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] = [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] = [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 in the following way
[math]\displaystyle{ A_{1, p}((\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, H}((\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 ((\pi+) - (\pi-)) }[/math] = [math]\displaystyle{ \frac {\sigma} {3} }[/math] A⁵