New insights into the two-decades old question from lattice QCD
Most of the mass in ordinary matter consists of hadrons: bound states of the fundamental particles quarks and gluons. Commonly-known examples of hadrons are the protons and neutrons (which are 3-quark states) in atomic nuclei, and the pions (which are quark-antiquark states, called mesons). There are also many other types of hadrons that do not play a role in atomic nuclei, but can be produced in particle colliders; some of these may be exotic and contain four or five quarks+antiquarks.
Quarks come in six “flavors”: the light quarks named up and down, the medium-mass strange quark, heavy quarks named charm and bottom, and the very heavy top quark. Quarks are Fermions like electrons, but in addition to carrying an electric charge, quarks also carry a different type of charge called “color.” Quarks come in “red”, “green”, and “blue” varieties. Particles with color charge interact with eight types of gluons that mediate the strong force. Gluons are analogous to photons, except that they interact among themselves. Hadrons are held together by gluon exchange between quarks, analogous to how photon exchange binds electrons to the nuclei of atoms. Quantum chromodynamics (QCD) is the theory of quarks and gluons.
The self-interaction of the gluons has the effect that quarks are not detected as isolated particles, but are confined within hadrons. If one tries to pull apart the quark and antiquark inside a meson, one must put so much energy into the system that new quarks will pop out of the vacuum and one ends up with two mesons. See this animation of the “color-confinement” feature of QCD.
Throughout the 20th century a simple quark-antiquark picture explained observed mesons well. Since 2003, however, mesons have been observed which do not fit this picture, and the charm-strange Ds0*(2317)± was among the first. This meson is much lighter and more long-lived than was expected. These features suggest that it is a ”DK molecule“ - a charm-light meson called D and a strange-light meson called kaon (K) loosely bound together, as shown on the right side of the picture.
Many particle-physics calculations use “perturbation theory”, where all interactions are treated as small corrections in an expansion about freely propagating particles—but the relevant corrections from the strong force are not small. A first-principles, nonperturbative approach is lattice QCD, where the QCD equations are solved directly in a “universe in a box” by statistical sampling on a discrete spacetime grid.
In a new paper, graduate student Forrest Guyton and professor Stefan Meinel study the lowest-lying heavy-strange mesons in lattice QCD. Using “universes” with different grid spacings, box sizes, and quark masses, results are extrapolated into the real universe. The way energy levels depend on box size tells us about scattering of a negative-parity charm-light or bottom-light meson (H = D(*), B(*)) with a negative-parity kaon in infinite volume. The features of this scattering tell how likely the particles will “stick together” to form a molecular positive-parity meson, as well as the mass of that meson in infinite volume.
Figure 1: Scattering a D meson and kaon K (left). Bound state “molecular” Ds∗0(2317)± (right).
Guyton and Meinel use a precise measure of “molecularity” that was introduced by Steven Weinberg. Imagine turning off the interactions between the positive-parity meson and the HK-scattering state. In that case, the positive-parity meson is only a compact quark-antiquark state. Such a “bare” state is an intuitively nice mathematical fiction. In the real world, a physical (“renormalized”) particle such as the Ds0*(2317)± is “dressed” by its interactions. Every quantum process where the Ds0*(2317)± can disintegrate into D and K, which then annihilate each other to turn back into the Ds0*(2317)± is included. The amount of bare state left inside the renormalized state is called the “field-strength renormalization” Z, which is determined from the scattering behavior. Z = 1 indicates a pure quark-antiquark state; Z = 0 indicates a pure molecule. The calculations of Guyton and Meinel show that all positive-parity heavy-strange mesons have some degree of molecular structure and are inconsistent with pure quark-antiquark states.
The authors also computed the “decay constants” of the heavy-strange mesons. Decay constants are sensitive to the meson’s structure—in a conventional quark-antiquark picture, and in the non-relativistic limit, they would be related to the probability for the heavy quark and the strange antiquark to be found on top of each other. This is the first time that the decay constants have been calculated on the lattice for the bottom-strange positive-parity mesons. The decay constants for the positive-parity charm-strange mesons are subject to much intrigue, as previous works have found a large difference between the values for the spin-0 Ds0*(2317)± and its spin-1 partner Ds1(2460)±, which violates expectations from leading-order heavy quark effective theory (HQET). In that approximation, the chromo-magnetic moment of the heavy quark vanishes, and the decay constants become independent of the meson spin. The calculation by Guyton and Meinel confirms large violations for the charm-meson mesons, showing that relativistic effects are important there, but the expectations from leading-order HQET hold in the bottom-strange case. The decay constants may also be used to improve predictions for rare decays involving the bottom quark → strange quark transition. Some measurements of such decay rates are in tension with Standard-Model predictions. More precise theory predictions can determine if this tension points towards undiscovered kinds of fundamental physics.
The lightest positive-parity bottom-strange mesons had not yet been observed experimentally when Guyton and Meinel finished their calculations. However, in August 2026, the LHCb experiment announced the discovery of a new bottom-strange particle with a mass consistent with the Guyton’s and Meinel’s prediction for the ground-state spin-0 positive-parity bottom-strange meson, the Bs0*.