The Sudakov form factor

The $t$ variable fills the function of a kind of time for the shower evolution. In final-state showers, $t$ 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 $t$ values: in the end, each parton is associated with a fixed $t$ 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 $t$, cf. the concept of parton distributions.

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

$${\cal I}_{a \to bc}(t) = \int_{z_{-}(t)}^{z_{+}(t)} \d z \, \frac{\alpha_{abc}}{2 \pi} \, P_{a \to bc}(z) ~.$$ (164)

The naïve probability that a branching occurs during a small range of $t$ values, $\delta t$, is given by $\sum_{b,c} {\cal I}_{a \to bc}(t) \, \delta t$, and thus the probability for no emission by $1 - \sum_{b,c} {\cal I}_{a \to bc}(t) \, \delta t$.

If the evolution of parton $a$ starts at a `time' $t_0$, the probability that the parton has not yet branched at a `later time' $t > t_0$ is given by the product of the probabilities that it did not branch in any of the small intervals $\delta t$ between $t_0$ and $t$. In other words, letting $\delta t \to 0$, the no-branching probability exponentiates:

$${\cal P}_{\mathrm{no-branching}}(t_0,t) = \exp \left\{ - \in... ... t' \, \sum_{b,c} {\cal I}_{a \to bc}(t') \right\} = S_a(t) ~.$$ (165)

Thus the actual probability that a branching of $a$ occurs at $t$ is given by

$$\frac{\d {\cal P}_a}{\d t} = - \frac{\d {\cal P}_{\mathrm{no... ...t_0}^t \d t' \, \sum_{b,c} {\cal I}_{a \to bc}(t') \right\} ~.$$ (166)

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' $t' < t$, it can no longer branch at $t$. This is nothing but the exponential factor that is familiar from radioactive decay. In parton-shower language the exponential factor $S_a(t) = {\cal P}_{\mathrm{no-branching}}(t_0,t)$ is referred to as the Sudakov form factor [Sud56].

The ordering in terms of increasing $t$ above is the appropriate one for initial-state showers. In final-state showers the evolution is from an initial $t_{\mathrm{max}}$ (set by the hard scattering) and towards smaller $t$. In that case the integral from $t_0$ to $t$ in eqs. () and () is replaced by an integral from $t$ to $t_{\mathrm{max}}$. Since, by convention, the Sudakov factor is still defined from the lower cut-off $t_0$, i.e. gives the probability that a parton starting at scale $t$ will not have branched by the lower cut-off scale $t_0$, the no-branching factor is actually ${\cal P}_{\mathrm{no-branching}}(t_{\mathrm{max}},t) = S_a(t_{\mathrm{max}})/S_a(t)$.

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.