Transversity and Collins functions from SIDIS and $e^{+}{e}^{-}$ data

A global analysis of the experimental data on azimuthal asymmetries in semi-inclusive deep inelastic scattering (SIDIS), from the HERMES and COMPASS Collaborations, and in $e^{+}{e}^{-}\to h_1{h}_{2}X$ processes, from the Belle Collaboration, is performed. It results in the extraction of the Collins fragmentation function and, for the first time, of the transversity distribution function for $u$ and $d$ quarks. These turn out to have opposite signs and to be sizably smaller than their positivity bounds. Predictions for the azimuthal asymmetry $A_{UT}^{\sin ({\varphi }_S+{\varphi }_h)}$, as will soon be measured at JLab and COMPASS operating on a transversely polarized proton target, are then presented.

Figure 1

Three-dimensional kinematics of the SIDIS process, according to Trento conventions [14]. The photon and the proton collide along the $\widehat{z}$-axis, while the leptonic plane defines the $\widehat{xz}$ plane. The fragmenting quark and the final hadron $h$ are emitted at azimuthal angles $\phi$ and ${\varphi }_h$, and the proton transverse spin direction is identified by ${\varphi }_S$.

Figure 2

Three-dimensional kinematics of the $e^{+}{e}^{-}\to h_1{h}_{2}X$ process, in the $q\overline{q}$ c.m. frame. In this configuration the reconstructed thrust axis identifies the $\widehat{z}$ direction; the lepton-quark scattering plane defines the $\widehat{xz}$ plane.

Figure 3

Three-dimensional kinematics of the $e^{+}{e}^{-}\to h_1{h}_{2}X$ process. In this configuration the $\widehat{z}$ direction is identified by the momentum of the final hadron $h_2$, while $h_1$ is emitted at an azimuthal angle ${\varphi }_1$ with respect to the lepton-$h_2$ plane, defined as the $\widehat{xz}$ plane.

Figure 4

HERMES experimental data [8, 9] on the azimuthal asymmetry $A_{UT}^{\sin ({\varphi }_S+{\varphi }_h)}$ for ${\pi }^{\pm }$ production are compared to the curves obtained from Eq. (20) with the parametrizations of Eqs. (13, 14, 15, 16, 17), and the parameter values, determined through our global best fit, given in Table . The shaded area corresponds to the theoretical uncertainty on the parameters, as explained in the text.

Figure 5

The measurements of $A_{UT}^{\sin ({\varphi }_S+{\varphi }_h)}$, for the production of positively and negatively charged hadrons, from the COMPASS experiment operating on a deuterium target [10] are compared to the curves obtained from Eq. (20) with the parametrizations of Eqs. (13, 14, 15, 16, 17), and the parameter values, determined through our global best fit, given in Table . The shaded area corresponds to the theoretical uncertainty on the parameters, as explained in the text. Notice the extra $\pi$ phase in addition to ${\varphi }_S+{\varphi }_h$ in the figure label, to keep into account the different choice of the Collins angle, with respect to Trento [14] and HERMES conventions, adopted by COMPASS Collaboration.

Figure 6

The experimental data on two different azimuthal correlations in unpolarized $e^{+}{e}^{-}\to h_1{h}_{2}X$ processes, as measured by Belle Collaboration [7], are compared to the curves obtained from Eqs. (44) [$A_{12}$] and (60) [$A_0$] with the parametrizations of Eqs. (14, 16, 17). The solid lines correspond to the parameters given in Table , obtained by fitting the $A_{12}$ asymmetry; the shaded area corresponds to the theoretical uncertainty on these parameters, as explained in the text. The dashed lines correspond to the parameters given in Table  obtained by fitting the $A_0$ asymmetry. The agreement between the results obtained from the two fits shows the consistency between the two sets of Belle data and the solidity of our analysis.

Figure 7

The transversity distribution functions for $u$ and $d$ quarks as determined through our global best fit. In the left panel, $x{\Delta }_{T}u(x)$ (upper plot) and $x{\Delta }_{T}d(x)$ (lower plot), see Eq. (5), are shown as functions of $x$ and $Q^2=2.4{\mathrm{GeV}}^2$. The Soffer bound [20] is also shown for comparison (bold line). In the right panel we present the unintegrated transversity distributions, $x{\Delta }_{T}u(x,k_{\perp })$ (upper plot) and $x{\Delta }_{T}d(x,k_{\perp })$ (lower plot), as defined in Eq. (13), as functions of $k_{\perp }$ at a fixed value of $x$. Notice that this $k_{\perp }$ dependence is not obtained from the fit, but it has been chosen to be the same as that of the unpolarized distribution functions: we plot it in order to show its uncertainty (shaded area), due to the uncertainty in the determination of the free parameters.

Figure 8

Favored and unfavored Collins fragmentation functions as determined through our global best fit. In the left panel we show the $z$ dependence of the $p_{\perp }$ integrated Collins functions defined in Eq. (38) and normalized to twice the corresponding unpolarized fragmentation functions; we compare them to the results of Refs. 24 (dashed line) and 25 (dotted line). In the right panel we show the $p_{\perp }$ dependence of the Collins functions defined in Eq. (14), at a fixed value of $z$. The $Q^2$ value is $2.4{\mathrm{GeV}}^2$, having assumed that the $Q^2$ evolution of ${\Delta }^{N}D$ is the same as that of $D$. The solid lines show the results based on the parameters of Table , while the dashed ones show the results corresponding to the parameters of Table . In all cases we also show the positivity bound (19) (upper lines).

Figure 9

Predictions for the single spin asymmetry $A_{UT}^{\sin ({\varphi }_S+{\varphi }_h)}$ as it will be measured by the COMPASS experiment operating with a transversely polarized hydrogen target. For the extra $\pi$ phase in the figure label see the caption of Fig. 5.

Figure 10

Predictions for the single spin asymmetry $A_{UT}^{\sin ({\varphi }_S+{\varphi }_h)}$ as it will be measured at JLab operating on polarized hydrogen (proton, upper plot) and ${\mathrm{He}}^3$ (neutron, lower plot) targets at a beam energy of 6 GeV.

Figure 11

Predictions for the single spin asymmetry $A_{UT}^{\sin ({\varphi }_S+{\varphi }_h)}$ as it will be measured at JLab operating on polarized hydrogen (proton, upper plot) and ${\mathrm{He}}^3$ (neutron, lower plot) targets at a beam energy of 12 GeV.

Figure 12

Our results, based on the extracted transversity and Collins functions, for the azimuthal asymmetry $A_{UT}^{\sin ({\varphi }_S+{\varphi }_h)}$ for $K^{\pm }$ production, compared with the HERMES experimental data [8, 9].