1. YetiSM 28 th March1 Higher Orders in Parton Shower Monte Carlos Peter Richardson IPPP, Durham University Durham University

2. YetiSM 28 th March2 Summary Introduction Leading Order calculations Simulation of Hard Radiation –Matrix Element Corrections –MC@NLO –CKKW Conclusions

3. YetiSM 28 th March3 Introduction Advances in QCD are usually in one of –Fixed order Calculations, either Processes with more external particles at some order Processes at higher orders. –Better resummation techniques –or better ways to combine resummation techniques and fixed order calculations. The last is often a Monte Carlo implementation.

4. YetiSM 28 th March4 Introduction Most of the recent work has been on calculating processes with more external particles at leading-order. I will concentrate on this at the beginning mainly to give my opinion on this work. I will then spend the rest of the talk on recent progress in simulating hard QCD radiation in Monte Carlo event generators.

5. YetiSM 28 th March5 High multiplicity final states There has recently been a renewal of interest in calculating matrix elements with high numbers of outgoing particles. The recent progress is all based on ideas of Witten, hep-th/0312171. While there has undoubtedly been progress it has yet to be used to give phenomenological results, i.e. numbers.

6. YetiSM 28 th March6 High multiplicity final states There are two parts to calculating the cross section for a high multiplicity final state. 1)Calculating the matrix element for a given point in phase space. 2)Integrating it over the phase space with whatever cuts are required. The new ideas aim to give better methods to calculate the matrix element.

7. YetiSM 28 th March7 Matrix Element Calculations There are a number of ways of calculating the matrix element 1)Trace Techniques – We all learnt this as students, but spend goes like N 2 with the number of diagrams A 1 n B 1 n + + 2 = 2 A 1 n + 2 B 1 n A 1 n B 1 n * * + + + B 1 n A 1 n * *

8. YetiSM 28 th March8 Matrix Element Calculations 2)Helicity Amplitudes –Work out matrix element as a complex number, sum up the diagrams and then square. – Numerically, grows like N with the number of diagrams. A 1 n B 1 n + + 2

9. YetiSM 28 th March9 Matrix Element Calculations 3)Off-Shell recursion relations (Berends- Giele, ALPHA, Schwinger-Dyson.) –Don’t draw diagrams at all use a recursion relation to get final states with more particles from those with fewer particles.

10. YetiSM 28 th March10 Matrix Element Calculations n 1 m l+1 = n m+1 m 1 n l 1 + Blue lines are off-shell gluons, other lines are on-shell. Full amplitudes are built up from simpler amplitudes with fewer particles. This is a recursion built from off-shell currents.

11. YetiSM 28 th March11 4)All the new techniques are essential on- shell recursion relations. The first results Cachazo, Svrcek and Witten were for the combination of maximum helicity violating (MHV) amplitudes. Has two negative helicity gluons (all others positive) MHV/BCFW Recursion + + + + - -

12. YetiSM 28 th March12 MHV Recursion Here all the particles on-shell, with rules for combining them. + + + - - - + + + - - = - - +

13. YetiSM 28 th March13 BCFW Recursion Initially only for gluons. Many developments since then culminating (for the moment) in the BCFW recursion relations. This has been generalised to massive particles.

14. YetiSM 28 th March14 Integration All of this is wonderful for evaluating the matrix element at one phase space point. However it has to be integrated. The problem is that the matrix element has a lot of peaks. Makes numerical integration tricky.

15. YetiSM 28 th March15 Integration There are essentially two techniques. 1)Adaptive integration VEGAS and variants –Automatically smooth out peaks to reduce error –Often not good if peaks not in one of the integration variables 2)Multi-Channel integration –Analytically smooth out the peaks in many different channels –Automatically optimise the different channels.

16. YetiSM 28 th March16 Integration In practice the best programs use a combination of the two. –Multichannel to get rid of the worst behaviour –Then adaptive to make it more efficient.

17. YetiSM 28 th March17 Programs TraceHelicity Amplitude Off-shell recursion On-Shell Recursion Adaptive COMPHEP CALCHEP Slowest NoneALPGEN Faster None Multi Channel NoneMADEVENT SHERPA Fastest None Faster

18. YetiSM 28 th March18 Integration The integration of the matrix elements is the problem not the calculation of the matrix element. The best programs have the best integration NOT the best calculation of the matrix element.

