PYTHIA 6.4 Physics and Manual

ments, thus giving a decent description over the whole p ⊥ range. This does not providethe first-order corrections to the total W production rate, however, nor the possibility toselect only a high-p ⊥ tail of events.2.2.2 Parton showersThe separation of radiation into initial- and final-state showers is arbitrary, but veryconvenient. There are also situations where it is appropriate: for instance, the processe + e − → Z 0 → qq only contains final-state QCD radiation (QED radiation, however, ispossible both in the initial and final state), while qq → Z 0 → e + e − only contains initialstateQCD one. Similarly, the distinction of emission as coming either from the q or fromthe q is arbitrary. In general, the assignment of radiation to a given mother parton is agood approximation for an emission close to the direction of motion of that parton, butnot for the wide-angle emission in between two jets, where interference terms are expectedto be important.In both initial- and final-state showers, the structure is given in terms of branchingsa → bc, specifically e → eγ, q → qg, q → qγ, g → gg, and g → qq. (Further branchings,like γ → e + e − and γ → qq, could also have been added, but have not yet been of interest.)Each of these processes is characterized by a splitting kernel P a→bc (z). The branching rateis proportional to the integral ∫ P a→bc (z) dz. The z value picked for a branching describesthe energy sharing, with daughter b taking a fraction z and daughter c the remaining 1−zof 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 approximatesense of time ordering to the cascade. We stress here that somewhat different definitionof Q 2 are possible, and that Pythia actually implements two distinct alternatives, asyou will see. In the initial-state shower, Q 2 values are gradually increasing as the hardscattering is approached, while Q 2 is decreasing in the final-state showers. Shower evolutionis cut off at some lower scale Q 0 , typically around 1 GeV for QCD branchings. Fromabove, a maximum scale Q max is introduced, where the showers are matched to the hardinteraction itself. The relation between Q max and the kinematics of the hard scattering isuncertain, 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 machineriesare in fact quite different, and are described separately below.Final-state showers are time-like, i.e. partons have m 2 = E 2 − p 2 ≥ 0. The evolutionvariable Q 2 of the cascade has therefore traditionally in Pythia been associated with them 2 of the branching parton. As discussed above, this choice is not unique, and in morerecent versions of Pythia, a p ⊥ -ordered shower algorithm, with Q 2 = p 2 ⊥ = z(1 − z)m 2 ,is available in addition to the mass-ordered one. Regardless of the exact definition ofthe ordering variable, the general strategy is the same: starting from some maximumscale Q 2 max, an original parton is evolved downwards in Q 2 until a branching occurs. Theselected Q 2 value defines the mass of the branching parton, or the p ⊥ of the branching,depending on whether the mass-ordering or the p ⊥ -ordering is used. In both cases, thez value obtained from the splitting kernel represents the parton energy division betweenthe daughters. These daughters may now, in turn, evolve downwards, in this case withmaximum virtuality already defined by the previous branching, and so on down to theQ 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 massorderingconstraint, this does not follow automatically, but is implemented as an additionalrequirement on allowed emissions. The p ⊥ -ordered shower leads to the correctbehaviour without such modifications [Gus86]. Photon emission is not affected by angularordering. It is also possible to obtain non-trivial correlations between azimuthal angles inthe various branchings, some of which are implemented as options. Finally, the theoretical16