Matrix-element matching

In PYTHIA 6.1, matrix-element matching was introduced for the initial-state
shower description of initial-state
radiation in the production of a single colour-singlet resonance, such
as
[Miu99] (for Higgs production, see
below). The basic idea is to map the
kinematics between the PS and ME descriptions, and to find a correction
factor that can be applied to hard emissions in the shower so as to bring
agreement with the matrix-element expression. The PYTHIA shower
kinematics definitions are based on as the space-like virtuality of
the parton produced in a branching and as the factor by which the
of the scattering subsystem is reduced by the branching.
Some simple algebra then shows that the two
emission rates disagree by a factor

The other possible
graph is
, where the corresponding correction factor is

(196) |

The reweighting procedure prompts some other changes in the shower.
In particular, translates into a constraint on the phase
space of allowed branchings, not implemented before PYTHIA 6.1. Here
, where the association with the
variable is relevant if the branching is reinterpreted in terms of a
scattering. Usually such a requirement comes out of the
kinematics, and therefore is imposed eventually anyway. The corner of
emissions that do not respect this requirement is that where the
value of the space-like emitting parton is little changed and the
value of the branching is close to unity. (That is, such branchings are
kinematically allowed, but since the mapping to matrix-element variables
would assume the first parton to have , this mapping gives an
unphysical , and hence no possibility to impose a matrix-element
correction factor.) The correct behaviour in this region is beyond
leading-log predictivity. It is mainly important for the hardest emission,
i.e. with largest . The effect of this change is to reduce the total
amount of emission by a non-negligible amount when no matrix-element
correction is applied. (This can be confirmed by using the special option
`MSTP(68) = -1`.) For matrix-element corrections to be applied, this
requirement must be used for the hardest branching, and then whether it
is used or not for the softer ones is less relevant.

Our published comparisons with data on the spectrum show quite a good agreement with this improved simulation [Miu99]. A worry was that an unexpectedly large primordial , around 4 GeV, was required to match the data in the low- region. However, at that time we had not realized that the data were not fully unsmeared. The required primordial therefore drops by about a factor of two [Bál01]. This number is still uncomfortably large, but not too dissimilar from what is required in various resummation descriptions.

The method can also be used for initial-state photon emission, e.g. in the process . There the old default allowed no emission at large , at LEP2. This is now corrected by the increased , and using the of eq. () with .

The above method does not address the issue of next-to-leading order corrections to the total cross section. Rather, the implicit assumption is that such corrections, coming mainly from soft- and virtual-gluon effects, largely factorize from the hard-emission effects. That is, that the shape obtained in our approach will be rather unaffected by next-to-leading order corrections (when used both for the total and the high- cross section). A rescaling by a common factor could then be applied by hand at the end of the day. However, the issue is not clear. Alternative approaches have been proposed, where more sophisticated matching procedures are used also to get the next-to-leading order corrections to the cross section integrated into the shower formalism [Mre99].

A matching can also be attempted for other processes than the ones above.
Currently a matrix-element correction factor is also used for
and
branchings in the
process, in order to match on to the
and
matrix elements [Ell88]. The
loop integrals of Higgs production are quite complex, however, and
therefore only the expressions obtained in the limit of a heavy top
quark is used as a starting point to define the ratios of
and
to
cross sections. (Whereas the
cross section by itself
contains the complete expressions.) In this limit, the compact
correction factors

can be derived. Even though they are clearly not as reliable as the above expressions for , they should hopefully represent an improved description relative to having no correction factor at all. For this reason they are applied not only for the Standard Model Higgs, but for all the three Higgs states , and . The Higgs correction factors are always in the comfortable range between and .

Note that a third process, does not fit into the pattern of the other two. The above process cannot be viewed as a showering correction to a lowest-order one: since the is assumed (essentially) massless there is no pointlike coupling. The graph above instead again involved a top loop, coupled to the initial state by a single s-channel gluon. The final-state gluon is necessary to balance colours in the process, and therefore the cross section is vanishing in the limit.