Monomial scalar interaction power-counting identity ($\Delta =d-(d-2)E/2-[\lambda ]V$)

For a connected scalar graph with nonderivative interaction $-\lambda {\varphi }^n$, $E$ external legs and $V$ vertices, $nV=2I+E$ and $L=I-V+1$. Since $[\varphi ]=(d-2)/2$ and $[\lambda ]=d-n(d-2)/2$, the superficial degree of divergence isThe graph contributes at order ${\lambda }^V$. This is simultaneous large-loop-momentum counting for an amputated graph; it does not test subdivergences or guarantee that the predicted pole coefficient is nonzero.

// Target: perturbative-quantum-field-theory.bigb