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.
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Parton level MC’s
Antenna-Dipole Shower MC’s
Leading Order Born
Exclusive final state
“Integrate”/Sum over unmeasured degrees of freedom (e.g. boost,…).Leading Order Monte Carlo’s
5Leading Order Monte Carlo’s
One can now construct a simple LO MC generator:
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’s
Example: 3 jet production through a virtual photon decay:
Depends on chosen value of close to resolution scale )
Higher multiplicity, larger uncertainty
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
Finite resolved (n+1)-cluster contribution
finiteNext-to-Leading Order MC’s
Still 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 contributionNext-to-Leading Order MC’s
Potential negative weights observable dependence)
Putting it all together gives the NLO MC master equation:
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.
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:
If the dipole (i-1,i,i+1) is unresolved we have a massless clustering
The phase space now factorizes using this mapping
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
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:
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).
Exclusive 2 jet fraction at NLO
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:
The Sudakov factor estimates the likelyhood of not resolving an additional cluster in the dipole at the resolution scale.
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:
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
Next we expand the Shower function:
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
H2,3 gluons + shower
H2 gluons + shower