Theory

Quark distribution Functions

define and describe [math] q_v (x)[/math] and [math]\Delta q_v (x)[/math] here

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]