Abstract
If known, the spectrum of heavy-hadron molecules will be a key tool to disentangle the nature of the exotic states that are being discovered in experiments. Here we argue that the general features of the molecular spectrum can be deduced from the idea that the short-range interaction between the heavy hadrons is effectively described by scalar and vector meson exchange, i.e. the $\sigma$, $\omega$ and $\rho$ mesons. By means of a contact-range theory where the couplings are saturated by the aforementioned light mesons we are indeed able to postdict the $X(3872)$ (as a $D^* \bar{D}$ molecule) from the three $P_c(4312)$, $P_c(4440)$ and $P_c(4457)$ pentaquarks (as $\bar{D}\Sigma_c$ and $\bar{D}^* \Sigma_c$ molecules). We predict a $J^{PC} = 1^{--}$ $D \bar{D}_1$ molecule at $4240-4260\,{\rm MeV}$ which might support the hypothesis that the $Y(4260)$ is at least partly molecular. The extension of these ideas to the light baryons requires minor modifications, after which we recover approximate SU(4)-Wigner symmetry in the two-nucleon system and approximately reproduce the masses of the deuteron and the virtual state.
Highlights
We argue that the general features of the molecular spectrum can be deduced from the idea that the short-range interaction between the heavy hadrons is effectively described by scalar- and vector-meson exchange, i.e., the σ, ρ, and ω mesons
For effectively combining the contribution from the saturation of scalarand vector-meson exchange, which happen at a different renormalization scale as the masses of these light mesons are different, we will follow a renormalization group equation (RGE)
We do not know the exact form of the short-distance wave function, but owing to the large mass of the heavy hadrons, it is sensible to assume that the semiclassical approximation applies
Summary
Theoretical predictions of the hadronic spectrum are fundamental for testing our understanding of strong interactions against experiments. The idea is as follows: first, we will describe the interaction between two heavy hadrons in terms of a contact-range potential. We will assume that the couplings in the contact-range potential are saturated by light-meson exchange (i.e., σ, ρ, and ω) within the saturation procedure of Ref. For effectively combining the contribution from the saturation of scalarand vector-meson exchange, which happen at a different renormalization scale as the masses of these light mesons are different, we will follow a renormalization group equation (RGE). This RGE will tell us what is the importance of scalar- and vector-meson contributions to saturation relative to each other. We can derive the predictions of this procedure for other molecular states
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