The Effects of Charged Charm Mesons on the Line Shapes of the X(3872)

The quantum numbers J^PC = 1++ of the X(3872) and the proximity of its mass to the D*0 anti-D0 threshold imply that it is either a loosely-bound hadronic molecule whose constituents are a superposition of D*0 anti-D0 and D0 anti-D*0 or it is a virtual state of charm mesons. The line shapes of the X(3872) can discriminate between these two possibilities. At energies within a few MeV of the D*0 anti-D0 threshold, the lines shapes of the X produced in B -> K transitions are determined by its binding energy and its width. Their normalizations are determined by a short-distance constant that is different for B+ -> K+ and B0 -> K0. At energies comparable to the 8 MeV splitting between the D*0 anti-D0 and D*+ D- thresholds, the charged meson channels D*+ D- and D+ D*- have a significant effect on the line shapes of the X. We calculate the line shapes taking into account the resonant coupling between the charged and neutral 1++ channels. The line shapes and their normalizations depend on one additional scattering parameter and two additional short-distance constants associated with the B -> K transitions. The line shapes of the X resonance depend on its decay channel; they are different for J/psi pi+ pi-, J/psi pi+ pi- pi0, and D0 anti-D0 pi0. The line shapes are also different for X produced in B+ decays and in B0 decays. Some conceptual errors in previous work on this problem are pointed out.Comment: 30 pages, 7 figure

Weakly-bound Hadronic Molecule near a 3-body Threshold

The X(3872) seems to be a loosely-bound hadronic molecule whose constituents are two charm mesons. A novel feature of this molecule is that the mass difference of the constituents is close to the mass of a lighter meson that can be exchanged between them, namely the pion. We analyze this feature in a simple model with spin-0 mesons only. Various observables are calculated to next-to-leading order in the interaction strength of the exchanged meson. Renormalization requires summing a geometric series of next-to-leading order corrections. The dependence of observables on the ultraviolet cutoff can be removed by renormalizations of the mass of the heaviest meson, the coupling constant for the contact interaction between the heavy mesons, and short-distance coefficients in the operator product expansion. The next-to-leading order correction has an unphysical infrared divergence at the threshold of the two heavier mesons that can be eliminated by a further resummation that takes into account the nonzero width of the heaviest meson.Comment: 37 pages, 13 figures, submitted to Phys.Rev.

An experimental study on learning about voting powers

We investigate experimentally whether subjects can learn, from their limited experiences, about relationships between the distribution of votes in a group and associated voting powers in weighted majority voting systems (WMV). Subjects are asked to play two-stage games repeatedly. In the second stage of the game, a group of four subjects bargains over how to divide fixed amount of resources among themselves through theWMV determined in the first stage. In the first stage, two out of four subjects in the group, independently and simultaneously, choose from two options that jointly determine the distribution of a given number of votes among four members. These two subjects face a 2 × 2 matrix that shows the distribution of votes, but not associated voting powers, among four members for each outcome. Therefore, to obtain higher rewards, subjects need to learn about the latter by actually playing the second stage. The matrix subjects face in the first stage changes during the experiment to test subjects' understanding of relationships between distribution of votes and voting power. The results of our experiments suggest that although (a) many subjects learn to choose, in the votes apportionment stage, the option associated with a higher voting power, (b) it is not easy for them to learn the underlying relationships between the two and correctly anticipate their voting powers when they face a new distribution of votes.experiment, learning, voting power, bargaining

An experimental study on learning about voting powers

We investigate experimentally whether subjects can learn, from their limited experiences, about relationships between the distribution of votes in a group and associated voting powers in weighted majority voting systems (WMV). Subjects are asked to play two-stage games repeatedly. In the second stage of the game, a group of four subjects bargains over how to divide fixed amount of resources among themselves through the WMV determined in the first stage. In the first stage, two out of four subjects in the group, independently and simultaneously, choose from two options that jointly determine the distribution of a given number of votes among four members. These two subjects face a 2 ~ 2 matrix that shows the distribution of votes, but not associated voting powers, among four members for each outcome. Therefore, to obtain higher rewards, subjects need to learn about the latter by actually playing the second stage. The matrix subjects face in the first stage changes during the experiment to test subjects' understanding of relationships between distribution of votes and voting power. The results of our experiments suggest that although (a) many subjects learn to choose, in the votes apportionment stage, the option associated with a higher voting power, (b) it is not easy for them to learn the underlying relationships between the two and correctly anticipate their voting powers when they face a new distribution of votes.