# Presentation: 519474

## Recent Developments in Monte Carlo Event Generators

- Peter Richardson
- IPPP, Durham University

**Durham University**

## Summary

- Introduction
- Basics of Monte Carlo Simulations
- Parton Shower Improvements
- Simulation of Hard Radiation
- Matrix Element Corrections
- MC@NLO
- CKKW
- Future

## Introduction

- Monte Carlo event generators are essential for experimental particle physics analysis.
- They are used for:
- Comparison of experimental results with theoretical predictions;
- Studies for future experiments.
- Often these programs are ignored by theorists and treated as black boxes by experimentalists.
- It is important to understand the assumptions and approximations involved.

## Introduction

- Experimental physicists need to be able to answer the following questions
- Should the simulation describe my data?
- Is the effect I’m seeing due to different models, or approximations, or is it a bug?
- Am I measuring a fundamental quantity or merely a parameter of the simulation code?
- Theorists need to understand enough to be able ask
- Have the experimentalists misused the Monte Carlo giving incorrect results?

## Introduction

- Often if an experimentalist is asked how a given background or process was calculated the answer will be
- Well I ran HERWIG
- However there have been a lot of developments in recent years.
- There are many improvements on the default general purpose generators which should be used.

- Well I ran PYTHIA

## Introduction

- Monte Carlo simulations can be used to simulate a wide range of processes.
- The experiments at the high-energy frontier for the next 10 years will be hadron colliders, Tevatron and LHC.
- For hadron collisions, regardless of what signal you are looking at, getting the QCD right is the most important thing.
- I will concentrate on recent progress in the simulation of processes which are important for the Tevatron and LHC.

## Monte Carlo Event Generators

- There are a number of Monte Carlo event generators in common use
- PYTHIA
- HERWIG
- SHERPA
- They all split the event generation up into the same pieces.
- The models and approximations they use for the different pieces are of course different.

## C++ Generators

- Most of these programs are written in Fortran 77
- There are ongoing projects to rewrite HERWIG and PYTHIA in C++.
- Some of the newer projects, SHERPA, are also in C++.

## A Monte Carlo Event

- Initial and Final State parton showers resum the large QCD logs.

- Hard Perturbative scattering:
- Usually calculated at leading order in QCD, electroweak theory or some BSM model.

- Perturbative Decays calculated in QCD, EW or some BSM theory.

- Multiple perturbative scattering.

- Non-perturbative modelling of the hadronization process.

- Modelling of the soft underlying event

- Finally the unstable hadrons are decayed.

## Monte Carlo Event Generators

- All the event generators split the simulation up into the same phases:
- Hard Process;
- Parton Shower;
- Secondary Decays;
- Multiple Scattering/Soft Underlying Event;
- Hadron Decays.
- I will briefly discuss the different models and approximations in the different programs.
- I will try and give a fair and objective comparision, but bear in mind that I’m one of the authors of HERWIG.

## QCD Radiation

- It is impossible to calculate and integrate the 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.
- The different generators differ in the sophistication of their simulation of this.

## Collinear Singularities

- In the collinear limit the cross section for a process factorizes
*P**ji**(z)* is the DGLAP splitting function
- This expression is singular as         .
- What is a parton? (or what is the difference between a collinear pair and a parton)

## Collinear Singularities

- Introduce a resolution criterion, e.g.
- Combine the virtual corrections and unresolvable emission

- Resolvable Emission
- Finite

- Unresolvable Emission
- Finite

- Unitarity: Unresolved + Resolved =1

## Monte Carlo Procedure

- Using this approach we can exponentiate the real emission piece.
- This gives the Sudakov form factor which is the probability of evolving between two scales and emitting no resolvable radiation.
- More strictly it is the probability of evolving from a high scale to the cut-off with no resolvable emission.

## Monte Carlo Procedure

- The key difference between the different Monte Carlo simulations is in the choice of the evolution variable.
- Evolution Scale
- Virtuality, *q**2*
- Transverse Momentum, *k**T**.*
- Angle, 
- ....
- Energy fraction, *z*
- Energy fraction
- Light-cone momentum fraction
- ....
- All are the same in the collinear limit.

