New findings in gluon TMD physics.

We revisit the model-independent decomposition of the gluon correlator, producing T even and T odd gluon transverse momentum distributions (TMDs), at leading twist-2. We propose an expansion of the gluon correlator, using a basis of four tensors (one antisymmetric and three symmetric), which are expressed through generators of the $U(2)$ group acting in the two-dimensional transverse plane. One can do clear interpretations of the two transversity T odd TMDs with linear polarization of gluons: symmetric and asymmetric under permutation of the transverse spin of nucleon and the transverse momentum of struck gluon. Using light-front wave function (LFWF) representation we also derive T even and T odd gluon TMDs in the nucleon at leading twist. The gluon-three quark Fock component in the nucleon is considered as bound state of struck gluon and three-quark core (spectator). The TMDs are constructed as factorized product of two LFWFs and gluonic matrix encoding information about both T even and T odd TMDs. In particular, T odd TMDs arise due to gluon re-scattering between the active gluon and three-quark spectator. Gluon re-scattering effects are parametrized by unknown scalar functions depending on the $x$ and ${\bf k}_\perp$ variables, which can be fixed from data. Our gluon TDMs obey the model-independent Mulders-Rodrigues inequalities. We also derive new sum rules (SRs) involving T even TMDs. One of the SRs states that the square of the unpolarized TMD is equal to a superposition of the squares of three polarized TMDs. Based on the SR derived for T even gluon TMDs we make a conjecture that there should two additional SRs involving T odd gluon TMDs, valid at orders $\alpha_s$ and $\alpha_s^2$. Then we check these SRs at small and large values of $x$. We think that our study could serve as useful input for future phenomenological studies of TMDs.