Theory

Quark distribution Functions

define and describe [math] q_v (x)[/math] and [math]\Delta q_v (x)[/math] 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,
[math]\int u(x)dx = 2[/math] (2)
[math]\int d(x)dx = 1[/math] (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,

[math]u(x) = u_v (x) + u_s (x)[/math]
[math]d(x) = d_v (x) + d_s (x)[/math]
q(x) is the unpolarized distribution function and [math]\Delta q(x)[/math] - the polarized.

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

[math]F_1 (x) = \frac {1} {2}[/math] [math]\sum[/math][math]e_q^{2} (q^+ (x) + q^- (x)) = \frac{1} {2} \sum e_p^{2} q(x)[/math] (4)

[math]g_1 (x) = \frac {1} {2}[/math] [math]\sum[/math][math]e_q^{2} (q^+ (x) - q^- (x)) = \frac{1} {2} \sum e_p^{2} \Delta q(x)[/math] (5)

The unpolarized structure function [math]F_1 (x)[/math] - measures the total quark number density in the nucleon, [math]g_1 (x)[/math] - the polarized structure function is helicity difference quark number density.

The unpolarized structure functions [math]F_i (x)[/math]( i= L, R, O ) should satisfy the following inequalities,
[math]xF_3 (x)\leq 2xF_1 (x)\leq F_2 (x)[/math](1)

Polarized

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

The inclusive double polarization asymmetries [math]A_N[/math] in the valence region, where the scaling variable [math]x_{Bj} \gt 0.3[/math]can be written in terms of polarized [math]\Delta q_v (x)[/math] and unpolarized [math] q_v (x)[/math] valence quark distributions,

[math]A_{1, p}^{I} = \frac {4\Delta u_v (x) + \Delta d_v (x)} {4 u_v (x) + d_v (x)} [/math] (1)
[math]A_{1, n}^{I} = \frac {\Delta u_v (x) + 4\Delta d_v (x)} {u_v (x) + 4d_v (x)} [/math] (2)

The semi-inclusive pion electro-production asymmetries can be written in terms of the valence quark distributions
[math]A_{1, p}[/math]^{[math]\pi^+ - \pi^-[/math]} = [math]\frac {4\Delta u_v (x) - \Delta d_v (x)} {4 u_v (x) - d_v (x)} [/math] (3)

[math]A_{1,2H}[/math]^{[math]\pi^+ - \pi^-[/math]} = [math]\frac {\Delta u_v (x) + \Delta d_v (x)} { u_v (x) + d_v (x)} [/math] (4)

where

[math]A[/math]^{[math]\pi^+ - \pi^-[/math]} =[math]\frac {\sigma^{\pi^+ - \pi^-}_{\uparrow \downarrow} - \sigma^{\pi^+ - \pi^-}_{\uparrow \uparrow}} {\sigma^{\pi^+ - \pi^-}_{\uparrow \downarrow} + \sigma^{\pi^+ - \pi^-}_{\uparrow \uparrow}} [/math] (5)

where [math]\sigma^{\pi^+ - \pi^-}[/math] 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 [math]\pi^+[/math] and [math]\pi^-[/math] asymmetries:

[math]A_{1,2H}^{\pi^+ - \pi^-} = \frac {A^{\pi^+}} {1 + \frac {1} {R_p^{{\pi^+}/{\pi^-}}} } [/math] - [math]\frac {A^{\pi^-}} {1 + R_p^{{\pi^+}/{\pi^-}} } [/math] (6)

where [math]R_{2H}^{\pi^+/\pi^-} = \frac{\sigma^{\pi^+}} {\sigma^{\pi^-}}[/math] and

[math]A^{\pi^+ (\pi^-)} = \frac {\sigma^{\pi^+ (\pi^-)}_{\uparrow \downarrow} - \sigma^{\pi^+(\pi^-)}_{\uparrow \uparrow}} {\sigma^{\pi^+ (\pi^-)}_{\uparrow \downarrow} + \sigma^{\pi^+(\pi^-)}_{\uparrow \uparrow}} [/math] (7)

An asymmetry [math]\Delta R_{np} ^{\pi^+ + \pi^-} = \frac {\Delta\sigma_p^{\pi^+ + \pi^-} - \Delta\sigma_n^{\pi^+ + \pi^-}} {\sigma_p^{\pi^+ + \pi^-} - \sigma_n^{\pi^+ + \pi^-}} = \frac {g_1^p - g_1^n} {F_1^p - F_1^n} (x, Q^2)[/math] (8)
where [math]F_1[/math] is the unpolarized structure function and [math]g_1[/math] the scaling polarized structure function.

The last equation can be expressed as
[math]\triangle R_{np} ^{\pi^+ + \pi^-} = R_{n/p}[\frac {A_p^{\pi^+}} {1 + \frac {1} {R_p^{{\pi^+}/{\pi^-}}} } + \frac {A_p^{\pi^-}} {1 + R_p^{{\pi^+}/{\pi^-}} } ] + R_{p/n}[\frac {A_n^{\pi^+}} {1 + \frac {1} {R_n^{{\pi^+}/{\pi^-}}} } + \frac {A_n^{\pi^-}} {1 + R_n^{{\pi^+}/{\pi^-}} } ][/math] (9)

using the nomenclature of (6) equation, we have

[math]R_{i/j} = \frac {\frac {1 + (1-y)^2} {2y(2 - y)} } {1 - \frac {R_{i/j}^{\pi^+}} {1 + \frac{1}{R_j^{\pi^+/\pi^-} }} - \frac {R_{i/j}^{\pi^+}} {1 + R_j^{\pi^+/\pi^-} }} [/math]

[math]R_{i/j}^{\pi^c} = \frac {\sigma_i ^{\pi^c}} {\sigma_j ^{\pi^c}} [/math]