- #1
omrihar
- 7
- 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:
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
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:
- Am I correct in stating that there is a branch-cut singularity here?
- Does it make sense to have such a singularity here?
- 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