Automatic Calculation of Particle Interaction or Decay

The automatic calculation of particle interaction or decay is part of the computational particle physics branch. It refers to computing tools that help calculating the complex particle interactions as studied in high-energy physics, astroparticle physics and cosmology. The goal of the automation is to handle the full sequence of calculations in an automatic (programmed) way: from the Lagrangian expression describing the physics model up to the cross-sections values and to the event generator software.

Particle accelerator or colliders produce collisions (interactions) of particle (like the electron or the proton). The colliding particles form the Initial State. In the collision, particles can be annihilated or/and exchanged producing possibly different sets of particles, the Final States. The Initial and Final States of the interaction relate through the so-called scattering matrix (S-matrix).

For example at LEP, e+ + e− → e+ + e−, or e+ + e− → μ+ + μ− are processes where the initial state is an electron and a positron colliding to produce an electron and a positron or two muons of opposite charge: the final states. In these simple cases, no automatic packages are needed and cross-section analytical expression can be easily derived at least for the lowest approximation: the Born approximation also called the leading order or the tree level (as Feynman diagrams have only trunk and branches, no loops).

But particle physics is now requiring much more complex calculations like at LHC where are protons and is the number of jets of particles initiated by proton constituents (quarks and gluons). The number of subprocesses describing a given process is so large that automatic tools have been developed to mitigate the burden of hand calculations.

Interactions at higher energies open a large spectrum of possible final states and consequently increase the number of processes to compute.

High precision experiments impose the calculation of higher order calculation, namely the inclusion of subprocesses where more than one virtual particle can be created and annihilated during the interaction lapse creating so-called loops which induce much more involved calculations.

Finally new theoretical models like the supersymmetry model (MSSM in its minimal version) predict a flurry of new processes.

The automatic packages, once seen as mere teaching support, have become, this last 10 years an essential component of the data simulation and analysis suite for all experiments. They help constructing event generators and are sometime viewed as generators of event generators or Meta-generators.

A particle physics model is essentially described by its Lagrangian. To simulate the production of events through event generators, 3 steps have to be taken. The Automatic Calculation project is to create the tools to make those steps as automatic (or programmed) as possible:

I Feynman rules, coupling and mass generation

• LanHEP is an example of Feynman rules generation.
• Some model needs an additional step to compute, based on some parameters, the mass and coupling of new predicted particles.

II Matrix element code generation: Various methods are used to automatically produce the Matrix element expression in a computer language (Fortran, C/C++). They use values (i.e. for the masses) or expressions (i.e. for the couplings) produced by step I or model specific libraries constructed by hands (usually heavily relying on Computer algebra languages). When this expression is integrated (usually numerically) over the internal degrees of freedom it will provide the total and differential cross-sections for a given set of initial parameters like the initial state particle energies and polarization.

III Event generator code generation: This code must them be interfaced to other packages to fully provide the actual final state. The various effects or phenomenon that need to be implemeted are:

• Initial state radiation and beamstrahlung for e+e− initial states.
• Parton distribution functions describing the actual content in terms of gluons and quarks of the p or p-bar initial state particles
• Parton showering describing the way final state quarks or gluons due to the QCD confinement generate additional quark/gluon pairs generating a so-called shower of partons before transforming into hadrons.
• Hadronization describing how the final quark pairs/triplets form the visible and detectable hadrons.
• Underlying event takes care of the way the rest, in term of constituent, of the initial protons also contribute to any given event.

The interplay or matching of the precise matrix element calculation and the approximations resulting from the simulation of the parton shower gives rise to further complications, either within a given level of precision like at leading order (LO) for the production of n jets or between two levels of precision when tempting to connect matrix element computed at next-to-leading (NLO) (1-loop) or next-to-next-leading order (NNLO) (2-loops) with LO partons shower package.

Several methods have been developed for this matching:

• Subtraction methods
• ...

But the only correct way is to match packages at the same level theoretical accuracy like the NLO matrix element calculation with NLO parton shower packages. This is currently in development.