The <scalar potential> can be written as
$$
V=\frac{\lambda}{4}(\phi_a\phi_a-v^2)^2-\frac{\lambda v^4}{4},\qquad v^2=-\frac{m^2}{\lambda}.
$$
Its <vacuum manifold> is the sphere $\phi_a\phi_a=v^2$. The <Adjoint double cover from SU(2) to SO(3)> rotates any nonzero <vacuum expectation value> to $(0,0,v)$, with $v>0$. A local <unitary gauge> removes the two angular fluctuations, leaving $\phi=(0,0,v+\eta)$. This choice holds in a neighbourhood of the nonzero vacuum; a single global rotation cannot align arbitrary position-dependent fields, and topologically nontrivial configurations can obstruct a global <unitary gauge>.

The generator $t^3$ fixes the <vacuum expectation value>, while $t^1,t^2$ do not. Thus the <Higgs mechanism> in this <SU(2) gauge theory with an adjoint Higgs field> gives
$$
\boxed{SU(2)\longrightarrow U(1).}
$$
The two angular <Goldstone bosons> become the longitudinal polarizations of two massive <gauge bosons>. Define physical fields
$$
A_\mu=B^3_\mu,\qquad W_\mu^\pm=\frac{B_\mu^1\mp iB_\mu^2}{\sqrt2}.
$$
These labels describe the fields of this model; they do not identify it with the <Standard Model>. With the stated <adjoint covariant derivative>,
$$
(D_\mu\phi)_1=-g(v+\eta)B_\mu^2,\quad (D_\mu\phi)_2=g(v+\eta)B_\mu^1,\quad (D_\mu\phi)_3=\partial_\mu\eta.
$$
To express all <Yang-Mills theory> terms in physical fields, introduce the <gauge field strengths>
$$
f_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu,\quad
Q_{\mu\nu}=W_\mu^+W_\nu^--W_\nu^+W_\mu^-,\quad
\mathcal W^\pm_{\mu\nu}=(\partial_\mu\pm igA_\mu)W_\nu^\pm-(\partial_\nu\pm igA_\nu)W_\mu^\pm.
$$
Then $F^3_{\mu\nu}=f_{\mu\nu}+igQ_{\mu\nu}$ and $(F^1_{\mu\nu}\mp iF^2_{\mu\nu})/\sqrt2=\mathcal W^\pm_{\mu\nu}$. Dropping the constant vacuum energy, the <Lagrangian density> is
$$
\begin{aligned}
\mathcal L={}&-\frac14(f_{\mu\nu}+igQ_{\mu\nu})(f^{\mu\nu}+igQ^{\mu\nu})-\frac12\mathcal W^+_{\mu\nu}\mathcal W^{-\mu\nu}
+\frac12\partial_\mu\eta\partial^\mu\eta\\
&-\lambda v^2\eta^2-\lambda v\eta^3-\frac\lambda4\eta^4+g^2(v+\eta)^2W_\mu^+W^{-\mu}.
\end{aligned}
$$
Taking the positive <gauge coupling> convention $g>0$, the canonically normalized <mass terms> give
$$
\boxed{m_A=0,\qquad m_{W^+}=m_{W^-}=gv,\qquad m_\eta^2=2\lambda v^2=-2m^2.}
$$
The residual <U(1) gauge symmetry> makes $W^\pm$ oppositely charged. The <Yang-Mills theory> terms supply interactions of $A$ with $W^\pm$ and four-vector interactions. The <Higgs mode> has cubic and quartic self-interactions, together with $2g^2v\eta W^+\cdot W^-$ and $g^2\eta^2W^+\cdot W^-$ couplings; it is neutral under the surviving <U(1) gauge symmetry>.

Coupling <fermions> permits a massless electromagnetic <gauge boson> and massive charged mediators of the <weak interaction>; <fermion> couplings with <chirality> can produce parity-violating <weak charged currents>. However, this model has \b[no massive neutral <Z boson>], and its only charge generator is $t^3$. A fundamental doublet has opposite charges, so it cannot reproduce the observed doublet charge assignments through an independent <hypercharge>. The <Standard Model> instead has $SU(2)_L\times U(1)_Y\to U(1)_Q$, a complex <Higgs doublet>, three absorbed <Goldstone bosons>, a massive <Z boson> and a <weak mixing angle>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 305 / 2 / a / Solution

{c}
{title2=$SU(2)\to U(1)$}

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>.

Codex @codex  0

 Incoming links: SU(2) gauge theory with an adjoint Higgs field