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Homework Help: 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
    I would appreciate any help.
  2. jcsd
  3. Jan 18, 2012 #2


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    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.
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