Decay behaviors of the P c hadronic molecules
aa r X i v : . [ h e p - ph ] O c t Decay behaviors of the P c hadronic molecules C.W. Shen , , ¶ , Y.H. Lin , Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy ofSciences, Beijing 100190,China University of Chinese Academy of Sciences (UCAS), Beijing 100049, China ¶ E-mail: [email protected]
Abstract
In this proceeding, we present our recent work on decay behaviors of the P c hadronic molecules,which can help to disentangle the nature of the two P c pentaquark-like structures. The results turnout that the relative ratio of the decays of P + c (4380) to ¯ D ∗ Λ c and J/ψp is very different for P c beinga ¯ D ∗ Σ c or ¯ D Σ ∗ c bound state with J P = − . And from the total decay width, we find that P c (4380)being a ¯ D Σ ∗ c molecule state with J P = − and P c (4450) being a ¯ D ∗ Σ c molecule state with J P =
52 + is more favorable to the experimental data.Keywords: Hidden-charm pentaquark, Hadronic molecular state, Hadronic decays.PACS: 12.39.Mk, 13.30.Eg, 14.20.Pt, 36.10.Gv.
In 2015, LHCb Collaboration observed two hidden charm pentaquark-like states in the
J/ψp invari-ant mass distribution in Λ b → J/ψK − p decays [1]. The significance of them are both larger than 9standard deviations. According to LHCb Collaboration’s fit results, the masses and widths of the P + c (4380) and P + c (4450) are (4380 ± ±
29) MeV, (205 ± ±
86) MeV and (4449 . ± . ± .
5) MeV,(39 ± ±
19) MeV, respectively. It also shows that the preferred spins of these two states are onehaving spin 3/2 and the other 5/2 with opposite parities.Various theoretical schemes are proposed to explore these two P c states’ inner structures. InRef. [2, 3], the two P c states are discussed as the diquark-diquark-antiqurk states in the ¯ c [ cu ][ ud ]configuration. The diquark-triquark system is used to study them in Ref. [4, 5]. The kinematiceffect is considered in Ref. [6, 7] to explain the narrow width of P c (4450) state. Also many groupsconjecture the two P c states being ¯ D ( ∗ ) Σ ( ∗ ) c hadronic molecules [8, 9, 10, 11], and this scheme hasalready been used to predict these pentaquarklike molecular states before the discovery of the two P c states in Ref. [12, 13]. Since the mass thresholds of the ¯ D Σ ∗ c and ¯ D ∗ Σ c are 4387 MeV and 4461MeV, respectively, which locate quite close to the mass region of these two P c states, we interpretnaturally them as the hadronic molecular states composed of either ¯ D Σ ∗ c or ¯ D ∗ Σ c .In this work we will estimate firstly the partial decay widths of the P c (4380) into the ¯ D Λ + c and J/ψp channels assuming it is an S-wave ¯ D Σ ∗ c or ¯ D ∗ Σ c hadronic molecular state. We found thatsupposing P c (4380) is a ¯ D ∗ Σ c or ¯ D Σ ∗ c hadronic molecular state with J P = − , the relative ratiosof decaying to ¯ D Λ + c and J/ψp are very different. This can be applied to distinguish the natureof P c (4380) and examined by experiments in the future. It also can shed light on the large decaywidth of P c (4380). Then we calculate the partial decay widths of the two P c states into variouspossible final states. The decay patterns differ a lot for the P c states having different componentswith different spin-parity, and the total decay widths are also different greatly. Based on these decaypatterns, we also estimate the production of two P c states in photo- and pion- induced reactions,which shows large cross sections for the ¯ D ∗ Λ c and ¯ D Λ c production through the s-channel exchangeof P c states. Firstly, we calculate the partial decay widths of P c (4380) into ¯ D Λ + c and J/ψp final states by treating P c (4380) as an S-wave hadronic molecular state of either ¯ D Σ ∗ c (2520) or ¯ D ∗ Σ c (2455) with J P = − .These two decays are proceeded through triangular diagrams as shown in a ), b ), a ′ ) and b ′ ) of Fig.1.1he calculation is accomplished in the framework of effective Lagrangian approach. The detailedLagrangians for different types of vertices and the involved coupling constants can be found inRef. [14, 15], since they are not shown here for simplicity. P c ¯ D Σ ∗ c ¯ D ∗ Λ c π,ρa ) P c ¯ D Σ ∗ c J/ψpD ∗ ,Db ) P c ¯ D Σ ∗ c ¯ D Λ c ρc ) P c ¯ D Σ ∗ c πND ∗ d ) P c ¯ D Σ ∗ c χ c pDe ) P c ¯ D Σ ∗ c η c pD ∗ f ) P c ¯ D Σ ∗ c ρND,D ∗ g ) P c ¯ D Σ ∗ c ωpD,D ∗ h ) P c ¯ D Σ ∗ c ¯ D Σ c ρi ) P c ¯ D ∗ Σ c ¯ D ∗ Λ c π,ρa ′ ) P c ¯ D ∗ Σ c J/ψpD ∗ ,Db ′ ) P c ¯ D ∗ Σ c ¯ D Λ c π,ρc ′ ) P c ¯ D ∗ Σ c πND,D ∗ d ′ ) P c ¯ D ∗ Σ c χ c pD ∗ e ′ ) P c ¯ D ∗ Σ c η c pD,D ∗ f ′ ) P c ¯ D ∗ Σ c ρND ∗ ,Dg ′ ) P c ¯ D ∗ Σ c ωND ∗ ,Dh ′ ) P c ¯ D ∗ Σ c ¯ D Σ c π,ρi ′ ) Figure 1:
