Theory

Quark distribution Functions

Polarized

The inclusive double polarization asymmetries [math]\displaystyle{ A_N }[/math] can be written in terms of polarized [math]\displaystyle{ \triangle q_v (x) }[/math] and unpolarized [math]\displaystyle{ q_v (x) }[/math] valence quark distributions,

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

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

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

where

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

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