# Quark distribution Functions

define and describe and here

Quark distribution function q(x) is the probability(density) of finding a quark with fraction x of the proton momentum. It can be expressed as
(1)
It is known that the proton contains up(u) and down(d) quarks. Accordingly, We have up u(x) and down d(x) quark distribution functions. u(x) is the probability that momentum fraction x is carried by a u type quark and d(x) - for a d type quark. Moreover,
(2)
(3)

u(x)dx ( d(x)dx ) is the average number of up (down) quarks which have a momentum fraction between x and x+dx.

## Polarized

Both models, pQCD and a hyperfine perturbed constituent quark model(CQD), show that as the scaling variable goes to one the double spin asymmetry is unity. On the other hand, CQM with SU(6) symmetry predicts that at = 1, = 5/9 for the proton, = 0 for the neutron and = 1/3 for the deuteron. The double spin asymmetry and the ratio of the polarized valence down quark distribution function to the unpolarized can give knowledge of these two different results.

The inclusive double polarization asymmetries in the valence region, where the scaling variable can be written in terms of polarized and unpolarized valence quark distributions,

(1)
(2)

The semi-inclusive pion electro-production asymmetries can be written in terms of the valence quark distributions
= (3)

= (4)

where

= (5)

where is the measured difference of the yield from oppositely charged pions. Using the first four equation (1), (2), (3) and (4) one can construct the valence quark distribution functions.
The semi - inclusive asymmetry can be rewritten in terms of the measured semi-inclusive and asymmetries:

- (6)

where and

(7)

An asymmetry (8)
where is the unpolarized structure function and the scaling polarized structure function.

The last equation can be expressed as
(9)

using the nomenclature of (6) equation, we have