X(3872) and its production at hadron colliders

We evaluate the production cross sections of $X(3872)$ at the LHC and Tevatron at NLO in $\alpha_s$ in NRQCD by assuming that the short-distance production proceeds dominantly through its $\chi_{c1}'$ component in our $\chi_{c1}'\mbox{-}D^0\bar{D}^{*0}$ mixing model for $X(3872)$. The outcomes of the fits to the CMS $p_T$ distribution can well account for the recent ATLAS data in a much larger range of transverse momenta ($10~\mbox{GeV}<p_T<70~\mbox{GeV}$), and the CDF total cross section data, and are also consistent with the value of $k=Z_{c\bar c}\cdot Br(X\to J/\psi\pi^+\pi^-)$ constrained by the $B$-meson decay data. %It can also well describe the behavior of the CDF $\psi(2S)$ data, which show a strong %resemblance to that of the X(3872). For LHCb the predicted X(3872) total cross section is larger than the data by a factor of 2, which is due to the problem of the fixed-order NRQCD calculation that may not be applicable for the region with small $p_T$ ($p_T\sim 5 ~\mbox{GeV}$) and large forward rapidity $(2.5<y<4.5)$. In comparison, the prediction of molecule production mechanism for $X(3872)$ is inconsistent with both $p_T$ distributions and total cross sections of CMS and ATLAS, and the total cross section of CDF.