Trouble Finding Renormalization Conditions in Yukawa Theory

  • #1
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1

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?
 
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Answers and Replies

  • #2
MathematicalPhysicist
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You got it wrong, ##p^2 = p^\mu p_\mu##, what you wrote is square of p slash, which I am not sure how to LaTexed it.
 
  • #3
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I'm posting only for helping with the LaTeX.
You got it wrong, ##p^2 = p^\mu p_\mu##, what you wrote is square of p slash, which I am not sure how to LaTexed it.
\not{\!p}
 
  • #4
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##\not{\!p}## works, thanks.
 
  • #5
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Thanks for the reply!
##\not{\!p}^2=p_\mu \gamma^\mu p_\nu \gamma^\nu=p_\mu p_\nu(2g^{\mu\nu}-\gamma^\nu \gamma^\mu)=2p^2-\not{\!p}^2##
and therefore
##p^2=\not{\!p}^2##
 
  • #6
MathematicalPhysicist
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Thank you for correcting me, forgot the anticommutator relation between the gammas.
 

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