Radiation of gluon jets - Peskin final project question

[omrihar]

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 $$\mu$$ to regulate IR singularities) which couples universally to quarks.

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

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

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

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

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

 

[AndyW]

Hello Omri,

Thank you for sharing your progress on the first final project in Peskin. It's great to see fellow scientists discussing and working on these problems together.

To answer your questions, yes, you are correct in stating that there is a branch-cut singularity in your expression. This is due to the fact that q^2 can take on positive values, leading to a square-root singularity. This is a common occurrence in calculations involving virtual particles.

While it may seem concerning, it is actually expected and does not invalidate your calculation. In fact, it is a sign that you are on the right track. In order to continue your calculation, you can use standard techniques such as contour integration or Cauchy's residue theorem to handle the branch-cut singularity and obtain a well-defined result.

As for whether it makes sense to have such a singularity, it is a natural consequence of the model you are working with. However, it is important to keep in mind that this is a simplified model and in reality, gluons do not have a mass and therefore do not have a branch-cut singularity. This is just a theoretical exercise to understand the behavior of gluon radiation.

I hope this helps and I wish you the best of luck in completing your project. If you have any further questions, please don't hesitate to ask. Keep up the good work!