Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Renormalization of phi^4 Theory d dimensional Integral, Derivation in Peskin Schröder

  1. Mar 13, 2009 #1
    Hello,
    to understand the renormalization of phi^4 theory, I read Peskin Schröder and Ryder. In both books important steps are left out. I found the following identity in Peskin Schöder "An Introduction to Quantum Field Theory" on page page 808, equation A.52 (Appendix)

    [tex] \frac{\Gamma(2 - \frac{d}{2})}{(4\pi)^{\frac{d}{2}}} \left( \frac{1}{\Delta} \right)^{2-\frac{d}{2}} = \frac{1}{(4\pi)^2} \left( \frac{2}{\epsilon} - log(\Delta) - \gamma + log(4\pi) + O(\epsilon)\right)[/tex]

    Now I want to prove that explicitly, but I don't know how to start and how the logarithm on the right hand side appears.

    Could anyone help me?

    Regards,
    Mr. Fogg
     
  2. jcsd
  3. Mar 13, 2009 #2

    Avodyne

    User Avatar
    Science Advisor

    Re: Renormalization of phi^4 Theory d dimensional Integral, Derivation in Peskin Schr

    That's why you should read a book like Srednicki that doesn't leave steps out.

    For any nonzero A and small x,

    [tex]A^x=\exp(x\ln A)=1+x\ln A + O(x^2)[/tex]

    [tex]\Gamma(x) = {1\over x}-\gamma+O(x)[/tex]

    These are equations 14.33 and 14.26 in Srednicki.
     
    Last edited: Mar 14, 2009
  4. Mar 14, 2009 #3
    Re: Renormalization of phi^4 Theory d dimensional Integral, Derivation in Peskin Schr

    Thank You,

    so I get this expression:

    [tex] \frac{1}{(4\pi)^2} \left( \frac{2}{\epsilon} - \gamma \right) \left(1 + \frac{\epsilon}{2} ln(\frac{4 \pi}{\Delta})\right) [/tex]

    But that's not the equation from Peskin & Schröder, isn't it?

    How do I go on to get it finally?

    Regards,
    Mr. Fogg
     
  5. Mar 14, 2009 #4

    Avodyne

    User Avatar
    Science Advisor

    Re: Renormalization of phi^4 Theory d dimensional Integral, Derivation in Peskin Schr

    It's the same. Just multiply it out, and use ln(a/b)=ln(a)-ln(b).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Renormalization of phi^4 Theory d dimensional Integral, Derivation in Peskin Schröder
Loading...