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I am trying to calculate the $\beta$ functions of the massless pseudoscalar Yukawa theory, following Peskin & Schroeder, chapter 12.2. The Lagrangian is $$\mathcal{L}=\frac{1}{2}(\partial_\mu \phi)^2-\frac{\lambda}{4!}\phi^4+\bar{\psi}(i\gamma^\mu \partial_\mu)\psi-ig\bar{\psi}\gamma^5\psi\phi.$$ When calculating the one-loop correction to the electron ($\psi$) propagator, there is one diagram, the expression for which is of the form $$g^2\gamma^\mu p_\mu\left[\mbox{logarithmic divergence} + \mbox{finite terms that depend on } \log(-p^2)\right].$$ In order to calculate the $\beta(g)$ function, we now need to find the counterterm $\delta_\psi$ at the renormalization conditions given at an unphysical momentum $p^2=-M^2$, where $M$ defines the scale we're working at. The renormalization conditions, if I understand right, are chosen to make the $\log(-p^2)$ term finite, but there is also the $\gamma^\mu p_\mu$ term which should be set. If I set $$\gamma^\mu p_\mu=M,$$ I would get $$p^2=(\gamma^\mu p_\mu)^2=M^2,$$ instead of $p^2=-M^2$, as required. The remaining thing to do is to set $\gamma^\mu p_\mu=iM$, but I'm having trouble justifying that.

What are the correct renormalization conditions in this case?

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The renormalization condition is exactly that the propagator has a pole and residue of 1 at \begin{equation} \gamma^\mu p_\mu=M \end{equation} which leads to \begin{equation} p^2=(\gamma p)^2=M^2 \end{equation} I think your problem here is that you mermorized the renormalization condition as $p^2=-M^2$ which is not true. To understand the condition properly, just bring up the free-field propagator \begin{equation} \frac{i}{p^2-M^2} \end{equation} or \begin{equation} \frac{i}{\gamma^\mu p_\mu-M} \end{equation} notice the real-corrected propagator behaves the same as the free-field near the pole, in this way you may not have a memoral mistake.

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