Collins effect in semiinclusive deeply inelastic scattering and in electron-positron-annihilation

The Collins fragmentation function is extracted from HERMES data on azimuthal single spin asymmetries in semi-inclusive deeply inelastic scattering, and BELLE data on azimuthal asymmetries in $e^{+}{e}^{-}$-annihilations. A Gaussian model is assumed for the distribution of transverse parton momenta and predictions are used from the chiral quark-soliton model for the transversity distribution function. We find that the HERMES and BELLE data yield a consistent picture of the Collins fragmentation function which is compatible with COMPASS data and the information previously obtained from an analysis of DELPHI data. Estimates for future experiments are made.

Figure 1

Kinematics of the SIDIS process $lp\to l^{\prime }hX$ and the definitions of azimuthal angles in the lab frame. The target polarization vector is transverse with respect to the beam.

Figure 2

The quantity $2B_{\mathrm{Gauss}}{H}_1^{\perp (1/2)a}$ averaged over $z$, i.e. practically the weight of $h_1^a(x)$ in the Collins SSA $A_{\mathrm{UT}}^{\sin (\varphi +{\varphi }_S)}(x)$ in Eq. (4), vs $x$ as extracted from the preliminary HERMES data [19]. This quantity does not show any significant $x$-dependence—as expected, see text.

Figure 3

The Collins SSA $A_{\mathrm{UT}}^{\sin (\varphi +{\varphi }_S)}(x)$ as function of $x$. The preliminary HERMES data are from 19, the COMPASS data are from 20. The theoretical curves are based on the fit in Eqs. (10, 11) and predictions for the transversity distribution from the chiral quark-soliton model [26]. Notice that different sign conventions are used in 19, 20: $A_{\mathrm{UT}}^{\sin {\varphi }_C}(x)=-A_{\mathrm{UT}}^{\sin (\varphi +{\varphi }_S)}(x)$.

Figure 4

Kinematics of the process $e^{+}{e}^{-}\to h_1{h}_{2}X$ and the definitions of azimuthal angles in the $e^{+}{e}^{-}$ rest frame.

Figure 5

(a) The two best fit solutions for the parameters $C_i$ in the Ansatz (18) (indicated as discrete points) and their respective $1\mathrm{\text{−}}\sigma$ regions as obtained from a fit to the BELLE data [21]. The solutions are symmetric with respect to the line $C_{\mathrm{fav}}=C_{\mathrm{unf}}$ indicated by a dashed line. (b) The best fit for $H_1^{\perp (1/2)a}(z)$ resulting from Fig. 5 with the choice $H_1^{\perp \mathrm{fav}}>0$ and $H_1^{\perp \mathrm{unf}}<0$ as suggested by the analysis of the HERMES experiment, see Sec. 2.

Figure 6

The observable $P_1(z_1,z_2)$ as defined in Eqs. (16, 17) for fixed $z_1$-bins as function of $z_2$. The data are from the BELLE experiment [21]. The theoretical curves are obtained on the basis of the fit result shown in Fig. 5.

Figure 7

The Collins SSA $A_{\mathrm{UT}}^{\sin (\varphi +{\varphi }_S)}(z)$ as function of $z$. The preliminary HERMES data are from 19. The theoretical curves are based on the fit of $H_1^{\perp }$ to the BELLE data shown in Fig. 5b under the assumption (21). For $h_1^a$ the model prediction [26] is used. The dashed lines indicate the sensitivity of the SSA to the unknown ratio of the Gaussian widths of $h_1^a$ and $H_1^{\perp }$ in the range (22). The description of the preliminary HERMES data [19] is satisfactory.

Figure 8

The Collins SSA $A_N\equiv -A_{\mathrm{UT}}^{\sin (\varphi +{\varphi }_S)}$ in the production of charged hadrons $h^{\pm }$ in SIDIS from different targets. The squares denote the preliminary SMC data [15]. The open circles show our rough estimates.

Figure 9

$⟨2B_{\mathrm{Gauss}}{H}_1^{\perp }⟩$ as function of the parameter $r$ defined in Eq. (30). The figure demonstrates that the SIDIS data on the proton Collins SSA [18, 19] are practically insensitive to $h_1^d(x)$.

Figure 10

$x{h}_1^q(x)$ vs $x$ in the kinematic range of the HERMES experiment. The dashed/dotted lines are predictions for $h_1^u/h_1^d$ from the chiral quark-soliton model [26]. The shaded range for the $u$-quark is bound from above by the Soffer bound and from below by the estimated theoretical uncertainty of the model result [26]. The shaded range for the $d$-quark is due to the Soffer bound. The figure demonstrates roughly the emerging picture of the transversity distribution.

Figure 11

$A_{\mathrm{UT}}^{\sin (\varphi +{\varphi }_S)}$ vs $x$ as predicted on the basis of our study of the HERMES data in Sec. 2 for: (a) ${\pi }^0$ production from proton, which could be measured at HERMES [17] and CLAS [75]. (b) ${\pi }^{-}$ production from a neutron (in practice ${}^3\mathrm{He}$) target, as planned in the HALL-A experiment at JLab. The error bars estimated for the HALL-A experiment are taken from Ref. 76. The dashed lines indicate tentatively the effects of a different scenario for $h_1^d(x)$, see text. (c) The same as (b) but for ${\pi }^{+}$.

Figure 12

The observable $P_c(z_1,z_2)$ as defined in Eqs. (35, 36) for fixed $z_1$-bins as function of $z_2$. The predictions are obtained on the basis of the fit result shown in Fig. 5, and could be tested in the BELLE experiment.