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.

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.