We briefly describe a new general algorithm for carrying out QCD calculations
to next-to-leading order in perturbation theory. The algorithm can be used for
computing arbitrary jet cross sections in arbitrary processes and can be
straightforwardly implemented in general purpose Monte Carlo programs. We show
numerical results for the specific case of jet cross sections in deep inelastic
scattering.Comment: 5 pages, 2 figures. Presented at the International Workshop on Deep
  Inelastic Scattering and Related Phenomena (DIS 96), Rome, Italy, 15-19 Apr
  1996. This paper can also be obtained from
  http://surya11.cern.ch/users/seymour/pubs/rome.ps.

Catani, Stefano

Seymour, Michael H.

English

arXiv

We briefly describe a new general algorithm for carrying out QCD calculations to next-to-leading order in perturbation theory. The algorithm can be used for computing arbitrary jet cross sections in arbitrary processes and can be straightforwardly implemented in general purpose Monte Carlo programs. We show numerical results for the specific case of jet cross sections in deep inelastic scattering.We briefly describe a new general algorithm for carrying out QCD calculations to next-to-leading order in perturbation theory. The algorithm can be used for computing arbitrary jet cross sections in arbitrary processes and can be straightforwardly implemented in general purpose Monte Carlo programs. We show numerical results for the specific case of jet cross sections in deep inelastic scattering

Catani, S.

Seymour, M.H.

