Improved Parton Showers at Large Transverse Momenta
aa r X i v : . [ h e p - ph ] M a r LU TP 10-07MCnet/10/04March 2010
Improved Parton Showersat Large Transverse Momenta ∗ R. Corke and T. Sj¨ostrand Theoretical High Energy Physics,Department of Astronomy and Theoretical Physics,Lund University,S¨olvegatan 14A,S-223 62 Lund, Sweden
Abstract
Several methods to improve the parton-shower description of hard processesby an injection of matrix-element-based information have been presented overthe years. In this article we study (re)weighting schemes for the first/hardestemission. One objective is to provide a consistent matching of the POWHEGnext-to-leading order generator to the
Pythia shower algorithms. Another isto correct the default behaviour of these showers at large transverse momenta,based on a comparison with real-emission matrix elements. ∗ Work supported by the Marie Curie Early Stage Training program “HEP-EST” (contract num-ber MEST-CT-2005-019626) and in part by the Marie Curie RTN “MCnet” (contract numberMRTN-CT-2006-035606) [email protected] [email protected] Introduction
With the start of the LHC in mind, there has been a recent focus on improving the descrip-tion of event topologies and cross sections, going beyond the Born level for many processesof interest. Firstly, it involves the effects of events with one or more extra jets in the finalstate, which may affect the impact of background processes and thereby the choice of anal-ysis strategies. Secondly, it includes next-to-leading order (NLO) corrections to productioncross sections, which are needed for precision tests of the Standard Model and, hopefully,of physics beyond it.For the first point, real-emission matrix element (ME) calculations give a good descrip-tion of hard and widely separated jets, while parton shower (PS) models give the correctbehaviour in the soft and collinear regions of phase space. The goal is to find a way tocombine these two methods, so that each is used in its region of validity, with a smoothtransition between the two in all physical distributions. This is not a trivial task. Onekey issue is that ME calculations describe inclusive events, while the PS generates exclu-sive ones. The CKKW [1] method solves this by using ME’s supplemented by analyticalSudakov form factors to go from an inclusive to an exclusive language in the hard region,and then switching to a PS below some ME cutoff scale. In CKKW-L [2] the Sudakovsare instead generated from fictitious showers using the same PS algorithm as in the softregion, to improve the consistency and continuity. These methods continue to evolve [3, 4]while other approaches include MLM [5] and Pseudo-showers [6]. Comparisons between themethods have been made [7, 8].For the second point, the virtual correction terms required at NLO make the calculationsmore complicated. Analytically the cancellation of real and virtual ME divergences occur inthe soft/collinear region, i.e. where we would rather want to use the PS description. The firstapproach to solve this issue for nontrivial cases was MC@NLO [9, 10]. Here the analyticalexpression is derived for the phase space population by the first shower emission, in theabsence of a Sudakov form-factor correction. The difference between the real-emission MEand this analytical PS expression, which should be finite in the soft and collinear limits,defines the differential cross section for events with one real “ME-based” emission, fromwhich the shower should start. The rest of the cross section, wherein the analytical PSdivergences and the virtual divergences cancel, gives the events where the shower shouldstart from the lowest-order process.The MC@NLO approach has the disadvantage that the PS emission rate may well behigher than the ME one in some parts of phase space, in which case one is forced to introducenegative-weight events. Furthermore, the analytical shower expressions are specific to aparticular PS algorithm, so lengthy work has to be redone not only between differentgenerators but also for minor changes inside a given generator [11]. Both of these issues aresolved with the POWHEG method [12, 13], where the ME’s themselves are exponentiatedto provide a process-dependent Sudakov form factor. Thereby, a positive-weight algorithmcan be obtained wherein POWHEG (almost) always generates one emission, chosen to bethe one with largest transverse momentum. It is then up to the subsequent shower torespect this constraint, but otherwise without the need for a tight connection between theME and PS stages.When POWHEG is used with showers that are not p ⊥ -ordered, then further thoughtmust be given to the interface. Specifically, in HERWIG, with its angular ordered showers114, 15], the first emission is at largest angle but not necessarily at largest p ⊥ . This leads tothe idea of a “truncated shower” [12], where the shower is modified such that the hardestemission is generated with a modified Sudakov form factor. Subsequent emissions are thengenerated with the usual algorithm, but with a p ⊥ veto so that they cannot be harder thanthe hardest emission.The POWHEG approach is especially convenient, then, if the shower itself is p ⊥ -ordered,as is now the case in the Pythia generator [16–18]. Nevertheless there are some subtletiesthat should be taken into account to optimise the interface. We discuss these issues inSection 2, and in particular interface to POWHEG-hvq [19], an event generator for heavyquark production in hadronic collisions at NLO QCD, where some simple comparisons arepresented for top and bottom production. We also introduce a “poor man’s POWHEG” forcases where NLO calculations are available, but only in the traditional phase space slicingapproach, where shower Sudakovs can be used to provide a smoother matching.One should note that the more sophisticated the description aimed for, the higher theprice in terms of manpower that goes into the detailed simulation of a specific process.The point of injecting ME information is precisely to move away from the universal showerbehavior, which means that each new process must be considered “from scratch”. Toprovide a sensible first estimate, however, it is useful if the shower can be improved toget at least the qualitative behaviour right for a wide range of processes. In Section 3 weuse the MadGraph/MadEvent [20] generator to look at further pair-production processeswith a jet in the final state. By adding a Sudakov to the cross-section for real emissionof a jet, to approximate the prescription used by the POWHEG method, we are able tomake comparisons to the
Pythia shower, and find modest changes that improve agreementsignificantly.Specifically, we will address the issue of “power showers” vs. “wimpy showers”. In [21],the authors compare two extreme choices for the maximum emission scale of the partonshower; either the full CM energy of the incoming hadrons (power) or the transverse massof the particles produced in the hard collision (wimpy), for both virtuality- and p ⊥ -orderedshowers. Their conclusion was that these options bracketed the matrix-element behaviourfor the top and SUSY production processes studied, but also that the spread of “predictions”is large.One must note that an ultimate goal would be to have a matching scheme that allowsfor both a matching to multiple real emissions and to virtual corrections. Some algorithmshave recently been proposed in this direction [22–24], but are not yet at a stage to be usedfor serious LHC studies. We will not discuss such issues further here.The outline of the article is to study the POWHEG approach and its relation to Pythia in Section 2, to use MadEvent to gain an improved understanding of sensible default showerbehaviour in Section 3, and to provide a summary and outlook in Section 4.2
POWHEG
Pythia merging strategy
The
