Possible precise measurements of the X(3872) mass with the e+e- → π0γX(3872) and p<(p)over bar> → γX(3872) reactions

It was recently proposed that the X(3872) binding energy, the difference between the D-0(D) over bar*(0) threshold and the X(3872) mass, can be precisely determined by measuring the gamma X(3872) line shape from a short-distance D*(0)(D) over bar*(0) source produced at high-energy experiments. Here, we investigate the feasibility of such a proposal by estimating the cross sections for the e+e- -> pi(0)gamma X(3872) and p (p) over bar -> gamma X(3872) processes considering the D*(0)*(D) over bar*D-0(0)/(D) over bar*D-0*(0)(D) over bar (0) triangle loops. These loops can produce a triangle singularity slightly above the D*(0)(D) over bar*(0) threshold. It is found that the peak structures originating from the D*(0)(D) over bar*(0) threshold cusp and the triangle singularity are not altered much by the energy dependence introduced by the e+e- -> pi D-0*(0)(D) over bar *0 and p (p) over bar -> (D) over bar*D-0*0 production parts or by considering a finite width for the X(3872). We find that sigma(e(+)e(-) -> pi(0)gamma X(3872)) x Br(X(3872) -> pi(+)pi-J/psi ) is O(0.1 fb) with the gamma X(3872) invariant mass integrated from 4.01 to 4.02 GeV and the c.m. energy of the e(+)e(-) pair fixed at 4.23 GeV. The cross section sigma(p (p) over bar -> gamma X(3872)) x Br(X(3872) -> pi(+)pi(-) J/psi) is estimated to be of O(10 pb). Our results suggest that a precise measurement of the X(3872) binding energy can be done at PANDA.