= Derivative-interaction scalar power counting
{title2=$D=d-(d-2)E/2-V[\lambda]$}
For interaction $\lambda\Phi^n(\partial\Phi)^r$ with a two-derivative kinetic term, each vertex has $n+r$ legs and $r$ momentum powers. Since $[\Phi]=(d-2)/2$, its coupling has dimension $[\lambda]=d-(n+r)(d-2)/2-r$. The graph identities $2I+E=(n+r)V$ and $L=I-V+1$, inserted into $D=dL-2I+rV$, prove the displayed formula. It is an upper bound on ultraviolet scaling; momentum contractions and symmetry can cause cancellations.
Back to article page