Pythia parton shower orders final-state radiation (FSR) emissions in terms of an evo-lution variable Q , such as m (previous Pythia versions) or p ⊥ ( Pythia nowadays), withan additional energy-sharing variable z in the branching. For QCD emissions, introducing t = ln (cid:16) Q / Λ QCD (cid:17) , the DGLAP evolution equations lead to the probability for a splitting a → bc d P = X b,c α s π P a → bc ( z ) d t d z , (1)where P a → bc are the DGLAP splitting kernels. This inclusive quantity can be turned intoan exclusive one by requiring that, for the first (“hardest”) emission, no emissions can haveoccurred at a larger Q . The probability that a branching occurs at t is now given byd P d t = X b,c I a → bc ( t ) exp − Z t max t d t ′ X b,c I a → bc ( t ′ ) , (2)with I a → bc ( t ) = Z z max ( t ) z min ( t ) d z α s ( t )2 π P a → bc ( z ) . The introduction of this Sudakov form factor turns the unnormalised distribution into anormalised one, i.e. with unit integral over the full phase space. In practice, a lower cutoff, t , is introduced to keep the shower away from soft/collinear regions, which leads to afraction of events with no emissions inside the allowed region.For initial-state radiation (ISR), the evolution is performed using backwards evolution[25], where a given parton b entering a hard scattering is unresolved into a parton a whichpreceded it. Here, the parton distribution functions, reflecting the contents of the incominghadron, must be taken into account. Such a change leads to a Sudakov with the form S b ( x, t max , t ) = exp − Z t max t d t ′ X a,c Z d z α s ( p ⊥ )2 π P a → bc ( z ) x ′ f a ( x ′ , t ′ ) xf b ( x, t ′ ) ! . (3)One feature of the above equations is the running renormalisation and factorisationshower scales, i.e. the scales at which α s and the PDF’s are evaluated. For both ISR andFSR, α s is evaluated at the p ⊥ scale of the emission (the definition of p ⊥ in this contextis discussed later in Section 2.3). For ISR, the flavour dependent ratio of PDF’s given ineq. (3) is evaluated at the selected t value which, for the current versions of Pythia , is p ⊥ .Thus, the renormalisation and factorisation scales are the same. Probably the first use of an explicit matching of PS to ME, the so-called merging strategy,was introduced to handle the case of three-jet events in e + e − annihilation [26]. An outlineis given below, starting from the Born cross section σ B for the lowest-order process e + e − → γ ∗ /Z → qq. 3n the ME side, the real-emission cross section e + e − → γ ∗ /Z → qqg is given, formassless quarks, by the well-known expression1 σ B d σ R d x d x = W ME = α s π x + x (1 − x )(1 − x ) , (4)with α s typically evaluated at a scale of s , the invariant mass of the system.The DGLAP inclusive q → qg emission probability in ( Q , z ) space can be mapped ontothe ( x , x ) space W PSq = α s ( p ⊥ )2 π
43 1 Q z − z d( Q , z )d( x , x ) . (5)The sum of emissions off the q and q gives W PS = W PSq + W PSq . With the addition of aSudakov form factor, as above, this becomes W PScorrected ( Q ) = W PS ( Q ) exp − Z Q Q d Q ′ W PS ( Q ′ ) ! . (6)For ease of notation we have here omitted the dependence of W PS on z and the need of anintegration d z ′ over a range z ′ min ( Q ′ ) < z ′ < z ′ max ( Q ′ ) in the exponent.It now so happens that the Pythia shower algorithm covers the full three-jet phasespace and that W PS > W ME everywhere (so long as the former is true, the latter canalways be achieved by a suitable rescaling). One can therefore use the veto algorithm [17]to correct down the emission rate. That is, whenever a trial Q has been selected accordingto eq. (6), the ratio W ME /W PS in the chosen phase space point is the probability that thischoice should be retained. If not, the evolution is continued downwards from the rejected Q scale ( not from Q ). This gives a change from eq. (6) to W PS+MEcorrected ( Q ) = W ME ( Q ) exp − Z Q Q d Q ′ W ME ( Q ′ ) ! . (7)Note that, while all explicit dependence on W PS is gone in eq. (7), an implicit dependenceon the shower remains in two respects. Firstly, the Sudakov-factor modification of thebasic ME shape reflects the order in which the shower algorithm sweeps over phase space,i.e. the shower Q definition. Secondly, if the α s factors are omitted from the W ME /W PS reweighting, the p ⊥ -dependent expression used in the shower is retained, instead of the fixedvalue normal for ME’s.Once the first emission has been considered, an uncorrected shower is allowed to continuedownwards from the chosen Q scale. Thus, in this algorithm there is no fixed scale for thetransition from ME to PS, but a smooth merging of one into the other. This cannot giverise to discontinuities in the behaviour at any phase space point, except of course at thesoft/collinear shower cutoff.The formalism so far has only considered real emissions. For the e + e − case, however,the cancellation of real (R) and virtual (V) divergences results in a particularly simpleexpression (when integrated over the possible orientations of the event) σ NLO = σ B + Z d σ R + σ V = (cid:18) α s π (cid:19) σ B . (8)4t is therefore trivial to retain eq. (7) as an NLO prescription for the distribution of events,just by raising the cross section associated with each event from σ B to σ NLO . Identifying W ME d Q = d σ R /σ B , we arrive at the final equation for the differential cross sectiond σ = σ NLO d σ R σ B exp − Z Q Q d σ R ( Q ′ ) σ B ! . (9)It is thus assumed that the O ( α s ) “new” part, σ NLO − σ B , of the total cross sectionshould be associated with the same radiation function as σ B . Alternative choices on thiscount would only show up in O ( α ), which is beyond the certified accuracy of the algorithm.Physicswise it is the minimal assumption, relative to having a different radiation functionfor the pure NLO part of the cross section.The merging formalism is easy to extend to the case of the emission of one extra gluonin any 1 → → → Z [28]. Here, however, NLO corrections are not so trivial. Experimentalpractice has been to rescale the total cross section to the NLO answer, as in eq. (9), therebyneglecting other potential kinematical differences between the LO and NLO answers.The merging approach has also been applied to HERWIG [29–31], with two main dif-ferences. Firstly, since the HERWIG shower is ordered in angle rather than hardness, theME/PS correction weight must be applied to any emission that is the hardest so far, ratherthan only to the first. Secondly, the HERWIG algorithm leaves holes in the phase spacecovered by the first shower emission, so it becomes necessary also to introduce a matchingprocedure, whereby such holes are filled directly by the ME rather than by the PS. A general NLO cross section with hadronic incoming states will also contain remnant(counter) terms from the subtraction of initial-state collinear singularities, which havealready been incorporated into the PDF’s. The complete differential cross section cantherefore be written as the sum of contributions from leading order (Born), virtual, realand counter terms d σ = d σ B + d σ V + d σ R − d σ C . (10)We note that the virtual term has the same n -body phase space as the Born term, whilethe real and counter terms have an ( n + 1)-body phase space. In the POWHEG method,the phase space kinematics are factorised in terms of Born ( v ) and radiation ( r ) kinematicvariables, such that the overall cross section may now be written asd σ = B ( v ) dΦ v + V ( v ) dΦ v + R ( v, r ) dΦ v dΦ r − C ( v, r ) dΦ v dΦ r . (11)Defining a function that integrates over the radiation variables¯ B ( v ) = B ( v ) + V ( v ) + Z d Φ r [ R ( v, r ) − C ( v, r )] , (12)once a Born event has been generated, distributed according to ¯ B ( v ) dΦ v , the differentialcross section for the hardest emission may be written asd σ = ¯ B ( v ) d Φ v " R ( v, r ) B ( v ) exp − Z p ⊥ R ( v, r ′ ) B ( v ) d Φ ′ r ! d Φ r . (13)5he evolution variable of the POWHEG “shower” is taken as the kinematical p ⊥ of theemission with respect to the parton that branches, identifying this highest p ⊥ branching asthe hardest. For ISR this is the p ⊥ with respect to the beam axis. In this way, the hardestradiation is generated according to exact NLO matrix elements, but in a probabilistic,exclusive language, which can be directly interfaced to a suitable shower program.This expression shares many features with eq. (9). The integral is from some upperscale associated with the Born event, with a lower cutoff to avoid soft/collinear regions. Inboth cases it is possible for an event to evolve down to this lower cutoff without radiating.Here, however, the constant NLO prefactor, σ NLO , is upgraded to ¯ B ( v ) d Φ v , a fully differ-ential quantity which will encapsulate all kinematical differences between the LO and NLOanswers. It is thus in this term that all the sophistication and hard work of a full NLOcalculation lies. As in the Pythia shower, the radiation is generated with a running α s expression, evaluated at the p ⊥ scale of the emission.This formalism has been used for the hadronic production of vector bosons [32,33], heavyquark pairs [34], single tops [35] and Higgs bosons via gluon/vector-boson fusion [36, 37].Codes for generating all these processes, except for vector boson pairs, is publicly available.The latest development is the POWHEG BOX; a general framework for implementing NLOcalculations with the POWHEG method [38]. Pythia transverse-momentum-ordered showers
