hep-ph/0401061 LUTP04–05 January 2004 ∗ NEW SHOWERS WITH TRANSVERSE-MOMENTUM-ORDERING T.Sjo¨strand 4 0 DepartmentofTheoretical Physics,LundUniversity,So¨lvegatan14A,S-22362Lund,Sweden 0 2 1. INTRODUCTION n a The initial- [1, 2, 3] and final-state [4, 5] showers in the PYTHIA event generator [6, 7] are based on J virtuality-ordering, i.e.usesspacelikeQ2andtimelikeM2,respectively,asevolutionvariables. Otheral- 0 1 gorithmsincommonusearetheangular-orderedonesinHERWIG[8,9]andthep⊥-ordereddipole-based onesinARIADNE/LDC[10,11]. Allthreehavebeencomparablysuccessful,intermsofabilitytopredict 1 or describe data, and therefore have offered useful cross-checks. Some shortcomings of the virtuality- v 1 ordering approach, with respect to coherence conditions, have been compensated (especially relative to 6 HERWIG) by a better coverage of phase space and more efficient possibilities to merge smoothly with 0 first-ordermatrixelements. 1 0 Recently, the possibility to combine matrix elements of several orders consistently with showers 4 has been raised [12, 13], e.g. W+n jets, n = 0,1,2,3,.... In such cases, a p⊥-ordering presumably 0 / offers the best chance to provide a sensible definition ofhardness. Itmay also tie in better e.g. withthe h p p⊥-ordered approach to multiple interactions [14]. This note therefore is a study of how the existing - PYTHIA algorithms canbereformulated inp⊥-ordered terms,whileretaining theirstrongpoints. p e The main trick that will be employed is to pick formal definitions of p⊥, that simply and unam- h biguously canbetranslated intotheolder virtuality variables, e.g.forstandard matrix-element merging. : v These definitions are based on lightcone kinematics, wherein a timelike branching into two massless i X daughterscorrespondstop2 = z(1−z)M2 andthebranchingofamasslessmotherintoaspacelikeand ⊥ r amasslessdaughter top2⊥ = (1−z)Q2. Theactualp⊥ ofabranching willbedifferent, ande.g.depend a onthesubsequent showerhistory, butshould normallynotdeviatebymuch. 2. TIMELIKESHOWERS The new timelike algorithm is a hybrid between the traditional parton-shower and dipole-emission ap- proaches, in the sense that the branching process is associated with the evolution of a single parton, like in a shower, but recoil effects occur inside dipoles. That is, a dipole partner is assigned for each branching,andenergyandmomentumis‘borrowed’fromthispartnertogivemasstothepartonaboutto branch,whilepreservingtheinvariantmassofthedipole. (Thusfour-momentumisnotpreservedlocally foreachpartonbranching a → bc. Itwasintheoldalgorithm, wherethekinematics ofabranching was notconstructed before the off-oron-shell daughter masseshad been found.) Oftenthe twopartners are colour-connected, i.e.the colour ofone matches theanticolour of theother, asdefined by thepreceding showering history, but this need not be the case. In particular, intermediate resonances normally have masses that should be preserved by the shower, e.g., in t → bW+ the W+ takes the recoil when the b radiatesagluon. The evolution variable is approximately the p2⊥ of a branching, where p⊥ is the transverse mo- mentum for each of the two daughters with respect to the direction of the mother, in the rest frame of ∗submittedtotheproceedingsoftheWorkshoponPhysicsatTeVColliders,LesHouches,France,26May–6June2003 thedipole. (Therecoiling dipole partner does notobtain anyp⊥ kickinthisframe;onlyitslongitudinal momentum is affected.) For the simple case of massless radiating partons and small virtualities rela- tiveto the kinematically possible ones, and in thelimit that recoil effects from further emissions can be neglected, itagreeswiththedij p⊥-clustering distancedefinedinthePYCLUSalgorithm [15]. All emissions are ordered in a single sequence p⊥max > p⊥1 > p⊥2 > ... > p⊥min. That