Light Standard Model Higgs

MSEL = 16, 17, 18
ISUB =

3	$\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{h}^0$
24	$\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{Z}^0 \mathrm{h}^0$
26	$\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{W}^+ \mathrm{h}^0$
32	$\mathrm{f}_i \mathrm{g}\to \mathrm{f}_i \mathrm{h}^0$
102	$\mathrm{g}\mathrm{g}\to \mathrm{h}^0$
103	$\gamma \gamma \to \mathrm{h}^0$
110	$\mathrm{f}_i \overline{\mathrm{f}}_i \to \gamma \mathrm{h}^0$
111	$\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{g}\mathrm{h}^0$
112	$\mathrm{f}_i \mathrm{g}\to \mathrm{f}_i \mathrm{h}^0$
113	$\mathrm{g}\mathrm{g}\to \mathrm{g}\mathrm{h}^0$
121	$\mathrm{g}\mathrm{g}\to \mathrm{Q}_k \overline{\mathrm{Q}}_k \mathrm{h}^0$
122	$\mathrm{q}_i \overline{\mathrm{q}}_i \to \mathrm{Q}_k \overline{\mathrm{Q}}_k \mathrm{h}^0$
123	$\mathrm{f}_i \mathrm{f}_j \to \mathrm{f}_i \mathrm{f}_j \mathrm{h}^0$ ( $\mathrm{Z}^0 \mathrm{Z}^0$ fusion)
124	$\mathrm{f}_i \mathrm{f}_j \to \mathrm{f}_k \mathrm{f}_l \mathrm{h}^0$ ( $\mathrm{W}^+ \mathrm{W}^-$ fusion)

In this section we discuss the production of a reasonably light Standard Model Higgs, below 700 GeV, say, so that the narrow width approximation can be used with some confidence. Below 400 GeV there would certainly be no trouble, while above that the narrow width approximation is gradually starting to break down.

In a hadron collider, the main production processes are 102, 123 and 124, i.e. $\mathrm{g}\mathrm{g}$, $\mathrm{Z}^0 \mathrm{Z}^0$ and $\mathrm{W}^+ \mathrm{W}^-$ fusion. In the latter two processes, it is also necessary to take into account the emission of the space-like $\mathrm{W}/ \mathrm{Z}$ bosons off quarks, which in total gives the $2 \to 3$ processes above.

Further processes of lower cross sections may be of interest because of easier signals. For instance, processes 24 and 26 give associated production of a $\mathrm{Z}$ or a $\mathrm{W}$ together with the $\mathrm{h}^0$. There is also the processes 3 (see below), 121 and 122, which involve production of heavy flavours.

Process 3 contains contributions from all flavours, but is completely dominated by the subprocess $\t\overline{\mathrm{t}}\to \mathrm{h}^0$, i.e. by the contribution from the top sea distributions. Assuming, of course, that parton densities for top quarks are available, which is no longer the case in current parameterizations. This process is by now known to overestimate the cross section for Higgs production as compared with a more careful calculation based on the subprocess $\mathrm{g}\mathrm{g}\to \t\overline{\mathrm{t}}\mathrm{h}^0$, process 121. The difference between the two is that in process 3 the $\t$ and $\overline{\mathrm{t}}$ are added by the initial-state shower, while in 121 the full matrix element is used. The price to be paid is that the complicated multibody phase space in process 121 makes the program run slower than with most other processes. As usual, it would be double-counting to include the same flavour both with 3 and 121. An intermediate step -- in practice probably not so useful -- is offered by process 32, $\mathrm{q}\mathrm{g}\to \mathrm{q}\mathrm{h}^0$, where the quark is assumed to be a $\b$ one, with the antiquark added by the showering activity.

Process 122 is similar in structure to 121, but is less important. In both process 121 and 122 the produced quark is assumed to be a $\t$; this can be changed in KFPR(121,2) and KFPR(122,2) before initialization, however. For $\b$ quarks it could well be that process 3 with $\b\overline{\mathrm{b}}\to \mathrm{h}^0$ is more reliable than process 121 with $\mathrm{g}\mathrm{g}\to \b\overline{\mathrm{b}}\mathrm{h}^0$ [Car00]; see the discussion on $\mathrm{Z}^0 \b\overline{\mathrm{b}}$ final states in section . Thus it would make sense to run with all quarks up to and including $\b$ simulated in process 3 and then consider $\t$ quarks separately in process 121. Assuming no $\t$ parton densities, this would actually be the default behaviour, meaning that the two could be combined in the same run without double counting.

