= Normal-ordering subtraction scheme in two dimensions
{title2=$V_B=e^{-T_\Lambda\partial_\phi^2/2}V$}
For a massive <real scalar field> in two spacetime dimensions with a polynomial potential and no derivative interactions, every connected subgraph has <superficial degree of divergence> $\omega=2(1-V)$, where $V$ counts <interaction vertices>. The only divergent loop subgraphs with external legs have one vertex and consist of self-contractions. <Normal ordering> removes them. With $T_\Lambda=\int_{|k|<\Lambda}d^2k/[(2\pi)^2(k^2+m^2)]$, the ordinary-field potential representing the normal-ordered interaction is $e^{-T_\Lambda\partial_\phi^2/2}V(\phi)$. The exponential terminates for a polynomial. After these subtractions all remaining loop subgraphs have negative <superficial degree of divergence>, so no additional divergent field-dependent <counterterms> occur. Constant <vacuum energy> subtractions and finite <on-shell renormalization scheme> conditions are separate.
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