Longitudinal evolution

The evolution equations, eq. (), express that, during a small increase $\d t$ there is a probability for parton $a$ with momentum fraction $x'$ to become resolved into parton $b$ at $x = z x'$ and another parton $c$ at $x' - x = (1-z) x'$. Correspondingly, in backwards evolution, during a decrease $\d t$ a parton $b$ may be `unresolved' into parton $a$. The relative probability $\d {\cal P}_b$ for this to happen is given by the ratio $\d f_b / f_b$. Using eq. () one obtains

$$\d {\cal P}_b = \frac{\d f_b(x,t)}{f_b(x,t)} = \vert\d t\ver... ...a_{abc}}{2 \pi} \, P_{a \to bc} \left( \frac{x}{x'} \right) ~.$$ (186)

Summing up the cumulative effect of many small changes $\d t$, the probability for no radiation exponentiates. Therefore one may define a form factor

$$S_b(x,t_{\mathrm{max}},t) = \exp \left\{ - \int_t^{t_{\mathrm{max}}} \d t' \, \sum_{a,c} \int... ...ac{\alpha_{abc}(t')}{2\pi} \, P_{a \to bc} \left( \frac{x}{x'} \right) \right\}$$

$$= \exp \left\{ - \int_t^{t_{\mathrm{max}}} \d t' \, \sum_{a,c} \int... ...c}(t')}{2\pi} \, P_{a \to bc}(z) \, \frac{x'f_a(x',t')}{xf_b(x,t')} \right\} ~,$$ (187)

giving the probability that a parton $b$ remains at $x$ from $t_{\mathrm{max}}$ to a $t < t_{\mathrm{max}}$.

It may be useful to compare this with the corresponding expression for forward evolution, i.e. with $S_a(t)$ in eq. (). The most obvious difference is the appearance of parton distributions in $S_b$. Parton distributions are absent in $S_a$: the probability for a given parton $a$ to branch, once it exists, is independent of the density of partons $a$ or $b$. The parton distributions in $S_b$, on the other hand, express the fact that the probability for a parton $b$ to come from the branching of a parton $a$ is proportional to the number of partons $a$ there are in the hadron, and inversely proportional to the number of partons $b$. Thus the numerator $f_a$ in the exponential of $S_b$ ensures that the parton composition of the hadron is properly reflected. As an example, when a gluon is chosen at the hard scattering and evolved backwards, this gluon is more likely to have been emitted by a $\u$ than by a $\d$ if the incoming hadron is a proton. Similarly, if a heavy flavour is chosen at the hard scattering, the denominator $f_b$ will vanish at the $Q^2$ threshold of the heavy-flavour production, which means that the integrand diverges and $S_b$ itself vanishes, so that no heavy flavour remain below threshold.

Another difference between $S_b$ and $S_a$, already touched upon, is that the $P_{\mathrm{g}\to \mathrm{g}\mathrm{g}}(z)$ splitting kernel appears with a normalization $2 N_C$ in $S_b$ but only with $N_C$ in $S_a$, since two gluons are produced but only one decays in a branching.

A knowledge of $S_b$ is enough to reconstruct the parton shower backwards. At each branching $a \to bc$, three quantities have to be found: the $t$ value of the branching (which defines the space-like virtuality $Q_b^2$ of parton $b$), the parton flavour $a$ and the splitting variable $z$. This information may be extracted as follows:

86.

If parton $b$ partook in the hard scattering or branched into other partons at a scale $t_{\mathrm{max}}$, the probability that $b$ was produced in a branching $a \to bc$ at a lower scale $t$ is

$$\frac{\d {\cal P}_b}{\d t} = - \frac{\d S_b(x,t_{\mathrm{max... ...x'f_a(x',t')}{xf_b(x,t')} \right) S_b(x,t_{\mathrm{max}},t) ~.$$ (188)

If no branching is found above the cut-off scale $t_0$ the iteration is stopped and parton $b$ is assumed to be massless.

87.

Given the $t$ of a branching, the relative probabilities for the different allowed branchings $a \to bc$ are given by the $z$ integrals above, i.e. by

$$\int \d z \, \frac{\alpha_{abc}(t)}{2\pi} \, P_{a \to bc}(z) \, \frac{x'f_a(x',t)}{xf_b(x,t)} ~.$$ (189)

88.

Finally, with $t$ and $a$ known, the probability distribution in the splitting variable $z = x/x' = x_b/x_a$ is given by the integrand in eq. ().

In addition, the azimuthal angle $\varphi$ of the branching is selected isotropically, i.e. no spin or coherence effects are included in this distribution.

