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Other initial-state shower aspects

In the formulae above, $Q^2$ has been used as argument for $\alpha_{\mathrm{s}}$, and not only as the space-like virtuality of partons. This is one possibility, but in fact loop calculations tend to indicate that the proper argument for $\alpha_{\mathrm{s}}$ is not $Q^2$ but $p_{\perp}^2 = (1-z) Q^2$ [Bas83]. The variable $p_{\perp}$ does have the interpretation of transverse momentum, although it is only exactly so for a branching $a \to bc$ with $a$ and $c$ massless and $Q^2 = - m_b^2$, and with $z$ interpreted as light-cone fraction of energy and momentum. The use of $\alpha_{\mathrm{s}}((1-z)Q^2)$ is default in the program.

Angular ordering is included in the shower evolution by default. However, as already mentioned, the physics is much more complicated than for time-like showers, and so this option should only be viewed as a first approximation. In the code the quantity ordered is an approximation of $p_{\perp}/p \approx \sin\theta$. (An alternative would have been $p_{\perp}/p_L \approx \tan\theta$, but this suffers from instability problems.)

In flavour excitation processes, a $\c $ (or $\b $) quark enters the hard scattering and should be reconstructed by the shower as coming from a $\mathrm{g}\to \c\overline{\mathrm{c}}$ (or $\mathrm{g}\to \b\overline{\mathrm{b}}$) branching. Here an $x$ value for the incoming $\c $ above $Q_{\c }^2/(Q_{\c }^2 + m_{\c }^2)$, where $Q_{\c }^2$ is the space-like virtuality of the $\c $, does not allow a kinematical reconstruction of the gluon branching with an $x_{\mathrm{g}} < 1$, and is thus outside the allowed phase space. Such events (with some safety margin) are rejected. Currently they will appear in PYSTAT(1) listings in the `Fraction of events that fail fragmentation cuts', which is partly misleading, but has the correct consequence of suppressing the physical cross section. Further, the $Q^2$ value of the backwards evolution of a $\c $ quark is by force kept above $m_{\c }^2$, so as to ensure that the branching $\mathrm{g}\to \c\overline{\mathrm{c}}$ is not `forgotten' by evolving $Q^2$ below $Q_0^2$. Thereby the possibility of having a $\c $ in the beam remnant proper is eliminated [Nor98]. Warning: as a consequence, flavour excitation is not at all possible too close to threshold. If the KFIN array in PYSUBS is set so as to require a $\c $ (or $\b $) on either side, and the phase space is closed for such a $\c $ to come from a $\mathrm{g}\to \c\overline{\mathrm{c}}$ branching, the program will enter an infinite loop.

For proton beams, say, any $\c $ or $\b $ quark entering the hard scattering has to come from a preceding gluon splitting. This is not the case for a photon beam, since a photon has a $\c $ and $\b $ valence quark content. Therefore the above procedure need not be pursued there, but $\c $ and $\b $ quarks may indeed appear as beam remnants.

As we see, the initial-state showering algorithm leads to a net boost and rotation of the hard-scattering subsystems. The overall final state is made even more complex by the additional final-state radiation. In principle, the complexity is very physical, but it may still have undesirable side effects. One such, discussed further in section [*], is that it is very difficult to generate events that fulfil specific kinematics conditions, since kinematics is smeared and even, at times, ambiguous.

A special case is encountered in Deeply Inelastic Scattering in $\mathrm{e}\mathrm{p}$ collisions. Here the DIS $x$ and $Q^2$ values are defined in terms of the scattered electron direction and energy, and therefore are unambiguous (except for issues of final-state photon radiation close to the electron direction). Neither initial- nor final-state showers preserve the kinematics of the scattered electron, however, and hence the DIS $x$ and $Q^2$ are changed. In principle, this is perfectly legitimate, with the caveat that one then also should use different sets of parton distributions than ones derived from DIS, since these are based on the kinematics of the scattered lepton and nothing else. Alternatively, one might consider showering schemes that leave $x$ and $Q^2$ unchanged. In [Ben88] detailed modifications are presented that make a preservation possible when radiation off the incoming and outgoing electron is neglected, but these are not included in the current version of PYTHIA. Instead the current 'gamma/lepton' machinery explicitly separates off the $\mathrm{e}\to \mathrm{e}\gamma$ vertex from the continued fate of the photon.

The only reason for using the older machinery, such as process 10, is that this is still the only place where weak charged and neutral current effects can be considered. What is available there, as an option, is a simple machinery which preserves $x$ and $Q^2$ from the effects of QCD radiation, and also from those of primordial $k_{\perp}$ and the beam-remnant treatment, as follows. After the showers have been generated, the four-momentum of the scattered lepton is changed to the expected one, based on the nominal $x$ and $Q^2$ values. The azimuthal angle of the lepton is maintained when the transverse momentum is adjusted. Photon radiation off the lepton leg is not fully accounted for, i.e. it is assumed that the energy of final-state photons is added to that of the scattered electron for the definition of $x$ and $Q^2$ (this is the normal procedure for parton-distribution definitions).

The change of three-momentum on the lepton side of the event is balanced by the final-state partons on the hadron side, excluding the beam remnant but including all the partons both from initial- and final-state showering. The fraction of three-momentum shift taken by each parton is proportional to its original light-cone momentum in the direction of the incoming lepton, i.e. to $E \mp p_z$ for a hadron moving in the $\pm$ direction. This procedure guarantees momentum but not energy conservation. For the latter, one additional degree of freedom is needed, which is taken to be the longitudinal momentum of the initial-state shower initiator. As this momentum is modified, the change is shared by the final-state partons on the hadron side, according to the same light-cone fractions as before (based on the original momenta). Energy conservation requires that the total change in final-state parton energies plus the change in lepton side energy equals the change in initiator energy. This condition can be turned into an iterative procedure to find the initiator momentum shift.

Sometimes the procedure may break down. For instance, an initiator with $x > 1$ may be reconstructed. If this should happen, the $x$ and $Q^2$ values of the event are preserved, but new initial- and final-state showers are generated. After five such failures, the event is completely discarded in favour of a new kinematical setup.

Kindly note that the four-momenta of intermediate partons in the shower history are not being adjusted. In a listing of the complete event history, energy and momentum need then not be conserved in shower branchings. This mismatch could be fixed up, if need be.

The scheme presented above should not be taken too literally, but is rather intended as a contrast to the more sophisticated schemes already on the market, if one would like to understand whether the kind of conservation scheme chosen does affect the observable physics.

next up previous contents
Next: Matrix-element matching Up: Initial-State Showers Previous: Transverse evolution   Contents
Stephen Mrenna 2007-10-30