Existence and spectrum of fully-heavy pentaquarks $QQQQ\bar{Q}$: a rigorous five-body determination of stable bound states

Fully-heavy pentaquarks are hypothetical five-quark states $QQQQ\bar{Q}'$ whose valence constituents are all charm or bottom quarks. A constituent-quark-model study (An, Luo, Liu, and Liu, Phys. Rev. D 105, 074032 (2022)) found no stable fully-heavy pentaquarks, but its verdict rested on a minimal single-Gaussian variational basis and an approximate color-spin treatment, and the paper itself notes that the bound states claimed by channel-coupling quark-model calculations "still need to be further confirmed by solving accurately the five-body problem." The subsequent literature: correlated-Gaussian, diffusion Monte Carlo, complex-scaling, and neural-network calculations have largely confirmed the absence of compact bound states within nonrelativistic potential models, yet channel-coupling studies still find weakly bound molecular candidates, QCD sum rules give conflicting masses, no first-principles lattice QCD calculation exists, and no experimental search has been published. The problem of determining whether the fully-heavy pentaquark systems contain stable bound states — and their masses and binding energies relative to the relevant baryon+meson thresholds — remains open.

Background

Quantum chromodynamics (QCD) confines quarks into color-singlet hadrons: mesons ($q\bar q$) and baryons ($qqq$) in the conventional quark model, but QCD also admits exotic combinations such as tetraquarks ($QQ\bar Q\bar Q$) and pentaquarks (four quarks plus one antiquark, $QQQQ\bar Q$). In 2020 the LHCb collaboration observed a structure, called $X(6900)$, in the invariant mass of $J/\psi J/\psi$ pairs, providing the first evidence for hadrons made purely of charm quarks and triggering systematic study of 'all-heavy' multiquark systems. The pentaquark analogue is the fully heavy pentaquark $QQQQ\bar Q'$, where every valence constituent is a heavy quark ($c$ or $b$) or heavy antiquark; ten flavor systems exist, from $cccc\bar c$ to $bbbb\bar b$ and the mixed $cccb\bar c$, $ccbb\bar b$, etc. Because such states contain no light quarks, pion-exchange dynamics are absent, so any binding must come from the fundamental color (strong) interaction, making them exceptionally clean probes of multiquark confinement.

The workhorse theoretical tool is the constituent quark model: a nonrelativistic potential model in which quarks of effective mass $m_c\approx 1.5$–$1.8$ GeV and $m_b\approx 4.7$–$5.2$ GeV interact through a confining potential plus one-gluon exchange (a color-Coulomb and hyperfine spin-spin interaction), with parameters fitted to conventional heavy quarkonia and baryons. The five-body Schrödinger equation is solved variationally: a trial spatial wave function is assumed, and the energy expectation value is minimized. A state is judged bound or unbound by its binding energy relative to the lowest baryon+meson threshold into which it can fall apart (for example $\eta_c+\Omega_{ccc}$ for $cccc\bar c$, where $\Omega_{ccc}$ is the triply-charm baryon): a mass below threshold means the state is stable against strong decay; above threshold it decays. Related frameworks include the chromomagnetic interaction (CMI) model, which parameterizes the hyperfine color-spin contact interaction to estimate masses from splittings; QCD sum rules, which extract masses from correlators of interpolating currents via quark–hadron duality; and lattice QCD, the first-principles numerical evaluation of the QCD path integral.

An, Luo, Liu, and Liu (Phys. Rev. D 105, 074032 (2022)) systematically computed the S-wave spectrum of all ten fully-heavy pentaquark systems by the variational method, using a single-Gaussian spatial trial function, evaluating matrix elements approximately with the dominant color-spin basis component (amplitude above 90%), and concluded that all lowest states lie well above their baryon-meson thresholds — no stable fully-heavy pentaquark exists, so any such state would decay strongly into a heavy baryon plus a heavy meson. The study's own comparison with the chromomagnetic-interaction model shows that the effective quark masses extracted from the constituent model are systematically larger than the CMI values, indicating environment-dependent effective masses as hadron multipartonicity grows. The question is live because, as the paper notes, other approaches — QCD sum rules, and chiral quark-model calculations with channel coupling — had predicted bound fully-heavy pentaquark states that "still need to be further confirmed by solving accurately the five-body problem," so the field faces a genuine disagreement rather than a settled negative.

Problem Statement

Determine whether the fully-heavy pentaquark systems $QQQQ\bar{Q}'$ — all ten flavor combinations of charm and bottom quarks with $Q'\in\{c,b\}$, namely $cccc\bar c$, $cccc\bar b$, $bbbb\bar b$, $bbbb\bar c$, $cccb\bar c$, $cccb\bar b$, $bbbc\bar b$, $bbbc\bar c$, $ccbb\bar c$, and $ccbb\bar b$ — contain, in their S-wave (orbital ground-state) five-quark configurations, any stable bound states, i.e. states whose mass lies below the lowest baryon+meson threshold with the same quantum numbers.

(1) For each system and each spin-parity $J^P$ allowed by the S-wave (orbital ground-state) five-quark configuration, solve the five-body Schrödinger problem with a treatment whose conclusion is converged rather than an artifact of the trial space or of a single dominant color-spin component.

(2) State, for each system and each S-wave-allowed $J^P$, the mass and the binding energy $B_T = M_{\rm pentaquark} - M_{\rm baryon} - M_{\rm meson}$ relative to the lowest baryon+meson threshold with the same quantum numbers as the pentaquark, together with the dominant fall-apart decay channels where these are determined.

References

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