I consider vacuum polarization diagram in two different Yukawa theories: with scalar coupling $g\bar{\psi}\phi\psi$ and pseudoscalar coupling $ig\bar{\psi}\gamma^5\phi\psi$. I am interested in imaginary part of vacuum polarization function $\Pi(k^2)$ (pseudoscalar theory): $$\Pi(k^2)=g^2\int\frac{d^4p}{(2\pi)^4}\frac{\mathrm{Tr}(i\gamma^{5}(\gamma^{\mu}p_{\mu}+M)i\gamma^{5}(\gamma^{\mu}p_{\mu}+\gamma^{\mu}k_{\mu}+M))}{(p^2-M^2+i\epsilon)((p+k)^2-M^2+i\epsilon)}.$$ So, I use Cutkosky's rule and replace propagators by $(2\pi i)\delta(p^2-M^2)$ and $(2\pi i)\delta((p+k)^2-M^2)$. I also simplify numerator and obtain the following results: $$4(p^2+(p\cdot k)+M^2); -4(p^2+(p\cdot k)-M^2)$$ for scalar and pseudoscalar theory respectively. Finally, for scalar theory I find: $$\mathrm{Im}\,\Pi(k^2)=\frac{g^2}{8\pi}\sqrt{1-\frac{4M^2}{k^2}}(4M^2-k^2).$$ But I notice that for pseudoscalar theory the answer does not have the similar structure due to the numerator, I find: $$\mathrm{Im}\,\Pi(k^2)\propto k^2\sqrt{1-\frac{4M^2}{k^2}}$$ without mass term.

Then, I also calculate full function $\Pi(k^2)$ for pseudoscalar theory with help of dimensional regularization and Feynman parameters. I obtain the answer for imaganiry part which can be written as: $$\mathrm{Im}\,\Pi(k^2)=\frac{g^2}{8\pi}\sqrt{1-\frac{4M^2}{k^2}}(4M^2+k^2).$$ I have checked all the derivations twice and totally do not understand where I made mistake. So, may be I misunderstand something conceptual, not technical?


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