Theory

Quark distribution Functions

describe $q_v (x)$ and $\Delta q_v (x)$ here

Polarized

Both models, pQCD and a hyperfine perturbed constituent quark model(CQD), show that as the scaling variable $x_{Bj}$ goes to one the double spin asymmetry $A_{1,N}$ is unity. On the other hand, CQM with SU(6) symmetry predicts that at $x_{Bj}$ = 1 $A_{1,n}$ = 5/9 for the proton, $A_{1,n}$ = 0 for the neutron and $A_{1,d}$ = 1/3 for the deuteron. The double spin asymmetry and the ratio of the polarized valence down quark distribution function to the unpolarized $({\triangle d_v} / {d_v})$ can give knowledge of this two different results
The inclusive double polarization asymmetries $A_N$ can be written in terms of polarized $\triangle q_v (x)$ and unpolarized $q_v (x)$ valence quark distributions,

$A_{1, p}$^I = $\frac {4\triangle u_v (x) + \triangle d_v (x)} {4 u_v (x) + d_v (x)}$
$A_{1, n}$^I = $\frac {\triangle u_v (x) + 4\triangle d_v (x)} {u_v (x) + 4d_v (x)}$

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

$A_{1,2H}$^{$\pi^+ - \pi^-$} = $\frac {\triangle u_v (x) + \triangle d_v (x)} { u_v (x) + d_v (x)}$

where

$A$^{$\pi^+ - \pi^-$} =$\frac {\sigma^{\pi^+ - \pi^-}_{\uparrow \downarrow} - \sigma^{\pi^+ - \pi^-}_{\uparrow \uparrow}} {\sigma^{\pi^+ - \pi^-}_{\uparrow \downarrow} + \sigma^{\pi^+ - \pi^-}_{\uparrow \uparrow}}$
where $\sigma^{\pi^+ - \pi^-}$ is the measured difference of the yield from oppositely charged pions.
The semi - inclusive asymmetry can be expressed in the following way

$A_{1,2H}^{\pi^+ - \pi^-} = \frac {A^{\pi^+}} {1 + \frac {1} {R_p^{{\pi^+}/{\pi^-}}} }$ - $\frac {A^{\pi^-}} {1 + R_p^{{\pi^+}/{\pi^-}} }$

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

$A^{\pi^+ (\pi^-)} = \frac {\sigma^{\pi^+ (\pi^-)}_{\uparrow \downarrow} - \sigma^{\pi^+(\pi^-)}_{\uparrow \uparrow}} {\sigma^{\pi^+ (\pi^-)}_{\uparrow \downarrow} + \sigma^{\pi^+(\pi^-)}_{\uparrow \uparrow}}$

An asymmetry $\triangle R_{np} ^{\pi^+ + \pi^-} = \frac {\triangle\sigma_p^{\pi^+ + \pi^-} - \triangle\sigma_n^{\pi^+ + \pi^-}} {\sigma_p^{\pi^+ + \pi^-} - \sigma_n^{\pi^+ + \pi^-}} = \frac {g_1^p - g_1^n} {F_1^p - F_1^n} (x, Q^2)$

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

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

$R_{i/j}^{\pi^c} = \frac {\sigma_i ^{\pi^c}} {\sigma_j ^{\pi^c}}$