How to measure the linear polarization of gluons in unpolarized proton using the heavy-quark pair leptoproduction

We study the azimuthal $\cos \varphi$ and $\cos 2\varphi$ asymmetries in heavy-quark pair leptoproduction, $lN\rightarrow l^{\prime}Q\bar{Q}X$, as probes of linearly polarized gluons inside unpolarized proton, where the azimuth $\varphi$ is the angle between the lepton scattering plane $(l,l^{\prime})$ and the heavy quark production plane $(N,Q)$. First, we determine the maximal values for the $\cos \varphi$ and $\cos 2\varphi$ asymmetries allowed by the photon-gluon fusion with unpolarized gluons; these predictions are large, $(\sqrt{3}-1)/2$ and $1/3$, respectively. Then we calculate the contribution of the transverse-momentum dependent gluonic counterpart of the Boer-Mulders function, $h_{1}^{\perp g}$, describing the linear polarization of gluons inside unpolarized proton. Our analysis shows that the maximum values of the azimuthal distributions depend strongly on the gluon polarization; they vary from 0 to 1 depending on $h_{1}^{\perp g}$. We conclude that the azimuthal $\cos \varphi$ and $\cos 2\varphi$ asymmetries in heavy-quark pair leptoproduction are predicted to be large and very sensitive to the contribution of linearly polarized gluons. For this reason, future measurements of the azimuthal distributions in charm and bottom production at the proposed EIC and LHeC colliders seem to be very promising for determination of the linear polarization of gluons inside unpolarized proton.