is, each initial parton is evolved from the input p⊥max scale downwards, and a hypothetical branching p⊥ is thereby found for it. The one with the largest p⊥ is chosen to undergo the first actual branching. Thereafter, all partons now existing are evolved downwards from p⊥1, and a p⊥2 is chosen, and so on, until p⊥min is reached. (Technically, the p⊥ values for partons not directly or indirectly affected by a branching need not be reselected.) The evolution of a gluon is split in evolution on two separate sides, withhalfthebranchingkerneleach,butwithdifferentkinematicalconstraintssincethetwodipoleshave differentmasses. Theevolution ofaquarkisalsosplit,intoonep⊥ scaleforgluonemissionandonefor photonone,ingeneral corresponding todifferentdipoles. Withthechoicesabove,theevolutionfactorizes. Thatis,asetofsuccessivecalls,wherethep⊥min of one call becomes the p⊥max of the next, gives the same result (on the average) as one single call for the full p⊥ range. This is the key element to allow Sudakovs to be conveniently obtained from trial showers [13], and to veto emissions above some p⊥ scale, as required to combine different n-parton configurations efficiently. Theformalp⊥ definitionisp2⊥evol = z(1−z)(M2−m20),wherep⊥evol istheevolution variable, z gives the energy sharing in the branching, as selected from the branching kernels, M is the off-shell massofthebranching partonandm0 itson-shellvalue. Thisp⊥evol isalsousedasαs scale. Whenap⊥evol hasbeenselected, thisistranslated toaM2 = m20+p2⊥evol/(z(1−z)). Notethat theJacobian factoristrivial: dM2/(M2−m2)dz = dp2 /p2 dz. Fromthereon,thethree-body 0 ⊥evol ⊥evol kinematicsofabranchingisconstructedasintheoldroutine. Thisincludesthedetailedinterpretation of z and the related handling of nonzero on-shell masses for branching and recoiling partons, which leads to the physical p⊥ not agreeing with the p⊥evol defined here. In this sense, p⊥evol becomes a formal variable, whileM reallyisawell-definedmassofaparton. Also the corrections to b → bg branchings (b being a generic coloured particle) by merging with first-ordera→ bcgmatrixelementscloselyfollowstheexistingmachinery[5],oncethep⊥evol hasbeen converted to a mass of the branching parton. In general, the other parton c used to define the matrix element need notbe thesame asthe recoiling partner. Toillustrate, consider aZ0 → qq decay. Saythe q branches first, q → qg . Obviously the q then takes the recoil, and the new q, g and q momenta are 1 1 used tomatch tothe Z0 → qqg matrix element. Thenexttime qbranches, q → qg ,the recoil is taken 2 by the colour-connected g gluon, but the matrix element corrections are based on the newly created q 1 and g momenta together with the q (not the g !) momentum. That wayone may expect toachieve the 2 1 mostrealisticdescription ofmasseffectsinthecollinear andsoftregions. Theshowerinheritssomefurtherelementsfromtheoldalgorithm,suchasazimuthalanisotropies ingluonbranchings frompolarization effects. The relevant parameters will have to be retuned, since the shower is quite different from the old mass-ordered one. Inparticular, itappears thatthefive-flavourΛ valuehastobereduced relativeto QCD thecurrentdefault, roughly byafactoroftwo(from0.29to0.14GeV). 