Modern Approach to Monte Carlo’s

1. Modern Approach to Monte Carlo’s • (L1) The role of resolution in Monte Carlo’s • (L1) Leading order Monte Carlo’s • (L1) Next-to-Leading order Monte Carlo’s • (L2) Parton Shower Monte Carlo’s Academic Lectures, Walter Giele, Fermilab 2006

2. Parton Shower MC’s • We want to go beyond NLO and reduce the resolution further all the way to the hadronization scale. • In principle we can do this by calculating NNLO,NNNLO,… • However in practice this proves out very complicated. Already a NNLO MC for 2 jet production is very complicated (not due to the 2-loop diagrams, but the master equation and its numerical implementation for NNLO MC’s) Exclusive 2 jet fraction at NLO • Exclusivity is important to us: • Detector response is at the hadron level (i.e. fully exclusive final state) • More detailed understanding of jet structure • exclusive jet final states are of importance for e.g. LHC physics (e.g. suppressing backgrounds by jet vetoing for Higgs produced by vector boson fusion)

3. Parton Shower MC’s • We need to approximate the higher order corrections which are associated with resolving additional clusters as we reduce the resolution scale. • For a single dipole we know the process independent soft/collinear function used in the NLO MC master equation: • This function gives the right description of additional radiation in the dipole color field at small resolution scales • For larger resolution scales there is arbitrariness in this approximation function. For harder resolutions we rely on the LO/NLO MC’s to give us the initial dipoles from which we start the shower MC. • We will use this soft/collinear approximation function as a probability density for radiating an additional parton in a dipole color field

4. Parton Shower MC’s • We can calculate of not resolving a new cluster at a resolution scale in the dipole: • This gives us • We forget multiple emissions! It is very likely to have multiple branching at small resolution scales…

5. Parton Shower MC’s The change in the Sudakov factor (i.e. the likelyhood of not resolving a new cluster in the dipole) by lowering the resolution scale is a product of no emission up to the resolution scale and the emission probability at the resolution scale: • Green line: approximating NLO • Blue line: approximating NNLO • Purple line: approximating NNNLO • Red line: All “Leading Logs” are resummed The Sudakov factor estimates the likelyhood of not resolving an additional cluster in the dipole at the resolution scale.

6. Parton Shower MC’s (log scale)

7. Parton Shower MC’s • From the dipole Sudakov factor we can construct the event dipole factorwhich gives us the likelyhood not to resolve a cluster anywhere in the event at a given resolution scale • The problem now is that the resolution criterion is a theoretical construct defined in a color ordered dipole phase space. • The solution is to turn the Sudakov calculation in a Monte Carlo such that the experimental cuts and jet definitions can be numerically implemented. • The shower MC will start from an ordered set of initial partons generated by a LO/NLO MC at the hard scattering resolution scale. • By lowering the resolution scale more and more additional clusters will be resolved based on the event Sudakov factor (which changes after each newly resolved cluster). • Eventually we reach the hadronization scale.

8. Parton Shower MC’s • We start with a set of n partons at a resolution scale • The probability density to resolve a cluster in the event is given by • solve for where r is an uniform random number between 0 and 1 • next pick according 1-dimensional probability density • Reconstruct the new momenta in the resolved dipole at the new resolution scale • We now have (n+1) partons and repeat step 2 until the resolution scale reaches the hadronization scale

9. Parton Shower MC’s • Starting from 2 gluon dipole • Angular distributions between the 3 leading jets (angle(j1,j2), angle(j1,j3), angle(j2,j3)) in 3,4,5,6,7,8 exclusive jet events. • Kt-jet algorithm used with Yr=0.001; M=500 GeV • 1,000,000 showered events (30 min to generate on laptop). • (stacked histograms) • (logarithmic vertical scale) • Distributions rich in structure (which are all explainable…)

10. Parton Shower MC’s To exactly formulate the shower MC we derive the shower MC master formula which can be implemented numerically. This is a Markov chain formulation… First we take a LO MC generator to predict an observable: Next we replace the delta-distribution with a shower function: where the shower function evolves the event resolution. The Markov master formula now is:

11. Parton Shower MC’s We want to match the parton shower to NLO MC’s and different multiplicity LO MC’s. For example This causes “double counting” issues. This means the MC’s have to use modified parton MC’s such that we do not double count. To investigate this we need to re-expand the Shower function:First we expand the event Sudakov

