Using PDFs to Compute the Total Cross Section

[Ajdin Palavric]

Homework Statement

I would like to compute the total cross section of a lepton pair production using parton distribution functions.
The main problem I am having is the numerical computation and the order of magnitude I am getting as a final answer, which so far definitely indicates that something is not okay.

Homework Equations

The relevant equations involve the total cross section of the lepton pair production at the parton level:

$\sigma(q_{f}\overline{q_{f}}\rightarrow l^{+}l^{-}) = \frac{4q_{f}^{2}\pi\alpha^{2}}{9s}$,

together with the expression relating parton distribution functions and the total cross section at the hardon level:

$\sigma(p(P_{1})+p(P_{2})\rightarrow X + Y) =\\= \int_{0}^{1}\int_{0}^{1}dx_{1}dx_{2}\sum_{f}f_{f}(x_{1})f_{\overline{f}}(x_{2})\sigma(q_{f}(x_{1}P)+\overline{q_{f}}(x_{2}P)\rightarrow Y).$

The Attempt at a Solution

My attempt is fairly straightforward. I tried to express the total 4-momentum of the virtual photon using $x_{1}$ and $x_{2}.$ The relation I got is the following:

$q^2 = 4x_{1}x_{2}E^2$.

Next, plugging the previous term into the integral I got:

$\sigma(p(P_{1})+p(P_{2})\rightarrow X + Y) =\frac{\pi\alpha^{2}}{9E^{2}}\int_{0}^{1}\int_{0}^{1}dx_{1}dx_{2}\sum_{f}f_{f}(x_{1})f_{\overline{f}}(x_{2})\frac{q_{f}^{2}}{x_{1} x_{2}}.$

Here is where the problem for the order of magnitude arises. The term in front of the integral can be computed and using conversion factor converting GeV to mb I got:

$c =\frac{7.2346 nb}{\eta^{2}}$,

where [itex]\eta[/itex] is the energy scale.

Now, I tried to load the PDFs in Mathematica and tried to compute the integrals. I used the PDFs from the following link,

http://www.hep.ucl.ac.uk/mmht/index.shtml[/B]

Once I loaded everything in Mathematica and performed the integration for u and u bar quark for x ranging from 1E-9 to 1, I got the following results:

https://imgur.com/wQrMdac

I would like to ask you what could possibly be wrong with this solution and if you could tell me where I am making mistakes. Thank you in advance!

 

[_coyote_]

The answer to your question is that you are missing some factors in your calculation. Firstly, you need to take into account the fact that the PDFs are defined as parton momentum fractions rather than energy fractions. This means that the integration should be done over x_{1} and x_{2} instead of x_{1} and x_{2}E. In addition, the factor \frac{q_{f}^2}{x_{1}x_{2}} should also be included in the integral calculation. Finally, you need to take into account the fact that the PDFs are defined as parton momentum fractions rather than energy fractions. This means that the integration should be done over x_{1} and x_{2} instead of x_{1} and x_{2}E. Finally, you need to include the factor \frac{q_{f}^2}{x_{1} x_{2}} in the integral calculation.