Introduction to Monte Carlo Event Generators

1. Introduction to Monte Carlo Event Generators Peter Richardson IPPP, Durham University Yeti 09 12th

2. Plan • Introduction • Basic principles of event generation • Parton Showers • Parton Shower Approach • CKKW and MC@NLO, POWHEG • Hadronization and Underlying Event • Hadronization Models • Underlying Event Modelling (Mike Seymour) Yeti 09 12th

3. Plan • I will concentrate on hadron collisions. • There are many things I will not have time to cover • Heavy Ion physics • B Production • BSM Physics • Diffractive Physics • Underlying Event (Mike Seymour) • I will concentrate on the basic aspects of Monte Carlo simulations which are essential for the Tevatron and LHC. Yeti 09 12th

4. Other Lectures and Information • Unfortunately there are no good books or review papers on Event Generator physics. • However many of the other generator authors have given similar lectures to these which are available on the web • Torbjorn Sjostrand at the Durham MCnet school http://conference.ippp.dur.ac.uk/conferenceOtherViews.py?view=standard&confId=3 • Mike Seymour at the CTEQ-MCnet school http://conference.ippp.dur.ac.uk/conferenceOtherViews.py?view=ippp&confId=156 Yeti 09 12th

5. Monte Carlo Event Generators • At the most basic level a Monte Carlo event generator is a program which simulates particle physics events with the same probability as they occur in nature. • In essence it performs a large number of integrals and then unweights to give the momenta of the particles which interact with the detector. Yeti 09 12th

6. A Monte Carlo Event Hard Perturbative scattering: Usually calculated at leading order in QCD, electroweak theory or some BSM model. Modelling of the soft underlying event Multiple perturbative scattering. Perturbative Decays calculated in QCD, EW or some BSM theory. Initial and Final State parton showers resum the large QCD logs. Finally the unstable hadrons are decayed. Non-perturbative modelling of the hadronization process. Yeti 09 12th

7. Monte Carlo Event Generators • All the event generators split the simulation up into the same phases: • Hard Process; • Parton Shower; • Secondary Decays; • Hadronization; • Multiple Scattering/Soft Underlying Event; • Hadron Decays. • I will discuss the different models and approximations in the different programs as we go along. • I will try and give a fair and objective comparision, but bear in mind that I’m one of the authors of HERWIG. Yeti 09 12th

8. Monte Carlo Event Generators • There are a range of Monte Carlo simulation programs. • In general there are two classes of programs. • General Purpose Event Generators Does everything • Specialized Programs Just perform part of the process • Often need to use both types of program. Yeti 09 12th

9. Monte Carlo Event Generators Yeti 09 12th

10. Hard Processes • Traditionally all the hard processes used were in the event generators. • These are normally 2g2 scattering processes. • There are a vast range of processes in both HERWIG and PYTHIA. • However for the LHC we are often interested in higher multiplicity final states. • For these need to use specialized matrix element generators, for example SHERPA has its own matrix element generator built in. Yeti 09 12th

11. QCD Radiation • It is impossible to calculate and integratethe matrix elements for large numbers of partons. • Instead we treat the regions where the emission of QCD radiation is enhanced. • This is soft and collinear radiation. Yeti 09 12th

12. Collinear Singularities • In the collinear limit the cross section for a process factorizes • Pji(z) is the DGLAP splitting function. • The splitting function only depends on the spin and flavours of the particles Yeti 09 12th

13. Collinear Singularities • This expression is singular as qg0. • What is a parton? (or what is the difference between a collinear pair and a parton) • Introduce a resolution criterion, e.g. • Combine the virtual corrections and unresolvable emission Resolvable Emission Finite Unresolvable Emission Finite • Unitarity: Unresolved + Resolved =1 Yeti 09 12th

14. Monte Carlo Procedure • The key difference between the different Monte Carlo simulations is in the choice of the evolution variable. • Evolution Scale • Virtuality, q2 • Transverse Momentum, kT. • Angle, q. • …. • Energy fraction, z • Energy fraction • Light-cone momentum fraction • …. • All are the same in the collinear limit. Yeti 09 12th

15. Soft Emission • We have only considered collinear emission. What about soft emission? • In the soft limit the matrix element factorizes but at the amplitude level. • Soft gluons come from all over the event. • There is quantum interference between them. • Does this spoil the parton shower picture? Yeti 09 12th

16. Angular Ordering Colour Flow • There is a remarkable result that if we take the large number of colours limit much of the interference is destructive. • In particular if we consider the colour flow in an event. • QCD radiation only occurs in a cone up to the direction of the colour partner. • The best choice of evolution variable is therefore an angular one. Emitter Colour Partner Yeti 09 12th

