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 amplitudes from 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

Decreasing resolution HadronsClustersJetsInclusive x-sections Exclusive Inclusive Parton level MC’s Shower MC’s Analytic calculations Hadronization Leading Order Born Exclusive final state PYTHIA/HERWIG Antenna-Dipole Shower MC’s

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).

Shower  NLO/LO Object reconstruction Analysis Hadronization

Leading Order Monte Carlo’s At Leading order (LO) the resolution scale is of the order of the jet energy/separation Note 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

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 scattering is now given by “Integrate”/Sum over unmeasured degrees of freedom (e.g. boost,…). Order

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.

Leading Order Monte Carlo’s Example: 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: Use to 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: 1.Chose a random pair. 2.Generate a set of momenta according to the phase space measure 3.Check resolution cuts on jet-momenta (i.e. partons) (e.g. ) 4.If event passes resolution cut calculate event weight and either Calculate the observable and “bin” the event weight Write the event record 5.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 if where 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.

Leading Order Monte Carlo’s Example: 3 jet production through a virtual photon decay: 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 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 Higher multiplicity, larger uncertainty

Leading Order Monte Carlo’s 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 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 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).

Divergent as 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 Next-to-Leading Order MC’s Divergent Finite resolved (n+1)- cluster contribution 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 PDF’s are easily included Color suppressed terms ignored

Can be calculated analytical Still too complicated to evaluate (phase space and observable dependence) Next-to-Leading Order MC’s Finite, can be evaluated in MC Can be combined with loop contribution

Next-to-Leading Order MC’s Putting it all together gives the NLO MC master equation: 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): Potential negative weights Positive weights

Next-to-Leading Order MC’s 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 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 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 and Next-to-Leading Order MC’s

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:

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). This term (partially) compensates the renormalization change

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