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## Main Question or Discussion Point

I am trying to calculate the ##\beta## functions of the massless pseudoscalar Yukawa theory, following Peskin & Schroeder, chapter 12.2. The Lagrangian is

##{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\not{\!p}\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 ##\not{\!p}## term which should be set. If I set

##\not{\!p}=M,##

I would get

##p^2=\not{\!p}^2=M^2,##

instead of ##p^2=−M^2##, as required. The remaining thing to do is to set ##\not{\!p}=iM##, but I'm having trouble justifying that.

What are the correct renormalization conditions in this case?

##{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\not{\!p}\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 ##\not{\!p}## term which should be set. If I set

##\not{\!p}=M,##

I would get

##p^2=\not{\!p}^2=M^2,##

instead of ##p^2=−M^2##, as required. The remaining thing to do is to set ##\not{\!p}=iM##, but I'm having trouble justifying that.

What are the correct renormalization conditions in this case?

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