Weighted sin ⁡ ( 3 ϕ h − ϕ S ) asymmetry from pretzelosity in SIDIS at electron ion colliders

We investigate the transverse momentum weighted azimuthal asymmetry with a $\sin (3{\varphi }_h-{\varphi }_S)$ modulation in electroproduction of charged pion production off the transversely polarized proton using the weighted function as $W=Q_T^3\sin (3{\varphi }_h-{\varphi }_S)/\left(6M^2{M}_h\right)$, with $Q_T=\left|P_{hT}\right|/z$, and $P_{hT}$ the transverse momentum of the produced pion. The weighted asymmetry is contributed by the product of the second transverse-moment of the pretzelosity distribution function and the first transverse-moment of the Collins function. Using the extracted result for the proton pretzelosity and the available parametrization for the pion Collins function, we numerically predict the weighted $\sin (3{\varphi }_h-{\varphi }_S)$ asymmetry for charged pion production at electron ion colliders (EICs). One can find that the weighted asymmetries for the ${\pi }^{+}$ production in SIDIS process are positive, while those for the ${\pi }^{-}$ production are negative in the kinematics regions of EICs. We emphasize the importance of the planned high luminosity EICs in the aspect of constraining pretzelosity in the near future.