Unravelling Pentaquarks with Born--Oppenheimer effective theory

Abhishek Mohapatra; Antonio Vairo; Nora Brambilla

REVIEW 4 major objections 4 minor 1 cited by

Unravelling Pentaquarks with Born--Oppenheimer effective theory

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The four observed hidden-charm pentaquark states are bound states of QCD's lowest Born–Oppenheimer static potentials, with quantum numbers fixed by spin-dependent corrections and adjoint-baryon masses predicted for lattice QCD.

desk verdict A serious BOEFT analysis that gives testable quantum-number assignments, but its central identification rests on an unvalidated potential model—worth refereeing, not worth betting on. read the letter →

arxiv 2508.13050 v1 pith:SD6L76XX submitted 2025-08-18 hep-ph hep-exhep-lat

Nora Brambilla , Abhishek Mohapatra , Antonio Vairo This is my paper · ORCID

classification hep-phhep-exhep-lat

keywords pentaquarksBorn–Oppenheimereffectivefieldtheorystaticpotentialsadjointbaryonsspin-dependentcorrectionsquantumnumberscharmbottom

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to show that the four observed hidden-charm pentaquark states—$P_c(4312)^+$, $P_c(4380)^+$, $P_c(4440)^+$, and $P_c(4457)^+$—are the lowest bound states of QCD's Born–Oppenheimer static potentials, not objects requiring a separate molecular or compact-quark model. In the Born–Oppenheimer effective field theory, the leading potentials at short distance are a repulsive color-octet piece plus the mass of an 'adjoint baryon' (three light quarks in the color adjoint); at long distance they approach the $\Sigma_c\bar D$ threshold, and bound states form where a potential dips below that threshold. First-order $\mathcal O(1/m_Q)$ spin corrections split the degenerate multiplets; two scenarios reproduce the four masses, and the semi-inclusive $J/\psi$ and $\eta_c$ widths favor scenario 2. If scenario 2 is right, the quantum numbers are $(1/2)^-$ for $P_c(4312)^+$ and $P_c(4457)^+$ and $(3/2)^-$ for $P_c(4380)^+$ and $P_c(4440)^+$, the adjoint-baryon masses are $\Lambda_{(1/2)^+,RS}=1.125$ GeV and $\Lambda_{(3/2)^+,RS}=1.152$ GeV, and three further charm states plus a bottom multiplet are predicted. The payoff is that the framework is grounded in QCD rather than an assumed internal quark arrangement, and it makes concrete lattice and decay predictions that can be checked.

What carries the argument

The central objects are the pentaquark Born–Oppenheimer potentials $E_{(1/2)_g}(r)$, $E_{(1/2)'_g}(r)$, and $E_{(3/2)_g}(r)$, modeled in Eq. (2.11) by a renormalon-subtracted color-octet potential plus an adjoint-baryon mass and an $A r^2$ term at short distances, and a one-pion-exchange tail $F e^{-r/d}/r$ at long distances, matched by continuity. The coupled radial Schrödinger equations (3.6) and (3.7) for the $k^P=(1/2)^+$ and $(3/2)^+$ multiplets carry the dynamics; the $3\times3$ spin mass matrices (3.14) and (3.15), built from the $\mathcal O(1/m_Q)$ potential $V_{SS}$ of Eq. (3.8), generate the splittings and the physical-state superpositions. The same transition amplitude cancels in

What would settle it

Compute on the lattice the two adjoint-baryon masses $\Lambda_{(1/2)^+}$ and $\Lambda_{(3/2)^+}$ and the pentaquark static potentials. If the masses differ from 1.125 GeV and 1.152 GeV beyond the fit uncertainties, or if the potentials $E_{(1/2)_g}$, $E_{(1/2)'_g}$, $E_{(3/2)_g}$ do not cross below the $\Sigma_c\bar D$ threshold, the identification fails; alternatively, measuring $\Gamma(P_c(4312)^+\to\eta_c(1S)+X)/\Gamma(P_c(4312)^+\to J/\psi+X)$ near 5 (scenario 2) versus about 0.23 (scenario 1) would discriminate the scenarios.

