Tree-level Decay Modes

$h\to f\bar{f}$

Mediated at tree-level by the Yukawa term in the Lagrangian.

(diagram)

$=i\mathcal{M}(h\to f\bar{f})=-i\frac{m_q}{v}[{\bar{u}}_f{v}_f]$.

Using Casimir’s trick,

$|\mathcal{M}{|}^2=\frac{N_C{m}_f^2}{v^2}\text{Tr}[/p_1+m_f][/p_2-m_f]=\frac{N_C{m}_f^2}{v^2}(4p_1.p_2-4m_f^2)$

$=\frac{2N_C{m}_f^2{M}_h^2}{v^2}\left(1-\frac{4m_f^2}{M_h^2}\right)$.

We use $N_C=3$ for quarks and $N_C=1$ for leptons. Further,

$M_h^2=P^2=(p_1+p_2{)}^2=2m_f^2+2p_1.p_2⟹p_1.p_2=\frac{M_h^2}{2}-m_f^2$.

Thus,

$\Gamma (h\to f\bar{f})=\frac{2N_C{m}_f^2}{16\pi v^2{M}_h}\left(1-\frac{4m_f^2}{M_h^2}\right){\lambda }^{1/2}(M_h^2,m_f^2,m_f^2)=\frac{N_C{m}_f^2{M}_h}{8\pi v^2}{\left(1-\frac{4m_f^2}{M_h^2}\right)}^{3/2}$.

Note:

• The decay width is proportional to two powers of the fermion mass but only one power of the Higgs mass.
• All SM fermions except the top quark are much lighter than the Higgs boson, i.e., $m_f\ll M_h$. In this limit, velocity term $1-4m_f^2/M_h^2$ is close to unity.
• The decay width is linearly depend on the colour factor $N_C\in \{1,3\}$ for leptons and quarks, respectively.
• All the above renders $\Gamma (h\to b\bar{b})$ the dominant decay to fermion-antifermion pairs in the SM.

$h\to W^{+}{W}^{-}$

Energetically not possible but if it were, it would be mediated by the kinetic terms and the Higgs potential in the Lagrangian.

(diagram)

$=i\mathcal{M}(h\to W^{+}{W}^{-})=2i\frac{M_W^2}{v}{g}^{\mu \nu }{\epsilon }_{\mu }^{\ast }({\lambda }_1){\epsilon }_{\nu }^{\ast }({\lambda }_2)$.

If one or both of the two W bosons is off-shell, then the decay is possible and the corresponding matrix element can be obtained from the above by replacing the polarization vectors with the corresponding propagators, with subsequent decay of the virtual W boson(s) into a suitable final state.

$\Gamma (h\to W^{+}{W}^{-})=\frac{M_h^3}{16\pi v^2}\sqrt{1-\frac{4M_W^2}{M_h^2}}\left[1-\frac{4M_W^2}{M_h^2}+\frac{3}{4}{\left(\frac{4M_W^2}{M_h^2}\right)}^2\right]$.

$h\to ZZ$

This decay is also energetically not possible. If it were, it would be mediated by the kinetic terms and the Higgs potential in the Lagrangian.

$\Gamma (h\to ZZ)=\frac{1}{2}\Gamma (h\to W^{+}{W}^{-})\text{replacing the mass}$

$=\frac{M_h^3}{32\pi v^2}\sqrt{1-\frac{4M_Z^2}{M_h^2}}\left[1-\frac{4M_Z^2}{M_h^2}+\frac{3}{4}{\left(\frac{4M_Z^2}{M_h^2}\right)}^2\right]$

Note:

• The decay width for on-shell processes is proportional to three powers of the Higgs mass. For decays to off-shell W and Z bosons, we expect the proportionality to shift to $M^3$ for $M=\sqrt{(p_1+p_2{)}^2}$.
• Since these processes are not energetically possible, they are not dominant processes; however, they would quickly take over the total decay rate of the Higgs boson if it were heavier than $2M_W$ or $2M_Z$ respectively.
• Thus, $\Gamma (h\to b\bar{b})$ remains the dominant decay mode of the Higgs boson in the SM at $M_h\simeq 125\text{ GeV}$.