9.5 Higgs sector
Throughout this section we assume again that \(G\) is either the unitary group \(\mathsf {U}(n)\) or the special unitary group \(\mathsf {SU}(n)\). A Higgs field is a complex scalar field
\begin{equation} \Phi \,:\,\bbR ^d\longrightarrow \bbC \otimes V~,~~x\longmapsto \Phi (x) \end{equation}
on the \(d\)-dimensional Minkowski spacetime \((\bbR ^d,\eta )\) that takes values in a unitary representation \(\rho \) of \(G\) on a complex vector space \(V\). We demand that this field transforms under gauge transformations \(U:\bbR ^d\to G\,,~x\mapsto U(x)\) according to the given representation, i.e.
which by unitarity of \(\rho \) implies that the adjoint Higgs field \(\Phi ^\dagger \) transforms as
\(\seteqnumber{0}{9.}{68}\)\begin{flalign} T_U\,:\, \Phi ^\dagger (x)\,\longmapsto \,(T_U\Phi ^\dagger )(x)\,=\,\Phi ^\dagger (x)\,\rho \big (U^{-1}(x)\big )\quad . \end{flalign} We shall again suppress most of the time the argument \(x\in \bbR ^d\) and simply write this as \(T_U\Phi = \rho (U)\,\Phi \) and \(T_U\Phi ^\dagger = \Phi ^\dagger \,\rho (U^{-1})\).
Our definition (9.51) of the gauge covariant derivative for Dirac fields applies ad verbum to the Higgs field \(\Phi \) and its adjoint \(\Phi ^\dagger \), which yields
\begin{flalign} \label {eqn:YMgaugecovderchiral} D_\mu \Phi (x) \,&:=\, \partial _\mu \Phi (x) + \underline {\rho }(A_\mu (x))\,\Phi (x)\quad ,\\ D_\mu \Phi ^\dagger (x) \,&:=\, \partial _\mu \Phi ^\dagger (x) - \Phi ^\dagger (x)\,\underline {\rho }(A_\mu (x))\quad . \end{flalign}
Note that these formulas are related by taking adjoints, i.e. \(\big (D_\mu \Phi \big )^\dagger = D_\mu \Phi ^\dagger \), because \(\rho \) is by hypothesis a unitary representation. By the same type of calculation as in (9.53), one checks that under a combined gauge transformation
\(\seteqnumber{0}{9.}{70}\)\begin{flalign} \label {eqn:YMHiggscombinedgauge} T_U\Phi \,=\,\rho (U)\,\Phi ~~,\quad T_U\Phi ^\dagger \,=\,\Phi ^\dagger \,\rho (U^{-1})~~,\quad (T_U A)_\mu \,=\,U\,A_\mu \, U^{-1} + U\,\partial _\mu U^{-1} \end{flalign} on \(\Phi \), \(\Phi ^\dagger \) and \(A_\mu \) these covariant derivatives transform according to
\(\seteqnumber{0}{9.}{71}\)\begin{flalign} T_U\big (D_\mu \Phi \big ) \,=\, \rho (U)\,D_\mu \Phi ~~,\quad T_U\big (D_\mu \Phi ^\dagger \big ) \,=\, D_\mu \Phi ^\dagger \,\rho (U^{-1})\quad . \end{flalign} From this observation it follows that a kinetic term of the form \(D^\mu \Phi ^\dagger \,D_{\mu }\Phi \) is gauge invariant, and so are potential terms of the form \(\big (\Phi ^\dagger \,\Phi \big )^n\).
With these building blocks we can construct a gauge invariant action functional for the Higgs field. The typical choice for the Higgs action functional is given by
\begin{equation} \label {eqn:Higgsaction} S_{\mathrm {Higgs}}[\Phi ,\Phi ^\dagger ,A] \,:=\,\int _{\bbR ^d} - \bigg (D^\mu \Phi ^\dagger \,D_{\mu }\Phi \,{\color {red}-}\, \mu ^2\, \Phi ^\dagger \,\Phi + \lambda \, \big (\Phi ^\dagger \,\Phi \big )^2\bigg )~\dd x\quad , \end{equation}
where \(\mu ^2>0\) and \(\lambda >0\) are positive real parameters. Comparing this with the usual complex Klein-Gordon action in (2.17), we note that the mass term comes with a “wrong sign”. It is important to stress that this is not a typo or an accident, but it is the most essential feature of the Higgs field! Let me explain what happens as a consequence of this “wrong sign” mass term: Note that the Higgs potential
\(\seteqnumber{1}{9.74}{0}\)\begin{flalign} V(\Phi ^\dagger \,\Phi ) \,=\, -\mu ^2\,\Phi ^\dagger \,\Phi + \lambda \, \big (\Phi ^\dagger \,\Phi \big )^2 \end{flalign} has the following shape
![(9.74b) \{begin}{flalign} \parbox {8cm}{ \begin{tikzpicture} \begin{axis}[ xmin=-2.5,xmax=2.5, ymin=-1.5,ymax=2, axis x line=middle, axis y line=middle, x label style={anchor=west}, y label
style={anchor=south}, xlabel={$\Phi $}, ylabel={$V(\Phi ^\dagger \,\Phi ) $}, ] \addplot [blue, very thick, domain=-1.7:1.7, samples =100] ({x}, {-2 * x^2+x^4}); \end {axis} \end {tikzpicture} } \{end}{flalign}](MATH4017-images/image-61.svg)
from which we see that it has multiple minima. This means that the Higgs field has multiple classical vacua, which are, by definition, constant field configurations \(\Phi (x) \equiv \Phi _0\in \bbC \otimes V\) that minimize the potential. Note that each of these vacua is in general not invariant under gauge transformations, i.e. \(\rho (U)\,\Phi _0 \neq \Phi _0\), which means that they break gauge symmetry. This phenomenon is called spontaneous gauge symmetry breaking, where the term spontaneous refers to the fact that, even though the action (9.73) is gauge invariant, the individual vacuum solutions of the resulting equations of motion are not. This mechanism of symmetry breaking is used in the standard model of particle physics to generate mass terms for matter fermions and for the \(W^+\), \(W^-\) and \(Z\) gauge bosons. This will be discussed in more detail in some of the student projects/presentations.
We finish this section with the observation that we can add to (9.73) the Yang-Mills action (9.43) and thereby obtain the action functional
\begin{flalign} \nn S_{\mathrm {YM}+\mathrm {Higgs}}&[\Phi ,\Phi ^\dagger ,A] \,:=\, S_{\mathrm {Higgs}}[\Phi ,\Phi ^\dagger ,A] + S_{\mathrm {YM}}[A] \\ \,&=\, \int _{\bbR ^d} \bigg (-D^\mu \Phi ^\dagger \,D_{\mu }\Phi + \mu ^2\, \Phi ^\dagger \,\Phi - \lambda \, \big (\Phi ^\dagger \,\Phi \big )^2 +\frac {1}{2\,g_{\mathrm {YM}}^2}\,\mathrm {Tr}\big (F^{\mu \nu }\,F_{\mu \nu }\big )\bigg )~\dd x \end{flalign}
for the so-called Yang-Mills-Higgs field theory. This combined action functional is invariant under the combined gauge transformations (9.71) acting on all fields.