexperiments [80–84].The theoretical understanding of diphoton production and precise measurementsof the differential cross section are therefore not only important for the discovery ofnew phenomena but also as a check of the validity of the predictions of perturbative There has also been some work combining either many NLO matrix elements [63] or the NLOmatrix elements with subsequent emissions matched to leading-order matrix elements [64, 65] withthe parton shower. quantum chromodynamics (pQCD) and soft-gluon resummation methods.The dominant production method for direct photon pairs is leading order q ¯ q scattering ( q ¯ q → γγ ), although the formally next-to-next-to-leading-order, O ( α s ),gluon-gluon fusion ( gg → γγ ) process via a quark-loop diagram [85] can be im-portant, and even comparable to the leading-order contribution at low diphotonmass ( M γγ ) [84], due to the large gluon parton distribution function.The O ( α s ) corrections to the q ¯ q → γγ process includes the q ¯ q → γγg , gq → γγq and g ¯ q → γγ ¯ q processes and corresponding virtual corrections. More-over, the contribution where the final parton is collinear to a photon is calculated interms of the quark and gluon fragmentation function into a photon [85, 86]. Giventhe behaviour of the latter functions, ∼ αα s , these terms contribute to the sameorder as q ¯ q → γγ . The QCD corrections to the process are well known in the lit-erature [66, 87–91]. Fixed-order Monte Carlo programs, such as JETPHOX [92] and
DIPHOX [93], provide simulation for direct photon production together with theimplementation of isolation cuts.The present paper is organized as follows. In Sect. 2 we introduce the POWHEGformulae useful for the description of our approach and our treatment of the photonfragmentation contribution. The calculation of the leading-order kinematics withNLO accuracy in the POWHEG approach is discussed in Sect. 3. In Sect. 4 wedescribe the procedure used to generate the hardest emission. We show our resultsin Sect. 5 and finally present our conclusions in Sect. 6.
In the POWHEG approach the NLO differential cross section for a given N-bodyprocess is d σ = ¯ B (Φ B )dΦ B (cid:20) ∆ R (0) + R (Φ B , Φ R ) B (Φ B ) ∆ R ( k T (Φ B , Φ R ))dΦ R (cid:21) , (1)where ¯ B (Φ B ) is defined as¯ B (Φ B ) = B (Φ B ) + V (Φ B ) + Z [ R (Φ B , Φ R ) − C (Φ B , Φ R )] dΦ R , (2) B (Φ B ) is the leading-order contribution, Φ B the N-body phase-space variables of theLO Born process whereas Φ R are the radiative variables describing the phase spacefor the emission of an extra parton. The real contribution, R (Φ B , Φ R ), is the matrixelement including the radiation of an additional parton multiplied by the relevantparton flux factors, and is regulated by subtracting the counter terms C (Φ B , Φ R )which contain the same singularities as R (Φ B , Φ R ). In practice the counter termis usually composed of a sum over a number of terms, D i (Φ B , Φ R ), each of whichregulates one of the singularities in the matrix element using approaches of eitherCatani and Seymour (CS) [94] or Frixione, Kunszt and Signer (FKS) [95], i.e. C (Φ B , Φ R ) = P i D i (Φ B , Φ R ). The finite contribution V (Φ B ) includes the virtualloop corrections and the counter terms integrated over the real emission variables,which cancel the singularities from the loop corrections, and the collinear remnantfrom absorbing the initial-state singularities into the parton distribution functions. The modified Sudakov form factor is defined in terms of the real emission matrixelement ∆ R ( p T ) = exp (cid:20) − Z dΦ R R (Φ B , Φ R ) B (Φ B ) θ ( k T (Φ B , Φ R ) − p T ) (cid:21) , (3)where k T (Φ B , Φ R ) is equal to the transverse momentum of the emitted parton inthe soft and collinear limits.The POWHEG method is based on two steps: the N-body configuration is gen-erated according to ¯ B (Φ B ) and then the hardest emission is generated using theSudakov form factor given in Eqn. 3. Since ¯ B (Φ B ) is defined as the NLO