Source-linked AI summary
Study of $P_c(4457)$, $P_c(4440)$, and $P_c(4312)$ in a quasipotential Bethe-Salpeter equation approach
Jun He
TL;DR
The new LHCb observations renewed the question of how to assign the split Pc(4450) peaks and Pc(4312), whose spin parities were not reported. The authors recalculate Σc(*)D̄(*) molecular states with a coupled-channel quasipotential Bethe-Salpeter approach and assign the observed states to particular molecular channels and spin parities. They emphasize that experimental spin-parity determination is needed to understand the states' internal structure.
Problem
The split Pc(4450) peaks and newly observed Pc(4312) lack reported spin-parity determinations, leaving their molecular assignments unsettled.
Source: §1 · p. 1Method
Results
The calculation assigns Pc(4440) and Pc(4457) as mainly ΣcD̄* states with spin parities 1/2− and 3/2−, respectively, and Pc(4312) as a 1/2− ΣcD̄ state.
Source: §4 · p. 4Takeaways & Limitations
Experimental partial-wave analysis is essential to clarify the pentaquarks' origin and internal structure.
Source: §4 · p. 4Because the reported spin parities are unavailable, other assignments cannot be excluded directly.
Source: §4 · p. 4
Abstract
from arXiv · showhide
Very recently, the LHCb Collaboration reported their new results about the pentaquarks at charm energy region. Based on the new experimental results, we recalculate the molecular states composed of a $\Sigma_c^{(*)}$ baryon and a $\bar{D}^{(*)}$ meson in a quasipotential Bethe-Salpeter equation approach. The two-peak structure around 4450 MeV can be interpreted as two $\Sigma_c \bar{D}^*$ bound states with spin parities $1/2^-$ and $3/2^-$. The newly observed pentaquark $P_c(4312)$ can be assigned as a $\Sigma_c\bar{D}$ bound state with spin parity $1/2^-$. The experimental determination of spin parities will be very helpful to understand the internal structure of these pentaquarks.
1 Introduction
New LHCb data resolved the former Pc(4450) peak into Pc(4457) and Pc(4440), and identified Pc(4312) near the ΣcD̄ threshold. The paper revisits molecular assignments for these states, including the possibility that the two higher states are lower-spin S-wave ΣcD̄* molecules.
1 Introduction
- The former Pc(4450) peak split into Pc(4457) and Pc(4440), while a new Pc(4312) appeared near the ΣcD̄ threshold.The higher states lie near the ΣcD̄* threshold, and Pc(4312) near the ΣcD threshold.Source: §1 · p. 1
- The two higher peaks may be S-wave ΣcD̄* molecules with spin parities 1/2− and 3/2−, rather than one large-spin state.The report did not provide their spin parities; the two-peak structure could account for the earlier 5/2 suggestion.Source: §1 · p. 1
- Earlier molecular assignments proposed 1/2− and 3/2− ΣcD̄* states, but the suggested 5/2+ parity for Pc(4450) prompted alternative assignments.Prior work considered P-wave ΣcD̄* and other interpretations; later studies also examined molecular assignments for the three states.Source: §1 · p. 1
- The study uses a coupled-channel quasipotential Bethe-Salpeter approach to examine Σc(*)D̄(*) molecules, including a ΣcD̄ candidate for Pc(4312).The authors expect the smaller cutoff to move a previous 3/2− ΣcD̄* bound state toward threshold and remove 5/2 states.Source: §1 · p. 2
2 Coupled-channel Σ(∗) c ¯D(∗) interactions and relevant Lagrangians
The section constructs coupled-channel charmed-baryon–anticharmed-meson interactions from effective light-meson couplings and uses them in a quasipotential Bethe-Salpeter calculation. It describes the potential construction, partial-wave treatment, and regularization used to search for bound-state poles.
2 Coupled-channel interactions and relevant Lagrangians
- The calculation considers coupled-channel interactions between charmed baryons and anticharmed mesons, with light-meson exchanges mediating the interactions.The potentials are formed from interaction vertices and exchanged-light-meson propagators.Source: §2 · pp. 2–3
- Effective Lagrangians specify light-meson couplings to charmed mesons and baryons using chiral symmetry, the heavy-quark limit, and hidden local symmetry.The coupling constants are taken from the literature, and Table 1 lists the adopted parameters and constants.Source: §2 · p. 2, Table 1 · p. 2
- Rather than entering 16 potential elements explicitly, the calculation inputs vertices and propagators directly to construct the coupled-channel potential.The stated potential forms combine the two vertices with the exchanged-meson propagator.Source: §2 · pp. 2–3
- Bound states are searched as poles of the scattering amplitude using a spectator quasipotential Bethe-Salpeter equation for fixed spin-parity partial waves.The partial-wave potential uses a Wigner d-matrix, and the pole condition is |1 − V(z)G(z)| = 0.Source: §2 · p. 3
- In the chiral unitary approach, this work regularizes the calculation by introducing an exponential form factor in the propagator, and states that the cutoff Λ should be about 1 GeV.For the exchanged light meson, the monopole form factor has cutoff Λe = m + α·0.22 GeV, with α fixed to 2.Source: §2 · p. 3
3 Numerical results
The coupled-channel calculation yields three poles: two 1/2− states near 4440 and 4320 MeV and one 3/2− state near 4460 MeV. Their channel attribution and cutoff behavior support the proposed molecular assignments, while Σ*D remains attractive without binding.
3 Numerical results
- The cutoff Λ is the only free parameter, varied from 0.8 to 1.8 GeV while searching for poles between 4.3 and 4.5 GeV.Source: §3 · p. 3
- The calculation finds two 1/2− poles near 4440 and 4320 MeV and one 3/2− pole near 4460 MeV.The 1/2− poles are listed in Table 2 and the 3/2− pole in Table 3.Source: §3 · pp. 3–4, Table 2 · p. 3, Table 3 · p. 4
- The upper and lower 1/2− poles are attributed mainly to ΣcD* and ΣcD, respectively; the 3/2− state is mainly from ΣcD*.The coupled-channel effect is described as generally small.Source: §3 · pp. 3–4
- In coupled-channel calculations, the 1/2− upper state at about 4440 MeV decreases in mass as cutoff increases, while the lower state at about 4320 MeV and the 3/2− state are not sensitive to cutoff.Source: §3 · pp. 3–4
- No bound state is found from the Σ*D interaction, although that channel remains attractive at the adopted cutoff.Source: §3 · p. 4
4 Summary and discussion
The paper reanalyzes hidden-charm pentaquarks as near-threshold molecular states using a coupled-channel quasipotential Bethe-Salpeter approach. It assigns the two higher states to ΣcD* molecules and P_c(4312) to a ΣcD molecule, while noting that experimental spin-parity measurements are needed to distinguish interpretations.
4 Summary and discussion
- The authors reanalyze Σ(*)cD(*) interactions using a quasipotential Bethe-Salpeter equation approach.Source: §4 · p. 4
- The coupled-channel calculation assigns P_c(4440) and P_c(4457) to ΣcD* states with 1/2− and 3/2−, and P_c(4312) to a 1/2− ΣcD state.Source: §4 · p. 4
- No Σ*cD interaction bound state appears at the adopted cutoff, though the weaker attraction may permit a smaller, broader structure around 4380.Source: §4 · p. 4
- Because the reported results do not determine the three states' spin parities, alternative assignments remain possible and experimental partial-wave analysis is essential.Source: §4 · p. 4