The selection of $t$, $a$ and $z$ is then a standard task of the kind than can be performed with the help of the veto algorithm. Specifically, upper and lower bounds for parton distributions are used to find simple functions that are everywhere larger than the integrands in eq. (). Based on these simple expressions, the integration over $z$ may be carried out, and $t$, $a$ and $z$ values selected. This set is then accepted with a weight given by a ratio of the correct integrand in eq. () to the simple approximation used, both evaluated for the given set. Since parton distributions, as a rule, are not in a simple analytical form, it may be tricky to find reasonably good bounds to parton distributions. It is necessary to make different assumptions for valence and sea quarks, and be especially attentive close to a flavour threshold ([Sjö85]). An electron distribution inside an electron behaves differently from parton distributions encountered in hadrons, and has to be considered separately.

A comment on soft gluon emission. Nominally the range of the $z$ integral in $S_b$ is $x \leq z \leq 1$. The lower limit corresponds to $x' = x/z = 1$, and parton distributions vanish in this limit, wherefore no problems are encountered here. At the upper cut-off $z=1$ the splitting kernels $P_{\mathrm{q}\to \mathrm{q}\mathrm{g}}(z)$ and $P_{\mathrm{g}\to \mathrm{g}\mathrm{g}}$ diverge. This is the soft gluon singularity: the energy carried by the emitted gluon is vanishing, $x_{\mathrm{g}} = x' - x = (1-z) x' = (1-z) x/z \to 0$ for $z \to 1$. In order to calculate the integral over $z$ in $S_b$, an upper cut-off $z_{\mathrm{max}} = x/(x + x_{\epsilon})$ is introduced, i.e. only branchings with $z \leq z_{\mathrm{max}}$ are included in $S_b$. Here $x_{\epsilon}$ is a small number, typically chosen so that the gluon energy is above 2 GeV when calculated in the rest frame of the hard scattering. That is, the gluon energy $x_{\mathrm{g}} \sqrt{s}/2 \geq x_{\epsilon} \sqrt{s}/2 = 2~\mathrm{GeV}/\gamma$, where $\gamma$ is the boost factor of the hard scattering. The average amount of energy carried away by gluons in the range $x_{g} < x_{\epsilon}$, over the given range of $t$ values from $t_a$ to $t_b$, may be estimated [Sjö85]. The finally selected $z$ value may thus be picked as $z = z_{\mathrm{hard}} \langle z_{\mathrm{soft}}(t_a, t_b) \rangle$, where $z_{\mathrm{hard}}$ is the originally selected $z$ value and $z_{\mathrm{soft}}$ is the correction factor for soft gluon emission.

In QED showers, the smallness of $\alpha_{\mathrm{em}}$ means that one can use rather smaller cut-off values without obtaining large amounts of emission. A fixed small cut-off $x_{\gamma} > 10^{-10}$ is therefore used to avoid the region of very soft photons. As has been discussed in section , the electron distribution inside the electron is cut off at $x_{\mathrm{e}} < 1 - 10^{-10}$, for numerical reasons, so the two cuts are closely matched.

The cut-off scale $Q_0$ may be chosen separately for QCD and QED showers, just as in final-state radiation. The defaults are 1 GeV and 0.001 GeV, respectively. The former is the typical hadronic mass scale, below which radiation is not expected resolvable; the latter is of the order of the electron mass. Photon emission is also allowed off quarks in hadronic interactions, with the same cut-off as for gluon emission, and also in other respects implemented in the same spirit, rather than according to the pure QED description.

Normally QED and QCD showers do not appear mixed. The most notable exception is resolved photoproduction (in $\mathrm{e}\mathrm{p}$) and resolved $\gamma\gamma$ events (in $\mathrm{e}^+\mathrm{e}^-$), i.e. shower histories of the type $\mathrm{e}\to \gamma \to \mathrm{q}$. Here the $Q^2$ scales need not be ordered at the interface, i.e. the last $\mathrm{e}\to \mathrm{e}\gamma$ branching may well have a larger $Q^2$ than the first $\mathrm{q}\to \mathrm{q}\mathrm{g}$ one, and the branching $\gamma \to \mathrm{q}$ does not even have a strict parton-shower interpretation for the vector dominance model part of the photon parton distribution. This kind of configurations is best described by the 'gamma/lepton' machinery for having a flux of virtual photons inisde the lepton, see section . In this case, no initial-state radiation has currently been implemented for the electron (or $\mu$ or $\tau$). The one inside the virtual-photon system is considered with the normal algorithm, but with the lower cut-off scale modified by the photon virtuality, see MSTP(66).

An older description still lives on, although no longer as the recommended one. There, these issues are currently not addressed in full. Rather, based on the $x$ selected for the parton (quark or gluon) at the hard scattering, the $x_{\gamma}$ is selected once and for all in the range $x < x_{\gamma} <1$, according to the distribution implied by eq. (). The QCD parton shower is then traced backwards from the hard scattering to the QCD shower initiator at $t_0$. No attempt is made to perform the full QED shower, but rather the beam remnant treatment (see section ) is used to find the $\overline{\mathrm{q}}$ (or $\mathrm{g}$) remnant that matches the $\mathrm{q}$ (or $\mathrm{g}$) QCD shower initiator, with the electron itself considered as a second beam remnant.