differentialcross section integrated over the radiative variables, the event weight will not benegative.If the parton shower algorithm is ordered in transverse momentum we wouldgenerate the hardest emission first and then evolve the N + 1 parton final-statesystem using the shower forbidding any emission with transverse momentum higherthan that of the hardest emission. On the contrary for shower simulations which areordered in other variables, such as angular ordering in Herwig++ [19,96] , the hardestemission is not necessarily the first one. For this reason the shower must be split intoa truncated shower describing soft emission at higher evolution scales, the highest p T emission and vetoed showers simulating emissions at lower evolution scales; however,constraints are imposed to guarantee that the transverse momentum of the emittedparticles is smaller than the one corresponding to the hardest emission [41, 42].In order to use this procedure for processes involving photons where the realemission matrix elements contain both QCD singularities from the emission of softand collinear gluons and QED singularities from the radiation of soft and collinearphotons we need to make some modifications to the approach. We start by writingthe real emission piece as R (Φ B , Φ R ) = R QED (Φ B , Φ R ) + R QCD (Φ B , Φ R ), (4)where R QED (Φ B , Φ R ) = P i D i QED P j D j QED + P j D j QCD R (Φ B , Φ R ) (5a)contains the collinear photon emission singularities and R QCD (Φ B , Φ R ) = P i D i QCD P j D j QED + P j D j QCD R (Φ B , Φ R ) (5b)contains the singularities associated with QCD radiation. Here the counter termshave been split into those D i QCD which regulate the singularities from QCD radiationand those D i QED which regulate the singularities due to photon radiation.We can regard the real QCD emission terms as part of the QCD corrections tothe leading-order process, whereas the QED contributions are part of the photonfragmentation contribution coming from a leading-order process with one less photon In practice the counter terms can be negative in some regions and we choose to use theirmagnitude in this separation in order to ensure that the real contributions are positive. and an extra parton. We therefore modify the next-to-leading-order cross sectionfor processes with photon production givingd σ = ( B (Φ B ) + V (Φ B ) + Z " R QCD (Φ B , Φ R ) − X i D i QCD (Φ B , Φ R ) dΦ R ) dΦ B + R QED (Φ B , Φ R )dΦ R dΦ B . (6)There should also be an additional non-perturbative contribution with the convo-lution of the photon fragmentation function and the leading-order process with oneless photon and an extra parton.We can now write the cross section for photon production processes in thePOWHEG approach in the same way as in Eqn. 1d σ = ¯ B (Φ B )dΦ B (cid:20) ∆ QCD (0) + R QCD (Φ B , Φ R ) B (Φ B ) ∆ QCD ( k T (Φ B , Φ R ))dΦ R (cid:21) (7)+ B ′ (Φ ′ B )dΦ ′ B (cid:20) ∆ QED (0) + R QED (Φ ′ B , Φ ′ R ) B ′ (Φ ′ B ) ∆ QED ( k T (Φ ′ B , Φ ′ R ))dΦ ′ R (cid:21) ,where ¯ B (Φ B ) is now defined as¯ B (Φ B ) = ( B (Φ B ) + V (Φ B ) + Z " R QCD (Φ B , Φ R ) − X i D i QCD (Φ B , Φ R ) dΦ R ) dΦ B (8)and B ′ (Φ ′ B ) is the leading-order contribution for the process with an extra par-ton and one less photon with Φ ′ B and Φ ′ R being the corresponding Born and realemission phase-space variables.The Sudakov form factor for QCD radiation is∆ QCD ( p T ) = exp (cid:20) − Z dΦ R R QCD (Φ B , Φ R ) B (Φ B ) θ ( k T (Φ B , Φ R ) − p T ) (cid:21) , (9a)and the Sudakov form factor for QED radiation is∆ QED ( p T ) = exp (cid:20) − Z dΦ ′ R R QED (Φ ′ B , Φ ′ R ) B ′ (Φ ′ B ) θ ( k T (Φ ′ B , Φ ′ R ) − p T ) (cid:21) . (9b)Both the direct photon production and the non-perturbative fragmentation con-tribution are correctly included. The non-perturbative fragmentation contributionis simulated by the parton shower from the B ′ (Φ ′ B ) contribution when there is nohard QED radiation.The POWHEG algorithm is implemented for photon production processes usingthe following