12.  0 as Parton Shower MC’s Next we expand the Shower function:

13. Parton Shower MC’s • Now we can expand the Shower function in the differential cross section using a LO MC: • The matching to LO MC’s is straightforward. The LO MC generates the partons and is subsequently showered to produce the fully exclusive partonic state. • For the inclusive n-jet LO MC at a large resolution scale replacing gives us the exclusive n-jet LL MC at a small resolution scale (and estimates of higher multiplicity jet contributions)

14. Parton Shower MC’s • Combining multiple LO MC’s makes no sense as each LO MC is an inclusive jet generator: • However combining multiple LL shower MC’s makes sense:as long as the matrix element is corrected (MEC) to avoid double counting • We can derive what the modified matrix element is by expanding the Shower function and matching to the LO MC generators

15. Parton Shower MC’s givingwhich is LL finite and removes the “double counting” terms !We generate events using the MEC LO MC’s (with no LL resolution cuts) and shower the dipoles

16. Parton Shower MC’s • For a NLO MC generator the matching follows a identical path: • Expanding out the shower function as before is a simple algebraic exercise(but lots of intermediate terms to deal with) • I give here the final results you will find the MEC finite matrix elements: • This matching is correct up to higher orders in and power suppressedterms • The resulting shower MC is much simpler than the NLO parton MC (withits complicated master equation) because all modified matrix elements (whichare subsequently showered) are already LL finite!

17. Parton Shower MC’s H2,3 gluons + shower H2 gluons + shower • 2, 3,… exclusive jet fractions as a function of the Kt-jet resolution parameter. • Matching shower with fixed order strongly reduces the dependence on the (arbitrary) hard part (non soft/collinear) of the antenna function. • Being able to change the shower hardness we can see the importance of matching • We can also estimate the residual uncertainties within the leading log approximation

18. Parton Shower MC’s • We see matching the LO/NLO MC generators to shower MC’s is quite straightforward provided we know the antenna functionused in the shower MC (this exact matching is crucial to reduce uncertainties). • For existing shower MC’s such as PYTHIA and HERWIG this is not that easy. These MC’s were written before we started constructing LO/NLO MC’s without matching in mind. The internal variables, evolution variables, momentum mapping after a branching,… do not easily match to the LO/NLO MC’s • However, it is highly desirable to perform the matching to the existing shower MC’s for the simple reason they are the only fully functional shower MC’s • The MC@NLO procedure uses the previous outlined procedure for matching to NLO calculations. However, the subtraction function needs to be constructed on a case-by-case basis (usually with assistance of one of the authors of HERWIG involved to trace what the Shower MC is doing). • The matching to LO can be done with less knowledge of the shower MC

19. Parton Shower MC’s • CKKW matching to a shower MC is achieved by re-weighting the matrix elements instead of modifying the matrix elements • The weight is calculated by 2 jet, 3 jet,… resolution scales of the matrix element using a “jet algorithm” (resolution scale used in the Shower MC you match to). • This gives us a series of merging scales and merged momenta lists. • The shower needs to be modified, i.e. branchings with a resolution scale larger than • Also the Sudakov used in the weight factor has to match. • The final result is independent on the scale

20. Parton Shower MC’s • MLM matching is even simpler. We can apply it to any shower MC. • After the showering each parton in the hard matrix element has to be within the “jet-cone” of a jet • This means the number of jets is equal to the initial partons. • Nothing needs to be modified in the shower MC • Nor do we need to know anything of the internal workings of the Shower MC

21. Parton Shower MC’s • A shower MC can evolve a LO/NLO generator down to the hadronization scale by resolving more and more clusters when reducing the resolution scale. • At Leading Order several procedures exist to match the shower to the LO matrix elements with varying precision • At Next-to-Leading order the only possible matching is using Matrix Element Corrections • New types of shower MC’s allow the LO/NLO matching in a very easy and generic manner (i.e. the corrections to the ME’s are trivial). • NLO matching to HERWIG/PYTHIA requires MC@NLO type procedures: The ME corrections are complicated and have to be determined on a case-by-case basis. • In the coming years the area of parton shower MC’s will develop quickly (including uncertainty estimates, automated matching to LO/NLO,…) • Hadronization models still remail an issue…