The
Pythia p ⊥ [39]. This allows a picturewhere MPI, ISR and FSR are interleaved in one common sequence of decreasing p ⊥ values[18]. This is most important for MPI and ISR, since they are in direct competition formomentum from the beams, while FSR (mainly) redistributes momenta between alreadykicked-out partons.The interleaving implies that there is one combined evolution equationd P d p ⊥ = d P MPI d p ⊥ + X d P ISR d p ⊥ + X d P FSR d p ⊥ ! × exp − Z p ⊥ max p ⊥ d P MPI d p ′⊥ + X d P ISR d p ′⊥ + X d P FSR d p ′⊥ ! d p ′⊥ ! (14)that probabilistically determines what the next step will be. Here the ISR sum runs overall incoming partons, two per already produced MPI, the FSR sum runs over all outgoingpartons, and p ⊥ max is the p ⊥ of the previous step. Starting from a single hard interaction,eq. (14) can be used repeatedly to construct a complete parton-level event of arbitrarycomplexity. Recently also rescattering has been included as a further (optional) componentof the MPI framework [40].The decreasing p ⊥ scale can be viewed as an evolution towards increasing resolution;given that the event has a particular structure when activity above some p ⊥ scale is resolved,how might that picture change when the resolution cutoff is reduced by some infinitesimald p ⊥ ? It does not have a simple interpretation in absolute time; all the MPI occur essentiallysimultaneously (in a simpleminded picture where the protons have been Lorentz contractedto pancakes), while ISR stretches backwards in time (and is handled by backwards evolution625]) and FSR forwards in time. The closest would be to view eq. (14) as an evolutiontowards increasing formation time.For the following, a relevant aspect is that the p ⊥ definition is not exactly the same forMPI, ISR and FSR in Pythia . For an MPI the p ⊥ is the expected one; the transversemomentum of the two scattered partons in a 2 → → qg branching, where the p ⊥ of the emitted gluon is definedwith respect to the direction of the initial quark. The p ⊥ as a function of the gluon emissionangle θ increases up till 90 ◦ , but then decreases again, p ⊥ → θ → ◦ . Thus, anordering in such a p ⊥ would classify a ∼ ◦ emission as collinear and occurring late in theevolution, although it would involve a more off-shell propagator than an emission at 90 ◦ .It could also erroneously associate a 1 /p ⊥ divergence with the θ → ◦ limit. Therefore itis natural to choose an evolution variable that does not turn over at 90 ◦ .To this end, consider a branching a → bc (e.g. q → qg), where z is defined as thelightcone (LC) momentum along the a axis that b takes. Then p ⊥ LC = z (1 − z ) m a − (1 − z ) m b − zm c , (15)and this equation can be used as inspiration to define evolution variablesISR : p ⊥ evol = (1 − z ) Q with m b = − Q and m a = m c = 0 , (16)FSR : p ⊥ evol = z (1 − z ) Q with m a = Q and m b = m c = 0 , (17)which are monotonous functions of the virtuality Q . However, once a branching has beenfound and the kinematics is to be reconstructed, the Q interpretation is retained, but z isnow replaced by a Lorentz invariant definition. For ISR this is chosen to be z = m br /m ar ,where r is the “recoiler”, i.e. the incoming parton from the other side of the subcollision.The actual p ⊥ of b and c then becomes p ⊥ b,c = (1 − z ) Q − Q m ar = p ⊥ evol − p ⊥ evol p ⊥ evol , max , (18)where p ⊥ evol , max = (1 − z ) Q = (1 − z ) m ar . Instead, for FSR, z = E b /E a in the rest frameof a and its colour-connected recoiler. The p ⊥ b,c expression in this case becomes somewhatlengthier than eq. (18), but shares the same physical properties; at small values p ⊥ evol and p ⊥ b,c follow suit, and so correspond to identical 1 /p ⊥ singular behaviours, but the latterthen turns around and vanishes when p ⊥ evol approaches the kinematical limit. Pythia
In this study, the interface from POWHEG-hvq to
Pythia was performed using the LesHouches Event File (LHEF) [41] format. Both programs come with a number of PDF setsavailable for selection and
Pythia gives access to external PDF sets through the LHAPDFlibrary [42]. For consistency, the CTEQ6L [43] PDF set has been used throughout thisstudy, as this set is available in all three programs used; no other differences in the PDF setitself have been taken into account. One caveat is the associated α s running expression fora given PDF set. Pythia by default will use the first order expression, and allow α s ( M Z )7o be set independently for spacelike and timelike showers. Here, α s ( M Z ) was set to matchthe one used in the PDF set, but with first order running.Historically the LHA conventions [44], used by LHEF, only encompass one event “scale”,specified to be the factorisation one. The α s and α em values at the renormalisation scaleare also supplied, but not that scale itself, and there is no provision for shower-matchingscales. A proposal for an extended standard that would store more scales has recently beenpresented [45]. When reading an LHEF into a shower program, the question then is: atwhat scale should the subsequent shower evolution begin?In Pythia , such choices need to be made for ISR and FSR separately. The basicprinciple should be to avoid doublecounting to the largest extent possible. With LHEFinput it would be natural to let the showers begin at the factorisation scale, the only knownone, and then proceed downwards. This is an allowed choice both for ISR and FSR, butfor ISR another possibility is default. Here, events are split into two kinds, based on theabsence or presence of particles that the shower can produce, i.e. d, u, s, c, b, g and γ . Ifthe LHEF final state contains no such particles then the shower can populate the full phasespace without any risk of doublecounting, which should give more realistic event shapes.This would be the case for many W / Z, top, Higgs and New Physics processes. If, on theother hand, the final state does contain particles that could be produced in the shower, thendoublecounting would be more likely than not, and the factorisation scale again becomesthe only reasonable choice.This more flexible attitude works well if the ME program does not mix different topolo-gies, but breaks down if, say, both tt and ttg / ttq events are supplied, with the latterintended to correspond to the fraction of tt events with an extra emission above the fac-torisation scale. When such a mixing is present, the showers off tt should not be allowedto populate the whole phase space. Thus, we conclude that interfacing to an LHEF cannotbe done completely automatically, but must be made with some knowledge of which ruleswere used to produce the LHEF.Therefore, to provide a consistent interface to POWHEG, we must both consider itsgeneration strategy and the information it stores in the LHEF. In events with an emissionabove the lower cutoff scale p ⊥ min , POWHEG chooses the factorisation scale to be the p ⊥ of the emission, p ⊥ POWHEG = p ⊥ (defined with respect to the beam axis in the case ofPOWHEG-hvq).When the POWHEG “shower” reaches p ⊥ min without any emissions, the factorisationscale is instead set equal to this p ⊥ min , p ⊥ POWHEG = p ⊥ min . Since p ⊥ min normally is rathersmall, of the order of 1 GeV, the fraction of no-emission events is also small.The LHEF choice of storing α s and α em rather than the renormalisation scale makessense for traditional ME calculations, where typically one fixed scale is used. The POWHEGcase is somewhat special, as it can have different renormalisation scales for the Born levelprocess and for the subsequent radiative emission.For