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Extended reading notes

Core claim

The paper claims the four observed charm pentaquarks are the ground-state multiplet of the QCD Born–Oppenheimer potentials $E_{(1/2)_g}$, $E_{(1/2)'_g}$, $E_{(3/2)_g}$. These start as a repulsive octet Coulomb potential plus adjoint-baryon mass and asymptotically reach the $\Sigma_c\bar D$ threshold; solving the coupled Schrödinger equations reproduces the four masses. With $\mathcal O(1/m_Q)$ spin corrections, the preferred scenario gives $J^P=(1/2)^-$ for $P_c(4312)^+$ and $P_c(4457)^+$, $(3/2)^-$ for $P_c(4380)^+$ and $P_c(4440)^+$, and adjoint-baryon masses $\Lambda_{(1/2)^+,RS}=1.125$ GeV, $\Lambda_{(3/2)^+,RS}=1.152$ GeV—first lattice-testable predictions, transferable to bottom pentaq

Load-bearing premise

The load-bearing premise is the modeled intermediate-distance shape of the pentaquark Born–Oppenheimer potentials in Eq. (2.11)—hybrid-inspired $A r^2$ terms plus a one-pion-exchange tail—and the assumption that the potential connecting to the $\Lambda_c\bar D$ threshold falls monotonically from above; the paper states that lattice QCD has not yet computed these potentials, so if the real shapes differ, the spectrum, masses, and $J^P$ assignments change.

Editorial extensions

If this is right

The four known charm pentaquarks are accounted for as bound states in the lowest BO potentials; no molecular or diquark assumption is needed.
In scenario 2 the quantum numbers are $(1/2)^-$ for $P_c(4312)^+$ and $P_c(4457)^+$, and $(3/2)^-$ for $P_c(4380)^+$ and $P_c(4440)^+$; if correct, this fixes the open $J^P$ question for the $P_c(4440)/P_c(4457)$ pair.
Three additional charm pentaquarks are predicted just below the $\Sigma_c^*\bar D^*$ threshold with masses near 4.51–4.53 GeV; the $(5/2)^-$ member decays only to $\eta_c(1S)$, giving a distinctive search signature.
The adjoint baryon masses $\Lambda_{(1/2)^+,RS}=1.125$ GeV and $\Lambda_{(3/2)^+,RS}=1.152$ GeV are concrete numbers that lattice QCD can confirm or exclude.
With the same inputs, seven bottom pentaquarks are predicted between about 11.04 and 11.13 GeV, more deeply bound than the charm multiplet, with the lowest several expected to be narrow.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The computed $\eta_c(1S)/J/\psi$ semi-inclusive ratio for $P_c(4312)^+$ differs by a factor of about 20 between the two scenarios (0.23 vs 5.0), so a measurement of that ratio would discriminate scenarios before any lattice input arrives.
Because the static potentials are heavy-flavor independent, the same adjoint-baryon masses transferred to bottom are a strong cross-check: if lattice QCD later finds a very different $\Lambda_{(3/2)^+}-\Lambda_{(1/2)^+}$ splitting, the charm assignments would need revision even if the charm masses fit.
The paper mentions the recently reported $P_c(4337)^+$ only briefly; applying the same machinery to it—a potential that dips below the $\Lambda_c\bar D$ threshold and supports a P-wave resonance—would be a natural extension of this analysis.
If the three predicted near-threshold charm states are discovered, their $\Lambda_c\bar D^{(*)}$ decay ratios could distinguish the BOEFT description from a pure molecular picture despite similar masses.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is worth reading, but don't mistake its confidence for certainty. The authors identify the four LHCb pentaquarks as bound states in their BOEFT framework, assign J^P quantum numbers, extract adjoint baryon masses, and predict a bottom pentaquark spectrum. The framework is coherent, and the paper is refreshingly transparent about what is fitted and what is assumed.

What is genuinely new: this is the first BOEFT treatment of pentaquarks that goes beyond the four-state scenario, and it gives concrete, falsifiable outputs—adjoint baryon masses, J^P assignments, and a bottom spectrum. The decay ratios to Lambda_c D and Lambda_c D* are useful because the unknown transition amplitude cancels, and the comparison between scenarios 1 and 2 via these ratios is a sensible discriminator. The bottom pentaquark predictions are a natural extension.

