Trouble Finding Renormalization Conditions in Yukawa Theory

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1. Jul 11, 2015

gobbles

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?

Last edited: Jul 11, 2015
2. Jul 11, 2015

MathematicalPhysicist

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. Jul 11, 2015

ShayanJ

I'm posting only for helping with the LaTeX.
\not{\!p}

4. Jul 11, 2015

MathematicalPhysicist

$\not{\!p}$ works, thanks.

5. Jul 11, 2015

gobbles

$\not{\!p}^2=p_\mu \gamma^\mu p_\nu \gamma^\nu=p_\mu p_\nu(2g^{\mu\nu}-\gamma^\nu \gamma^\mu)=2p^2-\not{\!p}^2$
$p^2=\not{\!p}^2$