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From lectures:L. Peak and K. Varvell. The Physics of the Standard Model.

Full Lagrangian for fermion and photon

Combine the gauge-invariant Lagrangian density describing a fermion field in the presence of an electromagnetic field with that for the EM field itself

[tex]

\begin{array}{l}

L=\bar {\psi }\left[ {\gamma ^\mu \left( {i\partial _\mu -qA_\mu }

\right)-m} \right]\psi -\frac{1}{4}F_{\mu \nu } F^{\mu \nu }-j^\mu A_\mu =

\\

=\bar {\psi }\left[ {\gamma ^\mu i\partial _\mu -m} \right]\psi

-\frac{1}{4}F_{\mu \nu } F^{\mu \nu }-\left( {j^\mu +q\bar {\psi }\gamma

^\mu \psi } \right)A_\mu \\

\end{array}

[/tex]

Note that the term coupling to the photon field [tex]A_\mu [/tex]consists of two parts:

1) The external current density [tex]j^\mu [/tex]

2) A term corresponding to the fermion field itself [tex]q\bar {\psi }\gamma ^\mu \psi [/tex]. This is called the electromagnetic current (think flow of the fermion charge) and when coupled to [tex]A_\mu [/tex] describes the interaction vertex.

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# On spontaneous symmetry breaking and Higgs’s mechanism of mass production

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