Polarized Parton Distribution Functions with Estimate of Uncertainties

Y. Goto$^{1}$, N. Hayashi$^{2}$, M. Hirai$^{3}$, H. Kobayashi$^{2}$, S. Kumano$^{3}$, M. Miyama$^{4}$, T. Morii$^{5}$,
N. Saito$^{1,2}$, T.-A. Shibata$^{6}$, and T. Yamanishi$^{7}$
(Asymmetry Analysis Collaboration)

$^{1}$RIKEN BNL Research Center, Upton, NY 11973, U.S.A.
$^{2}$Radiation Laboratory, RIKEN, Saitama 351-0198, Japan
$^{3}$Department of Physics, Saga University, Saga 840-8502, Japan
$^{4}$Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
$^{5}$Faculty of Human Development, Kobe University, Kobe 657-8501, Japan
$^{6}$Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
$^{7}$Department of Management Science, Fukui University of Technology, Gakuen, Fukui 910-0028, Japan

Experimental data on the structure function $g_1(x,Q^2)$ have been accumulated with the proton, deuteron and $^3$He targets. We study parameterization of the polarized parton distributions (pol-PDFs) [1] in the leading order (LO) of $\alpha_s$ and in the next-to-leading order (NLO) with estimate of uncertainties.

The polarized distributions are provided with a number of parameters at $Q^2$=1 GeV$^2$ ($\equiv Q_0^2$). Considering the positivity condition and the helicity retention property following functional form is proposed for the pol-PDFs:

$$\Delta f_i(x, Q^2_0) = A_i \, x^{\alpha_i} \, (1 + \gamma_i \, x^{\lambda_i}) \, f_i(x, Q^2_0) \, , \nonumber$$

where $f_i(x, Q^2_0)$ is an unpolarized parton distribution (GRV98). The subscript $i$ denotes the type of the parton distribution: $f_i \in \{u_v,d_v, \bar{q}, g\}$. The parameters $A_i$, $\alpha_i$, $\gamma_i$, and $\lambda_i$ are determined by fitting available experimental data on the inclusive spin asymmetry $A_1$ from experiments: EMC, SMC, SLAC (E130, E142, E143, E154, E155) and HERMES, where $A_1$ is expressed with the structure function $g_1$ and other known functions and $g_1$ is expressed as linear combination of pol-PDFs $\Delta f_i$. The first moments of $\Delta u_v$ and $\Delta d_v$ are fixed by the semi-leptonic decay data by assuming the SU(3) symmetry, so that the number of free parameters is reduced from 16 to 14.

The $\chi^2$ minimization is performed by using the CERN library subroutine MINUIT with the following definition,

$$\chi^2\, =\, \sum (A_1^{data}\, -\, A_1^{calc})^2 / (\sigma _{A_1^{data}})^2 \nonumber$$

comparing theoretical asymmetries $A_1$ with the experimental data weighted by square inverse of the error in each points. As a result, we obtain the minimum $\chi^2$: $\chi^2$/d.o.f=322.6/359 for the LO and $\chi^2$/d.o.f=300.4/359 for the NLO.

The uncertainties of pol-PDFs can be estimated from the errors of the parameters. However, previous attempts to quantify the uncertainties on the pol-PDFs have been rather unsatisfactory. This is due to that extracting the uncertainties of pol-PDFs is not easy because there are strong correlations between pol-PDFs of different flavors and from different values of $x$ and $Q^2$. We present preliminary results from an effort to quantify the uncertainties in polarized parton distribution functions.

[1] Y. Goto, N. Hayashi, M. Hirai, H. Horikawa, S. Kumano, M. Miyama, T. Morii, N. Saito,

T.-A. Shibata, E. Taniguchi, and T. Yamanishi, hep-ph/0001046,

Phys. Rev. D 62 34017 (2000).