Loop-induced Decay Modes

One-loop Functions

One-loop diagrams between a scalar and two vector bosons with three internal lines can be reduced to the two loop functions

$F_{\text{q-loop}}(\tau )=-2\tau [1+(1-\tau )f(\tau )]$ and $F_{\text{W-loop}}(\tau )=2+3\tau +3\tau (2-\tau )f(\tau )$,

where $\tau =4m^2/M_h^2$ with $m$ being the mass of the particle running in the loop and

$$f(\tau )=\{\begin{array}{ll}{\arcsin }^2\left(\frac{1}{\sqrt{\tau }}\right)& \text{for }\tau \ge 1\\ -\frac{1}{4}{\left[\log \left(\frac{1+\sqrt{1-\tau }}{1-\sqrt{1-\tau }}\right)-i\pi \right]}^2& \text{for }\tau <1\end{array}$$

.

The imaginary part in the auxiliary function $f(\tau )$ is caused if the intermediate particles can go on-shell. Let us consider the limiting cases for the two loop functions:

• When the mass of the intermediate particle is negligible compared to the Higgs mass, i.e., $\tau \to 0$, $F_{\text{q-loop}}(\tau )\to -2\tau$ and $F_{\text{W-loop}}(\tau )\to 2$.
• When the mass of the intermediate particle is much larger than the Higgs mass, i.e., $\tau \to \infty$, $F_{\text{q-loop}}(\tau )\to -\frac{4}{3}$ and $F_{\text{W-loop}}(\tau )\to 7$.

(diagrams)

$h\to gg$

The absence of a $hgg$ vertex in the SM Lagrangian implies that the decay $h\to gg$ is not possible at tree-level. Nevertheless, this decay mode exists. The decay is mediated by a quark loop. Due to the large top Yukawa coupling, $y_t\simeq 1$, we would assume that the top quark loop dominates the decay width.

At one-loop level, $\Gamma (h\to gg)=\frac{{\alpha }_s^2}{256{\pi }^3}\frac{M_h^3}{v^2}\left[2{\left|{\sum }_{q=t,b,\dots }{F}_{\text{q-loop}}({\tau }_q)\right|}^2\right]$.

The explicit factor of 2 arises from the colour structure of the decay amplitude.

$h\to \gamma \gamma$

Similar to the $h\to gg$ decay, this decay is not possible at tree-level; there is no $h\gamma \gamma$ vertex in the SM Lagrangian. Besides quark loops, loop diagrams with W bosons and charged fermions also contribute to the decay width. Due to different spin statistics of fermions and bosons, we expect them to partially cancel each other.

At one-loop level, $\Gamma (h\to \gamma \gamma )=\frac{{\alpha }_e^2}{256{\pi }^3}\frac{M_h^3}{v^2}{\left|{\sum }_{f=t,b,c,\tau ,\dots }{N}_f{Q}_f^2{F}_{\text{f-loop}}({\tau }_f)+F_{\text{W-loop}}({\tau }_f)\right|}^2$.

Here, $N_f$ is a colour factor (3 for quarks and 1 for leptons), and $Q_f$ is the electric charge of the fermion in units of e.

$h\to Z\gamma$

This decay is also not possible at tree-level. The decay is mediated by the same set of particles as $h\to \gamma \gamma$. However, the couplings between $Z$ and the intermediate particles are different from those between $\gamma$ and the intermediate particles.

• First, the $Z$ couples to fermions with both vector and axial currents, while the $\gamma$ couples only to the vector part of the fermion current with coupling strength $Q_{f}e$. This leads to a slight parity violation (for example).
• Second, the $Z$ couples to the $W$ bosons with a different strength than the $\gamma$.