Are there approximate relations among transverse momentum dependent distribution functions?

Certain exact relations among transverse momentum dependent parton distribution functions due to QCD equations of motion turn into approximate ones upon the neglect of pure twist-3 terms. On the basis of available data from HERMES, we test the practical usefulness of one such “Wandzura-Wilczek-type approximation,” namely, of that connecting $h_{1L}^{\perp (1)a}(x)$ to $h_L^a(x)$, and discuss how it can be further tested by future CLAS and COMPASS data.

Figure 1

(a) Transversity, $x{h}_1^q(x)$, vs $x$, from various models. (b) The ratio $h_{1L}^{\perp (1)q}(x)/h_1^q(x)$ vs $x$ in various models, with $h_{1L}^{\perp }$ estimated by means of the WW-type approximation (11). (c) $x{h}_{1L}^{\perp (1)q}(x)$ vs $x$ from the WW-type approximation (11) and $h_1^a(x)$ from $\chi \mathrm{QSM}$ [64], in comparison with $(-\frac{1}{10})x{h}_1^q(x)$ from that model. All results here refer to a scale of $2.5{\mathrm{GeV}}^2$.

Figure 2

Kinematics of the SIDIS process $lN\to l^{\prime }hX$ and the definitions of azimuthal angles in the lab frame. Here the target polarization is antiparallel to the beam (i.e. ${\varphi }_S=\pi$).

Figure 3

Longitudinal target SSA $A_{UL}^{\sin 2\varphi }$ as a function of $x$. The proton (a, b) and deuterium (c–f) target data are from HERMES [17, 19]. The theoretical curves are obtained using information on the Collins fragmentation function from 55, 56, predictions from the instanton vacuum model and chiral quark-soliton model for $h_L^a(x)$ and $h_1^a(x)$ [53, 64], and—this is crucial in our context—assuming the validity of the WW-type approximation (11). The shaded error bands are due to the uncertainties in (27, 28), see Appendix  for details.