The choice of evolution variable

In the PYTHIA shower algorithm, the evolution variable $Q^2$ is associated with the squared mass of the branching parton, $Q^2 = m_a^2$ for a branching $a \to bc$. As a consequence, $t = \ln(Q^2/\Lambda^2) = \ln(m_a^2/\Lambda^2)$. This $Q^2$ choice is not unique, and indeed other programs have other definitions: HERWIG uses $Q^2 \approx m^2/(2z(1-z))$ [Mar88] and ARIADNE $Q^2 = p_{\perp}^2 \approx z(1-z)m^2$ [Pet88]. Below we will also modify the $Q^2$ choice to give a better account of mass effects, e.g. for $\b$ quarks.

With $Q$ a mass scale, the lower cut-off $Q_0$ is one in mass. To be more precise, in a QCD shower, the $Q_0$ parameter is used to derive effective masses

$$m_{\mathrm{eff},\mathrm{g}} = \frac{1}{2} Q_0 ~,$$

$$m_{\mathrm{eff},\mathrm{q}} = \sqrt{ m_{\mathrm{q}}^2 + \frac{1}{4} Q_0^2 } ~,$$ (167)

where the $m_{\mathrm{q}}$ have been chosen as typical kinematical quark masses, see section . A parton cannot branch unless its mass is at least the sum of the lightest pair of allowed decay products, i.e. the minimum mass scale at which a branching is possible is

$$m_{\mathrm{min},\mathrm{g}} = 2 \, m_{\mathrm{eff},\mathrm{g}} = Q_0 ~,$$

$$m_{\mathrm{min},\mathrm{q}} = m_{\mathrm{eff},\mathrm{q}} + m_{\mathrm{eff},\mathrm{g}} \geq Q_0 ~.$$ (168)

The above masses are used to constrain the allowed range of $Q^2$ and $z$ values. However, once it has been decided that a parton cannot branch any further, that parton is put on the mass shell, i.e. `final-state' gluons are massless.

When also photon emission is included, a separate $Q_0$ scale is introduced for the QED part of the shower, and used to calculate cut-off masses by analogy with eqs. () and () above [Sjö92c]. By default the two $Q_0$ scales are chosen equal, and have the value 1 GeV. If anything, one would be inclined to allow a cut-off lower for photon emission than for gluon one. In that case the allowed $z$ range of photon emission would be larger than that of gluon emission, and at the end of the shower evolution only photon emission would be allowed.

Photon and gluon emission differ fundamentally in that photons appear as physical particles in the final state, while gluons are confined. For photon emission off quarks, however, the confinement forces acting on the quark may provide an effective photon emission cut-off at larger scales than the bare quark mass. Soft and collinear photons could also be emitted by the final-state charged hadrons [Bar94a]; the matching between emission off quarks and off hadrons is a delicate issue, and we therefore do not attempt to address the soft-photon region.

For photon emission off leptons, there is no need to introduce any collinear emission cut-off beyond what is given by the lepton mass, but we keep the same cut-off approach as for quarks, although at a smaller scale. However, note that, firstly, the program is not aimed at high-precision studies of lepton pairs (where interference terms between initial- and final-state radiation also would have to be included), and, secondly, most experimental procedures would include the energy of collinear photons into the effective energy of a final-state lepton.