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### The Sudakov form factor

The variable fills the function of a kind of time for the shower evolution. In final-state showers, is constrained to be gradually decreasing away from the hard scattering, in initial-state ones to be gradually increasing towards the hard scattering. This does not mean that an individual parton runs through a range of values: in the end, each branching is associated with a fixed value, and the evolution procedure is just a way of picking that value. It is only the ensemble of partons in many events that evolves continuously with , cf. the concept of parton distributions.

For a given value we define the integral of the branching probability over all allowed values,

 (163)

The naïve probability that a branching occurs during a small range of values, , is given by , and thus the probability for no emission by .

If the evolution of parton starts at a time' , the probability that the parton has not yet branched at a later time' is given by the product of the probabilities that it did not branch in any of the small intervals between and . In other words, letting , the no-branching probability exponentiates:

 (164)

Thus the actual probability that a branching of occurs at is given by
 (165)

The first factor is the naïve branching probability, the second the suppression due to the conservation of total probability: if a parton has already branched at a `time' , it can no longer branch at . This is nothing but the exponential factor that is familiar from radioactive decay. In parton-shower language the exponential factor is referred to as the Sudakov form factor [Sud56].

The ordering in terms of increasing above is the appropriate one for initial-state showers. In final-state showers the evolution is from an initial (set by the hard scattering) and towards smaller . In that case the integral from to in eqs. () and () is replaced by an integral from to . Since, by convention, the Sudakov factor is still defined from the lower cut-off , i.e. gives the probability that a parton starting at scale will not have branched by the lower cut-off scale , the no-branching factor is actually .

We note that the above structure is exactly of the kind discussed in section . The veto algorithm is therefore extensively used in the Monte Carlo simulation of parton showers.

Next: Matching to the hard Up: Shower Evolution Previous: The evolution equations   Contents
Stephen Mrenna 2007-10-30