The evolution equations

In the shower formulation, the kinematics of each branching is given in terms of two variables, $Q^2$ and $z$. Somewhat different interpretations may be given to these variables, and indeed this is one main area where the various programs on the market differ. $Q^2$ has dimensions of squared mass, and is related to the mass or transverse momentum scale of the branching. $z$ gives the sharing of the $a$ energy and momentum between the two daughters, with parton $b$ taking a fraction $z$ and parton $c$ a fraction $1-z$. To specify the kinematics, an azimuthal angle $\varphi$ of the $b$ around the $a$ direction is needed in addition; in the simple discussions $\varphi$ is chosen to be isotropically distributed, although options for non-isotropic distributions currently are the defaults.

The probability for a parton to branch is given by the evolution equations (also called DGLAP or Altarelli-Parisi [Gri72,Alt77]). It is convenient to introduce

$$t = \ln(Q^2/\Lambda^2) ~~~ \Rightarrow ~~~ \d t = \d\ln(Q^2) = \frac{\d Q^2}{Q^2} ~,$$ (160)

where $\Lambda$ is the QCD $\Lambda$ scale in $\alpha_{\mathrm{s}}$. Of course, this choice is more directed towards the QCD parts of the shower, but it can be used just as well for the QED ones. In terms of the two variables $t$ and $z$, the differential probability $\d {\cal P}$ for parton $a$ to branch is now

$$\d {\cal P}_a = \sum_{b,c} \frac{\alpha_{abc}}{2 \pi} \, P_{a \to bc}(z) \, \d t \, \d z ~.$$ (161)

Here the sum is supposed to run over all allowed branchings, for a quark $\mathrm{q}\to \mathrm{q}\mathrm{g}$ and $\mathrm{q}\to \mathrm{q}\gamma$, and so on. The $\alpha_{abc}$ factor is $\alpha_{\mathrm{em}}$ for QED branchings and $\alpha_{\mathrm{s}}$ for QCD ones (to be evaluated at some suitable scale, see below).

The splitting kernels $P_{a \to bc}(z)$ are

$$P_{\mathrm{q}\to \mathrm{q}\mathrm{g}}(z) = C_F \, \frac{1+z^2}{1-z} ~,$$

$$P_{\mathrm{g}\to \mathrm{g}\mathrm{g}}(z) = N_C \, \frac{(1-z(1-z))^2}{z(1-z)} ~,$$

$$P_{\mathrm{g}\to \mathrm{q}\overline{\mathrm{q}}}(z) = T_R \, (z^2 + (1-z)^2) ~,$$

$$P_{\mathrm{q}\to \mathrm{q}\gamma}(z) = e_{\mathrm{q}}^2 \, \frac{1+z^2}{1-z} ~,$$

$$P_{\ell \to \ell\gamma}(z) = e_{\ell}^2 \, \frac{1+z^2}{1-z} ~,$$ (162)

with $C_F = 4/3$, $N_C = 3$, $T_R = n_f/2$ (i.e. $T_R$ receives a contribution of $1/2$ for each allowed $\mathrm{q}\overline{\mathrm{q}}$ flavour), and $e_{\mathrm{q}}^2$ and $e_{\ell}^2$ the squared electric charge ($4/9$ for $\u$-type quarks, $1/9$ for $\d$-type ones, and 1 for leptons).

Persons familiar with analytical calculations may wonder why the `+ prescriptions' and $\delta(1-z)$ terms of the splitting kernels in eq. () are missing. These complications fulfil the task of ensuring flavour and energy conservation in the analytical equations. The corresponding problem is solved trivially in Monte Carlo programs, where the shower evolution is traced in detail, and flavour and four-momentum are conserved at each branching. The legacy left is the need to introduce a cut-off on the allowed range of $z$ in splittings, so as to avoid the singular regions corresponding to excessive production of very soft gluons.

Also note that $P_{\mathrm{g}\to \mathrm{g}\mathrm{g}}(z)$ is given here with a factor $N_C$ in front, while it is sometimes shown with $2 N_C$. The confusion arises because the final state contains two identical partons. With the normalization above, $P_{a \to bc}(z)$ is interpreted as the branching probability for the original parton $a$. On the other hand, one could also write down the probability that a parton $b$ is produced with a fractional energy $z$. Almost all the above kernels can be used unchanged also for this purpose, with the obvious symmetry $P_{a \to bc}(z) = P_{a \to cb}(1-z)$. For $\mathrm{g}\to \mathrm{g}\mathrm{g}$, however, the total probability to find a gluon with energy fraction $z$ is the sum of the probability to find either the first or the second daughter there, and that gives the factor of 2 enhancement.