- Extraction of [math]\frac{\Delta d_v}{d_v}[/math] from charged pion asymmetries
Cross-section
A leading order expression for charged pion semi-inclusive pion electro-production cross section, represented as a sum of the [math]\pi^+[/math] and [math]\pi^-[/math] cross sections, using proton or neutron targets can be written, using Eq. 9 & 10 from Ref.<ref name="Christova9907265"> Christova, E., & Leader, E. (1999). Semi-inclusive production-tests for independent fragmentation and for polarized quark densities. hep-ph/9907265.</ref>, as:
| [math]\sigma_p^{\pi^+ + \pi^-} = \frac{1}{9}[4( u + \bar{u}) + ( d + \bar{d})]D_u^{\pi^+ + \pi^-} + 2 s D_u^{\pi^+ + \pi^-} [/math]
|
| [math]\sigma_n^{\pi^+ + \pi^-} = \frac{1}{9}[4(d + \bar{d}) + (u + \bar{u})]D_u^{\pi^+ + \pi^-}+ 2 s D_u^{\pi^+ + \pi^-} [/math]
|
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 ([math]D_u^{\pi^+} = D_{\overline{u}}^{\pi^-}[/math] and [math]D_d^{\pi^-} = D_{\overline{d}}^{\pi^+}[/math] ) and charge ([math]D_u^{\pi^+} = D_d^{\pi^-}[/math]) conjugation invariance for the fragmentation functions, the following equality holds:
| [math]D_u^{\pi^+ \pm \pi^-} = D_u^{\pi^+} \pm D_u^{\pi^-} = D_d^{\pi^+ \pm \pi^-}[/math]
|
Strange quark contributions to the above SIDIS cross-section become ignorable due to the dominant contributions from the up and down quarks as x_{Bj} increases beyond 0.3
| [math]\sigma_p^{\pi^+ + \pi^-} = \frac{1}{9}[4( u + \bar{u}) + ( d + \bar{d})]D_u^{\pi^+ + \pi^-}[/math]
|
| [math]\sigma_n^{\pi^+ + \pi^-} = \frac{1}{9}[4(d + \bar{d}) + (u + \bar{u})]D_u^{\pi^+ + \pi^-}[/math]
|
Helicity Difference Cross Section
The polarized cross section difference is defined as :
| [math]\Delta \sigma = \sigma_{\uparrow \downarrow} - \sigma_{\uparrow \uparrow}[/math]
|
using the polarized cross section [math](\sigma_{\alpha \beta})[/math] where [math]\alpha[/math] refers to the lepton helicity and [math]\beta[/math] to the target helicity.
The charged pion helicity difference [math](\Delta \sigma_p^{\pi^+ + \pi^-})[/math] can be written using equations 6 and 7 from Reference <ref name="Christova9907265"> </ref> as
| [math]\Delta \sigma_p^{\pi^+ + \pi^-} = \frac{1}{9}[4(\Delta u + \Delta \bar{u}) + (\Delta d + \Delta \bar{d})]D_u^{\pi^+ + \pi^-}[/math]
|
| [math]\Delta \sigma_n^{\pi^+ + \pi^-} = \frac{1}{9}[4(\Delta d + \Delta d^-) + (\Delta u + \Delta u^-)]D_u^{\pi^+ + \pi^-}[/math]
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The analogous expressions for the case of a Deuteron target are
| [math]\sigma_{2H}^{\pi^+ \pm \pi^-} = \frac{5}{9}[( u + \bar{u}) \pm ( d + \bar{d})]D_u^{\pi^+ \pm \pi^-}[/math]
|
| [math]\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^-}[/math]
|
and unpolarized:
LO models for SIDIS cross section
GJR08FFNS
GSRV
The polarized cross section difference is defined as :
| [math]\Delta \sigma = \sigma_{\uparrow \downarrow} - \sigma_{\uparrow \uparrow}[/math]
|
using the polarized cross section [math](\sigma_{\alpha \beta})[/math] where [math]\alpha[/math] refers to the lepton helicity and [math]\beta[/math] to the target helicity.
