Final state interactions and the extraction of neutron single spin asymmetries from semi-inclusive deep-inelastic scattering by a transversely polarized ${}^3\mathrm{He}$ target

The semi-inclusive deep-inelastic electron scattering off transversely polarized ${}^3\mathrm{He}$, i.e., the process $e+\overrightarrow{{}^3\mathrm{He}}\to e^{\prime }+h+X$, with $h$ being a detected fast hadron, is studied beyond the plane-wave impulse approximation. To this end, a distorted spin-dependent spectral function of a nucleon inside an $A=3$ nucleus is actually evaluated through a generalized eikonal approximation, in order to take into account the final state interactions between the hadronizing system and the $\left(A-1\right)$ nucleon spectator one. Our realistic description of both nuclear target and final state is a substantial step forward for achieving a reliable extraction of the Sivers and Collins single spin asymmetries of the free neutron. To illustrate how and to what extent the model dependence due to the treatment of the nuclear effects is under control, we apply our approach to the extraction procedure of the neutron single spin asymmetries from those measured for ${}^3\mathrm{He}$ for values of the kinematical variables relevant both for forthcoming experiments at Jefferson Laboratory and, with an exploratory purpose, for the future Electron Ion Collider.

Figure 1

The SIDIS process $A(e,e^{\prime }h)X$, with final state interactions taken into account.

Figure 2

The ${}^3\mathrm{He}$ spectral function, for the neutron, in the unpolarized case, as a function of $p_{\mathrm{mis}}=\left|{\mathbf{p}}_{\mathrm{mis}}\right|$ and of the removal energy $E$, in PWIA (full lines) and with FSI taken into account within GEA framework (dotted lines), The kinematical ranges of $p_{\mathrm{mis}}$ and $E$ correspond to the ones relevant for the calculation of the unpolarized light-cone distribution for $\alpha =1,\mathcal{E}=11\mathrm{GeV}$ [cf. Eq. (71)], $x_{Bj}=0.48$, and $Q^2=7.6{\mathrm{GeV}}^2$.

Figure 3

The ratio between the unpolarized neutron spectral function with FSI interactions and the corresponding quantity in PWIA that are shown in Fig. 2.

Figure 4

Dependence on $\alpha$ of the integration limits $p_{min,max}=\left|{\mathbf{p}}_{min,max}\right|$ in Eqs. (69) and (71) for the 3bbu channel and two values of the removal energy $E$, in the kinematics of the forthcoming JLab experiments (corresponding to an initial electron energy $\mathcal{E}=8.8\mathrm{GeV})$.

Figure 5

The PWIA distribution functions $f_N^{\perp }(\alpha ,Q^2,\mathcal{E})$, Eq. (68), and $f_N(\alpha ,Q^2,\mathcal{E})$, Eq. (70), for the neutron (left panel) and the proton (right panel) at $\mathcal{E}=8.8\mathrm{GeV}$, and $Q^2=5.73{(\mathrm{GeV}/\mathrm{c})}^2$. For the polarized proton, in the 2bbu and 3bbu channels, these distributions are almost equal and opposite in sign, resulting in a very small total distribution.

Figure 6

The functions $f_N(\alpha =0.65,Q^2,\mathcal{E},E)$ and $f_N^{\perp }(\alpha =0.65,Q^2,\mathcal{E},E)$, Eqs. (71) and (69) respectively, evaluated in PWIA, for the following kinematics: (1) $\mathcal{E}=11\mathrm{GeV}$ and $Q^2=7.58{(\mathrm{GeV}/\mathrm{c})}^2$ (dashed lines); (2) $\mathcal{E}=8.8\mathrm{GeV}$, and $Q^2=5.73{(\mathrm{GeV}/\mathrm{c})}^2$ (solid lines). (a) The proton and neutron functions $f_N(\alpha =0.65,Q^2,\mathcal{E},E)$ in an unpolarized ${}^3\mathrm{He}$. (b) The functions $f_N^{\perp }(\alpha =0.65,Q^2,\mathcal{E},E)$ for a transversely polarized neutron in a transversely polarized ${}^3\mathrm{He}$. For a transversely polarized proton, the corresponding function, very small, is not shown.

