Gluon fusion

 The gluon fusion process for Higgs production, shown in fig. 2.1, can be calculated from the width of the H $\rightarrow$ gg decay (section 2.3.4) and the gluon structure function. The loop is totally dominated by the top quark because of the strong Higgs coupling to the heavy top quark. The cross section for the basic gluon to Higgs process is
 $$\begin{align} \hat{\sigma}({gg \rightarrow H}) &= \frac{(2\pi)^4}{2m_{H}^2} ... ...pi^2\Gamma_{H \rightarrow gg}}{N_{g}^2m_{H}} \delta(\hat{s}-m_{H}^2)\end{align}$$
with the width of the gluon decay inserted using (2.42) and (2.78), N_g = 8 the number of different gluons and $\hat{s}$ = x₁x₂s the squared energy of the gluon pair.

To get the full cross section the gluon cross section has to be integrated with the structure functions of the gluons

 

$$\sigma_{0}^{} \rightarrow \iint \hat{\sigma} \rightarrow$$ (22)

The lowest order cross section given in (2.34) has large corrections from higher order QCD diagrams. The increase in cross section from higher order diagrams is conventionally defined as the K-factor

 

$${\frac{\sigma_{HO}}{\sigma_{LO}}}$$ (23)

where LO (HO) refer to lowest (higher) order results. The K-factor for gluon fusion is in [8] evaluated in a next-to-leading order calculation and gives K = 1.5 almost independent of the Higgs mass.

The value of the cross section including the K-factor has two main uncertainties. The first is from the gluon structure function which still has a large uncertainty in the low x region. The cross section using a large set of todays best available structure functions was calculated in [9] and their relative differences shown in fig. 2.3. It can be seen that they differ by around 20% which can be taken as the theoretical uncertainty from the gluon structure function. At the time of data taking for the LHC it can be expected to have much better structure functions available with data from HERA, the Tevatron and even the LHC itself.

  

Figure 2.3: Ratios of Higgs production rate through gluon fusion for the structure functions MRS(A,A',G,R1,R2), CTEQ(2M,2MS,2MF,2ML,3M) and GRVHO94 as the function of the Higgs mass. All production rates are relative to the MRS(A) structure function.

The second uncertainty in the gluon fusion cross section is from corrections above the next-to-leading order. The cross section changes with the renormalisation scale $\mu$ as an effect of un-calculated higher order effects. By changing $\mu$ between m_H/2 and 2m_H it is in [8] concluded that the remaining uncertainties from higher order effects is below 15%.

The uncertainty in the cross section arising from uncertainties in the top quark mass are small and will be insignificant with an improved measurement of the top mass at the starting time of the LHC.

The production of the Higgs through gluon fusion is sensitive to a fourth generation of quarks. Because the Higgs couples in proportion to the fermion mass, a heavier generation of quarks is not suppressed in the process fig. 2.1a as would be expected for a loop process with a heavier particle in the loop. Including a fourth generation of very heavy quarks will more than double the cross section as shown in fig. 2.4. This has the consequence that the Higgs cross section is sensitive to a fourth generation of quarks even if the quarks are too heavy for a direct discovery at the LHC. The mass range is limited by the scale of new physics where the standard model breaks down.

  

Figure 2.4: The enhancement in the Higgs production cross section from gluon fusion if a fourth very heavy generation of quarks exists.