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Trouble Finding Renormalization Conditions in Yukawa Theory

  1. Jul 11, 2015 #1
    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. jcsd
  3. Jul 11, 2015 #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.
     
  4. Jul 11, 2015 #3

    ShayanJ

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    I'm posting only for helping with the LaTeX.
    \not{\!p}
     
  5. Jul 11, 2015 #4

    MathematicalPhysicist

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    ##\not{\!p}## works, thanks.
     
  6. Jul 11, 2015 #5
    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##
     
  7. Jul 12, 2015 #6

    MathematicalPhysicist

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    Thank you for correcting me, forgot the anticommutator relation between the gammas.
     
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