The soft spots are real and load-bearing. The intermediate-distance form of the pentaquark BO potentials is unknown; the paper borrows the A parameters from hybrid potentials and explicitly says a different choice could change the spectrum. The mass agreement is by construction: the adjoint baryon masses are fitted to reproduce the four observed pentaquarks. The spin-dependent potential in Eq. (3.8) is assumed rather than derived. The choice of scenario 2 is made post hoc using decay data. There is no sensitivity check on the A parameters or the potential model, and that uncertainty is not propagated into the quoted J^P assignments or adjoint masses. None of this is hidden—the paper flags it in Section VI—but it means the central identification is contingent, not settled.

I would send this to a serious referee. It is a well-posed framework from a reputable group, the paper gives lattice QCD specific targets, and the decay predictions are worth having on record even if the J^P assignment later changes. The referee should push for a sensitivity analysis on the potential parameters and for a clearer separation of fitted versus predicted results in the abstract and conclusions.

Referee Report

4 major / 4 minor

Summary. The paper applies the Born–Oppenheimer effective field theory (BOEFT) to hidden-charm and hidden-bottom pentaquarks Q\bar Q qqq. It identifies the four LHCb states P_c(4312), P_c(4380), P_c(4440), and P_c(4457) as bound states in the lowest BO potentials (1/2)_g and {(1/2)'_g, (3/2)_g} that asymptotically approach the \Sigma_c \bar D threshold. Leading-order Schr\"odinger equations are supplemented by an O(1/m_Q) spin-dependent potential modeled on heavy-light baryon-meson splittings. Two scenarios reproduce the four observed masses by adjusting the adjoint baryon masses; the decay analysis favors scenario 2, which assigns J^P = (1/2)^- to P_c(4312), (3/2)^- to P_c(4380), (3/2)^- to P_c(4440), (1/2)^- to P_c(4457), and yields adjoint baryon masses \Lambda(1/2)^+,RS = 1.125 GeV and \Lambda(3/2)^+,RS = 1.152 GeV. With the same inputs the paper predicts the seven lowest bottom pentaquarks and their decay patterns.

Significance. If the extracted adjoint baryon masses and the J^P assignments survive scrutiny, the paper provides a useful QCD-EFT organizing framework for pentaquark spectroscopy and explicit lattice-targeted definitions of the relevant operators and generalized Wilson loops. Its model-independent decay-ratio predictions, in which the unknown transition amplitude cancels, are a genuine strength, as is the extension to the bottom sector. The main numerical output, however, is conditional: the bound-state spectrum, the J^P assignment, and the quoted adjoint masses all depend on modeled intermediate-distance potentials and on an assumed spin-dependent interaction that have not yet been computed from lattice QCD.

major comments (4)

[II.D.2, Eqs. (2.11)–(2.12)] The curvature parameters A_{(1/2)g}, A_{(1/2)'g}, A_{(3/2)g} are borrowed from hybrid-potential fits of Ref. [94] and transplanted to pentaquark BO potentials without any sensitivity analysis. The short- and long-distance limits are constrained by BOEFT, but the intermediate region is precisely what controls whether the potentials bind, and the two viable scenarios correspond to nearly degenerate eigenenergies E_{1/2}, E_{3/2} of order a few MeV. Since the adjoint baryon masses are free, reproducing the four masses does not test this choice. The authors should vary the A coefficients within plausible ranges (or use alternative potential forms) and show that the J^P assignments and extracted adjoint masses are stable, or quantify the resulting uncertainty.
[III.C, Sec. III.C.2–III.C.3, Tables IV–V] The adjoint baryon masses \Lambda(1/2)^+,RS and \Lambda(3/2)^+,RS are treated as adjustable parameters and tuned to the observed pentaquark masses by selecting (E_{1/2}, E_{3/2}) = (-23,-1) MeV or (-0.5,-14) MeV. The resulting spectrum agreement is therefore a fit, not a prediction, and the abstract's claim of 'first theoretical predictions for the adjoint baryon masses' overstates the status of these numbers. They should be presented as values extracted under the adopted potential model, with an explicit statement that they are not independent of the potential assumptions. This is load-bearing because the masses are a central advertised result.
[III.B, Eq. (3.8), Eqs. (3.12)–(3.15)] The spin-dependent potential V_SS = (2\Delta_1/3) S_1\cdot K_1 + \Delta_2 S_2\cdot K_2 is an ansatz, not a BOEFT-derived O(1/m_Q) potential; the 2/3 prefactor is chosen so that \Delta_1 reproduces the \Sigma_c^* - \Sigma_c splitting. All J^P assignments in Tables IV and V are obtained by diagonalizing matrices built from this ansatz. If the true spin-dependent BO potentials differ, the assignments could change. A concrete test would be to compute the spin-dependent potentials from lattice generalized Wilson loops (as done for hybrids in Ref. [104]) or to vary the relative strength of the S_1\cdot K_1 and S_2\cdot K_2 terms and check the stability of the scenario-2 assignment.
[II.D.1 and Sec. VI] The assumption that the (1/2)_g potential connecting to the \Lambda_c \bar D threshold decreases monotonically from above and supports no bound states is inferred solely from the absence of observed states near that threshold. This assumption is decisive: it removes three low-lying pentaquark states and allows the four observed states to be matched to the seven-state multiplet. The paper itself acknowledges in Sec. VI that other behaviors are possible and even notes that P_c(4337) could be a resonance in a potential that dips below the \Lambda_c \bar D threshold. A lattice calculation of the (1/2)_g potential, or a coupled-channel analysis of \Lambda_c \bar D scattering, is needed to make this assumption falsifiable.

