T-odd Parton Distribution Functions and Azimuthal Anisotropy at High Transverse Momentum in $p$-$p$ and $p$-$A$ Collisions

Various azimuthal anisotropies ($v_1, v_2, v_3, v_4$), at high transverse momentum (high-$p_T$), are shown to arise from the asymmetric scattering of transverse polarized quarks and gluons, arising from unpolarized nucleons (the Boer-Mulders' effect) and resulting in unpolarized hadrons (the Collins effect). Combined with the asymmetric scattering of partons from polarization independent but transverse momentum dependent (TMD) distributions, we obtain a possible mechanism to understand the azimuthal anisotropy of hadrons at large transverse momentum observed in $p$-$p$ collisions. Constraining the ratio of polarization dependent TMD distributions to polarization independent distributions by comparing with the data from $p$-$p$ collisions, we find that scaling the acquired transverse momentum of the initial state partons from the proton, due to prescattering before the hard interaction, with the length of the nucleus ($k_\perp^2 \propto A^{1/3}$), straightforwardly yields the azimuthal anisotropy at high $p_T$ in $p$-$A$ collisions.