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Modern Approach to Monte Carlo’s

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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

The role of resolution in MC’s

- In collider experiments we measure a series of scattering amplitudesfrom which we want to derive an understanding of the underlying particle interactions.
- The measurement is only defined through a resolution scale , either determined by the detector or an analysis imposed resolution such as a jet algorithm.

Resolution in Monte Carlo’s

- Through a theoretical model we assign a probability density to each scattering which depends on the model and its parameters. (Or a probability of the scattering given a resolution.)
- For example the top mass:

- The Monte Carlo generates a series of simulated events depending on a resolution scale (e.g. jets).
- The size of the resolution scale determines the accuracy of the predictions (LO/NLO/Shower/…).
- The series of generated events can be weighted or at unit weight

Exclusive

Hadrons

Clusters

Jets

Inclusive x-sections

Hadronization

Shower MC’s

Parton level MC’s

Analytic calculations

Antenna-Dipole Shower MC’s

Leading Order Born

Exclusive final state

PYTHIA/HERWIG

Decreasing resolution

Resolution in Monte Carlo’s

- Perturbative QCD MC’s should describes the events from the low resolution scale of the hard scattering (jets) up to the highly resolved hadronization scale of order 1 GeV.
- Depending on the relevant resolution scale of the observable we need more orders in the perturbative expansion to make a reliable prediction
- Shower Monte Carlo’s evolve the resolution scale down to the hadronization scale in an approximate manner (without doing the explicit higher order calculation).

Leading Order Monte Carlo’s

- At Leading order (LO) the resolution scale is of the order of the jet energy/separationNote that the resolution scale is related to a dipole of jets.

- At LO each jet is represented by one parton
- The parton momentum jet axis momentum
- The jet structure is not resolved at LO
- Any forward or soft jet activity is not resolved

“Integrate”/Sum over unmeasured degrees of freedom (e.g. boost,…).

Leading Order Monte Carlo’s- At LO we replace:
- Proton (quark, gluon) with probability
- Jet axis (quark, gluon) momentum
- B-tagged jet axis b-quark momentum
- Missing energy neutrino momentum

- The probability density for the scatteringis now given by

Leading Order Monte Carlo’s

- What remains for the calculation of the probability density is the calculation of the matrix elementr (after that we can construct the LO Monte Carlo).
- All we need for the calculation of the matrix element is the Feynman rules.
- This is at LO straightforward but cumbersome and is best left to computer by developing algorithmic solutions.

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Leading Order Monte Carlo’sExample: given

- The colors, helicities and momenta are known at input
- The helicity vector is simply a 4-vector of complex numbers
- The 3- and 4-gluon vertices are simply tensors of real numbers
- By summing over all the indices to contract vectors and tensors we simply get a complex number
- This can be done efficiently using algorithms as implemented in programs such as ALPHGEN, MADGRAF, COMPHEP,…(n=10 gluons gives 10,525,900 diagrams!)

Leading Order Monte Carlo’s

- Some more considerations about the color factors:
- Useto extract the color factor from the amplitudes
- These are called ordered amplitudes and are gauge invariant, cyclic invariant and form an ordered set of dipole charges in phase space.
- These ordered amplitudes form the basic generators in modern NLO and shower MC’s

Leading Order Monte Carlo’s

- The LO matrix elements are simply rational functions of the invariants in the scattering. The denominator is a product of available multi-particle poles:

Leading Order Monte Carlo’s

One can now construct a simple LO MC generator:

- Chose a random pair.
- Generate a set of momenta according to the phase space measure
- Check resolution cuts on jet-momenta (i.e. partons)(e.g. )
- If event passes resolution cut calculate event weight and either
- Calculate the observable and “bin” the event weight
- Write the event record

- Restart at step 1

Leading Order Monte Carlo’s

- The event record can be “un-weighted” using a simple procedure:
- Determine
- Accept event (i) with unit weight ifwhere is a list of uniform random numbers (between 0 and 1)

- The efficiency (or fraction of accepted events) is given by (and is 1 for a unit-weight set).
- The larger the weight fluctuations the lower the efficiency.

Large weights for “soft/collinear” gluon (i.e. radiation close to resolution scale )

Rewrite phase space into resolution variables (invariants )

In new variables no large weight fluctuations when(Not all LO MC’s have same efficiency numerical consequences.)

