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Single $\mathrm{W}/ \mathrm{Z}$ production

MSEL = 11, 12, 13, 14, 15, (21)
ISUB =
1 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \gamma^* / \mathrm{Z}^0$
2 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{W}^+$
15 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{g}(\gamma^* / \mathrm{Z}^0)$
16 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{g}\mathrm{W}^+$
19 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \gamma (\gamma^* / \mathrm{Z}^0)$
20 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \gamma \mathrm{W}^+$
30 $\mathrm{f}_i \mathrm{g}\to \mathrm{f}_i (\gamma^* / \mathrm{Z}^0)$
31 $\mathrm{f}_i \mathrm{g}\to \mathrm{f}_k \mathrm{W}^+$
35 $\mathrm{f}_i \gamma \to \mathrm{f}_i (\gamma^* / \mathrm{Z}^0)$
36 $\mathrm{f}_i \gamma \to \mathrm{f}_k \mathrm{W}^+$
(141) $\mathrm{f}_i \overline{\mathrm{f}}_i \to \gamma/\mathrm{Z}^0/\mathrm{Z}'^0$
(142) $\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{W}'^+$

This group consists of $2 \to 1$ processes, i.e. production of a single resonance, and $2 \to 2$ processes, where the resonance is recoiling against a jet or a photon. The processes 141 and 142, which also are listed here, are described further elsewhere.

With initial-state showers turned on, the $2 \to 1$ processes also generate additional jets; in order to avoid double-counting, the corresponding $2 \to 2$ processes should therefore not be turned on simultaneously. The basic rule is to use the $2 \to 1$ processes for inclusive generation of $\mathrm{W}/ \mathrm{Z}$, i.e. where the bulk of the events studied have $p_{\perp}\ll m_{\mathrm{W}/\mathrm{Z}}$. With the introduction of explicit matrix-element-inspired corrections to the parton shower [Miu99], also the high-$p_{\perp}$ tail is well described in this approach, thus offering an overall good description of the full $p_{\perp}$ spectrum of gauge bosons [Bál01].

If one is interested in the high-$p_{\perp}$ tail only, however, the generation efficiency will be low. It is here better to start from the $2 \to 2$ matrix elements and add showers to these. However, the $2 \to 2$ matrix elements are divergent for $p_{\perp}\to 0$, and should not be used down to the low-$p_{\perp}$ region, or one may get unphysical cross sections. As soon as the generated $2 \to 2$ cross section corresponds to a non-negligible fraction of the total $2 \to 1$ one, say 10%-20%, Sudakov effects are likely to be affecting the shape of the $p_{\perp}$ spectrum to a corresponding extent, and results should not be trusted.

The problems of double-counting and Sudakov effects apply not only to $\mathrm{W}/ \mathrm{Z}$ production in hadron colliders, but also to a process like $\mathrm{e}^+\mathrm{e}^-\to \mathrm{Z}^0 \gamma$, which clearly is part of the initial-state radiation corrections to $\mathrm{e}^+\mathrm{e}^-\to \mathrm{Z}^0$ obtained for MSTP(11) = 1. As is the case for $\mathrm{Z}$ production in association with jets, the $2 \to 2$ process should therefore only be used for the high-$p_{\perp}$ region.

The $\mathrm{Z}^0$ of subprocess 1 includes the full interference structure $\gamma^* / \mathrm{Z}^0$; via MSTP(43) you can select to produce only $\gamma^*$, only $\mathrm{Z}^0$, or the full $\gamma^* / \mathrm{Z}^0$. The same holds true for the $\mathrm{Z}'^0$ of subprocess 141; via MSTP(44) any combination of $\gamma^*$, $\mathrm{Z}^0$ and $\mathrm{Z}'^0$ can be selected. Thus, subprocess 141 with MSTP(44) = 4 is essentially equivalent to subprocess 1 with MSTP(43) = 3; however, process 141 also includes the possibility of a decay into Higgs bosons. Also processes 15, 19, 30 and 35 contain the full mixture of $\gamma^* / \mathrm{Z}^0$, with MSTP(43) available to change this. Note that the $\gamma^* / \mathrm{Z}^0$ decay products can have an invariant mass as small as the program cutoff. This can be changed using CKIN.

Note that process 1, with only $\mathrm{q}\overline{\mathrm{q}}\to \gamma^* \to \ell^+ \ell^-$ allowed, and studied in the region well below the $\mathrm{Z}^0$ mass, is what is conventionally called Drell-Yan. This latter process therefore does not appear under a separate heading, but can be obtained by a suitable setting of switches and parameters.

A process like $\mathrm{f}_i \overline{\mathrm{f}}_j \to \gamma \mathrm{W}^+$ requires some comment. When the $\mathrm{W}$ boson decays, photons can be radiated off the decay products. The full interference between photon radiation off the incoming fermions, the intermediate $\mathrm{W}$ boson, and the decay products is not included in the $\textsc{Pythia}$ treatment. If such effects are important, a full matrix element calculation is preferred. Some caution must therefore be exercised; see also section [*] for related comments.

