Quark helicity distributions in transverse momentum space and transverse coordinate space

The transverse momentum dependent helicity distributions of valence quarks are calculated in the light-cone diquark model by adopting two different approaches. We use the model results to analyze the $P_{h\perp }$-dependent double spin asymmetries for ${\pi }^{+}$, ${\pi }^{-}$ and ${\pi }^0$ productions in semi-inclusive deep inelastic scattering, and find that the asymmetries agree with the CLAS data in one of the approaches. By taking the Fourier transform of the transverse momentum dependent helicity distributions, we obtain the helicity distributions of valence quarks in the transverse coordinate space, and then apply them further to predict the Bessel-weighted double spin asymmetries of ${\pi }^{+}$, ${\pi }^{-}$ and ${\pi }^0$ productions in semi-inclusive deep inelastic scattering at CLAS, COMPASS and HERMES for the first time. The shape of the Bessel-weighted double spin asymmetry thereby provides a direct probe on the transverse structure of longitudinally polarized quarks.