Transverse momentum dependence of the quark helicity distributions and the Cahn effect in double-spin asymmetry $A_{LL}$ in semiinclusive DIS

Within the LO QCD parton model of Semi-Inclusive Deep Inelastic Scattering, $\ell N\to \ell hX$, with unintegrated quark distribution and fragmentation functions, we study the ${\mathit{P}}_{hT}$ dependence of the double longitudinal-spin asymmetry $A_{LL}$. We include $1/Q$ kinematic corrections, which induce an azimuthal modulation of the asymmetry, analogous to the Cahn effect in unpolarized SIDIS. We show that a study of $A_{LL}$ and of the weighted DSA $A_{LL}^{\cos {\varphi }_h}$ allows to extract the transverse momentum dependence of the unintegrated helicity distribution function $g_{1L}^q(x,k_{\perp })$ [or $\Delta q(x,k_{\perp })]$. Predictions, based on some models for the unknown functions, are given for ongoing COMPASS, HERMES and JLab experiments.

Figure 1

Predicted dependence of $A_{LL}$ on $P_{hT}$, for scattering off a proton target, with different choices of ${\mu }_2^2$: $0.1(\mathrm{GeV}/c{)}^2$-continuous, $0.17(\mathrm{GeV}/c{)}^2$-dashed and $0.25(\mathrm{GeV}/c{)}^2$-dot-dashed lines.

Figure 2

Predicted dependence of $A_{LL}$ on $P_{hT}$, for scattering off a deuteron (and neutron for JLab) target, with different choices of ${\mu }_2^2$: $0.1(\mathrm{GeV}/c{)}^2$-continuous, $0.17(\mathrm{GeV}/c{)}^2$-dashed and $0.25(\mathrm{GeV}/c{)}^2$-dot-dashed lines.

Figure 3

Predicted dependence of $A_{LL}^{\cos {\varphi }_h}$ on $P_{hT}$ for scattering off a proton target with different choices of ${\mu }_2^2$: $0.1(\mathrm{GeV}/c{)}^2$-continuous, $0.17(\mathrm{GeV}/c{)}^2$-dashed and $0.25(\mathrm{GeV}/c{)}^2$-dot-dashed lines. Each line splits into three almost overlapping lines corresponding, for each value of ${\mu }_2^2$, to three different values of ${\mu }_1^2=(\mathrm{\text{up-down}})$ 0.1, 0.15 and $0.2(\mathrm{GeV}/c{)}^2$.

Figure 4

Predicted dependence of $A_{LL}^{\cos {\varphi }_h}$ on $P_{hT}$ for scattering off a deuteron (and neutron for JLab) target with different choices of ${\mu }_2^2$: $0.1(\mathrm{GeV}/c{)}^2$-continuous, $0.17(\mathrm{GeV}/c{)}^2$-dashed and $0.25(\mathrm{GeV}/c{)}^2$-dot-dashed lines. Each line splits into three almost overlapping lines corresponding, for each value of ${\mu }_2^2$, to three different values of ${\mu }_1^2=(\mathrm{\text{up-down}})$ 0.1, 0.15 and $0.2(\mathrm{GeV}/c{)}^2$.

Figure 5

Predicted dependence of $A_{LL}^{\cos {\varphi }_h}$ on $x$, $y$ and $z$, for proton and deuteron targets, for COMPASS.

Figure 6

Predicted dependence of $A_{LL}^{\cos {\varphi }_h}$ on $x$, $y$ and $z$, for proton and deuteron targets, for HERMES.

Figure 7

Predicted dependence of $A_{LL}^{\cos {\varphi }_h}$ on $x$, $y$ and $z$, for proton, neutron and deuteron targets, for JLab.