Feynman Rules - SM Higgs

Coupling to Gauge Bosons

For the coupling to W bosons, the relevant term in the Lagrangian density is

$(D_{\mu }\Phi {)}^{†}(D^{\mu }\Phi )\supset \frac{1}{4}{g}^2(v+h{)}^2{W}_{\mu }^{+}{W}^{-\mu }$.

Expanding the term in the Higgs field h, we get as the leading term, the W boson mass term $M_W^2=g^2{v}^2/4$. The hWW and hhWW vertices are

$=i\frac{g^{2}v}{2}{g}_{\mu \nu }=2i\frac{M_W^2}{v}{g}_{\mu \nu }$

$=i\frac{g^2}{4}\times 2g_{\mu \nu }=2i\frac{M_W^2}{v^2}{g}_{\mu \nu }$. The extra factor of 2 is a symmetry factor, which accounts for the interchange of two h bosons, since they cannot be distinguished.

For the coupling to Z bosons, the relevant term in the Lagrangian density is,

$(D_{\mu }\Phi {)}^{†}(D^{\mu }\Phi )\supset \frac{1}{8}(g^2+g^{\prime 2})(v+h{)}^2{Z}_{\mu }{Z}^{\mu }$.

Expanding this term in the Higgs field h yields as the leading term, the Z boson mass term, $M_Z^2=(g^2+g{’}^2)v^2/4$. The hZZ and hhZZ vertices are

$=i\frac{(g^2+g^{\prime 2})v}{4}\times 2\times g_{\mu \nu }=2i\frac{M_Z^2}{v}{g}_{\mu \nu }$

$=i\frac{(g^2+g^{\prime 2})}{8}\times 2\times 2\times g_{\mu \nu }=2i\frac{M_Z^2}{v^2}{g}_{\mu \nu }$.

The extra factors of 2 and 2x2 are symmetry factors, accounting for the interchange of the two Z bosons and both the two Z bosons and two h bosons since they cannot be distinguished.

Coupling to Fermions

For the coupling to quark bosons, the relevant term in the Lagrangian density is

${\mathcal{L}}_{\text{Yukawa}}\supset \left[y_u{\bar{u}}_R{\tilde{\Phi }}^{†}{Q}_L+y_d{\bar{d}}_R{\Phi }^{†}{Q}_L\right]+\text{h.c.}$

$\tilde{\Phi }=i{\sigma }^2{\Phi }^{\ast }$ is the charge conjugate of the Higgs doublet, with hypercharge $Y_{\tilde{\Phi }}=-1/2$. Moreover, $Q_L$ is the left-handed quark doublet with three generations of quarks, and $u_R$ and $d_R$ are the right-handed quark singlets with three generations of up-type and down-type quarks respectively. The quark masses arise as

$m_{d,i}=\frac{[y_d{]}_{i}v}{\sqrt{2}},m_{u,i}=\frac{[y_u{]}_{i}v}{\sqrt{2}}$

where $[y_d{]}_i$ and $[y_u{]}_i$ are the Yukawa couplings to the $i^{th}$ generation.

Expanding the term in the Higgs field h yields as the hqq vertex,

$=-i\frac{[y_q]}{\sqrt{2}}=-i\frac{m_q}{v}$.

But, the above holds good only if the Yukawa couplings are diagonal, which holds in the so-called mass basis. (See CKM matrix - SM)

Note: The couplings to leptons follow the pattern of the up-type quarks.

Higgs Self Interactions

The relevant term in the Lagrangian density is

$V(\Phi )\supset {\mu }^2(v+h{)}^2+\lambda (v+h{)}^4=\lambda v^2{h}^2+\lambda v{h}^3+\frac{\lambda }{4}{h}^4+\text{const}$.

Expanding this term in the Higgs field h yields as the first term, the Higgs boson mass term $m_h^2=2\lambda v$. The hhh and hhhh vertices can be obtained as

$=-i\lambda v\times 3!=-6i\lambda v=-3i\frac{m_h^2}{v}$

$=-i\frac{\lambda }{4}\times 4!=-6i\lambda =-3i\frac{m_h^2}{v^2}$.