procedure. • First select either a direct photon production or a fragmentation event using¯ B (Φ B ) and B ′ (Φ ′ B ) and the competition method to correctly generate therelative contributions of the two different processes. • For a direct photon production process: ¯ B (Φ B ) – generate the hardest emission using the Sudakov form factor in Eqn. 9a; – directly hadronize non-radiative events; – map the radiative variables parameterizing the emission into the evolutionscale, momentum fraction and azimuthal angle, (˜ q h , z h , φ h ), from whichthe parton shower would reconstruct identical momenta; – generate the N -body configuration from ¯ B (Φ B ) and evolve the radiatingparton from the starting scale down to ˜ q h using the truncated shower; – insert a branching with parameters (˜ q h , z h , φ h ) into the shower when theevolution scale reaches ˜ q h ; – generate p T vetoed showers from all the external legs. • For a fragmentation contribution: – generate the hardest QED emission using the Sudakov form factor inEqn. 9b; – directly shower and hadronize non-radiative events, forbidding any per-turbative QED radiation in the parton shower generating thenon-perturbative fragmentation contribution; – for events with QED radiation map the radiative variables parameterizingthe emission into the evolution scale, momentum fraction and azimuthalangle, (˜ q h , z h , φ h ), from which the parton shower would reconstruct iden-tical momenta; – generate the N -body configuration from B ′ (Φ ′ B ) and evolve the radiatingparton from the starting scale down to ˜ q h using the truncated shower, butallowing QCD radiation with p T greater than that of the hardest QEDemission; – insert a branching with parameters (˜ q h , z h , φ h ) into the shower when theevolution scale reaches ˜ q h ; – generate the shower from all external legs forbidding QED radiation, butnot QCD radiation, above the p T of the hardest emission.This procedure now includes the QCD corrections to the leading-order direct pho-ton production process and both the perturbative QED corrections to the photonfragmentation contribution and the non-perturbative contribution are simulated bythe parton shower.In the next two sections we will describe how we implement this approach in Herwig++ for photon pair production. ¯ B (Φ B ) In this section we describe the O ( α s ) corrections to diphoton production. At leading-order, γγ -production is described by the Feynman diagram illustrated in Fig. 1.Next-to-leading-order contributions yield O ( α s ) corrections coming from q ¯ q → γγg , gq → γγq and g ¯ q → γγ ¯ q , together with the corresponding virtual corrections, as .1 Real emission contribution q ¯ q γ γ +1 ↔ Fig. 1:
Diphoton production at leading-order.shown in Fig. 2. These subprocesses contain QED singularities, corresponding toconfigurations where the final-state parton becomes collinear to a photon, whichdo not cancel when summing up the real and the virtual pieces of the cross sec-tion. As described in the previous section they are formally absorbed into a quark( G γq ( z, µ )) or gluon ( G γg ( z, µ )) fragmentation function into photons, which definethe probability of finding a photon carrying longitudinal momentum fraction z in aquark or gluon jet at scale µ for a given factorization scheme. This QED singularcomponent is called the Bremsstrahlung or single fragmentation contribution. Inour approach it is treated separately and simulated by showering the gq → γq or g ¯ q → γ ¯ q within the Monte Carlo algorithm, see Fig. 3, as described in the previoussection. At next-to-leading-order the same configuration appears in any subprocessin which a quark (gluon) undergoes a cascade of successive collinear splittings endingup with a quark-photon (gluon-photon) splitting. These singularities are factorizedto all orders in α s , according to the factorization theorem. When the fragmenta-tion scale µ is chosen higher than any other hadronic scale, i.e. µ ∼ αα s ( µ ) and therefore they contribute