the case where POWHEG has already generated an emission, the obvious choiceis to begin the evolution at the factorisation scale p ⊥ POWHEG , such that the shower willnot generate harder emissions. However, there are then two potential complications: first,the mismatch between p ⊥ POWHEG and the lightcone-inspired p ⊥ evol scales of Pythia , andsecond, specifically for FSR, that p ⊥ is defined with respect to the direction of the emittingparton rather than to the beam axis. For ISR, eq. (18) shows that p ⊥ < p ⊥ evol , such thatstarting the shower from p ⊥ evol = p ⊥ POWHEG will lead to a small area of phase space not8 ( / N ) d N / d x x = p ⊥ shower / p ⊥ hard (a)Factorisation ScaleKinematical Limit + Veto 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 1.2 ( / N ) d N / d x x = p ⊥ shower / p ⊥ hard (b)Factorisation ScaleKinematical Limit + Veto Figure 1: Ratio of the kinematic p ⊥ of the first shower emission to the POWHEG emission,where the shower emission is (a) ISR or (b) FSR. In both cases, the results are shownwhen starting the shower at the factorisation scale and when starting the shower at thekinematical limit and vetoing above the POWHEG scalebeing covered. An FSR emission, on the other hand, may have a small p ⊥ evol with respectto the emitting parton but still a p ⊥ > p ⊥ POWHEG with respect to the beam axis.A simple solution to both these problems is instead to begin the shower at the largestpossible scale, and then veto any emissions with a kinematic p ⊥ > p ⊥ POWHEG . If we considerthe first shower emission, the multiplicative nature of the no-emission probability ensuresthat the emission rates below p ⊥ POWHEG will be correct, i.e. unaffected by the vetoes aboveit. The picture is slightly less clear for subsequent emissions; having accepted one showeremission below p ⊥ POWHEG , it is still possible for a later emission to be above it, since thefirst emission may well have had p ⊥ evol > p ⊥ POWHEG . The probability of such an occurrenceis small, and effects formally of NNLO character, unenhanced by any large logarithms.They mainly show up for low- p ⊥ first emissions, where their importance on the event as awhole is less, but still nonzero. Another NNLO issue is that recoil effects from one emissioncan shift the p ⊥ of the previous ones, along with the hard process itself, either to lower orhigher values.The current POWHEG-hvq generator uses a second order running α s expression, butwith a Λ fixed at n f = 5. Although slightly inconsistent, this only leads to changes beneaththe Bottom and Charm scales. The Λ value is taken from a selected PDF set and is modifiedas in [46]. In the LHEF output file, all incoming and radiated partons are massless and thevalues of the couplings, α s and α em , are set to zero in all events.To quantify how well the proposed interfacing works, we begin with top pair production( m t = 171GeV), where all results are generated at LHC energies (pp, √ s = 14TeV). In thiscase, the number of light flavours, which defines the content of the proton and the allowedradiation flavours, goes up to and includes the bottom quark ( n l = 5). To study the effectof the different shower starting scales, we examine the ratio of the first p ⊥ in the showerto the p ⊥ of the POWHEG emission (where the shower p ⊥ value is taken directly afterthe emission). This is shown in Fig. 1, split into contributions from (a) ISR and (b) FSR.For ISR, we note that the ratios do not become larger than unity, but that when startingthe shower at the factorisation scale, there is a region close to p ⊥ shower /p ⊥ hard = 1 wherethe phase space is not completely filled. This gap is filled when starting the shower at the9 R a t i o p ⊥ [GeV] 10 -6 -5 -4 -3 -2 -1 d σ / dp ⊥ [ nb / G e V ] Factorisation ScaleKinematical Limit + 1st Emission VetoKinematical Limit + Veto
Figure 2: Final p ⊥ of the top pair for three different cases (see text). In the upper plot, itis difficult to distinguish the different curves, but the results are clearer to see in the lowerplot, which shows the ratio to the “Kinematical Limit + Veto” resultkinematical limit and vetoing emissions above p ⊥ POWHEG = p ⊥ hard ; that is, the shower isrunning at “full steam” when it reaches the emission p ⊥ threshold. That the curve happensto be so flat near the endpoint is a coincidence, related to a cancellation between the blowupof the na¨ıve emission rate for smaller p ⊥ , with a Sudakov damping in the same limit, helpedalong by many events having a small p ⊥ hard in the first place. It does not happen for FSR,where the large top mass reduces radiation to a lower level overall. For FSR, when startingat the factorisation scale, there is a tail beyond unity, as discussed previously, while this nolonger happens with the veto scheme in place. Note that the FSR rate below p ⊥ POWHEG does come up, however, since emitted partons now are allowed at a larger separation fromtheir mother parton so long as they are still at small p ⊥ with respect to the beam. Asalready noted, the corrections are of higher order, but their inclusion is worthwhile foroverall consistency.We move on to study the effect of shower emissions beyond the first. As discussedpreviously, after a first allowed shower emission below p ⊥ POWHEG , a subsequent emissionmay be generated above p ⊥ POWHEG . To examine this effect, three different cases are con-sidered: (1) the showers are started at the factorisation scale, (2) the showers are startedat the kinematical limit and only the first shower emission is vetoed, and (3) the showersare started at the kinematical limit and all emissions above p ⊥ POWHEG are vetoed. As ameasure of this effect, one may use the final p ⊥ of the top pair, which is shown in Fig. 2for the three different cases. At first glance, all three approaches appear to give similarresults, but the ratio plot reveals that the difference between the factorisation scale andthe veto schemes is around 10%, while there is little difference between the two differentveto schemes. Instead, in Fig. 3, we study the smearing of the first shower emission p ⊥ dueto subsequent emissions. In this case only ISR is generated, so there is no ambiguity inpicking out the first shower emission in the final event, and again, we show the ratio of thefirst shower emission to the POWHEG one, p ⊥ shower /p ⊥ hard , (a) immediately after the firstemission and (b) after the shower evolution has finished. Fig. 3a shows the same features10 ( / N ) d N / d x x = p ⊥ shower / p ⊥ hard (a)Factorisation ScaleKinematical Limit + 1st Emission VetoKinematical Limit + Veto 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 ( / N ) d N / d x x = p ⊥ shower / p ⊥ hard (b)Factorisation ScaleKinematical Limit + 1st Emission VetoKinematical Limit + Veto Figure 3: Ratio of the kinematic p ⊥ of the first shower emission to the POWHEG emissionfor ISR only. (a) Shows the results immediately after the first emission, while (b) showsthe results after the full shower evolution. In both cases, the “Kinematical Limit + 1stEmission Veto” and “Kinematical Limit + Veto” curves lie on top of each other d N / d x x = p ⊥ shower / p ⊥ hard (a)Factorisation ScaleKinematical Limit + Veto 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.2 0.4 0.6 0.8 1 1.2 d N / d x x = p ⊥ shower / p ⊥ hard (b)Factorisation ScaleKinematical Limit + Veto Figure 4: Ratio of the kinematic p ⊥ of the first shower emission to the POWHEG emissionfor bottom pair production, where the shower emission is (a) ISR or (b) FSRas Fig. 1, although with different normalisation due to the lack of FSR. In Fig. 3b, the firstshower emission p ⊥ is smeared by small amounts due to subsequent emissions, but againthe difference between vetoing just the first emission and all emissions is negligible.For bottom production ( m b = 4 . n l = 4), meaning that there will be no incoming or radiated b quarks. This is differentfrom the default Pythia settings, where b quarks are both taken from the beam andallowed to be created in radiative emissions. The results are shown in Fig. 4, again splitinto contributions from (a) ISR and (b) FSR. Although the overall pattern of radiation isdifferent, due to the smaller bottom mass, the details with respect to the interface with
Pythia show the same features as Fig. 1.
