 Parity restriction on renormalizable pseudoscalar Yukawa counterterms

For an unbroken parity symmetry in quantum field theory with $\varphi$ a pseudoscalar, pure-scalar local counterterms of mass dimension at most four are even in $\varphi$. A four-vertex fermion loop requires a quartic counterterm, even when the initial Lagrangian density shows only a Yukawa interaction. The one-point and three-point pure-boson vertices vanish at all orders: their available external four-momenta cannot form a parity-odd Lorentz scalar.