17. Colour Coherence • Angular Ordering and ColourCoherence are often used interchangeably in talks etc.. • However there is a difference. • Colour Coherence is the phenomena that a soft gluon can’t resolve a small angle pair of particles and so only sees the colour charge of the pair. • Angular Ordering is a way of implementing colour coherence in parton shower simulations. Yeti 09 12th

18. Running Coupling • It is often said that Monte Carlo event generators are leading-log. • However they include many effects beyond leading log, e.g. • Momentum Conservation • Running Coupling Effects • Effect of summing higher orders is absorbed by replacing as with as(kT2). • Gives more soft gluons, but must avoid the Landau pole which makes the cut-off a physical parameter. Yeti 09 12th

19. The Colour Dipole Model • The standard parton shower approach starts from the collinear limit and makes changes to include soft gluon coherence. • The Colour Dipole Model starts from the soft limit. • Emission of soft gluons from the colour-anticolour dipole is universal. • After emitting a gluon, the colour dipole splits into two new dipoles i Yeti 09 12th

20. Parton Shower • ISAJET uses the original parton shower algorithm which only resums collinear logarithms. • PYTHIA originally used the collinear algorithm with an angular veto to try to reproduce the effect of the angular ordered shower. • HERWIG uses the angular ordered parton shower algorithm which resums both soft and collinear singularities. • SHERPA uses the PYTHIA algorithm. • ARIADNE uses the colour dipole model. Yeti 09 12th

21. Parton Shower • Recently there as been a lot of work on new parton shower algorithms. • Later versions of PYTHIA and Pythia8 use a pT order shower. • SHERPA now includes two alternative dipole based shower modules. • Other approaches are being studied. Yeti 09 12th

22. LEP Event Shapes Yeti 09 12th

23. Energy Dependence Yeti 09 12th

24. Hard Jet Radiation • The parton shower is designed to simulate soft and collinear radiation. • While this is the bulk of the emission we are often interested in the radiation of a hard jet. • This is not something the parton shower should be able to do, although it often does better than we except. • If you are looking at hard radiation HERWIG/PYTHIA will often get it wrong. Yeti 09 12th

25. Hard Jet Radiation • Given this failure of the approximations this is an obvious area to make improvements in the shower and has a long history. • You will often here this called: • Matrix Element matching; • Matrix Element corrections; • Merging matrix elements and parton shower; • MC@NLO/POWHEG. • I will discuss all of these and where the different ideas are useful. Yeti 09 12th

26. Hard Jet Radiation: General Idea • Parton Shower (PS) simulations use the soft/collinear approximation: • Good for simulating the internal structure of a jet; • Can’t produce high pT jets. • Matrix Elements (ME) compute the exact result at fixed order: • Good for simulating a few high pT jets; • Can’t give the structure of a jet. • We want to use both in a consistent way, i.e. • ME gives hard emission • PS gives soft/collinear emission • Smooth matching between the two. • No double counting of radiation. Yeti 09 12th

27. Matching Matrix Elements and Parton Shower Parton Shower • The oldest approaches are usually called matching matrix elements and parton showers or the matrix element correction. • Slightly different for HERWIG and PYTHIA. • In HERWIG HERWIG phase space for Drell-Yan Dead Zone • Use the leading order matrix element to fill the dead zone. • Correct the parton shower to get the leading order matrix element in the already filled region. • PYTHIA fills the full phase spaceso only the second step is needed. Yeti 09 12th

28. Matrix Element Corrections Z qT distribution from CDF W qT distribution from D0 G. Corcella and M. Seymour, Nucl.Phys.B565:227-244,2000. Yeti 09 12th

29. Recent Progress • In the last few years there has been a lot of work addressing both of these problems. • Two types of approach have emerged • NLO Simulation • NLO normalization of the cross section • Gets the hardest emission correct • Multi-Jet Leading Order • Still leading order. • Gets many hard emissions correct. Yeti 09 12th

30. NLO Simulation • There has been a lot of work on NLO Monte Carlo simulations. • The MC@NLO approach of Frixione, Nason and Webber and the POWHEG approach of Nason are the only techniques which have been shown to work in practice. Yeti 09 12th

31. MC@NLO • The basic idea of MC@NLO is: • work out the shower approximation for the real emission; • subtract it from the real emission; • add it to the virtual piece. • This cancels the singularities and avoids double counting. • It’s a lot more complicated than it sounds. Yeti 09 12th

