# How to integrate over rapidity for Parton Luminosities in LHC?

I want to make a comparison of parton luminosities between Tevatron and LHC. According to Factorization theorem the cross section in hadron colliders, as long as the partonic cross section has $\hat \sigma \sim \frac{1}{\hat s}$ dependence, could be written like that :

$$\sigma =\sum_{12}\int dx_1 dx_2 f_1 \left(x_1,\mu_F^2\right) f_2\left(x_2,\mu_F^2 \right) \hat{\sigma}_{12} \left(x_1 x_2 s,...\right)$$

here

$f_{i}(x_{i},\mu_F^2)$ represents the parton distribution function for $x_i$ fraction of the incoming hadron, $i$ represents the parton flavor and $\mu_F$ is the factorization scale.

I wanted to reproduce the Figure 4.4 (page 82) in book Physics at The Terascale or the Figures in LHC Physics Potential vs. Energy.

My Questions? I appreciate for any reply.

1. When I am scanning over the $\sqrt{\hat s }$ (partonic-center-of-mass energy) what value should I set the factorization scale? $\mu_F = \sqrt{\hat s }$ or $\mu_F = \sqrt{s}$ (center-of-mass energy)?
2. How could I integrate over the rapidity? What I mean is on page 82 it says that "The Luminosity $dL_{ij}/d\hat s$ integrated over y is shown in Figure 4.4a" and I don't understand what changes after integration over y. In Equation 4.11 the parton distribution functions are not function of rapidity thus I could simply move $dy$ to the right side and carry out the integration and it will contribute as a factor (though I don't know the bounds $-\infty<y<\infty$ or something specific for the collider $-5<y<5$ ) however there is relation between the rapidity($y$) and $x$ like this $y=\ln(x1/x2)/2$. Thus I don't understand that line?
3. If I just ignore the rapidity and use the Equation (1) in "LHC Physics Potential vs. Energy" I got some results but I am still off compared to the Figure 1 in the same paper, so I concluded that I don't know how the integration over rapidity affects the over all Parton Luminosity.

Equation 4.11

$$\frac{dL_{ij}}{d \hat{s} dy}=\frac{1}{s}\left(x_1,\mu\right) f_2\left(x_2,\mu \right)$$

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I don't have access to the book, but I'll see if I can check it out at the library tomorrow and look into this for you. As for part 3, just to check the obvious: you did use the CTEQ6L1 PDFs, right? It doesn't seem that the rapidity directly factors into that at all. –  David Z Dec 13 '11 at 0:14
I have found similar lines in this note as well, just check the equation 3 and below it says "Figure 2(left) shows a plot of the luminosity function integrated over rapidity, $dL_{ij}/d\hat{s}$ ,pa.msu.edu/~huston/leshouches/lum/les_houches_luminosity2.pdf I think the rapidity in those figures is not a factor at all, because the eq.3 has $dy$ in the numerator and denominator and it just cancels out –  user6629 Dec 13 '11 at 7:04