1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Problem with Dimensional Regularization

  1. Jan 18, 2012 #1
    Good day to everyone. I am trying to apply dimensional regularization to divergent integral
    [tex] \int\frac{d^{4}l}{\left(2\pi\right)^{4}}\frac{4\, l_{\mu}l_{\nu}}{\left[l^{2}-\triangle+i\epsilon\right]^{3}}.[/tex]
    I am very new to these thing. The first question is how should i apply Wicks rotation to the term [tex]l_{\mu}l_{\nu}[/tex]As i understand it should be done before going to d dimensions. I need to avoid substitution [tex]l_{\mu}l_{\nu}\to\frac{1}{4}g_{\mu \nu}l^2[/tex]
    Would it work to rewrite
    [tex]l_{\mu}l_{\nu}=\frac{g_{\mu \nu}}{g_{\mu \nu}g^{\mu \nu}}g^{\mu \nu}l_{\mu}l_{\nu}[/tex]
    Which gives in d dimensions
    [tex]\frac{g_{\mu\nu}}{d}l^2[/tex]
    I would appreciate any help.
     
  2. jcsd
  3. Jan 18, 2012 #2

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The way to usually determine these integrals is to start with the result for

    [tex]\int \frac{d^N l}{(l^2 + a^2)^A} = \frac{\Gamma(A-N/2)}{\Gamma(A)} \frac{\pi^{N/2}}{(a^2)^{A-N/2}}[/tex]

    shift [itex]l=l'+p[/itex], so that

    [tex]\int \frac{d^N l'}{((l')^2 + 2 p\cdot l' + a^2+p^2)^A} = \frac{\Gamma(A-N/2)}{\Gamma(A)} \frac{\pi^{N/2}}{(a^2+p^2)^{A-N/2}}.[/tex]

    Now we can generate factors of [itex]l'_\mu[/itex] in the numerator by differentiating with respect to p, setting [itex]p=0[/itex] at the end as needed.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Problem with Dimensional Regularization
Loading...