= Counterterm closure of a massive Yukawa theory
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 $M+y\phi$, and its divergent local <determinant> contribution contains $(M+y\phi)^4$. Linear and cubic terms are therefore generated when $M\ne0$. 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.
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