Parton distribution functions plot axes

In summary: If we plot this quantity against x, then we get a graph with a slope of -1, meaning that for every unit increase in x, the fraction of momentum carried by partons falls by 1.
  • #1
MarekS
34
0
I am slightly confused by the labelling of the vertical axis on parton distribution function plots.

Take the one here: http://www.hep.phy.cam.ac.uk/~wjs/partons2008nlo.jpg
as an example.

The vertical axis is labelled as xf(x, Q^2), where f is the probability density of finding a particular parton with a given longitudinal momentum fraction x.

I think it should simply be f(x, Q^2). Why the extra 'x' in front of f?
 
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  • #2
f and xf are two different things. f is the fraction of partons carrying a momentum fraction f, and xf is the fraction of the proton's momentum carried by partons that each carry a momentum fraction x.
 
  • #3
Is the purpose of plotting the product xf instead of f to compress the functions along the vertical direction? That is to say, f goes up for low x quite a bit, but the product xf increases less as one decreases x.
 
  • #4
It does that, but the purpose is to represent something different than the fraction of partons carrying a momentum fraction x. It's to represent the fraction of the proton's momentum carried by partons that each carry a momentum fraction x.
 
  • #5
Vanadium 50 said:
It's to represent the fraction of the proton's momentum carried by partons that each carry a momentum fraction x.

I am having difficulty parsing this sentence. "fraction of the proton's momentum" would imply the vertical axis wouldn't go above 1, but it does. Could you phrase it longer/differently?

If I think of the proton as a sack of balls with the sack moving with a large momentum, then all the balls must be moving at the same speed or they wouldn't stay together. In the approximation that the differences in mass are negligible then all the balls will have the same momentum. That is to say, if a ball has momentum fraction x, then there must be 1/x balls in total in the sack each carrying fraction x of the sack's momentum.

Being a quantum sack, the number of balls is not fixed and it is the task of the balls' distribution function to determine the number and kind of them (e.g 10 gluon balls, 2 up balls, 1 down ball).

Is this the right way to think about it?
 
  • #6
Maybe examples will help:

The integral of xf(x) must be 1. (The fraction of momentum carried by all of its constituents must sum to 1).

The integral of f(x) can sum to anything.

The integral of u(x)-ubar(x) must sum to 2. (2 up quarks)

The integral of d(x)-dbar(x) must sum to 1. (1 down quark)
 
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  • #7
If I think of the proton as a sack of balls with the sack moving with a large momentum, then all the balls must be moving at the same speed or they wouldn't stay together. In the approximation that the differences in mass are negligible then all the balls will have the same momentum.
The point of having a parton distribution function of course is that they do not all have the same momentum. Paraphrased from Halzen and Martin: fi(x) represents the probability that a particular parton carries a fraction x of the proton's total momentum. All the fractions have to add up to 1, therefore ∑i∫ xfi(x) dx = 1.

And thus good reason for focusing on the quantity xfi(x), which as 50V said, is "the fraction of the proton's momentum carried by partons that each carry a momentum fraction x."
 

FAQ: Parton distribution functions plot axes

What is a Parton Distribution Function (PDF)?

A Parton Distribution Function (PDF) is a mathematical function that describes the probability of finding a particular parton (quarks, antiquarks, and gluons) with a certain momentum fraction inside a proton or other hadron. It is an essential tool in understanding the structure of protons and other hadrons in high-energy particle collisions.

What do the axes of a Parton Distribution Function plot represent?

The x-axis of a PDF plot represents the momentum fraction (x) of the parton, while the y-axis represents the probability density. This means that the higher the y-value at a given x-value, the more likely it is to find a parton with that momentum fraction inside the hadron.

How are Parton Distribution Functions determined?

PDFs are determined through a combination of experimental data and theoretical calculations. Experimental data from high-energy particle collisions is used to constrain the PDFs, while theoretical models such as perturbative QCD calculations are used to calculate the PDFs at higher x-values where experimental data is not available.

What is the importance of Parton Distribution Functions in particle physics?

Parton Distribution Functions are crucial in understanding the behavior of particles at high energies. They are used in calculations of scattering cross-sections, which are essential for interpreting experimental data and testing theories. PDFs also provide valuable insight into the internal structure of hadrons.

Are Parton Distribution Functions universal?

No, PDFs are not universal. They depend on the type of hadron and the energy scale at which they are measured. PDFs for protons, for example, will be different from those for neutrons. Additionally, PDFs change with increasing energy, which is known as the "DGLAP evolution".

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