The POWHEG approach is a very powerful tool for NLO studies. Unfortunately,POWHEG-based implementations are not available for all relevant processes. Many older11LO calculations are instead available in terms of phase space slicing results (see e.g. [47]).That is, the ( n + 1)-body cross section is based on the pure real-emission cross section d σ R ,without any Sudakov corrections. Unresolved emissions, virtual corrections and countert-erms are lumped together with the Born cross section to provide the n -body cross section.The absence of a Sudakov factor means that soft/collinear divergences in d σ R are not damp-ened, thus requiring a reasonably large cutoff scale to stay away from the region where σ n would turn negative. On the other hand, d σ R should be used where it can. Therefore σ n and σ n +1 typically are chosen to be of the same order, while σ n ≪ σ n +1 in POWHEG,where a much smaller cutoff can be used without any risk of inconsistencies.In order to make these older NLO codes also useful to the experimental community theyneed to be interfaced to event generators, for further showering, MPI and hadronisation. Itis then convenient to add Sudakovs to d σ R , to bring it closer to the POWHEG approach.This would also smoothen out the transition between ( n + 1)- and n -body phase space,in the spirit of modern-day leading-order matching procedures, already mentioned in theintroduction.As a practical example we mention the Wgamma nlo package for W γ production toNLO [48], with up to one additional quark or gluon in the final state, and allowing foranomalous WW γ couplings. Here an implementation of the principles to be presented isalready available [49].Assume a lowest-order process that does not contain any coloured particles in the finalstate. The NLO processes thus include one quark or gluon extra, with divergences whenits p ⊥ →
0. A scale p ⊥ min is used to separate the ( n + 1)- and n -body phase space. An( n + 1)-body state is characterised by the transverse momentum of the quark or gluon, p ME ⊥ ,with p ME ⊥ > p ⊥ min . We now want to include the Sudakov to express that p ME ⊥ is the hardestjet of the event, i.e. that there are no jets at a larger scale.In the CKKW-L [2] approach a fictitious trial shower is used to provide this. The betterthis shower attaches to the correct ME behaviour, the more accurate the Sudakov will be(thus, an advantage to having a more accurate default behaviour of the shower, as we strivefor in this article). The point should not be overstressed, however; at large p ⊥ the Sudakovsuppression is negligible anyway, and at small p ⊥ the universal behaviour should dominate.The ( n + 1) topology has to be projected down to the core n process, without theextra parton, to provide the starting point for the trial shower. This requires a choiceof which side the emission occurs on. Sometimes flavours allow only one possibility; forug → W + γ d the hard subprocess must be ud → W + γ since the gluon cannot couple tothe W + γ state. For ud → W + γ g the emission could be on either side, and the relativeprobability for a shower emission is related to the respective 1 /p ⊥ evol . Since z = m n /m n +1 is independent of emission side, it follows that p ⊥ evol ∝ Q ∝ ∓ cos ˆ θ , where ˆ θ is the angleof the gluon in the rest frame of the subcollision. The constancy of z also implies thatthe splitting kernel values are the same on both sides, and so give no net contribution. Inanother case, where the core process would be different depending on the side of emission,the weight of the respective splitting kernel would have to be taken into account, but stillat the same z . In summary, the relative probability for a gluon emission from side 1 is Q − / ( Q − + Q − ) = Q / ( Q + Q ) = (1 + cos ˆ θ ) /
2. The choice of side determines a new x ′ = xz for it, while x on the other side is unchanged. A combined rotation and boost canbring the n particles to this new frame.From there, the fictitious shower is allowed to generate the first/hardest emission at a12cale p PS ⊥ . The probability for p ME ⊥ > p PS ⊥ is precisely the desired Sudakov, in the CKKW-Lspirit. That is(i) if p ME ⊥ > p PS ⊥ the original n + 1-body topology is retained;(ii) else the projected n -body topology is selected.In case (i), a normal shower can then be started up from p ME ⊥ and downwards; that is,further jets may be generated above the p ⊥ min scale, to give ( n + 2)-body topologies etc.The events in case (ii) are to be lumped together with the ones already originally classifiedas n -body, and allowed to shower from the p ⊥ min scale downwards.In this approach, there is not guaranteed to be a smooth matching at p ⊥ min , at leastfor the p ⊥ scale of the hardest emission. The hope is that this step will be smeared out bysubsequent showers and hadronization. Also note that, unlike POWHEG defined by eq.(13),but in line with MC@NLO, the high- p ⊥ tail is defined by the LO ( n + 1) expression, withoutany ¯ B ( v ) /B ( v ) “K factor”. Pythia shower
As will be shown in Section 3.3 for top pair production, an evolution with Q = s (power)overestimates the high- p ⊥ tail while Q = m ⊥ t (wimpy) underestimates it, cf. [21]. Thisis not so surprising, as follows. Let us recall that the matrix element for a QCD processsuch as gg → ggg scattering roughly behaves like [50, 51]d σ ∼ d p ⊥ p ⊥ d p ⊥ p ⊥ d p ⊥ p ⊥ δ (2) ( p ⊥ + p ⊥ + p ⊥ ) , (19)where the p ⊥ i are the transverse momenta of the three outgoing gluons. In the limit p ⊥ ≪ p ⊥ , where | p ⊥ | ≈ | p ⊥ | , this reduces to 1 /p ⊥ p ⊥ , i.e. the hard interaction behaveslike d p ⊥ /p ⊥ and the subsequent shower emission of an additional gluon like d p ⊥ /p ⊥ .Put another way, for a fixed p ⊥ , you may distinguish a low- p ⊥ region with a fall-off liked p ⊥ /p ⊥ and a high- p ⊥ region where the fall-off instead is like d p ⊥ /p ⊥ , with a smoothtransition when p ⊥ ≈ p ⊥ . In practice you would not want to simulate the process like this,of course, but reserve the hardest propagator to be described by matrix elements, whichalso would avoid doublecounting problems.Obviously the picture is not equivalent for the gg → ttg, but let us apply a similarreasoning to the extra gluon in this case. For small p ⊥ g the picture of an ISR branchingg → gg followed by a hard process gg → tt ought to be a valid approximation, and so weexpect a d p ⊥ g /p ⊥ g falloff. At large p ⊥ g , on the other hand, it would make more sense tothink in terms of an ISR branching g → tt followed by a hard process gt → gt, and thusa shape more like d p ⊥ g /p ⊥ g . Now, we don’t simulate the latter kind of hard processes, e.g.because the top is so heavy that a top PDF is not a fruitful approximation, and so wewould like to obtain these high- p ⊥ g configurations in the context of the simulation of thegg → tt process. This leads to an ansatz of the formd P ISR d p ⊥ ∝ p ⊥ k M k M + p ⊥ , (20)13here M is a reasonable scale to associate with the hard 2 → k is a fudge factor of order unity, parameterising at what scale the transitionfrom a 1 /p ⊥ to a 1 /p ⊥ behaviour occurs.How generic would such an ansatz be? First of all, for QCD processes involving lightquarks and gluons, the showers should be cut off at (around) the scale of the hard process,or else one would doublecount, since in this case all possible 2 → → → p ⊥ /p ⊥ ansatz works well up to the kinematicallimit [28]. A reasonable assumption is that this generalises to processes where two or morecolour singlet particles are produced in the core process, while a shape similar to eq. (20)should occur in processes that involve one or several coloured particles in the final state.The argument for such a difference is one of colour coherence; with colour charge inboth the initial and the final state one expects a destructive interference between ISR andFSR emissions that limits the radiation [52], while no such interference occurs with coloursonly in the initial state. One can then argue exactly what the scale M appearing ineq. (20) should be; for pair production of coloured particles, the factorisation or renormali-sation scale should be a reasonable choice, but when considering the production of a mixedcoloured/non-coloured final state, by the coherence argument, it is primarily the