= Renormalized quartic scalar vertex
{title2=$\Gamma_R^{(4)}$}
The renormalized quartic scalar vertex $\Gamma_R^{(4)}$ is the four-field proper vertex expressed in terms of renormalized fields and parameters. For a <quartic interaction>, its leading contribution is the tree-level coupling; loop corrections and <counterterms> modify its dependence on external <momenta> and the chosen renormalization scale.
A renormalization condition can define the coupling by prescribing the vertex at a specified momentum configuration. Changing that scale changes the coupling according to its <renormalization-group beta function>, while consistently computed physical predictions remain unchanged to the retained accuracy. This proper vertex is <one-particle irreducible>; it is not an arbitrary four-point <correlation function> with external propagation left attached. A comparison must state field normalization, momentum convention and whether an overall factor of $i$ is included in the definition.
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