Two photon decay

 Since the photon is massless there is no coupling between the standard model Higgs and the photon. However, the decay is possible through loop processes with either fermions or bosons in the loop. For an extended Higgs sector charged scalars can also enter. Feynman diagrams for the lowest order processes are shown in fig. 2.9.

  

Figure 2.9: Feynman diagrams in the Standard Model for the Higgs decay to two photons in lowest order.

The calculation of the matrix element for the decay is rather complicated and involves dimensional regularisation of the infinities arising from the loop. The end result for the squared matrix element is [20]

 

$${\frac{g^2m_{H}^4}{32\pi^2m_{W}^2}} \left\vert \sum_i \alpha N_c e_i^2 F_i \right\vert^$$ (34)

with the sum over all scalars, fermions and bosons in the loop with charge e_i and colour factor N_c (3 for quarks else 1). The factor F is given as
  $$\begin{align} F_{\text{boson}} &= 2 + 3\tau + 3\tau(2-\tau)f(\tau), \\ F_{\tex... ...cases} \\ \intertext{and} \tau &= \left(\frac{2m_i}{m_{H}}\right)^2.\end{align}$$
The term F_fermion disappears for small $\tau$ which means that light quarks and leptons are insignificant. The only fermion participating is the top quark. For bosons the loop only contains the charged W.

The F parameters for the top and W contributions scaled with the colour factor are plotted in fig. 2.10 for the Higgs mass range where the decay is of interest. It can be seen how the fermion and boson loops have opposite signs but with the W loop dominating. Adding a fourth family with the lepton and the two quarks having masses above 500 GeV will give a near cancellation in the matrix element. A possible scalar particle in the loop will only have a small influence following (2.74).

  

Figure: The contribution from the W and top quark loop to the matrix element in the H $\rightarrow$ $\gamma$$\gamma$ decay. The sum of the two and the influence of adding a 4th heavy family ( l',$\nu$',b',t') is also shown.

The width of the H $\rightarrow$ $\gamma$$\gamma$ decay channel is now easily obtained using (2.42) and performing the trivial angular integration. The result is

 

$$\Gamma_{H \rightarrow\gamma\gamma}^{} {\frac{\alpha^2 g^2m_{H}^3}{1024\pi^3m_{W}^2}} \left\vert \sum_i N_c e_i^2 F_i \right\vert^$$ (35)

which always gives a branching ratio below 0.3% due to the much larger width of the H $\rightarrow$ b$\bar{b}$ decay. The two photon signature is, however, clean and as will be seen in section 2.6.1 an important decay channel at the LHC. Adding a fourth family will reduce the branching ratio by more than a factor 8 if the Higgs mass is below 140 GeV.

The radiative corrections to the H $\rightarrow$ $\gamma$$\gamma$ decay width are relatively simple as they only affect the top quark loop and neither the W loop nor the final state photons. The corrections are below 3% [21] and thus of limited importance.