The charged pion helicity difference [math](\Delta \sigma_p^{\pi^+ + \pi^-})[/math] can be written using equations 6 and 7 from Reference <ref name="Christova9907265"> </ref> as
| [math]\Delta \sigma_p^{\pi^+ + \pi^-} = \frac{1}{9}[4(\Delta u + \Delta \bar{u}) + (\Delta d + \Delta \bar{d})]D_u^{\pi^+ + \pi^-}[/math]
|
| [math]\Delta \sigma_n^{\pi^+ + \pi^-} = \frac{1}{9}[4(\Delta d + \Delta d^-) + (\Delta u + \Delta u^-)]D_u^{\pi^+ + \pi^-}[/math]
|
The analogous expressions for the case of a Deuteron target are
| [math]\sigma_{2H}^{\pi^+ \pm \pi^-} = \frac{5}{9}[( u + \bar{u}) \pm ( d + \bar{d})]D_u^{\pi^+ \pm \pi^-}[/math]
|
| [math]\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^-}[/math]
|
and unpolarized:
The charged pion asymmetry may be defined as
| [math]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}]}[/math]
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| [math]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}]}[/math]
|
where the fragmentations functions [math]D[/math] do not contribute if independent fragmentation, and isospin and charge conjugation are invariant.
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 ([math]D_u^{\pi^+} = D_{\overline{u}}^{\pi^-}[/math] and [math]D_d^{\pi^-} = D_{\overline{d}}^{\pi^+}[/math] ) and charge ([math]D_u^{\pi^+} = D_d^{\pi^-}[/math]) conjugation invariance for the fragmentation functions, the following equality holds:
| [math]D_u^{\pi^+ \pm \pi^-} = D_u^{\pi^+} \pm D_u^{\pi^-} = D_d^{\pi^+ \pm \pi^-}[/math]
|
The polarized and unpolarized cross sections for pion electroproduction can be written in terms of valence quark distribution functions in the valence region as:
| [math]\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^-}[/math]
|
| [math]\Delta \sigma_n^{\pi^+ \pm \pi^-} = \frac{1}{9}[4(\Delta d + \Delta d^-) \pm (\Delta u + \Delta u^-)]D_u^{\pi^+ \pm \pi^-}[/math]
|
| [math]\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^-}[/math]
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and unpolarized:
| [math]\sigma_p^{\pi^+ \pm \pi^-} = \frac{1}{9}[4( u + \bar{u}) \pm ( d + \bar{d})]D_u^{\pi^+ \pm \pi^-}[/math]
|
| [math]\sigma_n^{\pi^+ \pm \pi^-} = \frac{1}{9}[4(d + \bar{d}) \pm (u + \bar{u})]D_u^{\pi^+ \pm \pi^-}[/math]
|
| [math]\sigma_{2H}^{\pi^+ \pm \pi^-} = \frac{5}{9}[( u + \bar{u}) \pm ( d + \bar{d})]D_u^{\pi^+ \pm \pi^-}[/math]
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In the valence region ([math]x_{B}\gt 0.3[/math]), where the sea quark contribution is minimized, the above asymmetries can be expressed in terms of polarized and unpolarized valence quark distributions:
| [math]A_{1,p}^{\pi^+ \pm \pi^-} = \frac{4 \Delta u_v(x) \pm \Delta d_v(x)}{4u_v(x) \pm d_v(x)}[/math]
|
| [math]A_{1,2H}^{\pi^+ \pm \pi^-} = \frac{\Delta u_v(x) + \Delta d_v(x)}{u_v(x) + d_v(x)}[/math]
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The ratio of polarized to unpolarized valence up and down quark distributions may then be written as
| [math]\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)[/math]
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and
| [math]\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)[/math]
|
The ratio of polarized to unpolarized valence quark distribution functions can be extracted using the last two equations.
Next to leading Order (NLO)
<ref name="Sissakian074032"> A.N. Sissakian, O. Yu. Shevchenko, and O.N. Ivanov Phys Rev D 70 (074032) 2004 https://arxiv.org/abs/hep-ph/0411243</ref>
Towards semi-inclusive deep inelastic scattering at next-to-next-to-leading order, Daniele Anderle https://arxiv.org/pdf/1612.01293.pdf
D. de Florian, R. Sassot, M. Epele, R. J. Herna ́ndez-Pinto, M. Stratmann. Phys. Rev. D 91 014035 (2015).
Global extraction of the parton-to-pion fragmentation functions at NLO accuracy in QCD R. J. Herna ́ndez-Pinto 2016 https://arxiv.org/pdf/1609.02455.pdf
References
2009 talk by Xiodong Jiang, kinematic cuts identified (Q2 >1GeV2,W >2GeV,W' (missing mass of undetected hadrons) >1.5GeV,xF >0,pπ >2GeV/c) , http://www.int.washington.edu/talks/WorkShops/int_09_3/People/Jiang_X/Jiang.pdf
Bibliography
<references/>
Documents
File:Christova Leader hep-ph-9907265.pdf
File:Sissakian PhysRevD70 074032 2004.pdf
https://twiki.cern.ch/twiki/bin/view/LHCPhysics/PDF
SIDIS cross sections
File:Asauryan nucle-ex1103.1649.pdf