Past exam of the particle and nuclear physics course of the University of Cambridge / 2022 / Section A / 3

Use the usual Feynman rule convention in which the diagonal quadratic terms of the free Lagrangian, including the physical W boson and Z boson masses, are incorporated into their propagators. Then there areA line joining two ends is a propagator, not an interaction vertex. Gauge boson self-interactions begin with cubic terms in the Yang-Mills theory Lagrangian. The question's $g,W,Z,\gamma$ name four families: they include eight gluon colour components and the two charged W bosons $W^{\pm }$.

There is a convention to distinguish here: if a calculation expands around massless free fields, quadratic $m_W^2{W}_{\mu }^{+}{W}^{-\mu }$ and $m_Z^2{Z}_{\mu }{Z}^{\mu }/2$ terms may be drawn as two-leg mass insertions. Those two insertion types belong to that bookkeeping convention. Similarly, two-point self-energy corrections occur in loop diagrams. Neither is an additional fundamental interaction in the standard physical-mass, tree-level rules used for the count above.

There are three vertex types, with all legs labeled in the following original drawings:

The gluon cubic interaction vertex follows from the non-Abelian gauge field strength of Quantum chromodynamics. The electroweak Yang-Mills theory Lagrangian, after expressing the fields in the physical charged and neutral basis, produces the two charged-W boson vertices. In particular there are no fundamental $\gamma \gamma \gamma$, $ZZZ$, $ZZ\gamma$, or $Z\gamma \gamma$ vertices, and the separate colour and electroweak gauge factors do not generate a mixed $gg\gamma$ or $ggZ$ vertex. The count is by the boson labels requested, not by separate assignments of gluon colour indices.

There are five vertex types:

The quartic terms arise from the square of the quadratic-field piece of the non-Abelian gauge field strength. Every electroweak quartic gauge boson self-interaction contains a $W^{+}{W}^{-}$ pair. The absence of a fundamental purely neutral quartic vertex does not forbid amplitudes such as light-by-light scattering: those can be generated by loop diagrams, rather than by a single fundamental interaction vertex.

There areFor a Yang-Mills theory, schematically $F=\partial A+gAA$. The gauge kinetic Lagrangian is quadratic in $F$, so its terms are at most quartic in the gauge fields:Thus there is no elementary five-leg pure-gauge interaction vertex in the renormalizable Standard Model. A five-boson scattering amplitude can still be assembled from several lower-valence vertices, or generated by loop diagrams. The number of external legs of a complete Feynman diagram is not the valence of a fundamental vertex.