Counterterm

A counterterm is a local term added to a regulated Lagrangian density to cancel ultraviolet divergences and impose chosen renormalization conditions. Field-strength, mass and coupling counterterms respectively adjust propagator normalization, pole position and interaction strength.

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.

A family of actions is closed under counterterms when all required divergent local terms can be absorbed into that family. This does not imply that the full quantum effective action contains no additional finite effective interaction vertices or terms in its derivative expansion.

Without a symmetry excluding odd scalar powers, a real-scalar Yukawa interaction with a massive Dirac field requires a general scalar potential through degree four. For a constant background, the fermion mass becomes $M+y\varphi$, and its divergent local determinant contribution contains $(M+y\varphi {)}^4$. Linear and cubic terms are therefore generated when $M\ne 0$. A zero renormalized tadpole is a renormalization condition, not a prohibition of its counterterm. An exact discrete chiral symmetry in a massless fermion theory can instead enforce an even scalar potential.

A mass counterterm cancels the divergent part of a self-energy that has the tensor structure of a mass term. Its finite part depends on the renormalization scheme, while an on-shell scheme chooses it so that the renormalized mass equals the pole mass.

With renormalized Dirac fields, the coefficient of $-\bar{\psi }\psi$ in the counterterm Lagrangian is $\delta m_L=Z_2{m}_0-m$, whereas the bare-parameter shift is $m_0-m=\delta m_L-m\delta Z_2$ at first order. In Feynman gauge and $d=4-ϵ$, QED has $\delta Z_2=-e^2/(8{\pi }^2ϵ)$ and $\delta m_L=-4m{e}^2/(8{\pi }^2ϵ)$. Their difference gives $m_0-m=-3m{e}^2/(8{\pi }^2ϵ)$. Defining the insertion and the mass coefficient before quoting its sign avoids confusing these two counterterms.