minor comments (4)

[II (text near Eq. (2.3))] There are typos: 'NQRCD' should be 'NRQCD', and 'They are are given' should read 'They are given'.
[Table VI and Sec. IV.A] The quoted decay-width uncertainties cover adjoint-mass and \alpha_s-scale variations but do not include the uncertainty from the modeled intermediate-distance potentials or from the assumed spin interaction in Eq. (3.8). This limitation should be stated explicitly where the scenario comparison is made.
[Eq. (4.7)] The notation in the coupling potential is very dense; the accompanying footnote helps, but a short worked example for one transition would improve readability and reduce the chance of misinterpreting the summed quantum numbers.
[Sec. III.C.1, Eq. (3.16)] The scenario-0 discussion includes states exactly at thresholds; it may be worth noting explicitly that these are not genuine bound states in the usual sense, since they have zero binding energy and are at the continuum edge.

Circularity Check

2 steps flagged · score 6.0 of 10

Adjoint-baryon masses are fitted parameters relabeled as predictions, and the intermediate-distance BO potentials are borrowed from an overlapping-author hybrid-potential paper; the charm spectrum is therefore a fit, with genuine but conditional bottom-sector predictions.

fitted input called prediction [Abstract; Sec. II D 2 (Eq. 2.11); Secs. III C and VI]
"Abstract: "we provide the first theoretical predictions for the adjoint baryon masses, which can be confirmed by future lattice QCD studies." Sec. II D 2: "We treat the adjoint baryon masses Λ_{(1/2)^+} and Λ_{(3/2)^+} as adjustable parameters to reproduce the P_{c\bar c}(4312)^+, P_{c\bar c}(4380)^+, P_{c\bar c}(4440)^+, and P_{c\bar c}(4457)^+ pentaquark masses within the experimental uncertainties.""

The adjoint baryon masses Λ_{(1/2)^+} and Λ_{(3/2)^+} enter the BO potentials in Eq. (2.11) as free constants. Section II D 2 states explicitly that they are adjusted to reproduce the four observed pentaquark masses, and Section III C then reports the values Λ_{(1/2)^+,RS}=1.125 GeV and Λ_{(3/2)^+,RS}=1.152 GeV in the preferred scenario. Calling these fitted values "first theoretical predictions for the adjoint baryon masses" is a relabeling of the fit parameters: by construction, solving the Schrödinger equations with potentials containing these Λ values returns the masses that were used to fix them. The adjoint-baryon-mass output carries no independent confirmation from the four input masses because it is the inverse of the fit, not a calculation from QCD input alone.
ansatz smuggled in via citation [Sec. II D 2, Eq. (2.12); Sec. VI]
"Sec. II D 2: "Due to the lack of lattice data, we choose the parameters A_{Λη} in Eq. (2.11) to be the same as the parameters for the lowest hybrid potentials in Ref. [94]: A_{(1/2)_g}=0.042 GeV^3, A_{(1/2)'_g}=0.0065 GeV^3, A_{(3/2)_g}=0.0726 GeV^3." Sec. VI: "The form of the potentials at intermediate distances is unknown, as they have not been computed in lattice QCD.""

