Extracting DeltaDoverD from PionAsym

Fragmentation Independence

$\frac{\Delta d_v}{d_v}(x,Q^2) = \frac{\Delta \sigma_p^{\pi^+ - \pi^-} - 4\Delta \sigma_{2H}^{\pi^+ - \pi^-}}{\sigma_p^{\pi^+ - \pi^-} - 4\sigma_{2H}^{\pi^+ - \pi^-}} (x,Q^2)$

The asymmetries from semi inclusive pion electroproduction using proton or deuteron targets can be written in terms of the difference of the yield from oppositely charged pions <ref name="Christova"> Christova, E., & Leader, E. (1999). Semi-inclusive production-tests for independent fragmentation and for polarized quark densities. hep-ph/9907265.</ref>:

$A_{1,p}^{\pi^+ \pm \pi^-} = \frac{\Delta \sigma_p^{\pi^+ \pm \pi^-}}{\sigma_p^{\pi^+ \pm \pi^-}} = \frac{[({\sigma_p}^{\pi^+})_{1/2}-({\sigma_p}^{\pi^+})_{3/2}] \pm [({\sigma_p}^{\pi^-})_{1/2}-({\sigma_p}^{\pi^-})_{3/2}]}{[({\sigma_p}^{\pi^+})_{1/2}+({\sigma_p}^{\pi^+})_{3/2}] \pm [({\sigma_p}^{\pi^-})_{1/2}+({\sigma_p}^{\pi^-})_{3/2}]}$

$A_{1,2H}^{\pi^+ \pm \pi^-} = \frac{\Delta \sigma_{2H}^{\pi^+ \pm \pi^-}}{\sigma_{2H}^{\pi^+ \pm \pi^-}} = \frac{[({\sigma_{2H}}^{\pi^+})_{1/2}-({\sigma_{2H}}^{\pi^+})_{3/2}] \pm [({\sigma_{2H}}^{\pi^-})_{1/2}-({\sigma_{2H}}^{\pi^-})_{3/2}]}{[({\sigma_{2H}}^{\pi^+})_{1/2}+({\sigma_{2H}}^{\pi^+})_{3/2}] \pm [({\sigma_{2H}}^{\pi^-})_{1/2}+({\sigma_{2H}}^{\pi^-})_{3/2}]}$

Independent fragmentation identifies the process in which quarks fragment into hadrons, independent of the photon-quark scattering process. In other words, the fragmentation process is independent of the initial quark environment, which initiates the hadronization process. Assuming independent fragmentation and using isospin ($D_u^{\pi^+} = D_{\overline{u}}^{\pi^-}$ and $D_d^{\pi^-} = D_{\overline{d}}^{\pi^+}$ ) and charge ($D_u^{\pi^+} = D_d^{\pi^-}$) conjugation invariance for the fragmentation functions, the following equality holds:

$D_u^{\pi^+ \pm \pi^-} = D_u^{\pi^+} \pm D_u^{\pi^-} = D_d^{\pi^+ \pm \pi^-}$

The polarized and unpolarized cross sections for pion electroproduction can be written in terms of valence quark distribution functions in the valence region as:

$\Delta \sigma_p^{\pi^+ \pm \pi^-} = \frac{1}{9}[4(\Delta u + \Delta \bar{u}) \pm (\Delta d + \Delta \bar{d})]D_u^{\pi^+ \pm \pi^-}$

$\Delta \sigma_n^{\pi^+ \pm \pi^-} = \frac{1}{9}[4(\Delta d + \Delta d^-) \pm (\Delta u + \Delta u^-)]D_u^{\pi^+ \pm \pi^-}$

$\Delta \sigma_{2H}^{\pi^+ \pm \pi^-} = \frac{5}{9}[(\Delta u + \Delta \bar{u}) \pm (\Delta d + \Delta \bar{d})]D_u^{\pi^+ \pm \pi^-}$

and unpolarized:

$\sigma_p^{\pi^+ \pm \pi^-} = \frac{1}{9}[4( u + \bar{u}) \pm ( d + \bar{d})]D_u^{\pi^+ \pm \pi^-}$

$\sigma_n^{\pi^+ \pm \pi^-} = \frac{1}{9}[4(d + \bar{d}) \pm (u + \bar{u})]D_u^{\pi^+ \pm \pi^-}$

$\sigma_{2H}^{\pi^+ \pm \pi^-} = \frac{5}{9}[( u + \bar{u}) \pm ( d + \bar{d})]D_u^{\pi^+ \pm \pi^-}$

In the valence region ($x_{B}\gt 0.3$), where the sea quark contribution is minimized, the above asymmetries can be expressed in terms of polarized and unpolarized valence quark distributions:

$A_{1,p}^{\pi^+ \pm \pi^-} = \frac{4 \Delta u_v(x) \pm \Delta d_v(x)}{4u_v(x) \pm d_v(x)}$

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

The ratio of polarized to unpolarized valence up and down quark distributions may then be written as

$\frac{\Delta u_v}{u_v}(x,Q^2) = \frac{\Delta \sigma_p^{\pi^+ - \pi^-} + \Delta \sigma_{2H}^{\pi^+ - \pi^-}}{\sigma_p^{\pi^+ - \pi^-} + \sigma_{2H}^{\pi^+ - \pi^-}} (x,Q^2)$

and

$\frac{\Delta d_v}{d_v}(x,Q^2) = \frac{\Delta \sigma_p^{\pi^+ - \pi^-} - 4\Delta \sigma_{2H}^{\pi^+ - \pi^-}}{\sigma_p^{\pi^+ - \pi^-} - 4\sigma_{2H}^{\pi^+ - \pi^-}} (x,Q^2)$

The ratio of polarized to unpolarized valence quark distribution functions can be extracted using the last two equations.