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Parton showers

The separation of radiation into initial- and final-state showers is arbitrary, but very convenient. There are also situations where it is appropriate: for instance, the process $\mathrm{e}^+\mathrm{e}^-\to \mathrm{Z}^0 \to \mathrm{q}\overline{\mathrm{q}}$ only contains final-state QCD radiation (QED radiation, however, is possible both in the initial and final state), while $\mathrm{q}\overline{\mathrm{q}}\to \mathrm{Z}^0 \to \mathrm{e}^+\mathrm{e}^-$ only contains initial-state QCD one. Similarly, the distinction of emission as coming either from the $\mathrm{q}$ or from the $\overline{\mathrm{q}}$ is arbitrary. In general, the assignment of radiation to a given mother parton is a good approximation for an emission close to the direction of motion of that parton, but not for the wide-angle emission in between two jets, where interference terms are expected to be important.

In both initial- and final-state showers, the structure is given in terms of branchings $a \to bc$, specifically $\mathrm{e}\to \mathrm{e}\gamma$, $\mathrm{q}\to \mathrm{q}\mathrm{g}$, $\mathrm{q}\to \mathrm{q}\gamma$, $\mathrm{g}\to \mathrm{g}\mathrm{g}$, and $\mathrm{g}\to \mathrm{q}\overline{\mathrm{q}}$. (Further branchings, like $\gamma \to \mathrm{e}^+ \mathrm{e}^-$ and $\gamma \to \mathrm{q}\overline{\mathrm{q}}$, could also have been added, but have not yet been of interest.) Each of these processes is characterized by a splitting kernel $P_{a \to bc}(z)$. The branching rate is proportional to the integral $\int P_{a \to bc}(z) \, \d z$. The $z$ value picked for a branching describes the energy sharing, with daughter $b$ taking a fraction $z$ and daughter $c$ the remaining $1-z$ of the mother energy. Once formed, the daughters $b$ and $c$ may in turn branch, and so on.

Each parton is characterized by some virtuality scale $Q^2$, which gives an approximate sense of time ordering to the cascade. We stress here that somewhat different definition of $Q^2$ are possible, and that PYTHIA actually implements two distinct alternatives, as you will see. In the initial-state shower, $Q^2$ values are gradually increasing as the hard scattering is approached, while $Q^2$ is decreasing in the final-state showers. Shower evolution is cut off at some lower scale $Q_0$, typically around 1 GeV for QCD branchings. From above, a maximum scale $Q_{\mathrm{max}}$ is introduced, where the showers are matched to the hard interaction itself. The relation between $Q_{\mathrm{max}}$ and the kinematics of the hard scattering is uncertain, and the choice made can strongly affect the amount of well-separated jets.

Despite a number of common traits, the initial- and final-state radiation machineries are in fact quite different, and are described separately below.

Final-state showers are time-like, i.e. partons have $m^2 = E^2 - \mathbf{p}^2 \geq 0$. The evolution variable $Q^2$ of the cascade has therefore traditionally in PYTHIA been associated with the $m^2$ of the branching parton. As discussed above, this choice is not unique, and in more recent versions of PYTHIA, a $p_{\perp}$-ordered shower algorithm, with $Q^2=p_{\perp}^2=z(1-z)m^2$, is available in addition to the mass-ordered one. Regardless of the exact definition of the ordering variable, the general strategy is the same: starting from some maximum scale $Q^2_{\mathrm{max}}$, an original parton is evolved downwards in $Q^2$ until a branching occurs. The selected $Q^2$ value defines the mass of the branching parton, or the $p_{\perp}$ of the branching, depending on whether the mass-ordering or the $p_{\perp}$-ordering is used. In both cases, the $z$ value obtained from the splitting kernel represents the parton energy division between the daughters. These daughters may now, in turn, evolve downwards, in this case with maximum virtuality already defined by the previous branching, and so on down to the $Q_0$ cut-off.

In QCD showers, corrections to the leading-log picture, so-called coherence effects, lead to an ordering of subsequent emissions in terms of decreasing angles. For the mass-ordering constraint, this does not follow automatically, but is implemented as an additional requirement on allowed emissions. The $p_{\perp}$-ordered shower leads to the correct behaviour without such modifications [Gus86]. Photon emission is not affected by angular ordering. It is also possible to obtain non-trivial correlations between azimuthal angles in the various branchings, some of which are implemented as options. Finally, the theoretical analysis strongly suggests the scale choice $\alpha_{\mathrm{s}}= \alpha_{\mathrm{s}}(p_{\perp}^2) =
\alpha_{\mathrm{s}}(z(1-z)m^2)$, and this is the default in the program, for both shower algorithms.

The final-state radiation machinery is normally applied in the c.m. frame of the hard scattering or a decaying resonance. The total energy and momentum of that subsystem is preserved, as is the direction of the outgoing partons (in their common rest frame), where applicable.

In contrast to final-state showers, initial-state ones are space-like. This means that, in the sequence of branchings $a \to bc$ that lead up from the shower initiator to the hard interaction, particles $a$ and $b$ have $m^2 = E^2 - \mathbf{p}^2 <0$. The `side branch' particle $c$, which does not participate in the hard scattering, may be on the mass shell, or have a time-like virtuality. In the latter case a time-like shower will evolve off it, rather like the final-state radiation described above. To first approximation, the evolution of the space-like main branch is characterized by the evolution variable $Q^2 = -m^2$, which is required to be strictly increasing along the shower, i.e. $Q_b^2 > Q_a^2$. Corrections to this picture have been calculated, but are basically absent in PYTHIA. Again, in more recent versions of PYTHIA, a $p_{\perp}$-ordered ISR algorithm is also available, with $Q^2=p_{\perp}^2=-(1-z)m^2$.

Initial-state radiation is handled within the backwards evolution scheme. In this approach, the choice of the hard scattering is based on the use of evolved parton distributions, which means that the inclusive effects of initial-state radiation are already included. What remains is therefore to construct the exclusive showers. This is done starting from the two incoming partons at the hard interaction, tracing the showers `backwards in time', back to the two shower initiators. In other words, given a parton $b$, one tries to find the parton $a$ that branched into $b$. The evolution in the Monte Carlo is therefore in terms of a sequence of decreasing $Q^2$ (space-like virtuality or transverse momentum, as applicable) and increasing momentum fractions $x$. Branchings on the two sides are interleaved in a common sequence of decreasing $Q^2$ values.

In the above formalism, there is no real distinction between gluon and photon emission. Some of the details actually do differ, as will be explained in the full description.

The initial- and final-state radiation shifts around the kinematics of the original hard interaction. In Deeply Inelastic Scattering, this means that the $x$ and $Q^2$ values that can be derived from the momentum of the scattered lepton do not automatically agree with the values originally picked. In high-$p_{\perp}$ processes, it means that one no longer has two jets with opposite and compensating $p_{\perp}$, but more complicated topologies. Effects of any original kinematics selection cuts are therefore smeared out, an unfortunate side-effect of the parton-shower approach.

next up previous contents
Next: Beam Remnants and Multiple Up: Initial- and Final-State Radiation Previous: Matrix elements   Contents
Stephen Mrenna 2007-10-30