The intermediate-distance curvature of the pentaquark BO potentials controls whether each potential dips below the Σ_c \bar D threshold, which bound states exist, and how E_{1/2} and E_{3/2} are ordered. Instead of computing these curvatures for the three-light-quark (qqq) LDF, the paper imports the A_{Λη} coefficients from hybrid (Q\bar Q g) potentials of Ref. [94], a work coauthored by one of the present authors. The paper itself admits that the intermediate-distance form is unknown. The spectrum and J^P ordering are thus contingent on an ansatz transferred by self-citation from the hybrid sector; the subsequent agreement with the observed masses cannot validate this transfer, because the Λ masses are independently fitted to those same masses.

full rationale

The paper's own text makes the central fit explicit: Λ_{(1/2)^+} and Λ_{(3/2)^+} are freely adjusted to reproduce the four PDG pentaquark masses (Sec. II D 2), and the two scenarios correspond to choosing E_{1/2}, E_{3/2} = (−23,−1) MeV or (−0.5,−14) MeV for that purpose. Solving Eqs. (3.6)–(3.7) with these values returns the input masses, so the charm "spectrum" is a reproduction of the data, not a prediction. The advertised "first theoretical predictions for the adjoint baryon masses" are exactly the tuned parameter values, which is the fitted-input-called-prediction pattern. A second load-bearing input is the intermediate-distance form of the BO potentials: the A_{Λη} coefficients are taken from the hybrid-potential paper Ref. [94] (with overlapping authorship) and are acknowledged in Sec. VI to be unknown for pentaquarks; this is an ansatz imported via citation rather than a QCD computation. There is, however, substantial independent content: the spin-splitting formulas use measured Δ^c_1, Δ^c_2 splittings from PDG; the Λ_c\bar D and Λ_c\bar D^* decay ratios are built from a transition amplitude that cancels in the ratio; and the bottom-sector masses are genuine predictions conditional on the charm-fitted parameters. Because the central charm identification partly reduces to a fit and to a borrowed ansatz, but the paper contains real predictive elements, the overall circularity score is 6.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles or forces. The adjoint baryons are expected QCD states (color-octet three-quark systems), but their masses are treated as free parameters fitted to the pentaquark data. The main modeling assumptions are the intermediate-distance potential shape and the spin-interaction form, both of which are needed for the central identification.

free parameters (5)

Lambda_(1/2)+,RS = 0.998 GeV (scenario 1), 1.125 GeV (scenario 2)
Adjoint baryon mass in Eq. (2.11), tuned to reproduce the observed pentaquark masses in Section III C.
Lambda_(3/2)+,RS = 1.209 GeV (scenario 1), 1.152 GeV (scenario 2)
Adjoint baryon mass in Eq. (2.11), tuned with Lambda_(1/2)+ to reproduce the observed masses.
A_(1/2)g = 0.042 GeV^3
Coefficient of r^2 term in Eq. (2.11), taken from hybrid potential calculations in Ref. [94], not from pentaquark data.
A_(1/2)'g = 0.0065 GeV^3
Same source as above; modeling choice for the excited (1/2)_g potential.
A_(3/2)g = 0.0726 GeV^3
Same source as above; modeling choice for the (3/2)_g potential.

assumptions (6)

