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Thermal green function and complex integration

  1. Jan 19, 2010 #1
    green function and complex integration

    1. The problem statement, all variables and given/known data

    By reading this paper http://arxiv.org/pdf/hep-ph/0610391v4 I cannot proof the following relation on page 9 equation (23) , by a suitable choice of a contour in the complex omega plane:

    [tex]\int\frac{d^3 p}{(2\pi)^2} \coth(\frac{|\vec{p}|}{2T})\frac{\cos(|\vec{p}| t-\vec{p}\vec{x})}{|\vec{p}|}=
    \int\frac{d^3 p}{(2\pi)^3} e^{i\vec{p}\vec{x}}\int\frac{d\omega}{2\pi} \coth(\frac{\omega}{2T})\frac{e^{-i\omega t}}{\omega^2-|\vec{p}|^2-i \epsilon}}[/tex]

    T stands for the temperature, therefore T>0.

    2. Relevant equations

    [tex]\cos(|\vec{p}| t-\vec{p}\vec{x})=\frac{1}{2} e^{i \vec{p}| t-i \vec{p}\vec{x}}+\frac{1}{2} e^{-i \vec{p}| t+i \vec{p}\vec{x}}[/tex]

    3. The attempt at a solution

    First of all I calculated the residues at [tex]\omega=-|{\vec{p}}|[/tex] and [tex]\omega=|{\vec{p}}|[/tex]. This gives me:

    [tex]Res(\omega=-|{\vec{p}}|)=\frac{coth(-|{\vec{p}}|/2T)}{-2|{\vec{p}}|} e^{i |{\vec{p}}|t}=\frac{coth(|{\vec{p}}|/2T)}{2|{\vec{p}}|} e^{i |{\vec{p}}|t}[/tex] since the coth is an odd function.
    [tex]Res(\omega=|{\vec{p}}|)=\frac{coth(|{\vec{p}}|/2T)}{2|{\vec{p}}|} e^{-i |{\vec{p}}|t}[/tex]

    The -i epsilon term in the denominator says one should circumvent the pole at [tex]\omega=|{\vec{p}}|[/tex] anti-clockwise and at [tex]\omega=-|{\vec{p}}|[/tex] clockwise (Feynman prescription). By suming up these contributions i get almost the result (missing just a numerical factor [tex]-2\pi i[/tex]).

    Then I investigated the behaviour of the integrand for [tex]|\omega|\rightarrow \infty[/tex] and found out that
    [tex]coth(\omega /2T)\rightarrow 1, |\omega|\rightarrow \infty[/tex] and
    [tex]e^{-i\omega t} \rightarrow 0, |\omega|\rightarrow \infty[/tex] only if t<0 in the upper omega halfplane and t>0 in the lower halfplane. Else the contributions at infinity to the contour integral would not vanish (there would not be an damping factor exp(-Im(w)t).

    The Problem is that I do not really know how to choose the contour in the complex plane, since additionally the coth has poles on the imaginary axis at [tex]\omega=i 2\pi nT[/tex] for every integer n too. Especially coth has a simple singularity at [tex]\omega=0[/tex].
    I asked my professor and he told me to try the following contour (look at the attachment). But even he does not know what to do with the pole at zero.
    I am trying to figure this out now for more than a week, but no success.

    I hope one of you guys can help me out. Thanks in advance!

    Attached Files:

    Last edited: Jan 20, 2010
  2. jcsd
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