Generalized disconnection exponents

We introduce and compute the generalized disconnection exponents $\eta_\kappa(\beta)$ which depend on $\kappa\in(0,4]$ and another real parameter $\beta$, extending the Brownian disconnection exponents (corresponding to $\kappa=8/3$) computed by Lawler, Schramm and Werner 2001 (conjectured by Duplantier and Kwon 1988). For $\kappa\in(8/3,4]$, the generalized disconnection exponents have a physical interpretation in terms of planar Brownian loop-soups with intensity $c\in (0,1]$, which allows us to obtain the first prediction of the dimension of multiple points on the cluster boundaries of these loop-soups. In particular, according to our prediction, the dimension of double points on the cluster boundaries is strictly positive for $c\in(0,1)$ and equal to zero for the critical intensity $c=1$, leading to an interesting open question of whether such points exist for the critical loop-soup. Our definition of the exponents is based on a certain general version of radial restriction measures that we construct and study. As an important tool, we introduce a new family of radial SLEs depending on $\kappa$ and two additional parameters $\mu, \nu$, that we call radial hypergeometric SLEs. This is a natural but substantial extension of the family of radial SLE$_\kappa(\rho)s$.Comment: 43 pages, 9 figures. Final version, to appear in Probab. Theory Relat. Fields. Contains a clarification about the terminology 'hypergeometric SLE' inappropriately used in other work

Is Zc(3900)Z_c(3900) a molecular state

Assuming the newly observed $Z_c(3900)$ to be a molecular state of $D\bar D^*(D^{*} \bar D)$, we calculate the partial widths of $Z_c(3900)\to J/\psi+\pi;\; \psi'+\pi;\; \eta_c+\rho$ and $D\bar D^*$ within the light front model (LFM). $Z_c(3900)\to J/\psi+\pi$ is the channel by which $Z_c(3900)$ was observed, our calculation indicates that it is indeed one of the dominant modes whose width can be in the range of a few MeV depending on the model parameters. Similar to $Z_b$ and $Z_b'$, Voloshin suggested that there should be a resonance $Z_c'$ at 4030 MeV which can be a molecular state of $D^*\bar D^*$. Then we go on calculating its decay rates to all the aforementioned final states and as well the $D^*\bar D^*$. It is found that if $Z_c(3900)$ is a molecular state of ${1\over\sqrt 2}(D\bar D^*+D^*\bar D)$, the partial width of $Z_c(3900)\to D\bar D^*$ is rather small, but the rate of $Z_c(3900)\to\psi(2s)\pi$ is even larger than $Z_c(3900)\to J/\psi\pi$. The implications are discussed and it is indicated that with the luminosity of BES and BELLE, the experiments may finally determine if $Z_c(3900)$ is a molecular state or a tetraquark.Comment: 17 pages, 6 figures, 3 table