domain assumption The BOEFT hierarchy m_Q v >> Lambda_QCD >> m_Q v^2 and the factorization of heavy and light dynamics are valid for pentaquarks.
Invoked in Section II to justify the Born-Oppenheimer approximation and the potential description.
standard math The static energies at short distance have the form V_o(r) + Lambda_kappa + O(r^2), from the pNRQCD multipole expansion.
Used in Eq. (2.10) and discussed around it.
domain assumption The BO quantum numbers are conserved between the adjoint baryon sector and the baryon-meson thresholds, so the pentaquark potentials connect smoothly to the Sigma_c Dbar thresholds.
Stated in Section II D 1 and used to restrict which potentials can support bound states.
ad hoc to paper The (1/2)_g potential connecting to the Lambda_c Dbar threshold decreases monotonically from above and does not support bound states.
Assumed in Section II D 1 based on the absence of observed states near that threshold; crucial for the number of predicted states.
ad hoc to paper The spin-dependent BO potential in Eq. (3.8) has the same form as the spin splittings in the heavy-light BM thresholds, with the 2/3 prefactor chosen so that Delta_1 equals the Sigma_c* - Sigma_c splitting.
Assumed in Section III B; not derived from QCD in this work and directly determines the J^P ordering in the multiplets.
domain assumption The long-distance potentials in Eq. (2.11) have a one-pion-exchange form F e^{-r/d}/r.
Motivated by molecular models, cited in Section II D 2; this determines the tail behavior and hence the bound-state wavefunctions.

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Pith. "Pith review of Unravelling Pentaquarks with Born--Oppenheimer effective theory." pith.science (2026). https://pith.science/paper/SD6L76XX

@misc{pith2026250813050,
  author       = {Pith},
  title        = {Pith review of: Unravelling Pentaquarks with Born--Oppenheimer effective theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SD6L76XX}},
  note         = {Machine review of arXiv:2508.13050}
}

abstract

The hidden-charm pentaquark states $P_{c\bar{c}}\left(4312\right)^+$, $P_{c\bar{c}}\left(4380\right)^+$, $P_{c\bar{c}}\left(4440\right)^+$, and $P_{c\bar{c}}\left(4457\right)^+$, all with isospin $I = 1/2$, were discovered by the LHCb collaboration in the decay process $\Lambda_b^0 \to J/\psi p K^-$. Although their quantum numbers remain undetermined, these states have generated significant theoretical interest. We analyze their spectrum and decay patterns-including those of their spin partners-within the Born--Oppenheimer effective field theory (BOEFT), a framework grounded in QCD. At leading order in BOEFT, we identify these pentaquark states as bound states in BO potentials that exhibit at short-distance a repulsive octet behavior and a nonperturbative shift due to the adjoint baryons masses, while asymptotically approaching the $\Sigma_c\bar{D}$ threshold. We further incorporate ${\cal O}(1/m_Q)$ spin-dependent corrections to compute pentaquark multiplet spin splittings. Based on the spectrum, semi-inclusive decay widths to $J/\psi$ and $\eta_c$, and the decay width ratios to $\Lambda_c\bar{D}$ and $\Lambda_c\bar{D}^*$, we provide the first theoretical predictions for the adjoint baryon masses, which can be confirmed by future lattice QCD studies. Moreover, our analysis supports the quantum number assignments: $J^{P} = (1/2)^-$ for $P_{c\bar{c}}\left(4312\right)^+$, $(3/2)^-$ for $P_{c\bar{c}}\left(4380\right)^+$, $(1/2)^-$ for $P_{c\bar{c}}\left(4457\right)^+$, and $(3/2)^-$ for $P_{c\bar{c}}\left(4440\right)^+$. We also present results for the lowest bottom pentaquarks.

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1 extracted references · cited by 1 Pith paper

[1]

This scenario was considered by Voloshin to study the decay properties ofP c¯cstates in the molecular picture in Ref

Scenario 0:E 1/2 =E 3/2 = 0 For illustration, we first discuss the simplest scenario where the adjoint baryon masses Λ(1/2)+ and Λ (3/2)+ are chosen in such a way to set the eigenenergies of the Schr¨ odinger equations (3.6) and (3.7) to zero:E 1/2 =E 3/2 = 0. This scenario was considered by Voloshin to study the decay properties ofP c¯cstates in the mole...

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[1] [1]

This scenario was considered by Voloshin to study the decay properties ofP c¯cstates in the molecular picture in Ref

Scenario 0:E 1/2 =E 3/2 = 0 For illustration, we first discuss the simplest scenario where the adjoint baryon masses Λ(1/2)+ and Λ (3/2)+ are chosen in such a way to set the eigenenergies of the Schr¨ odinger equations (3.6) and (3.7) to zero:E 1/2 =E 3/2 = 0. This scenario was considered by Voloshin to study the decay properties ofP c¯cstates in the mole...