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 $\gamma^*/\mathrm{Z}^0/\mathrm{W}^{\pm}$ [Miu99]. 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 $Q^2$ as the spacelike virtuality of the parton produced in a branching and $z$ as the factor by which the $\hat{s}$ of the scattering subsystem is reduced by the branching. Some simple algebra then shows that the two $\mathrm{q}\overline{\mathrm{q}}' \to \mathrm{g}\mathrm{W}^{\pm}$ emission rates disagree by a factor

$$R_{\mathrm{q}\overline{\mathrm{q}}' \to \mathrm{g}\mathrm{W}... ...u}^2+2 m_{\mathrm{W}}^2\hat{s}}{\hat{s}^2+m_{\mathrm{W}}^4} ~,$$ (195)

which is always between $1/2$ and 1. The shower can therefore be improved in two ways, relative to the old description. Firstly, the maximum virtuality of emissions is raised from $Q^2_{\mathrm{max}} \approx m_{\mathrm{W}}^2$ to $Q^2_{\mathrm{max}} = s$, i.e. the shower is allowed to populate the full phase space. Secondly, the emission rate for the final (which normally also is the hardest) $\mathrm{q}\to \mathrm{q}\mathrm{g}$ emission on each side is corrected by the factor $R(\hat{s},\hat{t})$ above, so as to bring agreement with the matrix-element rate in the hard-emission region. In the backwards evolution shower algorithm [Sjö85], this is the first branching considered.

The other possible ${\mathcal{O}}(\alpha_{\mathrm{s}})$ graph is $\mathrm{q}\mathrm{g}\to \mathrm{q}'\mathrm{W}^{\pm}$, where the corresponding correction factor is

$$R_{\mathrm{q}\mathrm{g}\to \mathrm{q}'\mathrm{W}}(\hat{s},\h... ... \hat{t}}{(\hat{s}-m_{\mathrm{W}}^2)^2 + m_{\mathrm{W}}^4} ~,$$ (196)

which lies between 1 and 3. A probable reason for the lower shower rate here is that the shower does not explicitly simulate the $s$-channel graph $\mathrm{q}\mathrm{g}\to \mathrm{q}^* \to \mathrm{q}'\mathrm{W}$. The $\mathrm{g}\to \mathrm{q}\overline{\mathrm{q}}$ branching therefore has to be preweighted by a factor of 3 in the shower, but otherwise the method works the same as above. Obviously, the shower will mix the two alternative branchings, and the correction factor for a final branching is based on the current type.

The reweighting procedure prompts some other changes in the shower. In particular, $\hat{u} < 0$ translates into a constraint on the phase space of allowed branchings, not previously implemented. Here $\hat{u} = Q^2 - \hat{s}_{\mathrm{old}} (1-z)/z = Q^2 - \hat{s}_{\mathrm{new}} (1-z)$, where the association with the $\hat{u}$ variable is relevant if the branching is reinterpreted in terms of a $2 \to 2$ 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 $Q^2$ value of the spacelike emitting parton is little changed and the $z$ 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 $Q^2 = 0$, this mapping gives an unphysical $\hat{u}$, 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 $Q^2$. 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 $p_{\perp\mathrm{W}}$ spectrum show quite a good agreement with this improved simulation [Miu99]. A worry was that an unexpectedly large primordial $k_{\perp}$, around 4 GeV, was required to match the data in the low- $p_{\perp\mathrm{Z}}$ region. However, at that time we had not realized that the data were not fully unsmeared. The required primordial $k_{\perp}$ 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 $\mathrm{e}^+\mathrm{e}^-\to \gamma^* / \mathrm{Z}^0$. There the old default $Q^2_{\mathrm{max}} = m_{\mathrm{Z}}^2$ allowed no emission at large $p_{\perp}$, $p_{\perp}\raisebox{-0.8mm}{\hspace{1mm}$\stackrel{>}{\sim}$\hspace{1mm}}m_{\mathrm{Z}}$ at LEP2. This is now corrected by the increased $Q^2_{\mathrm{max}} = s$, and using the $R$ of eq. () with $m_{\mathrm{W}} \to m_{\mathrm{Z}}$.

The above method does not address the issue of next-to-leading order corrections to the total $\mathrm{W}$ 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 $p_{\perp}$ shape obtained in our approach will be rather unaffected by next-to-leading order corrections (when used both for the total and the high-$p_{\perp}$ cross section). A rescaling by a common $K$ 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 $\mathrm{g}\to \mathrm{g}\mathrm{g}$ and $\mathrm{q}\to \mathrm{g}\mathrm{q}$ branchings in the $\mathrm{g}\mathrm{g}\to \mathrm{h}^0$ process, in order to match on to the $\mathrm{g}\mathrm{g}\to \mathrm{g}\mathrm{h}^0$ and $\mathrm{q}\mathrm{g}\to \mathrm{q}\mathrm{h}^0$ 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 $\mathrm{g}\mathrm{g}\to \mathrm{g}\mathrm{h}^0$ and $\mathrm{q}\mathrm{g}\to \mathrm{q}\mathrm{h}^0$ to $\mathrm{g}\mathrm{g}\to \mathrm{h}^0$ cross sections. (Whereas the $\mathrm{g}\mathrm{g}\to \mathrm{h}^0$ cross section by itself contains the complete expressions.) In this limit, the compact correction factors

$$R_{\mathrm{g}\mathrm{g}\to \mathrm{g}\mathrm{h}^0}(\hat{s},\... ...%% {2(\hat{s}^2-m_{\mathrm{h}}^2(\hat{s}-m_{\mathrm{h}}^2))^2}$$ (197)

and

$$R_{\mathrm{q}\mathrm{g}\to \mathrm{q}\mathrm{h}^0}(\hat{s},\... ...c{\hat{s}^2+\hat{u}^2}{\hat{s}^2+(\hat{s}-m_{\mathrm{h}}^2)^2}$$ (198)

can be derived. Even though they are clearly not as reliable as the above expressions for $\gamma^*/\mathrm{Z}^0/\mathrm{W}^{\pm}$, 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 $\mathrm{h}^0$, $\H ^0$ and $\mathrm{A}^0$. The Higgs correction factors are always in the comfortable range between $1/2$ and 1.

Note that a third process, $\mathrm{q}\overline{\mathrm{q}}\to \mathrm{g}\mathrm{h}^0$ does not fit into the pattern of the other two. The above process cannot be viewed as a showering correction to a lowest-order $\mathrm{q}\overline{\mathrm{q}}\to \mathrm{h}^0$ one: since the $\mathrm{q}$ 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 $p_{\perp}\to 0$ limit.