Leading Order Monte Carlo’sExample: 3 jet production through a virtual photon decay:

Leading Order Monte Carlo’s close to resolution scale )

- LO MC’s should be able to describe the data at large resolution scales: i.e. hard, well separated jets.
- Good at shapes
- Not good at normalization

Depends on chosen value of close to resolution scale )

Higher multiplicity, larger uncertainty

Leading Order Monte Carlo’s close to resolution scale )

- Summary
- In the lowest order estimate of a (multi-) jet cross section each jet is modeled by a single parton (which represents the jet axis: i.e. the average direction of the hadrons in the jet-cone).
- At large resolution scale LO should give a reasonable estimate of event shapes.
- Nowadays many LO MC’s exist. All these MC’s must give identical results for each phase space point provided:
- Same renormalization/factorization scale used
- Same PDF’s
- Same evolution of and PDF’s

Next-to-Leading Order MC’s close to resolution scale )

- Next order in -expansion gives uncertainty estimate on LO shapes
- First estimate of normalization of jet cross sections
- Resolution scale pushed beyond jet resolution:
- Some estimates of jet shapes.
- Some limited information about exclusivity.
- Resolving some initial state radiation.

Next-to-Leading Order MC’s close to resolution scale )

- At NLO a jet can be modeled by two partons
- Sensitivity to jet scales and jet algorithm (by increasing the resolution scale we can resolve another cluster).
- Better modeling of jet axis (cluster dependence and 2 terms in expansion).

- At NLO we are sensitive to initial state resolution (the incoming “parton” can be an unresolved cluster of two partons).

PDF’s are easily included close to resolution scale )

Color suppressed terms ignored

Each contribution is “unphysical” as they exist in an infinite resolved world.

(Infrared safety of the observable guarantees finiteness of sum.)

We cannot Monte Carlo this as is, we need the (theoretical) resolution

Divergent

Divergent as

- Unresolved and “observed” as a n-cluster contribution
- Needs to be combined with virtual contribution
- We need to analytically integrate over this region (because it is divergent)
- We want a simplified universal function which encapsulate the soft/collinear function.

Finite resolved (n+1)-cluster contribution

finite

Next-to-Leading Order MC’sStill too complicated to evaluate (phase space and observable dependence)

Finite, can be evaluated in MC

Finite, can be evaluated in MC

Can be calculated analytical

Can be combined with loop contribution

Next-to-Leading Order MC’sPotential negative weights observable dependence)

Positive weights

- The term is finite
- Each term is finite and can be evaluated using a MC.
- The NLO MC generated weighted events containing clusters (i.e. 4-vectors) which need to be combined into observables (jets, applied cuts,…)
- The observable does not depend on the resolution variable.
- Two limits are often used (leads to simplifications):
- Slicing (large weight fluctuations due to -cancelation):
- Subtraction (uncanceled weights through phase space shifts):

Putting it all together gives the NLO MC master equation:

Next-to-Leading Order MC’s observable dependence)

Most of the previous derivation of the NLO master equation is adding and subtracting terms and reshuffling them.

However, one step is important to understand in more detail as it forms the connection to Shower MC’s. This is the soft/collinear approximation of a LO matrix element and the accompanying phase space factorization:

such that for each phase space point

To achieve this we go back to the ordered amplitudes

giving us ordered dipoles and resolution functions. We can now look at each dipole.

Next-to-Leading Order MC’s observable dependence)

The ordered dipoles introduce an ordered resolution concept (which only exists in color space and is not accessible for experiments)

Ordered amplitudes have only soft/collinear divergences within the dipole ordering:

This gives the resolution function for a dipole:

Next-to-Leading Order MC’s observable dependence)

If the dipole (i-1,i,i+1) is unresolved we have a massless clustering

The phase space now factorizes using this mapping

Next-to-Leading Order MC’s observable dependence)

Finally the ordered amplitude factorizes per ordered dipole in the soft/collinear limit:

where e.g. for three gluons

Now we can define such that

and

Next-to-Leading Order MC’s observable dependence)

We can now construct a NLO MC program according to the master equation (each contribution is finite)

For a NLO n-jet cross section we get positive/negative weighted n-parton and (n+1)-parton events.

The jet algorithm combines the events to n-jets and (n+1)-jets events which can be used to calculate jet observables.

Finding less than n-jets are part of the NNLO corrections of (n-1)-jet production and should be vetoed:

This term (partially) compensates the renormalization change

Next-to-Leading Order MC’s- Having a NLO prediction is a great asset
- Normalization prediction
- Uncertainty estimate on shape of distribution(e.g. the uncertainty from extrapolating the background into a signal region).

Next-to-Leading Order MC’s

NLO analysis show the great success of QCD to predict inclusive collider observables (in this case the inclusive jet transverse momentum differential cross sections at different rapidity intervals).

- Uses the JETRAD MC (“slicing” MC)
- Small scale uncertainties compared to dominant PDF uncertainties!
- Works well for inclusive jets!

Part 2

- Shower Monte Carlo’s and matching to the LO/NLO event genetators

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)

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

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! That is it is very likely to have multiple branching probabilities…

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.

Parton Shower MC’s

(log scale)

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.

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

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…)

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:

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

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)

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

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

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!

Parton Shower MC’s

H2,3 gluons + shower

H2 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

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

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 reconstructing a “shower history” of the matrix element using a “jet algorithm” (resolution scale used in the Shower MC you match to) which inverts the shower.
- 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

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

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…

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