Baryon-baryon, meson-meson, and meson-baryon interactions in nonrelativistic QCD

TL;DR

This work analyzes hadron-hadron interactions in the heavy-quark limit of QCD using the pNRQCD framework. It proves, using all-orders structure and Gauss's law, that van der Waals forces between color-singlet hadrons vanish at leading order and remain too weak to bind in the NNLO regime for equal-mass heavy quarks, with higher-order effects unlikely to change this in perturbation theory. The authors perform GFMC calculations for four-, five-, and six-quark systems in SU(3), finding that exotic color configurations are energetically disfavored relative to color-singlet product states, thus ruling out equal-mass tetraquark, pentaquark, and hexaquark bound states at LO/NLO in this limit. They also discuss possible binding mechanisms beyond weakly-coupled pNRQCD, such as large mass hierarchies, relativistic light-quark effects, or nonperturbative dynamics, which could enable hadron-hadron bound states in realistic QCD or dark sectors.

Abstract

Van der Waals potentials describing interactions between color-singlet mesons and/or baryons vanish at leading order in potential nonrelativistic quantum chromodynamics (pNRQCD). This result and constraints from Gauss's law are used to prove that weakly-coupled pNRQCD van der Waals potentials in generic non-Abelian gauge theories with only heavy quarks are too weak to form bound states whose color state is a product of color-singlets. Quantum Monte Carlo calculations of four, five, and six quarks with equal masses provide numerical evidence that exotic color configurations are higher energy than products of color-singlet hadrons, suggesting that equal-mass fully-heavy tetraquark, pentaquark, and hexaquark bound states do not exist at next-to-leading order in pNRQCD and at all orders in QCD-like theories in which all quark masses are asymptotically large. Mechanisms for generating hadron-hadron bound states are identified, which necessarily involve large quark-mass hierarchies, relativistic effects arising from the presence of sufficiently light quarks, or nonperturbative effects outside the scope of weakly-coupled pNRQCD.

Figures (6)

• Figure 1: Effective energies for equal‑mass hexaquark trial states with imaginary-time evolution computed using GFMC in "atomic units" where the prefactor $(2/3)\alpha_s$ of the color-antisymmetric quark-quark potential is set to unity. Top: leading‑order (LO) evolution of the compact $\mathbf{1\otimes1}$ color‑singlet state, the elongated singlet ($b/a\gg1$), and the ${AAA}$, ${AAS}$, and ${SSS}$ color structures defined in the text. Bottom: next‑to‑leading‑order (NLO) evolution using $\alpha_s(\mu=m_b)=0.227325$ at $m_b=4.86831\;\mathrm{GeV}$. In both panels, the dashed horizontal line marks the open two‑baryon threshold.
• Figure 2: LO results for a subset of the trial states in \ref{['fig:hexa_equalmass_effmass']} compared to those including explicit quark antisymmetry, suitable for a product of two identical spin-1/2 (heavy mass) $(uds)$-flavored baryons.
• Figure 3: Top: effective energies for equal‑mass pentaquarks at LO in pNRQCD. Colors and markers distinguish the three trial wavefunctions discussed in the text: the compact color‑singlet configuration $\mathbf{1\otimes 1}$ (red circles), an elongated $\mathbf{1\otimes 1}$ state with aspect ratio $b/a\gg1$ (orange triangles), and the color‑octet configuration $\mathbf{8\otimes 8}$ (green squares). Bottom: analogous results at NLO, obtained with $m_Q = m_b$ and the renormalisation scale from Ref. Assi:2023cfo and $\mu_p$ defined in Eq. \ref{['eq:mup']}. In both panels, the dashed horizontal line marks the open meson‑baryon threshold.
• Figure 4: Feynman diagrams with two active quarks that lead to non-vanishing interactions between color-singlet meson-meson (left), meson-baryon (center), and baryon-baryon systems.
• Figure 5: Feynman diagrams with three active quarks that lead to non-vanishing interactions between color-singlet baryon-baryon systems.