at leading-order.For a full study at NLO accuracy, the O ( α s ) corrections to the Bremsstrahlungcontribution need to be calculated. Moreover, these corrections in their turn yieldthe leading-order contribution of the double fragmentation type process; in the lattercase, both photons result from the collinear fragmentation of a parton. However,these corrections are out of the scope of the present work and are not consideredhere. In order to calculate the real emission contribution to ¯ B (Φ B ) we need to specify boththe radiative phase space, Φ R , and the subtraction counter terms. We choose to usethe dipole subtraction algorithm of Catani and Seymour [94] to specify the counterterms and the associated definition of the real emission phase space as follows.In the centre-of-mass frame the incoming hadronic momenta are, P ⊕ and P ⊖ ,respectively for the hadrons traveling in the positive and negative z -directions. Sim-ilarly the momenta of the incoming partons in the Born process are ¯ p ⊕ = ¯ x ⊕ P ⊕ and ¯ B (Φ B ) +1 ↔ ↔ q q qqqqq q qq q ′ q q ′ q q ′ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ g g gg g g ¯ q ¯ q ¯ q ¯ q ¯ q ¯ q ¯ q ¯ q ¯ q ( a )( b ) Fig. 2:
Diphoton production at next-to-leading-order. In (a) the real and virtualFeynman diagrams contributing to the q ¯ q → γγ subprocess are shown whilein (b) the real diagrams for gq initiated process are given.¯ p ⊖ = ¯ x ⊖ P ⊖ , respectively. The momenta of the photons in the Born process are ¯ k , respectively. The corresponding momenta in the real emission process are p ⊕ and p ⊖ for the incoming partons and k , , for the outgoing particles which are chosensuch that k , are the momenta of the photons and k that of the radiated final-stateparton.In the CS approach the real phase space depends on which parton is the emitter ofthe radiation and which the associated spectator defining the dipole [94]. When theparton with momentum ¯ p ⊕ is the emitter and that with momenta ¯ p ⊖ the spectatorthe full phase space is [94]dΦ = dΦ B dΦ R = dΦ B ( k + k ) π dφ ⊕ π dv ⊕ dxx θ ( v ⊕ ) θ (cid:18) − v ⊕ − x (cid:19) θ ( x (1 − x )) θ ( x − ¯ x ⊕ ),(10)where the radiative phase space variables are x = 1 − ( p ⊕ + p ⊖ ) · k p ⊕ · p ⊖ , v ⊕ = p ⊕ · k p ⊕ · p ⊖ , φ ⊕ , (11) .1 Real emission contribution +1 ↔ q qγ γ γ g ¯ qq γ γ γ ¯ q γ γ +1 ↔ qg Fig. 3:
Bremsstrahlung contribution for diphoton production. φ ⊕ is the azimuthal angle of the emitted particle around the ˆ ⊕ -direction and x ∈ [ x ⊕ , , v ⊕ ∈ [0 , − x ]. (12)In terms of these variables p ⊕ = ¯ p ⊕ /x , p ⊖ = ¯ p ⊖ , (13a) x ⊕ = ¯ x ⊕ /x , x ⊖ = ¯ x ⊖ . (13b)It is useful to specify the momentum of the radiated parton in terms of itstransverse momentum, p T , and rapidity, y , such that k = p T (cosh y ; cos φ ⊕ , sin φ ⊕ , sinh y ) . (14)Using the definition of x and v ⊕ we have k = v ⊕ p ⊖ + (1 − x − v ⊕ ) p ⊕ + q ⊥ , (15)where q ⊥ is the component of the 4-momenta transverse to the beam direction. Theon-shell condition, k = 0, gives − q ⊥ = p T = 2 p ⊕ · p ⊖ (1 − x − v ⊕ ) v ⊕ . (16)From Eqn. 15 and the definition of rapidity y = 12 ln (cid:20) k E + k z k E − k z (cid:21) = 12 ln (cid:20) (1 − x − v ⊕ ) x ⊕ v ⊕ xx ⊖ (cid:21) , (17) ¯ B (Φ B ) the CS variables are v ⊕ = x ⊖ √ s p T e − y ,x = − pTx ⊖√ s e − y pTx ⊕√ s e y . (18)This is sufficient to calculate the momentum of the radiated parton, however,rather than implementing the real emission variables in the Sudakov form factor inthis way and then imposing the θ ( k T (Φ B , Φ R ) − p T ) function it is easier to considerthe real emission in terms of the transverse momentum, rapidity and azimuthal angleof the emitted parton.The Jacobian for this transformation is (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( x, v ) ∂p T ∂y (cid:12)(cid:12)(cid:12)(cid:12) = p T sx ⊕ x ⊖ (cid:16) − p T √ sx ⊖ e − y (cid:17)(cid:16) p T √ sx ⊕ e y (cid:17) = 2 p T x sx ⊕ x ⊖ (1 − v ⊕ ) . (19)The momenta of the photons in the real emission process can then be calculatedfrom the Born