19. YetiSM 28 th March19 Integration I found a military saying –Amateurs study tactics, professionals study logistics. I think the appropriate Monte Carlo equivalent would be –Amateurs study matrix elements, professionals study phase space.

20. YetiSM 28 th March20 Calculation of Matrix Elements The MHV … work is exciting, it may turn out to be useful in useful calculations, particularly if it can be extended to loops in real QCD. However at the moment calculating the matrix element it not the real bottleneck for tree-level calculations, the phase space is.

21. YetiSM 28 th March21 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.

22. YetiSM 28 th March22 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 I will discuss all of these and where the different ideas are useful.

23. YetiSM 28 th March23 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 p T jets. Matrix Elements (ME) compute the exact result at fixed order: –Good for simulating a few high p T 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.

24. YetiSM 28 th March24 Matching Matrix Elements and 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 –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 space so only the second step is needed. Parton Shower Dead Zone HERWIG phase space for Drell- Yan

25. YetiSM 28 th March25 Matrix Element Corrections G. Corcella and M. Seymour, Nucl.Phys.B565:227-244,2000. W q T distribution from D0 Z q T distribution from CDF

26. YetiSM 28 th March26 Matrix Element Corrections There was a lot of work for both HERWIG and PYTHIA. The corrections for –e + e - to hadrons –DIS –Drell-Yan –Top Decay –Higgs Production were included. There are problems with this –Only the hardest emission was correctly described –The leading order normalization was retained.

27. YetiSM 28 th March27 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 1)NLO Simulation NLO normalization of the cross section Gets the hardest emission correct 2)Multi-Jet Leading Order Still leading order. Gets many hard emissions correct.

28. YetiSM 28 th March28 NLO Simulation There has been a lot of work on NLO Monte Carlo simulations. However apart from some early work by Dobbs the only Frixione, Nason and Webber have produced code which can be used to generate results. I will therefore only talk about the work of Frixione, Nason and Webber.

29. YetiSM 28 th March29 MC@NLO S. Frixione and B.R. Webber JHEP 0206(2002) 029, hep-ph/0204244, hep- ph/0309186 S. Frixione, P. Nason and B.R. Webber, JHEP 0308(2003) 007, hep-ph/0305252. http://www.hep.phy.cam.ac.uk/theory/webb er/MCatNLO/

30. YetiSM 28 th March30 MC@NLO MC@NLO was designed to have the following features. –The output is a set of fully exclusive events. –The total rate is accurate to NLO –NLO results for observables are recovered when expanded in  s. –Hard emissions are treated as in NLO calculations. –Soft/Collinear emission are treated as in the parton shower. –The matching between hard emission and the parton shower is smooth. –MC hadronization models are used.

31. YetiSM 28 th March31 Basic Idea The basic idea of MC@NLO is –Work out the shower approximation for the real emission. –Subtract it from the real emission from – Add it to the virtual piece. This cancels the singularities and avoids double counting. It’s a lot more complicated than it sounds.

32. YetiSM 28 th March32 Toy Model I will start with Bryan Webber’s toy model to explain MC@NLO to discuss the key features of NLO, MC and the matching. Consider a system which can radiate photons with energy with energy with where is the energy of the system before radiation. After radiation the energy of the system Further radiation is possible but photons don’t radiate.

33. YetiSM 28 th March33 Toy Model Calculating an observable at NLO gives where the Born, Virtual and Real contributions are  is the coupling constant and

34. YetiSM 28 th March34 Toy Model In a subtraction method the real contribution is written as The second integral is finite so we can set The NLO prediction is therefore

35. YetiSM 28 th March35 Toy Monte Carlo In a MC treatment the system can emit many photons with the probability controlled by the Sudakov form factor, defined here as where is a monotonic function which has is the probability that no photon can be emitted with energy such that.

