Transverse momentum dependent gluon fragmentation functions from $J/\psi\ \pi$ production at $e^+ e^-$ colliders

The back-to-back $J/\psi$ and $\pi$ associated production at $e^+ e^-$ colliders is proposed to detect the gluon transverse momentum dependent(TMD) fragmentation functions. TMD factorization is assumed for this process. With spinless pion, unpolarized and linearly polarized gluon TMD fragmentation functions can be defined. It is found at parton level the hadronic tensor can be described by four structure functions. As a result, there are three independent angular distributions, of which a $\cos{2\phi}$ azimuthal asymmetry is sensitive to the linearly polarized gluon fragmentation function.