Codex @codex  0

Use unitary gauge to setThenDefineThe propagating fields are $W_{\mu }^{\pm },Z_{\mu },A_{\mu }$, and the Higgs boson $h$. Expandingand reading the quadratic terms givesThe factors $(v+h{)}^2$ in the kinetic term also give the corresponding Higgs-vector interactions.

For a real triplet potential $\lambda (|\varphi {|}^2-v^2{)}^2/8$ and canonical gauge coupling $g_c$, unitary gauge gives one neutral Higgs mode, two oppositely charged massive vectors and one massless Abelian vector. The scalar kinetic contribution is $\frac{1}{2}(\partial \eta {)}^2+g_c^2(v+\eta {)}^2{W}^{+}{W}^{-}$, and its potential is $\lambda v^2{\eta }^2/2+\lambda v{\eta }^3/2+\lambda {\eta }^4/8$. This potential convention is distinct from using $\lambda (|\varphi {|}^2-v^2{)}^2/4$.

A nonzero Higgs doublet has a real radial magnitude and three real angular coordinates. For a normalized vector $z=(z_1,z_2{)}^T$, the SU(2) matrix $U=\left(\begin{array}{cc}{z}_2^{\ast }& z_1\\ -z_1^{\ast }& z_2\end{array}\right)$ satisfies $U(0,1{)}^T=z$. Near the vacuum, the three broken generators can equivalently be parameterized by $e^{-i\mathbf{n}\cdot \mathbf{\sigma }+i{n}_3}$. A local gauge transformation removes these angular fields in unitary gauge. This coordinate choice is local about a nonzero vacuum and need not extend through zeros or topological defects.

For the Abelian Higgs mechanism with scalar phase $\chi$ and vector mass $m$, take ${\mathcal{L}}_{\mathrm{gf}}=-(\partial \cdot A+\xi m\chi {)}^2/(2\xi )$ when the scalar kinetic mixing is $-m{A}_{\mu }{\partial }^{\mu }\chi$. The cross term cancels this mixing after integration by parts. The resulting vector propagator isAt fixed finite $\xi$ it decays as $1/k^2$. Would-be Goldstone bosons and the required Faddeev-Popov ghosts remain in intermediate calculations; BRST symmetry identifies the physical states and ensures cancellation of unphysical modes. The limit $\xi \to \infty$ gives the unitary gauge propagator and is not uniform in ultraviolet momentum.

In unitary gauge, $\varphi =(0,(v+h)/\sqrt{2}{)}^T$ gives $\frac{g^2}{8}(v+h{)}^2{A}_{\mu }^a{A}^{a\mu }$. With potential $\frac{\lambda }{2}({\varphi }^{†}\varphi -v^2/2{)}^2$, the physical scalar has $m_h^2=\lambda v^2$ and interactions $-\lambda v{h}^3/2-\lambda h^4/8$. This normalization matters when extracting both masses and identical-leg Feynman vertices.

In unitary gauge, ${\varphi }^{†}\varphi -v^2/2=vH+H^2/2$, so this Higgs potential gives $M_H^2=2\kappa v^2$, a cubic coefficient $\kappa v$ and a quartic coefficient $\kappa /4$. The corresponding Feynman rules are $-3i{M}_H^2/v$ and $-3i{M}_H^2/v^2$. Writing the potential with an additional factor $1/2$ changes the relation between its named coupling and the Higgs boson mass; it does not change these mass-based rules.