SU(2) gauge theory ($SU(2)$ gauge theory)

An SU(2) gauge theory is a Yang-Mills theory with gauge group $SU(2)$. In its defining representation the generators are $T^a={\sigma }^a/2$, and its structure constants are the Levi-Civita symbols ${ϵ}^{abc}$.

A single complex scalar field in the two-dimensional fundamental representation with potential $\lambda ({\varphi }^{†}\varphi -v^2/2{)}^2$, $\lambda ,v>0$, has a three-sphere vacuum manifold. The stabilizer in $SU(2)$ of a chosen nonzero vector is trivial, so all three gauge bosons acquire the common mass $gv/2$. In unitary gauge, $\varphi =(0,v+h{)}^T/\sqrt{2}$ leaves one physical radial scalar of squared mass $2\lambda v^2$. Its couplings are $g^{2}vh{A}^2/4+g^2{h}^2{A}^2/8-\lambda v{h}^3-\lambda h^4/4$. The three angular fields become longitudinal vector polarizations through the Higgs mechanism.

For field strength $F_{\mu \nu }^a=f_{\mu \nu }^a-g{ϵ}^{abc}{A}_{\mu }^b{A}_{\nu }^c$, the square produces ${\mathcal{L}}_3=g{ϵ}^{abc}({\partial }_{\mu }{A}_{\nu }^a)A^{b\mu }{A}^{c\nu }$ and ${\mathcal{L}}_4=-g^2{ϵ}^{abc}{ϵ}^{ade}{A}_{\mu }^b{A}_{\nu }^c{A}^{d\mu }{A}^{e\nu }/4$. The cubic Feynman vertex has one derivative; the quartic one has none. No higher pure-gauge vertex occurs.

A nonzero complex fundamental doublet has trivial stabilizer in the gauged $SU(2)$. Its three angular directions are gauge directions, absorbed by the Higgs mechanism. For vacuum ${\varphi }_0=(0,v/\sqrt{2}{)}^T$, the three gauge bosons have equal mass $gv/2$; one radial scalar remains. There is no unbroken photon-like gauge generator in this theory.

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.

A real scalar field in the Adjoint representation can have a nonzero vacuum expectation value, leaving one massless and two massive gauge bosons. This breaking differs from electroweak symmetry breaking by a complex Higgs doublet.

A real scalar triplet of weak hypercharge zero with vacuum expectation value $(0,0,v)$ breaks $T^1,T^2$ but preserves both $T^3$ and the independent weak hypercharge generator. With canonical kinetic terms its gauge-boson mass matrix in $(W^1,W^2,W^3,B)$ is $g^2{v}^2\mathrm{diag}(1,1,0,0)$. The two massless neutral gauge bosons prevent this field alone from reproducing Standard Model electroweak symmetry breaking.

For a nonvanishing adjoint Higgs field, put $n=\Phi /|\Phi |$. If an oriented orthonormal internal basis satisfies $[e_a,e_b]=\kappa {\epsilon }_{abc}{e}_c$, the displayed tensor is gauge invariant. In a gauge with $n=e_3$, write $A_j=a_j^a{e}_a$; the transverse quadratic terms cancel, leaving ${\mathcal{F}}_{jk}={\partial }_j{a}_k^3-{\partial }_k{a}_j^3$. Hence it is a closed differential form wherever $\Phi \ne 0$. Under trace normalization of su(2) with $-\mathrm{tr}(XY)$, ${\kappa }^2=2$ and the commutator term has coefficient $1/2$. With $-2\mathrm{tr}(XY)$, its coefficient is one. Omitting this normalization factor generally destroys closure.

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 $\lambda (|\varphi {|}^2-v^2{)}^2/4$.

For a real triplet with potential $\lambda ({\Phi }^2-v^2{)}^2/8$, unitary gauge gives $\Phi =(0,0,v+h)$ and scalar interaction terms $-\lambda v{h}^3/2-\lambda h^4/8+2e^{2}vh{W}^{+}{W}^{-}+e^2{h}^2{W}^{+}{W}^{-}$. Functional differentiation, including identical-field multiplicities, gives vertices $-3i\lambda v$, $-3i\lambda$, $2i{e}^{2}v{g}_{\mu \nu }$ and $2i{e}^2{g}_{\mu \nu }$ respectively. The physical scalar is neutral under the residual U(1), so there are no $hAA$ or $hhAA$ vertices.