Figure 7

The neutron Sivers (left panel) and Collins (right panel) asymmetries for the JLab kinematics at an initial electron energy of $\mathcal{E}=8.8\mathrm{GeV}$. Full line, the model for the neutron asymmetry used in the calculation; dashed line, the neutron asymmetry extracted from the PWIA calculation using Eq. (82). Calculations have been performed at $Q^2=5.73{(\mathrm{GeV}/\mathrm{c})}^2$, i.e., the central $Q^2$ value for an energy beam $\mathcal{E}=8.8\mathrm{GeV}$ (see text).

Figure 8

The neutron unpolarized (a) and transversely polarized (b) distributions, Eqs. (70) and (68). Solid lines are PWIA results, while dashed lines include effects of FSI. JLab kinematics has been assumed, i.e., the initial electron energy is $\mathcal{E}=8.8\mathrm{GeV}$ and $Q^2=5.73{(\mathrm{GeV}/\mathrm{c})}^2$, which is the central $Q^2$ value for an energy beam $\mathcal{E}=8.8\mathrm{GeV}$ (see text).

Figure 9

The same as in Fig. 8, but for the proton distributions.

Figure 10

The ratio of the light-cone spin-independent momentum distribution evaluated taking into account FSI to the corresponding quantity obtained in PWIA. The ratio is shown in the neutron case, for $\alpha =1$, namely the value where the distributions reach their maximum value, as a function of the momentum transfer, $Q^2$, corresponding to four different kinematical conditions: the ones with $Q^2<9{\left(\mathrm{GeV}/\mathrm{c}\right)}^2$ have been evaluated by using JLab kinematical conditions, while the rightmost diamonds are appropriate for EIC kinematics (see text).

Figure 11

Left panel: The ${}^3\mathrm{He}$ Sivers asymmetry [see Eqs. (67) and (73)], evaluated taking into account FSI effects (full line) and in PWIA (dashed line). Right panel: the same, but for the ${}^3\mathrm{He}$ Collins asymmetry [see Eqs. (67) and (72)]. Calculations have been performed at $Q^2=5.73{(\mathrm{GeV}/\mathrm{c})}^2$, i.e., the central $Q^2$ value for an energy beam $\mathcal{E}=8.8\mathrm{GeV}$ (see text).

Figure 12

The neutron Sivers (left panel) and Collins (right panel) asymmetries for the JLab kinematics at an initial electron energy of $\mathcal{E}=8.8\mathrm{GeV}$. Full line, the model for the neutron asymmetry used in the calculation; dot-dashed line, the neutron asymmetry extracted from the full calculation of $A_3^j$ with FSI taken into account, using the extraction formula Eq. (84); dashed line, the result obtained using Eq. (84) to extract the neutron asymmetries from PWIA results. Calculations have been performed at $Q^2=5.73{(\mathrm{GeV}/\mathrm{c})}^2$, i.e., the central $Q^2$ value for an energy beam $\mathcal{E}=8.8\mathrm{GeV}$ (see text).

Figure 13

The neutron Sivers (left panel) and Collins (right panel) asymmetries, for the JLab kinematics at an initial electron energy of $\mathcal{E}=8.8\mathrm{GeV}$. Full line, the model for the neutron asymmetry used in the calculation; dot-dashed line, the neutron asymmetry extracted from the full calculation of $A_3^j$ with FSI taken into account, using the extraction formula Eq. (84); dashed line, the result obtained using Eq. (85) to extract the neutron asymmetries from the same calculation. Calculations have been performed at $Q^2=5.73{(\mathrm{GeV}/\mathrm{c})}^2$, i.e., the central $Q^2$ value for an energy beam $\mathcal{E}=8.8\mathrm{GeV}$ (see text).