The two subprocess 112 and 113, with a Higgs recoiling against a quark or gluon jet, are also effectively generated by initial-state corrections to subprocess 102. Thus, in order to avoid double-counting, just as for the case of $\mathrm{Z}^0/\mathrm{W}^+$ production, section , these subprocesses should not be switched on simultaneously. Process 111, $\mathrm{q}\overline{\mathrm{q}}\to \mathrm{g}\mathrm{h}^0$ is different, in the sense that it proceeds through an $s$-channel gluon coupling to a heavy-quark loop, and that therefore the emitted gluon is necessary in the final state in order to conserve colours. It is not to be confused with a gluon-radiation correction to the Born-level process 3, like in process 32, since processes 3 and 32 vanish for massless quarks while process 111 is mainly intended for such. The lack of a matching Born-level process shows up by process 111 being vanishing in the $p_{\perp}\to 0$ limit. Numerically it is of negligible importance, except at very large $p_{\perp}$ values. Process 102, possibly augmented by 111, should thus be used for inclusive production of Higgs, and 111-113 for the study of the Higgs subsample with high transverse momentum.

A warning is that the matrix-element expressions for processes 111-113 are very lengthy and the coding therefore more likely to contain some errors and numerical instabilities than for most other processes. Therefore the full expressions are only available by setting the non-default value MSTP(38)=0. Instead the default is based on the simplified expressions obtainable if only the top quark contribution is considered, in the $m_{\t } \to \infty$ limit [Ell88]. As a slight improvement, this expression is rescaled by the ratio of the $\mathrm{g}\mathrm{g}\to \mathrm{h}^0$ cross sections (or, equivalently, the $\mathrm{h}\to \mathrm{g}\mathrm{g}$ partial widths) of the full calculation and that in the $m_{\t } \to \infty$ limit. Simple checks show that this approach normally agrees with the full expressions to within $\sim 20$%, which is small compared with other uncertainties. The agreement is worse for process 111 alone, about a factor of 2, but this process is small anyway. We also note that the matrix element correction factors, used in the initial-state parton shower for process 102, subsection , are based on the same $m_{\t } \to \infty$ limit expressions, so that the high-$p_{\perp}$ tail of process 102 is well matched to the simple description of process 112 and 113.

In $\mathrm{e}^+\mathrm{e}^-$ annihilation, associated production of an $\mathrm{h}^0$ with a $\mathrm{Z}^0$, process 24, is usually the dominant one close to threshold, while the $\mathrm{Z}^0 \mathrm{Z}^0$ and $\mathrm{W}^+ \mathrm{W}^-$ fusion processes 123 and 124 win out at high energies. Process 103, $\gamma\gamma$ fusion, may also be of interest, in particular when the possibilities of beamstrahlung photons and backscattered photons are included (see subsection ). Process 110, which gives an $\mathrm{h}^0$ in association with a $\gamma$, is a loop process and is therefore suppressed in rate. It would have been of interest for a $\mathrm{h}^0$ mass above 60 GeV at LEP 1, since its phase space suppression there is less severe than for the associated production with a $\mathrm{Z}^0$. Now it is not likely to be of any further interest.

The branching ratios of the Higgs are very strongly dependent on the mass. In principle, the program is set up to calculate these correctly, as a function of the actual Higgs mass, i.e. not just at the nominal mass. However, higher-order corrections may at times be important and not fully unambiguous; see for instance MSTP(37).

Since the Higgs is a spin-0 particle it decays isotropically. In decay processes such as $\mathrm{h}^0 \to \mathrm{W}^+ \mathrm{W}^- / \mathrm{Z}^0 \mathrm{Z}^0 \to 4$ fermions angular correlations are included [Lin97]. Also in processes 24 and 26, $\mathrm{Z}^0$ and $\mathrm{W}^{\pm}$ decay angular distributions are correctly taken into account.