momenta using k µr = Λ µν ¯ k νr r = 1 , , (20)where the Lorentz transformation isΛ µν = g µν − K + ¯ K ) µ ( K + ¯ K ) ν ( K + ¯ K ) + 2 K µ ¯ K ν K , (21)with K = p ⊕ + p ⊖ − k = k + k , (22a)¯ K = ¯ p ⊕ + ¯ p ⊖ . (22b)The condition K = ¯ K is compatible with the definition of x given in Eqn. 11. Thekinematic variables for the ˆ ⊖ collinear direction are calculated in a similar way andthey provide a radiative phase space as in Eqn. 10. Moreover, given the x ⊕ ↔ x ⊖ asymmetry of the rapidity in Eqn. 17, it is [ y ] ⊖ = − [ y ] ⊕ . In the rest of the paperwe refer to the collinear direction as ˆO = { ˆ ⊖ , ˆ ⊕} , when both components need tobe included.In addition to the real emission variables we need the dipole subtraction terms ofRef. [94]. In the following B (Φ B ) and B ′ (Φ ′ B ) are computed in terms of the reducedmomenta defined in terms of the momenta for the real emission process in Ref. [94].The QCD singularities from q ¯ q → γγg are absorbed by the dipoles D qg, ¯ q ≡ D qg QCD = 8 πC F α s ( µ R ) 12¯ p ⊕ k (cid:26) − x − (1 + x ) (cid:27) B (Φ B ), (23a) D ¯ qg,q ≡ D ¯ qg QCD = 8 πC F α s ( µ R ) 12¯ p ⊖ k (cid:26) − x − (1 + x ) (cid:27) B (Φ B ), (23b)where the dipoles D ij,k denote the emitter i , emitted parton j and spectator k . .1 Real emission contribution The gq → γγq subprocess involves the QCD dipoles D gq,q ≡ D gq QCD = 8 πT F α s ( µ R ) 12¯ p ⊕ k { − x (1 − x ) } B (Φ B ). (24)In order to separate the QCD and QED emission we also need the QED dipoles D qqγ ≡ D qγF QED = 8 παe q k k ξ (cid:26) − ξ + z − z (cid:27) B ′ (Φ ′ B ), (25a) D qγq ≡ D qγI QED = 8 παe q p ⊖ k ξ (cid:26) − ξ + z − (1 + x ) (cid:27) B ′ (Φ ′ B ), (25b)where ξ = 1 − k k ( k + k ) p ⊕ , (26a) z = p ⊕ k ( k + k ) p ⊕ , (26b)and e q is the charge of the quark q in units of the electron charge. In this case, theradiative phase space is dΦ ′ R ( ξ, z, φ ′ ). Similar dipoles are included for the g ¯ q → γγ ¯ q subprocess. We do not include perturbative QED radiation from the q ¯ q → γg subprocess as it does not give a perturbative correction to G γg ( z, µ ).In practice we generate the real emission piece as a contribution from each ofthe incoming partons as Z " R QCD (Φ B , Φ R ) − X i D i QCD (Φ B , Φ R ) dΦ iR = (27) X i = ⊕ , ⊖ Z " | D i QCD | P j | D j QED | + P j | D j QCD | R (Φ B , Φ iR ) − D i QCD (Φ B , Φ R ) dΦ iR . For the later generation of the Sudakov form factor it is useful to express thedipoles as D I QCD ≡ C I α s ( µ R )2 π D I B (Φ B ), (28)where I = { qg ; ¯ qg ; gq ; g ¯ q } , C qg = C ¯ qg = C F , (29a) C gq = C g ¯ q = T F , (29b)and D J QED ≡ α π e q D J B (Φ ′ B ), (30)where J = { qγF, qγI, ¯ qγF, ¯ qγI } . ¯ B (Φ B ) The finite piece of the virtual correction isd σ V = C F α s ( µ R )2 π V ( w ) B (Φ B ). (31)where the finite contribution of I ( ǫ ) [94] and the virtual correction [89] is V ( w ) = (cid:0) w + ln (1 − w ) + 3ln(1 − w ) (cid:1) + F ( w ) (cid:0) − ww + w − w (cid:1) , (32)where e q is the electric charge of quark q , and F ( w ) = 2ln w + 2ln(1 − w ) + 3(1 − w ) w (ln w − ln(1 − w ))+ (cid:18) w − w (cid:19) ln w + (cid:18) − ww (cid:19) ln (1 − w ), (33)with w = 1 + ˆ t ˆ s , where ˆ s and ˆ t are the usual Mandelstam variables.The collinear remainders ared σ coll = C F α s ( µ R )2 π f m ( x O , µ F ) f ( x O , µ F ) B (Φ B ), (34)where the modified PDF is f mq ( x O , µ F ) = Z x O d xx n f g (cid:16) x O x , µ F (cid:17) A ( x )+ h f q (cid:16) x O x , µ F (cid:17) − xf q ( x O , µ F ) i B ( x )+ f q (cid:16) x O x , µ F (cid:17) C ( x ) o + f q ( x O , µ F ) D ( x O ), (35) f q and f g are the quark and gluon PDFs respectively, and A ( x ) = T F C F (cid:20) x (1 − x ) + ( x + (1 − x ) )ln Q (1 − x ) µ F x (cid:21) , (36) B ( x ) = (cid:20) − x ln Q (1 − x ) µ F (cid:21) , (37) C ( x ) = (cid:20) − x − − x ln x − (1 + x )ln Q (1 − x ) µ F x (cid:21) , (38) D ( x O ) = (cid:20)