For the $2 \to 1$ processes, the Breit-Wigner includes an $\hat{s}$-dependent width, which should provide an improved description of line shapes. In fact, from a line-shape point of view, process 1 should provide a more accurate simulation of $\mathrm{e}^+\mathrm{e}^-$ annihilation events than the dedicated $\mathrm{e}^+\mathrm{e}^-$ generation scheme of PYEEVT (see section [*]). Another difference is that PYEEVT only allows the generation of $\gamma^* / \mathrm{Z}^0\to \mathrm{q}\overline{\mathrm{q}}$, while process 1 additionally contains $\gamma^* / \mathrm{Z}^0\to \ell^+ \ell^-$ and $\nu \overline{\nu}$. The parton-shower and fragmentation descriptions are the same, but the process 1 implementation only contains a partial interface to the first- and second-order matrix-element options available in PYEEVT, see MSTP(48).

All processes in this group have been included with the correct angular distribution in the subsequent $\mathrm{W}/\mathrm{Z}\to \mathrm{f}\overline{\mathrm{f}}$ decays. In process 1 also fermion mass effects have been included in the angular distributions, while this is not the case for the other ones. Normally mass effects are not large anyway.

As noted earlier, some approximations can be used to simulate higher-order processes. The process $\mathrm{e}^+\mathrm{e}^-\to \mathrm{e}^+ \mathrm{e}^- \mathrm{Z}^0$ can be simulated in two different ways. One is to make use of the $\mathrm{e}$ `sea' distribution inside $\mathrm{e}$, i.e. have splittings $\mathrm{e}\to \gamma \to \mathrm{e}$. This can be obtained, together with ordinary $\mathrm{Z}^0$ production, by using subprocess 1, with MSTP(11) = 1 and MSTP(12) = 1. Then the contribution of the type above is 5.0 pb for a 500 GeV $\mathrm{e}^+\mathrm{e}^-$ collider, compared with the correct 6.2 pb [Hag91]. Alternatively one may use process 35, with MSTP(11) = 1 and MSTP(12) = 0, relying on the splitting $\mathrm{e}\to \gamma$. This process has a singularity in the forward direction, regularized by the electron mass and also sensitive to the virtuality of the photon. It is therefore among the few where the incoming masses have been included in the matrix element expression. Nevertheless, it may be advisable to set small lower cut-offs, e.g. CKIN(3) = CKIN(5) = 0.01, if one should experience problems (e.g. at higher energies).

Process 36, $\mathrm{f}\gamma \to \mathrm{f}' \mathrm{W}^{\pm}$ may have corresponding problems; except that in $\mathrm{e}^+\mathrm{e}^-$ the forward scattering amplitude for $\mathrm{e}\gamma \to \nu \mathrm{W}$ is killed (radiation zero), which means that the differential cross section is vanishing for $p_{\perp}\to 0$. It is therefore feasible to use the default CKIN(3) and CKIN(5) values in $\mathrm{e}^+\mathrm{e}^-$, and one also comes closer to the correct cross section.

The process $\mathrm{g}\mathrm{g}\to \mathrm{Z}^0 \b\overline{\mathrm{b}}$, formerly available as process 131, has been removed from the current version, since the implementation turned out to be slow and unstable. However, process 1 with incoming flavours set to be $\b\overline{\mathrm{b}}$ (by KFIN(1,5) = KFIN(1,-5) = KFIN(2,5) = KFIN(2,-5) = 1 and everything else = 0) provides an alternative description, where the additional $\b\overline{\mathrm{b}}$ are generated by $\mathrm{g}\to \b\overline{\mathrm{b}}$ branchings in the initial-state showers. (Away from the low-$p_{\perp}$ region, process 30 with KFIN values as above except that also incoming gluons are allowed, offers yet another description. Here it is in terms of $\mathrm{g}\b\to \mathrm{Z}^0 \b $, with only one further $\mathrm{g}\to \b\overline{\mathrm{b}}$ branching constructed by the shower.) At first glance, the shower approach would seem less reliable than the full $2 \to 3$ matrix element. The relative lightness of the $\b $ quark will generate large logs of the type $\ln(m_{\mathrm{Z}}^2/m_{\b }^2)$, however, that ought to be resummed [Car00]. This is implicit in the parton-density approach of incoming $\b $ quarks but absent from the lowest-order $\mathrm{g}\mathrm{g}\to \mathrm{Z}^0 \b\overline{\mathrm{b}}$ matrix elements. Therefore actually the shower approach may be the more accurate of the two in the region of intermediate transverse momenta.


next up previous contents
Next: pair production Up: Electroweak Gauge Bosons Previous: Prompt photon production   Contents
Stephen Mrenna 2007-10-30