36. YetiSM 28 th March36 Toy MC@NLO We want to interface NLO to MC. Naïve first try – start MC with 0 real emissions: – start MC with 1 real emission at x: So that the overall generating functional is This is wrong because MC with no emissions will generate emission with NLO distribution

37. YetiSM 28 th March37 Toy MC@NLO We must subtract this from the second term This prescription has many good features: –The added and subtracted terms are equal to –The coefficients of and are separately finite. –The resummation of large logs is the same as for the Monte Carlo renormalized to the correct NLO cross section. However some events may have negative weight.

38. YetiSM 28 th March38 Toy MC@NLO Observables As an example of an “exclusive” observable consider the energy y of the hardest photon in each event. As an “inclusive” observable consider the fully inclusive distributions of photon energies, z Toy model results shown are for

39. YetiSM 28 th March39 Toy MC@NLO Observables

40. YetiSM 28 th March40 Real QCD For normal QCD the principle is the same we subtract the shower approximation to the real emission and add it to the virtual piece. This cancels the singularities and avoids double counting. It’s a lot more complicated.

41. YetiSM 28 th March41 Problems 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. It could be worked out for PYTHIA or Herwig++ but this remains to be done.

42. YetiSM 28 th March42 W + W - Observables MC@NLO HERWIG NLO MC@NLO gives the correct high P T result and soft resummation. P T of W+W-  of W+W- S. Frixione and B.R. Webber JHEP 0206(2002) 029, hep-ph/0204244, hep-ph/0309186

43. YetiSM 28 th March43 W + W - Jet Observables MC@NLO HERWIG NLO S. Frixione and B.R. Webber JHEP 0206(2002) 029, hep-ph/0204244, hep-ph/0309186

44. YetiSM 28 th March44 Top Production MC@NLO HERWIG NLO S. Frixione, P. Nason and B.R. Webber, JHEP 0308(2003) 007, hep-ph/0305252.

45. YetiSM 28 th March45 Top Production at the LHC S. Frixione, P. Nason and B.R. Webber, JHEP 0308(2003) 007, hep-ph/0305252. MC@NLO HERWIG NLO

46. YetiSM 28 th March46 B Production at the Tevatron S. Frixione, P. Nason and B.R. Webber, JHEP 0308(2003) 007, hep-ph/0305252.

47. YetiSM 28 th March47 Higgs Production at LHC S. Frixione and B.R. Webber JHEP 0206(2002) 029, hep-ph/0204244, hep-ph/0309186

48. YetiSM 28 th March48 NLO Simulation So far MC@NLO is the only implementation of a NLO Monte Carlo simulation. Recently there have been some ideas by Paulo Nason JHEP 0411:040,2004. Here there would be no negative weights but more terms would be exponentiated beyond leading log. This could be an improvement but we will need to see physical results.

49. YetiSM 28 th March49 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 this approach and then show some results.

50. YetiSM 28 th March50 CKKW Procedure Catani, Krauss, Kuhn and Webber JHEP 0111:063,2001. In order to match the ME and PS we need to separate the phase space: One region contains the soft/collinear region and is filled by the PS; The other is filled by the matrix element. In these approaches the phase space is separated using in k T -type jet algorithm.

51. YetiSM 28 th March51 Durham Jet Algorithm For all final-state particles compute the resolution variables The smallest of these is selected. If is the smallest the two particles are merged. If is the smallest the particle is merged with the beam. This procedure is repeated until the minimum value is above some stopping parameter. The remaining particles and pseudo-particles are then the hard jets.

52. YetiSM 28 th March52 CKKW Procedure Radiation above a cut-off value of the jet measure is simulated by the matrix element and radiation below the cut-off by the parton shower. 1)Select the jet multiplicity with probability where is the n-jet matrix element evaluated at resolution using as the scale for the PDFs and  S, n is the jet of jets 2)Distribute the jet momenta according the ME.

53. YetiSM 28 th March53 CKKW Procedure 3)Cluster the partons to determine the values at which 1,2,..n-jets are resolved. These give the nodal scales for a tree diagram. 4)Apply a coupling constant reweighting.

54. YetiSM 28 th March54 CKKW Procedure 5)Reweight the lines by a Sudakov factor 6)Accept the configuration if the product of the  S and Sudakov weight is less than otherwise return to step 1.