32 ln (cid:18) Q µ F (cid:19) + 2ln(1 − x O )ln (cid:18) Q µ F (cid:19) + 2ln (1 − x O ) + π − (cid:21) . (39)The combined contribution of the finite virtual term and collinear remnants isd σ V +coll = C F α s ( µ R )2 π V (Φ B ) B (Φ B ), (40)where V (Φ B ) ≡ V ( w ) + ˜ V ( x O , µ F ), (41)with ˜ V ( x O , µ F ) = f m ( x O ,µ F ) f ( x O ,µ F ) . We write the modified PDF for the quark q , but a similar expression is valid for an incomingantiquark ¯ q . .3 Generation of the hard process The next-to-leading-order simulation of photon pair production in
Herwig++ usesthe standard
Herwig++ machinery to generate photon pair and photon plus jetproduction in competition. The ¯ B function is implemented as a reweighting of theleading-order matrix element as follows:1. the radiative variables Φ R { x, v, φ } and Φ ′ R { ξ, z, φ ′ } are transformed into anew set such that the radiative phase space is a unit volume;2. using the standard Herwig++ leading-order matrix element generator, we gen-erate a leading-order configuration and provide the Born variables Φ B with anassociated weight B (Φ B );3. the radiative variables Φ R are generated and ¯ B (Φ B ) sampled in terms of theunit cube (˜ x, ˜ v, ˜ φ ), using the Auto-Compensating Divide-and-Conquer (ACDC)phase-space generator [97];4. the leading-order configuration is accepted with a probability proportional tothe integrand of Eqn. 8 evaluated at { Φ B , Φ R } . Following the generation of the Born kinematics with next-to-leading-order accuracythe hardest QCD or QED emission must be generated according to Eqns. 9a or 9b,respectively depending on whether a direct or photon fragmentation contributionwas selected.
The hardest QED emission is generated by using the modified Sudakov form factordefined in Eqn. 9b. We generate it in terms of the variables Φ ′ R ( x p , z p , φ ), withdΦ ′ R = 12 π d x p d z p d φ , (42)defined in [9, 13], where x p ∈ [ x o , z p ∈ [0 ,
1] and the azimuthal angle φ ∈ [0 , π ].The invariant mass of the initial-final dipole q = ( p i − p k ) = − Q is preserved bythe photon radiation. It is easiest to generate the hardest emission by introducing x ⊥ such that the transverse momentum of the emission relative to the direction ofthe partons in the Breit frame of the dipole is p T = Q x ⊥ , where x ⊥ = 4(1 − x p )(1 − z p ) z p x p . (43)The Sudakov form factor can then be calculated in terms of ˜Φ ′ R ( x ⊥ , z p , φ ), such thatthe θ -function simply gives x ⊥ as integration limits and Eqn. 9b becomes∆ J QED ( x ⊥ ) = exp − Z x max ⊥ x ⊥ d x ′⊥ x ′ ⊥ d φ d z p α π W A J QED B ! , (44) where α π A J QED = | D J QED | P j | D j QED | + P j | D j QCD | R (Φ B , Φ JR ), (45)the Jacobian, W , is W = 4 z p (1 − z p )(1 − x p ) , (46)and Q x max ⊥ is the maximum value for the transverse momentum.It is impossible to generate the hardest emission directly using Eqn. 44 insteadwe use an overestimate g ( x ⊥ ) = ax ⊥ , (47)of the integrand in Eqn. 44 so that∆ overQED ( x ⊥ ) = exp (cid:18) − Z x max ⊥ x ⊥ d x ′⊥ x ′ ⊥ d φ d z p a (cid:19) (48)can be easily integrated in { x ⊥ , x max ⊥ } . This allows us to solve R = ∆ overQED ( x ⊥ )where R is a random number in [0 ,
1] to get the transverse momentum of a trialhard emission x ⊥ ( R ) = 1 x max ⊥ ) − a ln R . (49)This trial hard emission is then accepted or rejected using a probability given by theratio of the true integrand to the overestimated value. If the emission is rejected theprocedure is repeated with x max ⊥ set to the rejected x ⊥ value until the generated valueis below the cut-off. This procedure, called the veto algorithm , correctly generatesthe hardest emission according to Eqn. 44 [98]. The hardest QCD emission is generated in terms of the variables Φ R ( p T , y, φ ) definedin Sect. 3.1. Eqn. 9a then becomes∆ I QCD ( p T ) = exp − Z p max T p T d p ′⊥ d φ d y C I α s π W I A I QCD B ! , (50)where C I α s π A I QCD = | D I QCD | P j | D j QED | + P j | D j QCD | R (Φ B , Φ IR ), (51)the Jacobian is W I = x − v O , (52)where we mean to use v ⊕ for I = { qg ; gq ; g ¯ q } and v ⊖ for I = { ¯ qg } .As before we use the veto algorithm to generate