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Radiation of gluon jets - Peskin final project question

  1. Aug 17, 2010 #1
    Hello,
    this is the first time I post here, so if this is not in the correct section please let me know...

    I'm working on solving the first final project in peskin - Radiation of gluon jets.
    In this project we assume a simplified model for the gluon - it is a massive vector boson (with a small mass [tex]\mu[/tex] to regulate IR singularities) which couples universally to quarks.

    The first task is to compute the diagram contributing to [tex]e^+e^-\rightarrow \bar{q}q[/tex] with one virtual gluon. This gives a contribution to [tex]F_1(q^2)[/tex] of the gluon. After quite a lengthy calculation I reached the following correction (omitting constants):

    [tex]
    \delta F_1(q^2) = \int_0^1 dxdydz\delta(x+y+z-1)\int\frac{d^4 w}{(2\pi)^4}\frac{4q^2-2w^2}{[w^2-z\mu^2+xyq^2+i\epsilon]^3}
    [/tex]

    where I have defined q to be the sum of the electron and positron incoming momenta ([tex]q = p+p'[/tex]).

    In the instructions provided in the text it is said to regulate this integral by doing the substitution [tex]\delta F_1(q^2) \rightarrow \delta F_1(q^2)-\delta F_1(q^2=0)[/tex].
    My problem is that it seems to me that there is a branch-cut singularity in this expression, since [tex]q^2 = E_{CM}^2 > 0[/tex].

    So my questions are as follows:
    1. Am I correct in stating that there is a branch-cut singularity here?
    2. Does it make sense to have such a singularity here?
    3. What would you suggest to do in order to continue this calculation?

    Thanks in advance to any help and insights, and again if this is the wrong place to post I apologize and hope you can point me out to the right direction...

    Omri
     
  2. jcsd
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