55. YetiSM 28 th March55 CKKW Procedure 7)Generate the parton shower from the event starting the evolution of each parton at the scale at which it was created and vetoing emission above the scale.

56. YetiSM 28 th March56 CKKW Procedure Although this procedure ensures smooth matching at the NLL log level are still choices to be made: –Exact definition of the Sudakov form factors. –Scales in the strong coupling and  S. –Treatment of the highest Multiplicity matrix element. –Choice of the k T algorithm. In practice the problem is understanding what the shower is doing and treating the matrix element in the same way.

57. YetiSM 28 th March57 CKKW Procedure A lot of work has been done mainly by –Frank Krauss et. al. (SHERPA) –Leif Lonnblad (ARIADNE) –Steve Mrenna (PYTHIA) –Peter Richardson (HERWIG)

58. YetiSM 28 th March58 e + e - Results from SHERPA

59. YetiSM 28 th March59 p T of the W at the Tevatron from HERWIG ME HW 0 jets 1 jets 2 jets 3 jets 4 jets

60. YetiSM 28 th March60 p T of the W at the Tevatron with SHERPA

61. YetiSM 28 th March61 p T of the hardest jet at the Tevatron in W production from HERWIG ME HW 0 jets 1 jets 2 jets 3 jets 4 jets

62. YetiSM 28 th March62 p T of the hardest jet at the Tevatron in W production from SHERPA

63. YetiSM 28 th March63 WW at the Tevatron with SHERPA

64. YetiSM 28 th March64 Tevatron p T of the 4th jet from HERWIG ME HW 0 jets 1 jets 2 jets 3 jets 4 jets

65. YetiSM 28 th March65 LHC pt of W from HERWIG ME HW 0 jets 1 jets 2 jets 3 jets 4 jets

66. YetiSM 28 th March66 p T of the Z at the LHC with SHERPA

67. YetiSM 28 th March67 p T of the hardest jet at the LHC in Z production from SHERPA

68. YetiSM 28 th March68 p T of the third jet at the LHC in Z production from SHERPA

69. YetiSM 28 th March69 LHC E T of the 4th jet from HERWIG ME HW 0 jets 1 jets 2 jets 3 jets 4 jets

70. YetiSM 28 th March70 What Should I use? Hopefully this talk will help you decide which of the many different tools is most suitable for a given analysis. –Only soft jets relative to hard scale MC –Only one hard jet MC@NLO or old style ME correction –Many hard jets CKKW. The most important thing is to think first before running the simulation.

71. YetiSM 28 th March71 Future There has been a renewal of interest in the physics in the parton shower. There has been a lot of theoretical work. I will used briefly talk about a few of the more realistic ideas.

72. YetiSM 28 th March72 Large angle gluon Emission There has been a lot of interest in wide angle soft emission. Still unclear how to treat it. However it has been pointed out there are related problems, Nason JHEP 0411: 040, 2004 Although the average is correct. Some ideas about fixing this. –50% of events have lots of radiation –50% of events have no radiation

73. YetiSM 28 th March73 Alternative approach to MC@NLO Another approach for MC@NLO has been suggested, Nason JHEP 0411: 040, 2004. First reorganise the angular ordered-shower so the hardest emission is generated first. Wouldn’t be needed for p T ordered showers. Then generate this emission with a modified Sudakov form factor. Exponentiates different terms beyond the accuracy of the approximations. Remains to see what would happen in practice.

74. YetiSM 28 th March74 CKKW Progress has been made. Main problem is understanding the shower approximation. Hopefully the new Herwig++ and p T ordered PYTHIA showers will have better properties for the matching. Some problems with the treatment of the colour flow were noted in Nason JHEP 0411: 040, 2004, and have given problems with hadronization. Need improved approaches for handling the colour flows.

75. YetiSM 28 th March75 CKKW CKKW gives the right amount of radiation But puts some of it in the wrong place with the wrong colour flow S. Mrenna and PR JHEP 0405: 04 (2004)

76. YetiSM 28 th March76 Conclusions There has been progress in new ways of calculation matrix elements with many external particles. This is not really useful for phenomenology yet. In recent years new techniques have been developed for the simulation of hard radiation. You should be using these where appropriate rather than just relying on HERWIG and PYTHIA.