Counterterm Lagrangian (${\mathcal{L}}_{\mathrm{ct}}$)

The local Lagrangian density whose coefficients absorb the regulated ultraviolet divergences and implement chosen renormalization conditions. In a renormalizable four-dimensional theory, power counting permits operators of mass dimension at most four, subject to the actual symmetries and up to total derivatives. Operators absent from the initial displayed interaction can still be required for counterterm closure.

For a real scalar field in six spacetime dimensions with interaction $-g{\varphi }^3/6$, the mass dimension of $g$ is zero. With Euclidean sharp cutoff regularization, $C=1/(128{\pi }^3)$ and $L=\log ({\Lambda }^2/m^2)$, choose the Minkowski counterterm Lagrangian ${\delta }_Z(\partial \varphi {)}^2/2-{\delta }_m{\varphi }^2/2-{\delta }_g{\varphi }^3/6-{\delta }_h\varphi$. Its one-loop divergent coefficients areThe proof uses the Euclidean effective action $\frac{1}{2}\mathrm{Tr}\log (-{\partial }_E^2+m^2+g\varphi )$. Its first three background derivatives have divergent parts $g{I}_1/2$, $-g^{2}B(P)/2$, $g^3{I}_3$. Radial integration gives $I_1=C({\Lambda }^4/2-m^2{\Lambda }^2+m^{4}L)$, $I_3=CL$, and $B(P)=C({\Lambda }^2-2m^{2}L-P^{2}L/3)$ modulo finite terms. Subtraction proves the coefficients. The tadpole counterterm is essential despite the absence of a linear term in the original Lagrangian density. A separate constant handles the vacuum energy density.

A linear scalar counterterm fixes the one-point function or the expansion point of the scalar potential. An unbroken parity symmetry in quantum field theory forbids this term when the scalar is a pseudoscalar. A vanishing one-loop tadpole diagram alone would not prove the all-orders restriction without the symmetry argument.