Transversity Distribution and Polarized Fragmentation Function from Semi-inclusive Pion Electroproduction

TL;DR

The paper proposes a method to extract the hitherto unknown u-quark transversity distribution $\delta u(x)$ and the ratio of polarized to unpolarized fragmentation functions $H_1^{\perp(1)u}(z)/D_1^u(z)$ from a single-spin asymmetry in semi-inclusive pion electroproduction measured by HERMES with a transversely polarized target. It leverages the Collins effect in SIDIS and assumes $u$-quark dominance for $\pi^+$ production, fixing the overall normalization by identifying $\delta u(x_0)$ with the helicity distribution $\Delta u(x_0)$ at large $x$. The analysis models $H_1^{\perp(1)q}$ via Collins-type analyzing power with a Gaussian $k_T$ distribution, and demonstrates through HERMES Monte Carlo studies that both functions can be reconstructed from 25 data points in $(x,z)$ space, though normalization remains a key systematic. The main systematic uncertainties arise from the $u$-quark dominance assumption and the normalization condition, yielding a 2–5% level impact at $x\gtrsim 0.2$–0.3; deuteron data provide access to $\delta u+\delta d$ with reduced precision.

Abstract

A method is discussed to determine the hitherto unknown u-quark transversity distribution from a planned HERMES measurement of a single-spin asymmetry in semi-inclusive pion electroproduction off a transversely polarized target. Assuming u-quark dominance, the measurement yields the shapes of the transversity distribution and of the ratio of a polarized and the unpolarized u-quark fragmentation functions. The unknown relative normalization can be obtained by identifying the transversity distribution with the well-known helicity distribution at large x. The systematic uncertainty of the method is dominated by the assumption of u-quark dominance.

Figures (5)

• Figure 1: The transversity distribution $h_1^p(x,Q_0^2)$ (continuous line) which coincides with the helicity distribution $g_1^p(x,Q_0^2)$ at the scale $Q_0^2 =0.4\ GeV^2$ (as given by the GRSV LO parameterization grsv96 in the 'standard' scenario). Their evolved LO distributions $h_1^p(x,Q^2)$ (dotted) and $g_1^p(x,Q^2)$ (dot-dashed) are shown at $Q^2= 10\ GeV^2$.
• Figure 2: Proton target. a) The weighted asymmetry $A_T^{\pi^+}(x)$ in different intervals of $z$; b) the function $K(x, z)$.
• Figure 3: a) The transversity distribution $\delta u(x)$, and b) the ratio of the fragmentation functions $H_1^{\perp (1) u} (z)$ and $D_1^u (z)$ as would be measured by HERMES with a proton target. The asterisk in a) shows the normalization point.
• Figure 4: a) The transversity distribution $\delta u(x) + \delta d(x)$, and b) the ratio of the fragmentation functions $H_1^{\perp (1) u} (z)$ and $D_1^u (z)$ as would be measured by HERMES with a deuteron target. The asterisk in a) shows the normalization point.
• Figure 5: Relative difference between transversity distribution $\delta u(x, Q^2)$ and helicity distribution $\Delta u(x, Q^2)$ as a function of $x$ and $Q^2$ in the kinematical region accessible to the HERMES experiment ($< Q^2 > \simeq 2.5$ GeV$^2$).