Other final-state shower aspects

The electromagnetic coupling constant for the emission of photons on the mass shell is $\alpha_{\mathrm{em}}= \alpha_{\mathrm{em}}(Q^2 = 0) \approx 1/137$. For the strong coupling constant several alternatives are available, the default being the first-order expression $\alpha_{\mathrm{s}}(p_{\perp}^2)$, where $p_{\perp}^2$ is defined by the approximate expression $p_{\perp}^2 \approx z(1-z) m^2$. Studies of next-to-leading-order corrections favour this choice [Ama80]. The other alternatives are a fixed $\alpha_{\mathrm{s}}$ and an $\alpha_{\mathrm{s}}(m^2)$.

With the default choice of $p_{\perp}^2$ as scale in $\alpha_{\mathrm{s}}$, a further cut-off is introduced on the allowed phase space of gluon emission, not present in the options with fixed $\alpha_{\mathrm{s}}$ or with $\alpha_{\mathrm{s}}(m^2)$, nor in the QED shower. A minimum requirement, to ensure a well-defined $\alpha_{\mathrm{s}}$, is that $p_{\perp}/ \Lambda > 1.1$, but additionally PYTHIA requires that $p_{\perp}> Q_0/2$. This latter requirement is not a necessity, but it makes sense when $p_{\perp}$ is taken to be the preferred scale of the branching process, rather than e.g. $m$. It reduces the allowed $z$ range, compared with the purely kinematical constraints. Since the $p_{\perp}$ cut is not present for photon emission, the relative ratio of photon to gluon emission off a quark is enhanced at small virtualities compared with naïve expectations; in actual fact this enhancement is largely compensated by the running of $\alpha_{\mathrm{s}}$, which acts in the opposite direction. The main consequence, however, is that the gluon energy spectrum is peaked at around $Q_0$ and rapidly vanishes for energies below that, whilst the photon spectrum extends all the way to zero energy.

Previously it was said that azimuthal angles in branchings are chosen isotropically. In fact, it is possible to include some effects of gluon polarization, which correlate the production and the decay planes of a gluon, such that a $\mathrm{g}\to \mathrm{g}\mathrm{g}$ branching tends to take place in the production plane of the gluon, while a decay out of the plane is favoured for $\mathrm{g}\to \mathrm{q}\overline{\mathrm{q}}$. The formulae are given e.g. in ref. [Web86], as simple functions of the $z$ value at the vertex where the gluon is produced and of the $z$ value when it branches. Also coherence phenomena lead to non-isotropic azimuthal distributions [Web86]. In either case the $\varphi$ azimuthal variable is first chosen isotropically, then the weight factor due to polarization times coherence is evaluated, and the $\varphi$ value is accepted or rejected. In case of rejection, a new $\varphi$ is generated, and so on.

While the rule is to have an initial pair of partons, there are a few examples where one or three partons have to be allowed to shower. If only one parton is given, it is not possible to conserve both energy and momentum. The choice has been made to conserve energy and jet direction, but the momentum vector is scaled down when the radiating parton acquires a mass. The `rest frame of the system', used e.g. in the $z$ definition, is taken to be whatever frame the jet is given in.

In $\Upsilon \to \mathrm{g}\mathrm{g}\mathrm{g}$ decays and other configurations (e.g. from external processes) with three or more primary parton, one is left with the issue how the kinematics from the on-shell matrix elements should be reinterpreted for an off-shell multi-parton configuration. We have made the arbitrary choice of preserving the direction of motion of each parton in the rest frame of the system, which means that all three-momenta are scaled down by the same amount, and that some particles gain energy at the expense of others. Mass multiplets outside the allowed phase space are rejected and the evolution continued.

Finally, it should be noted that two toy shower models are included as options. One is a scalar gluon model, in which the $\mathrm{q}\to \mathrm{q}\mathrm{g}$ branching kernel is replaced by $P_{\mathrm{q}\to \mathrm{q}\mathrm{g}}(z) = \frac{2}{3} (1-z)$. The couplings of the gluon, $\mathrm{g}\to \mathrm{g}\mathrm{g}$ and $\mathrm{g}\to \mathrm{q}\overline{\mathrm{q}}$, have been left as free parameters, since they depend on the colour structure assumed in the model. The spectra are flat in $z$ for a spin 0 gluon. Higher-order couplings of the type $\mathrm{g}\to \mathrm{g}\mathrm{g}\mathrm{g}$ could well contribute significantly, but are not included. The second toy model is an Abelian vector one. In this option $\mathrm{g}\to \mathrm{g}\mathrm{g}$ branchings are absent, and $\mathrm{g}\to \mathrm{q}\overline{\mathrm{q}}$ ones enhanced. More precisely, in the splitting kernels, eq. (), the Casimir factors are changed as follows: $C_F = 4/3 \to 1$, $N_C = 3 \to 0$, $T_R = n_f/2 \to 3n_f$. When using either of these options, one should be aware that also a number of other components in principle should be changed, from the running of $\alpha_{\mathrm{s}}$ to the whole concept of fragmentation. One should therefore not take them too seriously.