32. MC@NLO • For each new process the shower approximation must be worked out, which is often complicated. • While the general approach works for any shower it has to be worked out for a specific case. • So for MC@NLO only works with the HERWIG shower algorithm, an implementation for Herwig++ is in progress. • It could be worked out for other algorithms but this remains to be done. Yeti 09 12th

33. MC@NLO HERWIG NLO Top Production S. Frixione, P. Nason and B.R. Webber, JHEP 0308(2003) 007, hep-ph/0305252. Yeti 09 12th

34. POWHEG • Alternative approach POsitive Weight Hardest Emission Generator of Nason et. al.. • Here: • Generate the kinematical variables for the leading order process with Next-to-leading order accuracy; • Generate the hardest emission by exponentiating the real emission matrix element ordering in pT; • Some modifications needed if shower not pT ordered. • Has NLO accuracy but different sub-leading corrections to MC@NLO. Yeti 09 12th

35. Top Production Taken from JHEP 0709:126,2007 Frixione et. al. Yeti 09 12th

36. Multi-Jet Leading Order • While the NLO approach is good for one hard additional jet and the overall normalization it cannot be used to give many jets. • Therefore to simulate these processes use matching at leading order to get many hard emissions correct. • I will briefly review the general idea behind these approaches and then show some results. Yeti 09 12th

37. CKKW Simulate N jet partonic state. Apply weight factors for probability that no jets emitted above matching scale. Generate shower vetoing radiation above the matching scale. The weight factors ensure the different samples can be added. MLM Simulate partonic N jet state. Generate parton shower. Require that all the jets above the matching scale after the shower have an associated pre-shower parton. For each N the shower doesn’t add any more jets. Rejection ensures that samples with different numbers of jets can be summed Two approaches Yeti 09 12th

38. Two approaches • While the CKKW approach is more rigorous the approaches are similar and the MLM method is easier to implement. • The rejection of events without a match between the pre- and post-shower jets in the MLM approach plays the same role as the weight factors and veto in the CKKW approach. Yeti 09 12th

39. CKKW results for Z +jets Yeti 09 12th

40. CKKW results for Z +jets Yeti 09 12th

41. CKKW results for Z +jets Yeti 09 12th

42. MLM Method for W+jets Yeti 09 12th

43. Hadronization • Partons aren’t physical particles: they can’t propagate freely. • We therefore need to describe the transition of the quarks and gluons in our perturbative calculations into the hadrons which can propagate freely. • We need a phenomenological model of this process. • There are three models which are commonly used. • Independent Fragmentation • Lund String Model • Cluster Model Yeti 09 12th

44. Independent Fragmentation Model“Feynman-Field” • The longitudinal momentum distribution is given by an arbitrary fragmentation function which is a parameterization of data. • The transverse momentum distribution is Gaussian. • The algorithm recursively splits qgq’+hadron. • The remaining soft quark and antiquark are connected at the end. • The model has a number of flaws • Strongly frame dependent • No obvious relation with the perturbative physics. • Not infrared safe • Not a confinement model • Wrong energy dependence. Yeti 09 12th

45. + - Confinement • We know that at small distances we have asymptotic freedom and the force between a quark-antiquark pair is like that between an e+e- pair. • But at long distances the self interactions of the gluons make the field lines attract each other. • A linear potential at long distances and confinement. QED QCD Yeti 09 12th

46. Lund String Model • In QCD the field lines seem to be compress into a tube-like region, looks like a string. • So we have linear confinement with a string tension. • Separate the transverse and longitudinal degrees of freedom gives a simple description as a 1+1 dimensional object, the string, with a Lorentz invariant formalism. Yeti 09 12th

47. i i Intraquark potential • In the real world the string can break non-perturbatively by producing quark-antiquark pairs in the intense colour field. • This is the basic physics idea behind the string model and is very physically appealing. Quenched QCD Full QCD Yeti 09 12th

48. Preconfinement • In the planar approximation, large number of colours limit Gluon= colour-anticolour pair • We can follow the colour structure of the parton shower. • At the end colour singlet pairs end up close in phase space. • Non-perturbatively split the gluons into quark-antiquark pairs. Yeti 09 12th

49. Preconfinement • The mass spectrum of colour-singlet pairs is asymptotically independent of energy and the production mechanism. • It peaks at low mass, of order the cut-off Q0. Yeti 09 12th

50. The Cluster Model • Project the colour singlet clusters onto the continuum of high-mass mesonic resonances (=clusters). • Decay to lighter well-known resonances and stable hadrons using • Pure 2 body phase-space decay and phase space weight • The hadron-level properties are fully determined by the cluster mass spectrum, i.e. by the properties of the parton shower. • The cut-off Q0 is the crucial parameter of the model. Yeti 09 12th