colouredparticles that should play a role in this scale. We note that the default choice for internal Pythia → In order to study processes for which an implementation of POWHEG is not yet available,we turn to the MadEvent matrix element generator. In eq. (13) we see that the POWHEGSudakov, used to generate hard emissions, contains only Born and real terms; all NLOcorrections are contained in a separate prefactor.We therefore use MadEvent to generate events with an extra jet in the final state in orderto extract the cross section for real jet emission, d σ R . An approximate “POWHEG” styleprobability for emission is then formed by normalising to the overall lowest order (Born)cross section (generated by simulating the corresponding 2 → P = d σ R σ B exp − Z p ⊥ max p ⊥ d σ R σ B ! , (21)where the p ⊥ integral begins at the kinematical limit. As noted previously, the NLO prefac-tor can lead to kinematical differences in the resulting distribution. Under the assumptionthat such kinematical differences are small (to be addressed further in the case of top pairproduction), a qualitative comparison to the Pythia shower can be made. The effect ofthe Sudakov will be most visible in the low- p ⊥ region, where the ME d σ R calculation is14ivergent. Here the integrand will blow up and the Sudakov will make the distributionturn over, such that it will have a unit integral (up to cutoff effects). Given that the low- p ⊥ region is where the parton shower should be most accurate, we expect the turnoverin the Sudakov-modified MadEvent distribution to roughly correspond with the Pythia distribution.All Standard Model masses in MadEvent are set equal to the default
Pythia values.Variations in particle widths between MadEvent and
Pythia are not expected to play alarge role and are therefore neglected. The definition of both the proton and jets includesbottom quarks, as would be included in the
Pythia default settings. In what follows, wewill only consider heavy final states (top mass and higher), such that we can neglect FSRand only consider ISR. In this case, there are only soft and collinear singularities from thebranching of incoming partons. A low- p ⊥ cutoff of 2 GeV is introduced on real-jet emission,such that these divergences are avoided. No further cuts are applied. In generating events with MadEvent, there are again some different choices available relatingto running α s expressions and PDF’s. The choice of PDF determines the running of α s andthe value α s ( M z ) and as before, the CTEQ6L PDF set was chosen. The ME calculationswere generated with fixed renormalisation and factorisation scales, taken to be the geometricmean of the masses of the two heavy final-state particles. The results of these choices remainevident in the ME-derived jet distribution. In eqs. (2) and (3) it was shown that the Pythia shower algorithm instead picks scales related to the the evolution variable when generatingemissions. These differences can be non-negligible, especially in the low- p ⊥ region, where α s will become large and we expect a similar turnover to the Pythia distribution. To accountfor these effects, additional weights are applied to MadEvent distributions after generation,with the new scale taken to be the kinematic p ⊥ of the jet.A correction for α s is simple to achieve, weighting events by the ratio of the new andold α s values. With M defined as the fixed renormalisation/factorisation scale used in theME calculations, we have a weight α s ( p ⊥ ) α s ( M ) . (22)With ISR, correcting for differences in the factorisation scale is more difficult. The distribu-tion for real emission from the ME calculation contains two PDF factors from the incomingpartons, unlike ISR generated from the shower algorithm. This difference makes it unclearwhat correction should be applied, especially as in our ME calculation, we do not knowwhich incoming parton has branched. Taking an exclusive point of view, there should havebeen no emissions between M and p ⊥ on either side of the event, and so both should bereweighted, overall giving a correction factor x f ( x , p ⊥ ) x f ( x , M ) x f ( x , p ⊥ ) x f ( x , M ) . (23)This may be a slight over-correction, however, and one would expect the results with justthe α s factor and those with this additional PDF factor to bracket the “correct” distribution.We examine this further in Sec. 3.3 for the case of top pair production.15 ul ul ul ul graph 712 3 4 5 (a) u ul ul u ul go graph 1812 3 4 5 (b) Figure 5: Two possible graphs for ˜u L squark pair production with an additional jet whichdo not directly correspond to a 2 → Pythia shower splitting
When moving to a final state containing an extra jet, MadEvent will correctly generate allpossible topologies, but when comparing against
Pythia , some of these graphs may notcorrespond to a shower history. Two examples are given in Fig. 5 for the case of up squarkpair production in the MSSM.In (a), the two t-channel squark propagators cannot directly be reproduced by a 2 → → Pythia works in the narrow width approximation, where the gluino productionand decay are described as part of the separate 2 → → ˜u˜g, it would bedoublecounting to include the same graph as a 2 → We begin by comparing the jet emission probability in top pair production of POWHEG,MadEvent and the first
Pythia shower emission (ISR only). In all studies that follow,events are generated at LHC energies (pp, √ s = 14 TeV) and, additionally, when thedamping ansatz is used, the scale M of eq. (20) is set to be the factorisation scale unlessstated otherwise. For top production, we restrict both Pythia and MadEvent to QCDproduction only, as this is the dominant contribution to the cross section, and also allows aconsistent comparison to POWHEG. As noted previously, the default
Pythia
ISR showerwill begin evolution at the kinematical limit, given that a top cannot be produced in the16 R a t i o p ⊥ [GeV] 10 -3 -2 d P / dp ⊥ [ G e V - ] (a) POWHEGMadEventMadEvent ( α s corrected)MadEvent ( α s & µ corrected) 0.8 1 1.2 100 200 300 400 500 600 700 800 900 1000 R a t i o p ⊥ [GeV] 10 -7 -6 -5 -4 -3 -2 d P / dp ⊥ [ G e V - ] (b) POWHEGMadEventMadEvent ( α s corrected)MadEvent ( α s & µ corrected) Figure 6: First emission probability as a function of p ⊥ for top pair production in POWHEGand MadEvent. Ratio plots are normalised to the POWHEG resultshower. The POWHEG to MadEvent comparison allows us to check the validity of theMadEvent approximation method outlined in Sec. 3.2.1. The comparison to Pythia thenlets us examine the effectiveness of the ansatz described in Sec. 3.1.The probability distributions for jet emission in top pair production are expected to beroughly in agreement over the entire p ⊥ range for POWHEG and MadEvent, as long as thekinematical differences that come from the full NLO prefactor of eq. (13) are small. Thisis shown in Fig. 6, separately for low and high p ⊥ regions, for three different sets of theMadEvent results. All three have the Sudakov correction applied, but one set additionallyhas the α s scale correction (“ α s corrected”), and the other both the α s and factorisationscale corrections (“ α s & µ corrected”). These will all meet when the p ⊥ scale matches thatof the fixed factorisation/renormalisation scale used in the MadEvent generation, but beginto diverge as the p ⊥ rises and falls away from this value. In the low- p ⊥ region, the Sudakovsmake the distributions turn over, while in the high p ⊥ tail, there is good agreement betweenthe distributions of POWHEG and the approximation from MadEvent; in the ratio plotthe POWHEG curve sits between the two different corrected MadEvent curves, but overallcloser to the α s & µ -corrected one.We now move on to compare POWHEG against the Pythia shower and the ansatzof Sec. 3.1. The expectation is that the default