CERN Document Server

arXiv:hep-ph/9609237v1  3 Sep 1996CERN-TH/96-239hep-ph/9609237QCD jet calculations in DIS based on thesubtraction method and dipole formalism∗S. CataniI.N.F.N., Sezione di Firenzeand Dipartimento di Fisica, Universita` di FirenzeLargo E. Fermi 2, I-50125 Florence, ItalyM.H. SeymourTheory Division, CERNCH-1211 Geneva 23, SwitzerlandAbstractWe briefly describe a new general algorithm for carrying out QCDcalculations to next-to-leading order in perturbation theory. Thealgorithm can be used for computing arbitrary jet cross sectionsin arbitrary processes and can be straightforwardly implementedin general purpose Monte Carlo programs. We show numericalresults for the specific case of jet cross sections in deep inelasticscattering.CERN-TH/96-239September 1996∗Invited talk presented by S. Catani at the International Workshop on Deep InelasticScattering and Related Phenomena (DIS 96), Rome, Italy, 15-19 Apr 1996. To appear inthe Proceedings.QCD JET CALCULATIONS IN DIS BASED ON THESUBTRACTION METHOD AND DIPOLE FORMALISMS. CATANII.N.F.N., Sezione di Firenze, and Dipartimento di Fisica, Universita` di Firenze,Largo E. Fermi 2, I-50125 Florence, ItalyM.H. SEYMOURTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandWe briefly describe a new general algorithm for carrying out QCD calculations tonext-to-leading order in perturbation theory. The algorithm can be used for com-puting arbitrary jet cross sections in arbitrary processes and can be straightfor-wardly implemented in general purpose Monte Carlo programs. We show numericalresults for the specific case of jet cross sections in deep inelastic scattering.1 IntroductionIn order to make quantitative predictions in perturbative QCD, it is essen-tial to work to (at least) next-to-leading order (NLO). However, this is farfrom straightforward because for all but the simplest quantities, the necessaryphase-space integrals are too difficult to do analytically, making numericalmethods essential. But the individual integrals are divergent, and only afterthey have been regularized and combined is the result finite. The usual pre-scription, dimensional regularization, involves working in a fractional numberof dimensions, making analytical methods essential.To avoid this dilemma, one must somehow set up the calculation such thatthe singular parts can be treated analytically, while the full complexity of theintegrals can be treated numerically. Efficient techniques have been set up todo this, at least to NLO, during the last few years.A new general algorithm was recently presented,1 which can be used tocompute arbitrary jet cross sections in arbitrary processes. It is based on twokey ingredients: the subtraction method for cancelling the divergences betweendifferent contributions; and the dipole factorization theorems (which generalizethe usual soft and collinear factorization theorems) for the universal (process-independent) analytical treatment of individual divergent terms. These aresufficient to write a general-purpose Monte Carlo program in which any jetquantity can be calculated simply by making the appropriate histogram in auser routine.In this contribution we give a brief summary of these two ingredients (moredetails and references to other general methods can be found in Refs.1−3) and1show numerical results for the specific case of jets in deep-inelastic lepton-hadron scattering (DIS).2 The Subtraction MethodThe general structure of a QCD cross section in NLO is σ = σLO+σNLO, wherethe leading-order (LO) cross section σLO is obtained by integrating the fullyexclusive Born cross section dσB over the phase space for the corresponding jetquantity. We suppose that this LO calculation involves m partons, and write:σLO =∫mdσB . (1)At NLO, we receive contributions from real and virtual processes (we assumethat the ultraviolet divergences of the virtual term are already renormalized):σNLO =∫m+1dσR +∫mdσV . (2)As is well known, each of these is separately divergent, although their sum isfinite. These divergences are regulated by working in d = 4 − 2ǫ dimensions,where they are replaced by singularities in 1/ǫ. Their cancellation only becomesmanifest once the separate phase space integrals have been performed.The essence of the subtraction method is to use the exact identityσNLO =∫m+1[dσR − dσA]+∫mdσV +∫m+1dσA , (3)which is obtained by subtracting and adding back the ‘approximate’ (or ‘fake’)cross section contribution dσA, which has to fulfil two main properties. Firstly,it must exactly match the singular behaviour (in d dimensions) of dσR itself.Thus it acts as a local counterterm for dσR and one can safely perform thelimit ǫ → 0 under the integral sign in the first term on the right-hand side ofEq. (3). Secondly, dσA must be analytically integrable (in d dimensions) overthe one-parton subspace leading to the divergences. Thus we can rewrite theintegral in the last term of Eq. (3), to obtainσNLO =∫m+1[(dσR)ǫ=0−(dσA)ǫ=0]+∫m[dσV +∫1dσA]ǫ=0. (4)Performing the analytic integration∫1dσA, one obtains ǫ-pole contributionsthat can be combined with those in dσV , thus cancelling all the divergences.Equation (4) can be easily implemented in a ‘partonic Monte Carlo’ programthat generates appropriately weighted partonic events with m + 1 final-statepartons and events with m partons.23 The Dipole FormalismThe fake cross section dσA can be constructed in a fully process-independentway, by using the factorizing properties of gauge theories. Specifically, in thesoft and collinear limits, which give rise to the divergences, the factorizationtheorems can be used to write the cross section as the contraction of the Borncross section with universal soft and collinear factors (provided that colourand spin correlations are retained). However, these theorems are only validin the exactly singular limits, and great care should be used in extrapolatingthem away from these limits. In particular, a careful treatment of momen-tum conservation is required. Care has also to be taken in order to avoiddouble counting the soft and collinear divergences in their overlapping region(e.g. when a gluon is both soft and collinear to another parton). The use ofthe dipole factorization theorem