the hardest QCD emissionaccording to Eqn. 50. In this case we introduce the overestimate function g I ( p T ) = a I p T , (53) so that ∆ overQCD ( p T ) = exp (cid:18) − Z p max T p T d p ′ T p ′ T d φ d ya I (cid:19) (54)is easily integrable in { p T , p max T } and R = ∆ overQCD ( p T ) can be solved giving p T ( R ) = R a . (55)As before this trial hard emission is then accepted or rejected using a probabilitygiven by the ratio of the true integrand to the overestimated value. If the emissionis rejected the procedure is repeated with p max T set to the rejected p T value until thegenerated value is below the cut-off. Unlike the implementations of many other processes in the POWHEG formalism itis impossible to directly compare our results for any quantities directly with next-to-leading-order simulations in order to test the implementation due to the verydifferent treatment of the photon fragmentation contribution. Instead we comparea simple observable, the rapidity of the photons, with the next-to-leading-orderprogram
DIPHOX [93] as a sanity check of our results not expecting exact agreement,although the PDFs and electroweak parameters were chosen to give exact agreementfor the leading order q ¯ q → γγ process.For proton-proton collisions at a centre-of-mass energy of 14 TeV, we used thefollowing set of cuts on p T and rapidity of photons p γT >
25 GeV , | y γ | < . , (56)together with a cut on the invariant mass of the γγ -pair80 GeV < M γγ < E had T ,released in the cone, centred around the photon direction in the rapidity and az-imuthal angle plane, is smaller than 15 GeV, i.e. ( y − y γ ) + ( φ − φ γ ) ≤ R (58) E had T ≤
15 GeV, (59)where R = 0 . DIPHOX at NLO(reddashed line) and LO (red dash-dotted line), together with LO
Herwig++ (dottedblack line) and
Herwig++ with POWHEG corrections (solid black line) do not in-clude the gluon-gluon channel. At LO the
Herwig++ and
DIPHOX distributions areindistinguishable. At NLO they show a difference that is very small compared tothe correction from LO to NLO, which means that the NLO curves are in reasonableagreement given the sizable contribution of the fragmentation contribution that istreated differently in the two approaches. Fig. 4:
Rapidity of the γγ -pair at NLO. The distribution from the Herwig++ partonshower with POWHEG correction (solid black line) is compared with NLOcross section from
DIPHOX (dashed red line). At LO the
Herwig++ distri-bution is given by the dotted black line while the cross section from
DIPHOX by the dash-dotted red line.In Fig. 5a we compare the results from
Herwig++ with the data of Ref. [83],a fixed next-to-leading-order calculation from
DIPHOX (dotted magenta line) and
RESBOS (dashed-dotted green line) [100–104], which performs an analytic resum-mation of the logarithmically enhanced contributions. Here and in the following theLO
Herwig++ parton shower (red dashed line) includes the q ¯ q → γγ , qg → γ jetand gg → γγ contribution. The implementation of POWHEG correction improvesthe description and this results in a distribution (solid blue line) that is in goodagreement with the data. Here, as in the following, the NLO curve includes the gg → γγ subprocess. In the lower frame, we plot the ratio MC/data and the yellowband gives the one sigma variation of data. All the plots comparing the results of Herwig++ with experimental results were made using the
Rivet [105] package.It is of interest to study the transverse momentum of the γγ -pair, because itis not infrared safe for p γγ ⊥ →
0. The q ¯ q → γγ and gg → γγ processes present aloss of balance between the corresponding real emission and virtual contribution,which results in large logarithms at every order in perturbation theory. In addition,the fragmentation components introduce an extra convolution that smears out thissingularity. Since DIPHOX is based on a fixed, finite order calculation it is notsuitable for the study of infrared sensitive observables and it fails in the descriptionof these observables at low p γγ ⊥ , as it is shown in Fig. 5b (dotted magenta line).Resummation for diphoton production in hadron-hadron collision has been providedat all orders in α s in Ref. [106] and implemented in RESBOS , as the correspondingdistribution (dashed-dotted green line) shows in the same figure. The
