# Understanding the Parton Distribution Functions

Here I have a PDFs for a proton-proton collision.

I have few questions about the plot:

i) What are we plotting along y-axis? Is it the probability density? ii) What does the width of the plot of individual parton tell me? What I mean is, let's say for $Q^2 = 10$, the width of s is very thick for a very small value of x and when $Q^2 = 10^4$, it becomes very sharp. What does it tell me? iii) Also, is it because this is a p-p collision, and thus, the plot of u has its peak at almost twice the height as that of d? Does this particular value of x give me any physical significance?

P.S. I did check the website for similar kind of questions, and there were some good answers, but I could not find the one that suits to my level of understanding. Any reference to a useful material would be helpful.

• there should always be a link for the original publication of the figure, (which should have a definition for the variables in the plot). Commented Nov 16, 2017 at 5:30

The vertical axis is the number density of the parton. Effectively it's a probability density, but normalized to the expected number of a given parton in the proton. This is why the $u$ line is universally higher: there are more up quarks than down quarks in a proton, and also why in general the partons pretty clearly do not all integrate to the same value.
The width in each line is the uncertainty in the parton distribution function for that parton. Essentially in your example this would mean that at low $Q^2$ the PDF is not known precisely for small $x$.
• Actually, I think the $y$-axis in the plot is $x \cdot f(x)$, i.e. the product of $x$ and the density $f(x)$. This is the usual way to plot parton distribution functions, since otherwise, the gluon PDF rises too rapidly towards zero. See slide 6 in this talk by James Stirling. Commented Aug 1, 2019 at 9:00