introduced in Ref. 2 allows one to overcomethese difficulties in a straightforward way.The dipole factorization formulae relate the singular behaviour ofMm+1,the tree-level matrix element with m + 1 partons, to Mm. They have thefollowing symbolic structure:|Mm+1(p1, ..., pm+1)|2 = |Mm(p˜1, ..., p˜m)|2 ⊗ V ij + . . . . (5)The dots on the right-hand side stand for contributions that are not singularwhen pi · pj → 0. The dipole splitting functions V ij are universal (process-independent) singular factors that depend on the momenta and quantum num-bers of the m partons in the tree-level matrix element |Mm|2. Colour andhelicity correlations are denoted by the symbol ⊗. The set p˜1, ..., p˜m of mod-ified momenta on the right-hand side of Eq. (5) is defined starting from theoriginal m + 1 parton momenta in such a way that the m partons in |Mm|2are physical, that is, they are on-shell and energy-momentum conservation isimplemented exactly. The detailed expressions for these parton momenta andfor the dipole splitting functions are given in Ref. 1.Equation (5) provides a single formula that approximates the real matrixelement |Mm+1|2 for an arbitrary process, in all of its singular limits. Theselimits are approached smoothly, avoiding double counting of overlapping softand collinear singularities. Furthermore, the precise definition of the m modi-fied momenta allows an exact factorization of the m+1-parton phase space, sothat the universal dipole splitting function can be integrated once and for all.This factorization, which is valid for the total phase space, is not sufficientto provide a universal fake cross section however, as its phase space shoulddepend on the particular jet observable being considered. The fact that the mparton momenta are physical provides a simple way to implement this depen-3dence. We construct dσA by adding the dipole contributions on the right-handside of Eq. (5) and for each contribution we calculate the jet observable notfrom the originalm+1 parton momenta, but from the correspondingm partonmomenta, p˜1, ..., p˜m. Since these are fixed during the analytical integration, itcan be performed without any knowledge of the jet observable.4 Final ResultsRefering to Eq. (4), the final procedure is then straightforward. The cal-culation of any jet quantity to NLO consists of anm+1-parton integral and anm-parton integral. These can be performed separately using standard MonteCarlo methods.For the m + 1-parton integral, a phase-space point is generated and thecorresponding real matrix element in dσR is calculated. These are passedto a user routine, which can analyse the event in any way and histogram anyquantities of interest. Next, for each dipole term (there are about m(m2−1)/2of them) in dσA, the set of m parton momenta is derived from the same phase-space point and the corresponding dipole contribution is calculated. These arealso given to the user routine. They are such that for any singular m + 1-parton configuration, one or more of the m-parton configurations becomesindistinguishable from it, so that they fall in the same bin of any histogram.Simultaneously, the real matrix element and dipole term will have equal andopposite weights, so that the total contribution to that histogram bin is finite.Thus the first integral of Eq. (4) is finite.The m-parton integral in Eq. (4) has a simpler structure: it is identical tothe LO integration in Eq. (1), but with the Born term replaced by the finitesum of the virtual matrix element in dσV and the analytical integral of thedipole contributions in dσA.In addition to the above considerations, there are slight extra complica-tions for processes involving incoming partons, like DIS, or identified outgoingpartons, like fragmentation-function calculations. However, these can be over-come in an analogous way, as discussed in Ref. 1.For the specific case of jets in DIS, we have implemented the algorithmas a Monte Carlo program, which can be obtained from the world wide web,at http://surya11.cern.ch/users/seymour/nlo/. In Fig. 1a we show as anexample the differential jet rate as a function of jet resolution parameter, fcut,using the k⊥ jet algorithm4. We see that the NLO corrections are generallysmall and positive, except at very small fcut. In Fig. 1b, we show the variationof the jet rate at a fixed fcut with factorization and renormalization scales.The scale dependence is considerably smaller at NLO.A Monte Carlo program based on a different method is presented in Ref.5.4Figure 1: Jet cross sections in ep collisions at HERA energies (√s = 300 GeV). (a) Thedistribution of resolution parameter fcut at which DIS events are resolved into (2 + 1) jetsaccording to the k⊥ jet algorithm. Curves are LO (dashed) and NLO (solid) using factoriza-tion and renormalization scales equal to Q2, and the MRS D−′ distribution functions. Bothcurves are normalized to the LO cross section. (b) The rate of events with exactly (2+1) jetsat fcut = 0.25 with variation of renormalization (solid) and factorization (dashed) scales.Normalization is again the LO cross section with fixed factorization scale.5 ConclusionThe subtraction method provides an exact way to calculate arbitrary quan-tities in a given process using a general purpose Monte Carlo program. Thedipole formalism provides a way to construct such a program from process-independent components. Recent applications have included jets in DIS. Moredetails of the program, and its results, will be given elsewhere.Acknowledgments. This research is supported in part by EEC ProgrammeHuman Capital and Mobility, Network Physics at High Energy Colliders, con-tract CHRX-CT93-0357 (DG 12 COMA).References1. S. Catani and M.H. Seymour, preprint CERN-TH/96-29 (hep-ph/9605323).2. S. Catani and M.H. Seymour, Phys. Lett. 378B (1996) 287.3. S. Catani and M.H. Seymour, preprint CERN-TH/96-181 (hep-ph/9607318).4. S. Catani, Yu.L. Dokshitzer and B.R. Webber, Phys. Lett. 285B (1992)291, Phys. Lett. 322B (1994) 263; S. Catani, Yu.L. Dokshitzer, M.H.Seymour and B.R. Webber, Nucl. Phys. B406 (1993) 187.5. E. Mirkes and D. Zeppenfeld, Phys. Lett. 380B (1996) 205, and theseproceedings (hep-ph/9608201).5

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