3. SPACELIKESHOWERS Initial-state showersareconstructed bybackwardsevolution [1],startingatthehardinteraction andsuc- cessivelyreconstructing precedingbranchings. Tosimplifythemergingwithfirst-ordermatrixelements, z is defined by the ratio of sˆbefore and after an emission. For a massless parton branching into one spacelike with virtuality Q2 and one with mass m, this gives p2 = Q2 −z(sˆ+Q2)(Q2 +m2)/sˆ, or ⊥ p2 = (1 − z)Q2 − zQ4/sˆ for m = 0. Here sˆ is the squared invariant mass after the emission, i.e. ⊥ excluding theemittedon-mass-shell parton. Thelastterm,zQ4/sˆ,whilenormallyexpectedtobesmall,givesanontrivialrelationshipbetween p2 andQ2,e.g.withtwopossibleQ2solutionsforagivenp2. Toavoidtheresultingtechnicalproblems, ⊥ ⊥ theevolutionvariableispickedtobep2⊥evol = (1−z)Q2. Alsoherep⊥evol setsthescalefortherunning α . Onceselected,thep2 istranslatedintoanactualQ2 bytheinverserelationQ2 =p2 /(1−z), s ⊥evol ⊥evol withtrivial Jacobian: dQ2/Q2 dz = dp2 /p2 dz. From Q2 the correct p2, including the zQ4/sˆ ⊥evol ⊥evol ⊥ term,canbeconstructed. Emissions on the two incoming sides are interspersed to form a single falling p⊥ sequence, p⊥max > p⊥1 > p⊥2 > ... > p⊥min. That is, the p⊥ of the latest branching considered sets the starting scale of the downwards evolution on both sides, with the next branching occurring at the side thatgivesthelargestsuchevolvedp⊥. In a branching a → bc, the newly reconstructed mother a is assumed to have vanishing mass — a heavy quark would have to be virtual to exist inside a proton, so it makes no sense to put it on mass shell. The previous mother b, which used to be massless, now acquires the spacelike virtuality Q2 and thecorrectp⊥ previously mentioned, andkinematics hastobeadjusted accordingly. In the old algorithm, the b kinematics was not constructed until its spacelike virtuality had been set, and so four-momentum was explicitly conserved at each shower branching. In the new algorithm, thisisnolongerthecase. (Acorresponding changeoccursbetweentheoldandnewtimelikeshowers,as noted above.) Instead itis the set of partons produced by this mother b and the current mother don the other side of the event that collectively acquire the p⊥ of the new a → bc branching. Explicitly, when thebispushed off-shell, thedfour-momentum ismodifiedaccordingly, suchthattheirinvariant massis retained. Thereafter asetof rotations and boosts ofthe wholeb+d-produced system bring them tothe framewherebhasthedesiredp⊥ anddisrestored toitscorrectfour-momentum. Matrix-element corrections can be applied to the first, i.e. hardest in p⊥, branching on both sides ofthe event, toimprove the accuracy ofthehigh-p⊥ description. Alsoseveral other aspects are directly inherited fromtheoldalgorithm. Workonthealgorithm isongoing. Inparticular, anoptimaldescription ofkinematics formassive quarksintheshower,i.e.candbquarks,remainstobeworkedout. Some first tests of the algorithm are reported elsewhere [16]. In general, its behaviour appears rathersimilartothatoftheoldalgorithm. 4. OUTLOOK The algorithms introduced above are still in a development stage. In particular, it remains to combine the two. One possibility would be to construct the spacelike shower first, thereby providing a list of emitted partons with their respective emission p⊥ scales. This list would then be used as input for the timelike shower,whereeachemission p⊥ setstheupper evolution scale oftherespective parton. Thisis straightforward, butdoesnotallowafullyfactorizedevolution, i.e.itisnotfeasibletostoptheevolution atsome p⊥ value and continue downwards from there inasubsequent call. Thealternative would be to intersperse spacelike andtimelikebranchings, inonecommonp⊥-ordered sequence. Obviously the finished algorithms have to be compared with data, to understand how well they do. Oneshouldnotexpectanymajorupheavals, sincechecksshowthattheyperformsimilarlytotheold ones atcurrent energies, butthehope isforasomewhatimproved andmoreconsistent description. The stepthereafterwouldbetostudyspecificprocesses,suchasW+njets,tofindhowgoodamatchingcan beobtainedbetweenthedifferentn-jetmultiplicities, wheninitialpartonconfigurationsareclassifiedby their p⊥-clustering properties. The PYCLUS algorithm here needs to be extended to cluster also beam jets. 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