Matter content

The model consists of a complex scalar field ${\displaystyle \phi (x)}$ minimally coupled to a gauge field ${\displaystyle A_{\mu }(x)}$.

This article discusses the theory on flat spacetime ${\displaystyle \mathbb {R} ^{1,3}}$ (Minkowski space) so these fields can be treated (naïvely) as functions ${\displaystyle \phi$ :\mathbb {R} ^{1,3}\rightarrow \mathbb {C} } , and ${\displaystyle A_{\mu }:\mathbb {R} ^{1,3}\rightarrow (\mathbb {R} ^{1,3})^{*}}$. The theory can also be defined for curved spacetime but these definitions must be replaced with a more subtle one. The gauge field is also known as a principal connection, specifically a principal ${\displaystyle {\text{U}}(1)}$ connection.

Lagrangian

The dynamics is given by the Lagrangian density

${\displaystyle {\begin{array}{lcl}{\mathcal {L}}&=&(D_{\mu }\phi )^{*}D^{\mu }\phi -V(\phi ^{*}\phi )-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\\&=&(\partial _{\mu }\phi )^{*}(\partial ^{\mu }\phi )-ie((\partial _{\mu }\phi )^{*}\phi -\phi ^{*}(\partial _{\mu }\phi ))A^{\mu }+e^{2}A_{\mu }A^{\mu }\phi ^{*}\phi \end{array}},}$

where

• ${\displaystyle F_{\mu \nu }=(\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu })}$ is the electromagnetic field strength, or curvature of the connection.
• ${\displaystyle D_{\mu }\phi =(\partial _{\mu }\phi -ieA_{\mu }\phi )}$ is the covariant derivative of the field ${\displaystyle \phi }$
• ${\displaystyle e=-|e|<0}$ is the electric charge
• ${\displaystyle V(\phi ^{*}\phi )}$ is the potential for the complex scalar field.

Gauge-invariance

This model is invariant under gauge transformations parameterized by ${\displaystyle \lambda (x)}$. This is a real-valued function ${\displaystyle \lambda$ :\mathbb {R} ^{1,3}\rightarrow \mathbb {R} .}

${\displaystyle \phi '(x)=e^{ie\lambda (x)}\phi (x)\quad {\textrm {and}}\quad A_{\mu }'(x)=A_{\mu }(x)+\partial _{\mu }\lambda (x).}$

Example

A simple choice of potential for demonstrating the Higgs mechanism is

${\displaystyle V(|\phi |^{2})=\lambda (|\phi |^{2}-\Phi ^{2})^{2}.}$

The potential is minimized at ${\displaystyle |\phi |=\Phi }$, which is chosen to be greater than zero. This produces a circle of minima, with values ${\displaystyle \Phi e^{i\theta }}$, for ${\displaystyle \theta }$ a real number.