Difference between revisions of "Theory"

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==Unpolarized==
 
==Unpolarized==
 
==Polarized==
 
==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>({\triangle d_v} / {d_v}) </math><br>   
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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>({\triangle d_v} / {d_v}) can give knowledge of this two different results </math><br>   
 
The inclusive double polarization asymmetries <math>A_N</math> can be written in terms of polarized <math>\triangle q_v (x)</math> and unpolarized <math> q_v (x)</math> valence quark distributions,
 
The inclusive double polarization asymmetries <math>A_N</math> can be written in terms of polarized <math>\triangle q_v (x)</math> and unpolarized <math> q_v (x)</math> valence quark distributions,
 
      
 
      

Revision as of 20:29, 18 July 2007

Inclusive Scattering

W

Semi-Inclusive Scattering

Quark distribution Functions

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

Unpolarized

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]({\triangle d_v} / {d_v}) can give knowledge of this two different results [/math]
The inclusive double polarization asymmetries [math]A_N[/math] can be written in terms of polarized [math]\triangle q_v (x)[/math] and unpolarized [math] q_v (x)[/math] valence quark distributions,


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


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


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]
where [math]\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]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]

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]


An asymmetry [math]\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)[/math]


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]


[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]