Herwig++ parton shower resums the effect of enhanced collinear emission to all orders in α s in the leading-logarithmic (LL) approximation and results in a finite behaviour for p γγ ⊥ → p γγ ⊥ and is in good agreement with the CDF data [83].In addition, Herwig++ distributions, with and without POWHEG corrections, b b b b b b CDF data b RESBOSDIPHOXHw++ POWHEGHw++ LO − − − (a) d σ / d M γγ [ p b / G e V ]
10 20 30 40 50 60 70 80 90 100 - σ - σ - σ σ σ σ σ M γγ [GeV] ( M C - d a t a ) b b b b b b b b b − (b) d σ / d p γγ ⊥ [ p b / G e V ] - σ - σ - σ σ σ σ σ p γγ ⊥ [GeV] ( M C - d a t a ) Fig. 5:
The (a) invariant mass and (b) transverse momentum of the γγ -pair. Thesolid blue line shows the POWHEG approach, while the dashed red curveshows the result of the Herwig++ shower at LO. We show the NLO cross sec-tion provided by
DIPHOX (magenta dotted line) and
RESBOS (green dashed-dotted line). The data are from Ref. [83] and the curves are plotted with
Rivet [105]. In the lower panel, the yellow band describes the one sigmavariation of data. b b b b b b
DØ data b Hw++ POWHEGHw++ LO − − − (a) GeV < M γγ < GeV d σ / d p γγ ⊥ / d M γγ [ p b / G e V ]
10 20 30 40 50 60 70 80 - σ - σ - σ σ σ σ σ p γγ ⊥ [GeV] ( M C - d a t a ) b b b b − − (b) GeV < M γγ < GeV d σ / d p γγ ⊥ / d M γγ [ p b / G e V ]
10 20 30 40 50 60 70 80 90 100 - σ - σ - σ σ σ σ σ p γγ ⊥ [GeV] ( M C - d a t a ) Fig. 6:
Transverse momentum of the diphoton system for (a) 50 GeV < M γγ <
80 GeV and (b) 80 GeV < M γγ <
350 GeV. The distribution for thePOWHEG formalism (solid blue line) is plotted together with the distri-bution for the
Herwig++ parton shower (dashed red line). The data arefrom Ref. [83] and the lower frame is as described in Fig. 5 b b b b b b DØ data b Hw++ POWHEGHw++ LO − − (a) GeV < M γγ < GeV d σ / d ∆ φ γγ / d M γγ [ p b / r a d / G e V ] . . . . . . - σ - σ - σ σ σ σ σ ∆ φ γγ [rad] ( M C - d a t a ) b b b − − (b) GeV < M γγ < GeV d σ / d ∆ φ γγ / d M γγ [ p b / r a d / G e V ] . . . . . . - σ - σ - σ σ σ σ σ ∆ φ γγ [rad] ( M C - d a t a ) Fig. 7:
Azimuthal angle between the photons for (a) 50 GeV < M γγ <
80 GeV and(b) 80 GeV < M γγ <
350 GeV. The solid blue line shows the result for the
Herwig++ shower with POWHEG corrections, while the red dashed line givesthe result from the
Herwig++ parton shower. The data are from Ref. [83]and the lower frame is as described in Fig. 5 b b b b b b
DØ data b Hw++ POWHEGHw++ LO − (a) GeV < M γγ < GeV d σ / d | c o s θ ∗ | / d M γγ [ p b / G e V ] . . . . . . . - σ - σ - σ σ σ σ σ | cos θ ∗ | ( M C - d a t a ) b b b − (b) GeV < M γγ < GeV d σ / d | c o s θ ∗ | / d M γγ [ p b / G e V ] . . . . . . . - σ - σ - σ σ σ σ σ | cos θ ∗ | ( M C - d a t a ) Fig. 8:
Polar scattering angle between the photons for two ranges of M γγ : 50 GeV 80 GeV (a) and 80 GeV < M γγ < 350 GeV (b). The solid blue linedescribes the Herwig++ result with POWHEG corrections, the dashed redline does not include matrix element corrections. The data are from Ref. [83]and the lower frame is as described in Fig. 5. are compared to the data of Ref. [84]. In Fig. 6, we show the transverse momentumof the diphoton pair for two ranges of invariant mass of the γγ -pair, M γγ ; in Fig. 6a50 GeV < M γγ < 80 GeV and in Fig. 6b 80 GeV < M γγ < 350 GeV. For thesame ranges of M γγ we plot the azimuthal angle distribution between the photonsin Fig. 7a and Fig. 7b respectively and the polar angle between the photons in Fig. 8aand Fig. 8b. For all distributions we see that the LO Herwig++ ditributions (reddashed line) do not correctly describe the data. The POWHEG approach improvesthe simulation and provides a good description of D0 data [84]. 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