Lam-Tung relation breaking in $Z^0$ hadroproduction as a probe of parton transverse momentum

The Lam-Tung relation breaking coefficient $A_{\mathrm{LT}} = A_0 - A_2$ in the Drell-Yan dilepton angular distributions in the $Z^0$ boson mass region at the LHC is analyzed in the $k_T$-factorization approach. This observable was recently measured with high precision by ATLAS collaboration. Within the $k_T$-factorization approach we perform an approximate ${\cal O}(\alpha_{\mathrm{em}}\alpha_s^2)$ calculation of the off-shell parton hard matrix elements in which we include the leading tree level contributions of valence quarks and off-shell gluons: the $q_{\mathrm{val}} g^* \to qZ^0$ channel and the $g^*g^* \to q\bar q Z^0$ channel. The resulting $A_{\mathrm{LT}}$ exhibits high sensitivity to the gluon transverse momentum distribution (TMD). Several gluon TMDs are probed derived from the CCFM and BFKL evolution equations, and given by QCD-inspired phenomenological parameterizations. The ATLAS data favor a simple "Weizs\"acker-Williams" (WW) hard gluon TMD with the asymptotic behavior of one-gluon exchange at large gluon transverse momenta and moderate $x$. It is verified that the proposed approach with the WW gluon TMD describes well also the $A_0$ and $A_2$ angular coefficients at the $Z^0$ peak, as well as the Drell-Yan dilepton mass distribution at lower masses. We conclude that inclusion of gluon transverse momentum effects improves description of the angular distributions of Drell-Yan dileptons and that the Drell-Yan scattering provides an excellent probe of the parton TMDs.