Pythia power shower will not fall offquickly enough and therefore overestimate the emission probability in the high p ⊥ tail. InFig. 7 we show, again separately for low and high p ⊥ regions, the POWHEG results againstfour variations of the Pythia shower (wimpy, power, damped with k = 1 and dampedwith k = 2). Again, there is good agreement in the low- p ⊥ region. In the high- p ⊥ tail,the wimpy shower sits clearly below the POWHEG result, while the Pythia power showerdoes indeed overestimate the jet emission probability. The damping procedure, particularlywith k = 2, then brings the Pythia distribution into closer agreement with POWHEG.It is interesting to study the kinematical differences that come from the full NLO pref-actor in POWHEG by considering the top pair rapidity both with and without such aprefactor. When full NLO corrections are included, we not only expect an overall changein cross section, but also that there may be kinematic differences in the distributions. InFig. 8a, the ratio of two POWHEG pair rapidity distributions is shown. The first is fromthe default POWHEG-hvq generator, with a full NLO prefactor, while the second is gen-17 -4 -3 -2 -1
0 20 40 60 80 100 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](a) POWHEGPythia Default (Power)Pythia Damp, k = 2Pythia Damp, k = 1Pythia Wimpy 10 -7 -6 -5 -4 -3 -2
100 200 300 400 500 600 700 800 900 1000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](b) POWHEGPythia Default (Power)Pythia Damp, k = 2Pythia Damp, k = 1Pythia Wimpy Figure 7: First emission probability as a function of p ⊥ for top pair production in POWHEGand Pythia , split into (a) the low- p ⊥ region where the Sudakov makes the distributionsturn over and (b) the high- p ⊥ tail R a t i o y(a) − B / B 0.4 0.6 0.8 1 1.2 1.4 1.6 -4 -3 -2 -1 0 1 2 3 4 R a t i o y(b)Pythia Default (Power)Pythia Damp, k = 1Pythia Damp, k = 2MadEvent Figure 8: (a) Ratio of top pair rapidity between POWHEG with the full NLO prefactor, ¯ B ,of eq. (13) and B only. Note the suppressed zero on the y axis. (b) Jet rapidity distributionsof Pythia and MadEvent for p ⊥ jet > B term of eq. (13) with just the Born contribution B . The figureshows both features; an overall shift in cross section (a k -factor of around 1.5) and a smallshift in pair rapidity to more central regions.We finally compare the jet rapidity distributions from POWHEG, MadEvent and Pythia . In
Pythia , the rapidity of the extra jet is taken immediately after the firstshower emission (if present). For MadEvent, however, we can not apply the Sudakov cor-rection in this case, and the direct output of the generator is taken. We restrict ourselvesto p ⊥ jet > Pythia shower, it isclear that by only reducing the high- p ⊥ tail relative to the power shower, events will betaken out of the central region. These distributions are shown in Fig. 8b, normalised tothe POWHEG result. MadEvent and POWHEG are in good agreement, although with anoverall shift in cross section. The default Pythia shower is strongly peaked in the centralregion, but comes into better agreement when damped.18 -10 -9 -8 -7 -6 -5 -4 -3 -2
100 200 300 400 500 600 700 800 900 1000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV]MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 2Pythia Wimpy Figure 9: First emission probability as a function of p ⊥ for Z pair production. Results show Pythia compared to the approximate MadEvent ( α s & µ corrected) prescription ( / N ) d N / d m * [ G e V - ] Invariant Mass [GeV](a) MadEventMadEvent + VetoMadEvent + Graph Removal 10 -9 -8 -7 -6 -5 -4 -3 -2
100 200 300 400 500 600 700 800 900 1000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](b)MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 2Pythia Wimpy Figure 10: W pair production. (a) Shows the invariant mass of the jet and W boson whenthe jet is a bottom quark for three different MadEvent runs (see text). The large resonantpeak in the “MadEvent” sample has been cut off above 0.15. (b) Shows the first emissionprobability as a function of p ⊥ We now study processes where there is not (yet) a “correct” POWHEG result to comparewith, but instead rely on the MadEvent approximation. If the ansatz of Sec. 3.1 is correct,the damping of the
Pythia shower is not expected to help improve the p ⊥ distribution in Zand W pair production in a significant way. Here, there are no coloured final-state particlesto use as a guide in setting the M scale of eq. (20), so we continue to use the factorisationscale when generating the damped results. Again, in this case, the Pythia default is tostart ISR shower at the kinematical limit. Fig. 9 shows the results for Z pair productionfor the default, damped ( k = 2) and wimpy shower against MadEvent ( α s & µ corrected).The results show that the default shower, although giving a slightly too hard p ⊥ tail, stilldoes a reasonable job of reproducing the MadEvent curve without additional damping.The case of W pair production is useful as a check on the MadEvent resonance vetomethod (see Sec. 3.2.3). When bottom quarks are allowed in both the incoming beamsand in the definition of the extra jet, there is now a resonant contribution coming fromt → bW. In Fig. 10a, the invariant mass of the jet and the matching W are shown whenthe jet is a bottom quark. A clear peak is visible when those events with a resonance are19 article Mass (GeV) ˜u L χ χ ± p ⊥ tail for the MadEventvetoed sample. As a final check, the radiation pattern was compared to a sample notincluding bottom quarks, and found to contain no large differences. To further study the production of heavy final states, MSSM processes were chosen split intothree groups: coloured final states, non-coloured final states, and coloured/non-coloured fi-nal states. All events were generated with the SPS1a [54] set of parameters (the relevantfinal-state masses used are shown in Tab. 1). Here, also the lowest order 2 → Pythia for showering, again with the fac-torisation and renormalisation scales fixed at the geometric mean of the masses of the twoheavy final-state particles. Note that this is slightly different from the default
Pythia internal 2 → M scale selectionof the damping ansatz, eq. 20. For the fully coloured final states, all events were generated with QCD only; as with toppair production, this is where the dominant contribution to the cross section lies. As before,the expectation is that the coloured final states will benefit from a damping of the high- p ⊥ tail. Figure 11 shows the tail of the p ⊥ distributions, all compared to the MadEvent α s & µ corrected data, for (a) ˜u L ¯˜u L , (b) ˜u L ˜g and (c) ˜g ˜g. Similarly to top pair production, thepower shower overestimates the high- p ⊥ tail, while the damping ansatz brings the curvesinto better agreement. For ˜u L ¯˜u L production, k = 2 leads to the best agreement, while for˜u L ˜g and ˜g ˜g production, the MadEvent curve lies between the damped k = 1 and k = 2curves. We move on to study the non-coloured final states, ˜ χ ˜ χ and ˜ χ +1 ˜ χ − , where we expect thedamping ansatz not to improve the p ⊥ tail of the parton shower. Here, for both processes, atlowest order, there are large resonant H /A contributions. These are relatively long-livedintermediate states which we expect to follow the rules for 2 → -8 -7 -6 -5 -4 -3 -2
200 400 600 800 1000 1200 1400 1600 1800 2000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](a)MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 1Pythia Damp, k = 210 -7 -6 -5 -4 -3 -2
200 400 600 800 1000 1200 1400 1600 1800 2000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](b)MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 1Pythia Damp, k = 2 10 -8 -7 -6 -5 -4 -3 -2
200 400 600 800 1000 1200 1400 1600 1800 2000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](c)MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 1Pythia Damp, k = 2 Figure 11: First emission probability as a function of p ⊥ for (a) ˜u L ¯˜u L , (b) ˜u L ˜g and (c) ˜g ˜gproduction. In (a) the k = 2 and MadEvent curves lie on top of each other -9 -8 -7 -6 -5 -4 -3 -2
200 400 600 800 1000 1200 1400 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](a)MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 1Pythia Damp, k = 2 10 -9 -8 -7 -6 -5 -4 -3 -2
200 400 600 800 1000 1200 1400 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](b)MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 1Pythia Damp, k = 2 Figure 12: First emission probability as a function of p ⊥ for (a) ˜ χ ˜ χ and (b) ˜ χ +1 ˜ χ − productionthe shower already does a good job in covering the entire phase space. For both processesthen, the MadEvent veto scheme is used to remove events of this type.The results are shown in Fig. 12 for (a) ˜ χ ˜ χ and (b) ˜ χ +1 ˜ χ − production. The resultsfollow the pattern for Z/W ± pair production; in both cases, the default power shower doesa reasonable job in the high- p ⊥ tail. 21 -8 -7 -6 -5 -4 -3 -2
