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Gluon-gluon OPI GF

  1. May 10, 2006 #1

    dextercioby

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    Basically it's the diagram attached below. If one uses the Feynman rules for QCD, he gets the expression

    [tex] \begin{array}{c} \left(\tilde{\Gamma}^{(2)}_{2,\mu\nu}\right)\left(p,-p\right)

    = \frac{1}{2}\hat{g}^{2}M^{\varepsilon }f^{d}{}_{ac}f^{c}{}_{bd}\int \frac{d^{2\omega }q}{(2\pi)^{2\omega }}\left( \frac{-g^{\lambda \rho }q^{2}+\eta q^{\lambda }q^{\rho }}{\left( q^{2}+i\epsilon \right) ^{2}}\right) \left( \frac{-g^{\sigma \tau }\left( p+q\right) ^{2}+\eta \left( p+q\right) ^{\sigma }\left( p+q\right) ^{\tau }}{\left[ \left( p+q\right) ^{2}+i\epsilon \right] ^{2}}\right) \\

    \times \left[ \left( q-p\right) _{\sigma }g_{\rho \mu }-\left( p+2q\right) _{\mu }g_{\sigma \rho }+\left( 2p+q\right) _{\rho }g_{\mu \sigma }\right] \left[ \left( 2p+q\right) _{\lambda }g_{\tau \nu }-\left( p+2q\right) _{\nu }g_{\lambda \tau }+\left( q-p\right) _{\tau }g_{\nu \lambda }\right] \end{array}

    [/tex].
     
    Last edited: May 10, 2006
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  3. May 10, 2006 #2

    dextercioby

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    My question involves solving that integral.

    [itex] \eta=1-\xi [/itex]

    and usually QFT books deal only with the Feynman gauge in which [itex] \eta =0 [/itex].

    I'm interested only in the [itex] \eta [/itex] dependent part (funny, right ?) in which, after doing all the possible contractions and multiplications (using that D=dim.space-time= 2 \omega), i got 18 terms leading me to 18 Feynman integrals.

    An example of an integral that is over my head is

    [tex] \frac{1}{2}\hat{g}^{2}M^{\varepsilon }f^{d}{}_{ac}f^{c}{}_{bd}\eta $\int \frac{d^{2\omega }q}{\left( 2\pi \right) ^{2\omega }} \left(\frac{\left( p+2q\right) _{\mu }\left( p+q\right) _{\nu }\left( q^{2}-p^{2}\right) q^{2}}{\left( q^{2}+i\epsilon \right) ^{2}\left[ \left( p+q\right) ^{2}+i\epsilon \right] ^{2}}\right) [/tex]

    So can anyone help...?

    Daniel.
     
    Last edited: Nov 22, 2006
  4. May 13, 2006 #3

    dextercioby

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    Well, the good news is that i finally bumped into Bailin & Love's book which have an interesting table of Feynman integrals, which can be used especially for QCD.

    Daniel.
     
  5. May 14, 2006 #4

    Physics Monkey

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    Ok, so you're good then? I'm happy to say something about the integral if you still need help.
     
  6. May 16, 2006 #5

    dextercioby

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    Thx for the help offer, but i could manage on my own. As a bonus, i could actually rigorously prove the first formula of that appendix:

    [tex] \int \frac{d^{2\omega}k}{(2\pi)^{2\omega}} \left(k^2 \right)^{-n} \left[(k+p)^2 \right]^{-m} =\frac{i (-)^{n+m}}{(4\pi)^{\omega}} \frac{\Gamma (\omega-n-m) B(\omega-n, \omega-m)}{\Gamma (n)\Gamma (m)} \left(-p^2\right)^{\omega-n-m} [/tex]

    Daniel.
     
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