# Inclusive Scattering

Kinematic variables in deep inelastic scattering

Kinematic variable description
, 4 - momenta of the initial and final state leptons
, Polar and azimuthal angle of the scattered lepton
4 - momentum of the initial target nucleon
4 - momentum of the virtual photon

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# 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 in the proton. 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. Actually, the proton can contain an extra pair of quark - anti quarks. The original(u, d) quarks are called valence quarks and the extra ones sea quarks.we are allowed to separate the quark distribution function into a valence and a sea part,

q(x) is the unpolarized distribution function and - the polarized.

The structure functions in the quark parton model can be written in terms of quark distribution functions,

  (4)
(5)

The unpolarized structure function  - measures the total quark number density in the nucleon,  - the polarized structure function is helicity difference quark number density.


The unpolarized structure functions ( i= L, R, O ) should satisfy the following inequalities,
(1)

If is increased so that the weak part of the natural current will be included, that means we have -exchange, Z-exchange and -Z interference. The cross-section can be expressed as

(2)

The structure functions in terms of the parton distributions can be written as,

(3)
(4)

where

(5)

(6)
where , , are the charge, left- and right-handed weak couplings of a ith type quark and , corresponding couplings for the electron.

## 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