## Soft Emission

- However 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?

## Angular Ordering

- 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.

- Colour Flow

- Emitter

- Colour Partner

## Parton Shower

- HERWIG uses the angular ordered parton shower algorithm which resums both soft and collinear singularities.
- PYTHIA uses the collinear algorithm with an angular veto to try to reproduce the effect of the angular ordered shower.
- SHERPA uses the PYTHIA algorithm.

## LEP Event Shapes

## Dipole Showers

- The best agreement with the LEP data was obtained using ARIADNE which is based on the dipole approach.
- This is based on 2      3 splittings rather than 1   2 which makes it easier to conserve momentum.
- The soft and collinear are included in a consistent way.
- The initial state shower is more difficult in this approach though.

## Third Jet Distribution at the Tevatron

## Parton Shower

- In general event generators which include angular ordering, colour coherence, give the best agreement with data.
- There however remain a number of problems
- Treatment of radiation from massive particles, particularly the top.
- The angular veto used in PYTHIA is not exactly the same as angular ordering.
- The angular-ordered algorithm is highly non-Lorentz invariant.
- Problems with wide angle gluon emission.
- ....

## Parton Showers- Recent Developments

- There are other problems which have hindered some of the other developments I will talk about later.
- Much of the recent work on parton showers has been on simulating hard radiation which I will talk about later.
- There are however some other improvements.
- The major new ideas are
- An improved coherent parton shower using massive splitting functions.
- A transverse momentum ordered shower.
- In practice these have implications for the simulation of hard radiation as well.

## Herwig++ Shower

- Based on using the massive splitting functions of Catani et.al. Phys.Lett.B500:149-160,2001.
- Also a new parameterization of the momenta of the particles produced in the shower.
- Improves
- Treatment of radiation from heavy particles.
- Lorentz invariance of the algorithm.

## Herwig++ Shower

- Gieseke et. al.,
-     JHEP 0402:005,2004 JHEP 0312:045,2003.
- Previously no radiation with angle less than ~m/E, the ‘dead-cone’
- This approximation was too severe.
- The new approach allows some radiation inside the ‘dead-cone’.

## Event Shapes

- Momentum transverse to the thrust axis in the event plane.

- Momentum transverse to the thrust axis out of the event plane.

## Hadron Collisions

- S. Gieseke et. al. hep-ph/0602069

## PT ordered shower

- T. Sjostrand, P.Z. Skands, hep-ph/0408302.
- Order the shower in transverse momentum rather than angle or virtuality.
- Still remains to shown that the coherence properties are correct, although should be O.K..
- Can be used in new ideas in multiple scattering and the underlying event.
- Has coherence and allows easier matching with hard matrix elements.

## PT of Z compared to CDF data

- Low PT

- All PT

## Hadronization

- As the hadronization is less important for what I will say later and there’s been less progress I will only briefly mention the different models.
- PYTHIA uses the Lund string model.
- HERWIG uses the cluster hadronization model.
- ARIADNE and SHERPA use the Lund model from PYTHIA.
- The cluster model gives good agreement with LEP data on event shapes but doesn’t fit the identified particle spectrum as well.
- The Lund string model gives the best agreement with data.

## Hadronization

- There have been some developments
- Improved treatment of the junction of three strings, which happens in baryon formation in the string model, T. Sjostrand and P. Skands NPB 659: 243, 2003.
- Improvements to the cluster model in Herwig++, S. Gieseke et. al. JHEP 0402: 005, 2004, and in the SHERPA implementation, Winter et. al., EPJ C36: 381, 2004.

## Signal Simulation

- In general we have become very good at simulating signals, be that top quark production, SUSY or other BSM physics.
- In many cases the simulations are very detailed including correlation effects.
- This should be good enough for top and is certainly good enough for things that haven’t been seen yet.

## Signal Simulation

- Angle between the lepton in top decay and the beam for top pair production at a 500 GeV linear collider.