200 400 600 800 1000 1200 1400 1600 1800 2000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV]Default (Power)Damp, k = 2, M = M u~ L Damp, k = 2, M = µ Figure 13:
Pythia first emission probability for the damped shower ( k = 2) in ˜u L ˜ χ production, where M is set either to the factorisation scale or to the mass of the heavyoutgoing squark. The default power shower curve is shown for comparison Finally, we study mixed coloured/non-coloured final states. Here, we would expect somemeasure of damping to improve the shower description due to the presence of colour inthe final state. As discussed in Sec. 3.1, one issue here is the meaning of the M scale ofeq. (20). With the mixed final state, the difference between the factorisation scale and e.g.just the mass of the final-state coloured object is large enough to give noticeable differencesin the damped shower tail. As an example, Fig. 13 shows the difference in the dampedshower ( k = 2) in ˜u L ˜ χ production, when M is set to the factorisation scale and when itis instead set to the mass of the outgoing squark. In the remaining results of this section,we retain the choice of setting M to the mass of the outgoing coloured state. Also notethat all of the processes studied in this section have strong resonant contributions, wherea squark decays into a neutralino/chargino and a jet, making it difficult to generate largestatistics when using the MadEvent veto scheme.Results are shown in Fig. 14 for (a) ˜u L ˜ χ and (b) ˜u L ˜ χ − . While the curve for ˜u L ˜ χ − does show the expected behaviour, with the damping improving the shower description, for˜u L ˜ χ we do not obtain the expected results. Instead, Fig. 15a shows the results for ˜u L ˜ χ when resonant graphs are manually set to zero in the MadEvent generation code. Here,beyond p ⊥ ∼ L ˜ χ against those of ˜u L ˜ χ − , a key difference is theappearance of right-handed intermediate squarks. In Fig. 15b, we show the results whenthe right-handed squark masses (˜u R , ˜d R , ˜s R and ˜c R ) are set high. In this case, we do recoverthe behaviour of the ˜u L ˜ χ − result.Finally, in Fig. 16, we study the processes (a) ˜g ˜ χ and (b) ˜g ˜ χ − . Again the expectedbehaviour is not apparent; both sit closest to the default power shower curve.In summary, for the mixed processes studied in this section, our ansatz is not an obviousimprovement relative to the power shower. There is a nontrivial dependence of the emissionpattern on the SUSY parameter choices that we do not understand, and do not go on tostudy further at this time. 22 -8 -7 -6 -5 -4 -3 -2
200 400 600 800 1000 1200 1400 1600 1800 2000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](a)MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 1Pythia Damp, k = 2 10 -8 -7 -6 -5 -4 -3 -2
200 400 600 800 1000 1200 1400 1600 1800 2000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](b)MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 1Pythia Damp, k = 2 Figure 14: First emission probability as a function of p ⊥ for (a) ˜u L ˜ χ and (b) ˜u L ˜ χ − -8 -7 -6 -5 -4 -3 -2
200 400 600 800 1000 1200 1400 1600 1800 2000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](a)MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 1Pythia Damp, k = 2 10 -8 -7 -6 -5 -4 -3 -2
200 400 600 800 1000 1200 1400 1600 1800 2000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](b)MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 1Pythia Damp, k = 2 Figure 15: First emission probability as a function of p ⊥ for ˜u L ˜ χ , where in (a) resonantgraphs have been removed and (b) the right handed squark masses have been set high -10 -9 -8 -7 -6 -5 -4 -3 -2
200 400 600 800 1000 1200 1400 1600 1800 2000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](b)MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 1Pythia Damp, k = 2 10 -9 -8 -7 -6 -5 -4 -3 -2
200 400 600 800 1000 1200 1400 1600 1800 2000 d P / dp ⊥ [ G e V - ] p ⊥ [GeV](b)MadEvent ( α s & µ corrected)Pythia Default (Power)Pythia Damp, k = 1Pythia Damp, k = 2 Figure 16: First emission probability as a function of p ⊥ for (a) ˜g ˜ χ and (b) ˜g ˜ χ − Summary and Outlook
One direct outcome of this study is an improved matching between the existing POWHEGprogram family and the
Pythia p ⊥ -ordered showers. The main point is that the Pythia p ⊥ definition does not quite agree with the normal kinematical one, although the two doagree closely over most of the phase space populated by showers. In light of this, thenumerical effect of these improvements are fairly modest, but, in view of the important rolethat we foresee for POWHEG-based studies at the LHC, there is every reason to have theinterface to Pythia well understood.The POWHEG approach, in complete or in simplified form, also allows us to test the de-fault behaviour of the p ⊥ -ordered showers. In particular we address the issue of “power” vs.“wimpy” showers, i.e. what starting scale to use for the downwards evolution, specificallyfor pair production of heavy particles. Here we show that many processes obey an inter-mediate behaviour, where the characteristic d p ⊥ /p ⊥ shower fall-off is replaced by a steeperd p ⊥ /p ⊥ fall-off for large p ⊥ values. That is, emissions are allowed up to the kinematicallimit, but at a dampened rate. The damping can be approximated by a factor P dampen = k M k M + p ⊥ , (24)for the emission of a parton of transverse momentum p ⊥ , where M is a characteristic scaleof the hard process and k is a fudge factor.More specifically, we have seen that for coloured pair production, with M set equal tothe factorisation scale (noting the slightly different scale choice used in the Pythia internalprocesses and the external MSSM ones) and k = 2, we get a reasonable shower behaviourfor the different final states studied here. Instead, when the particles in the final state ofthe hard process are colour singlets only, we have seen that there is no need to imposea damping, consistent with previous results for single W / Z production. The differencesbetween these two cases can be understood based on the destructive interference betweenISR and FSR that is expected for coloured particles but not for the uncoloured ones. In nocase is the wimpy shower a good choice.For the mixed coloured/non-coloured final states of Sec. 3.4.3, we have argued that M should instead be related to the coloured partons only, due to this coherence argument.Unfortunately, for these processes, the results are not as expected. We do not currently un-derstand the structure of the real-emission matrix elements that gives rise to the behaviourseen here and, specifically, whether it is an artifact of the complicated SUSY structure oran inherent property of the radiation pattern.Not studied here is the case of the production of light coloured particles, i.e. of normalQCD jets. There, the hard process and the showers produce the same kind of partons, anddoublecounting becomes a main concern. We intend to return to this class of event.With the advances in computational tools, allowing automatised (Born-level) higher-order calculations, one may question the need for more accurate showers. We believe thereare two main points in favour of improving showers, especially when this can be done withonly a modest effort. One is that new physics scenarios are continuously being proposed,where higher-order calculations may be overkill for first studies, but nevertheless a realisticpopulation over all possible event topologies is useful, even if off by a factor of two or soin some tails. The other is that trial showers as a means of obtaining Sudakov factors24s a crucial ingredient of CKKW-L matching schemes, so that the quality of matching isimproved if the quality of the shower is also improved. Acknowledgments
This work was supported in by the Marie Curie Early Stage Training program “HEP-EST”(contract number MEST-CT-2005-019626) and in part by the Marie Curie research trainingnetwork “MCnet” (contract number MRTN-CT-2006-035606).
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