On the precise measurement of the $X(3872)$ mass and its counting rate

Abstract

The lineshapes of specific production experiments of the exotic state such as $X(3872)$ with $J^{PC}=1^{++}$ quantum numbers involving triangle singularities have been found to become highly sensitive to the binding energy of weakly bound states, thus offering in principle the opportunity of benchmark determinations. We critically analyze recent proposals to extract accurately and precisely the $X(3872)$ mass, which overlook an important physical effect by regarding their corresponding production lineshapes as a sharp mass distribution and, thus, neglecting the influence of initial nearby continuum states in the $1^{++}$ channel. The inclusion of these states implies an effective cancellation mechanism which operates at the current and finite experimental resolution of the detectors so that one cannot distinguish between the $1^{++}$ bound-state and nearby $D \bar D^*$ continuum states with the same quantum numbers. In particular, we show that the lineshape for resolutions above 1 MeV becomes rather insensitive to the binding energy unless high statistics is considered. The very existence of the observed bumps is a mere consequence of short distance correlated $\bar D D^*$ pairs, bound or unbound. The cancellation also provides a natural explanation for a recent study reporting missing but unknown decay channels in an absolute branching ratio global analysis of the $X(3872)$.