## Hard Jet Radiation

- I’ve tried to show you that 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 and PYTHIA will often get it wrong.

## 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.

## 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.

## 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

## Matrix Element Corrections

- G. Corcella and M. Seymour, Nucl.Phys.B565:227-244,2000.

- W qT distribution from D0

- Z qT distribution from CDF

## 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.

## 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.

## 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.
- 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/webber/MCatNLO/

## MC@NLO

- MC@NLO was designed to have the following features.
- The total rate is accurate to NLO
- 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.

## 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.

## Toy Model

- Calculating an observable at NLO gives
- 	where the Born, Virtual and Real contributions are
- 	 is the coupling constant and

## 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                   .

## Toy MC@NLO

- The Monte Carlo simulation will generate photons with the distribution
- Main idea of MC@NLO is to subtract this from the real emission and add it back to the leading order.

## Toy MC@NLO

- 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.

## 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.
- 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, (work in progress for Herwig++).

## W+W- Observables

- MC@NLO gives the correct high PT  result and soft resummation.

- PT of W+W-

-  of W+W-

- S. Frixione and B.R. Webber JHEP 0206(2002) 029, hep-ph/0204244, hep-ph/0309186

## W+W- Jet Observables

- S. Frixione and B.R. Webber JHEP 0206(2002) 029, hep-ph/0204244, hep-ph/0309186

## Top Production

- S. Frixione, P. Nason and B.R. Webber, JHEP 0308(2003) 007, hep-ph/0305252.

## Top Production at the LHC

- S. Frixione, P. Nason and B.R. Webber, JHEP 0308(2003) 007, hep-ph/0305252.

## B Production at the Tevatron

- S. Frixione, P. Nason and B.R. Webber, JHEP 0308(2003) 007, hep-ph/0305252.

## 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.

## 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 kT-type jet algorithm.

## 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.
- 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
- Distribute the jet momenta according the ME.

## CKKW Procedure

- Cluster the partons to determine the values at which 1,2,..*n*-jets are resolved. These give the nodal scales for a tree diagram.
- Apply a coupling constant reweighting.

## CKKW Procedure

- Reweight the lines by a Sudakov factor
- Accept the configuration if the product of the S and Sudakov weight is less than            otherwise return to step 1.

## CKKW Procedure

- 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       .

## CKKW Procedure

- The procedure allows different jet multiplicity matrix elements to be consistently added together.
- 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 kT algorithm.
- In practice the problem is understanding what the shower is doing and treating the matrix element in the same way.

## 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)

## pT of the W at the Tevatron

- S. Mrenna and PR JHEP 0405: 04 (2004)

## pT of the W at the Tevatron with SHERPA

## LHC ET of the 4th jet

- S. Mrenna and PR JHEP 0405: 04 (2004)

## 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.

## 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.

## 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

## 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 pT 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.

## CKKW

- Progress has been made.
- Main problem is understanding the shower approximation.
- Hopefully the new Herwig++ and pT 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.

## 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)

## Future

- A lot of progress has been made in improving Monte Carlo simulations in the last five years.
- We know how to make a lot more.
- However Monte Carlo community is very small.
- There are three major projects
- HERWIG (3 permanent staff, 5 postdocs, 1 student, ~3FTE)
- PYTHIA (3 permanent staff, 1 postdoc,~2FTE)
- SHERPA (1 permanent staff, 4 students,~4FTE)

## Monte Carlo Collaborations

- Pythia/Ariadne
- L. Lonnblad
- S. Mrenna
- T. Sjostrand
- P. Skands
- +1 student

- Herwig
- P. Richardson, M. Seymour, B. Webber
- S. Gieseke, D. Grellscheid, K. Hamilton,  A. Ribon, P. Stephens
- +3 students

- SHERPA
- F. Krauss  +4 students

-    Permanent Staff
-    Postdoc

## Future

- Given the large demand for both support and development this is not sustainable in the long term.
- We know how to construct the tools for the LHC.
- However it is likely that not all the improvements we now know how to make will be available for the start of the LHC.