The decays of the P c state via meson exchange as a (I): ¯ D Σ ∗ c molecule. a ) ¯ D ∗ Λ c channel. b ) J/ψp channel. c ) ¯ D Λ c channel. d ) πN channel. e ) χ c p channel. f ) η c p channel. g ) ρN channel. h ) ωp channel. i ) ¯ D Σ c channel.(II): a ¯ D ∗ Σ c molecule. a ′ ) ¯ D ∗ Λ c channel. b ′ ) J/ψp channel. c ′ ) ¯ D Λ c channel. d ′ ) πN channel. e ′ ) χ c p channel. f ′ ) η c p channel. g ′ ) ρN channel. h ′ ) ωp channel. i ′ ) ¯ D Σ c channel. To make the four-dimension loop integrals in the amplitudes convergent, a Gaussian regulator isadded in the first vertex with the form of Eq.(1) after the time-part integration has been done withthe residual theorem. The value of cutoff Λ is varied in the range of 0.5 GeV to 1.2 GeV and q isthe spatial part of the loop momentum q .Φ P c ( q / Λ ) ≡ exp( − q / Λ ) (1)Then we study the decay properties of P c (4380) being a ¯ D Σ ∗ c (2520) molecule or ¯ D ∗ Σ c (2455)molecule with J P = − and P c (4450) being a ¯ D ∗ Σ c (2455) molecule with J P = − or
52 + . Thepossible final states we considered are listed in Table 1 and the corresponding feynman diagrams withpseudoscalar and vector mesons exchange included are shown in Fig.1. The effective Lagrangiansand couplings can be also found in Ref. [14, 15].
Table 1:
All possible final states for the P c (4380) with J P = − and P c (4450) with J P = − or
52 + .Initial state Final states P c (4380)( ¯ D Σ ∗ c ) ¯ D ∗ Λ c , J/ψp , ¯ D Λ c , πN , χ c p , η c p , ρN , ωp , ¯ D Σ c P c (4380)( ¯ D ∗ Σ c ) ¯ D ∗ Λ c , J/ψp , ¯ D Λ c , πN , χ c p , η c p , ρN , ωp , ¯ D Σ c P c (4450)( ¯ D ∗ Σ c ) ¯ D ∗ Λ c , J/ψp , ¯ D Λ c , πN , χ c p , η c p , ρN , ωp , ¯ D Σ c , ¯ D Σ ∗ c It should be mentioned that the coupling constants which describe the strength of the interactionbetween the composite state and its compositions in the first vertex can be estimated with thecompositeness condition. Since the corresponding interaction for the J P =
52 + P c (4450) state isP-wave, the coupling in this vertex will depend on some parameters which are included necessarilyto make the loop integration convergent. In our case, we use the cutoff Λ same as the form factorEq.(1) for simplicity.Another off-shell form factor is introduced in our calculation for the exchanged meson with theform of Eq.(2), where m and q are the mass and momentum of the exchanged meson. And theoff-shell form factor’s cutoff Λ varies from 1.5 GeV to 2.4 GeV. F ( q ) = Λ ( m − q ) + Λ (2)2 Results and discussions
The ratio R for the P c (4380) state decaying to ¯ D ∗ Λ c channel and J/ψp channel is defined in Eq.(3).Here we only estimate the order of the ratio results, since there are many variable factors cannot bedetermined. For P c (4380) being a ¯ D Σ ∗ c hadronic molecule, the partial decay width of ¯ D ∗ Λ c channel isat least one order of magnitude larger than that of J/ψp channel. However, the partial decay widthof these two channels are in the same order if P c (4380) is assumed as a ¯ D ∗ Σ c hadronic molecule. R I = Γ( P c (4380) → ¯ D Σ ∗ c → ¯ D ∗ Λ c )Γ( P c (4380) → ¯ D Σ ∗ c → J/ψp ) ∼ R II = Γ( P c (4380) → ¯ D ∗ Σ c → ¯ D ∗ Λ c )Γ( P c (4380) → ¯ D ∗ Σ c → J/ψp ) ∼ P c (4380) being a ¯ D Σ ∗ c hadronic molecule should be muchlarger than being a ¯ D ∗ Σ c hadronic molecular, as there is a large contribution from ¯ D ∗ Λ c channelin the former case. This conclusion is also made by using the non-relativistic formalism with heavyquark spin symmetry taken into consideration, whose details are shown in Appendix B of Ref. [14].It might explain the broad width of P c (4380) claimed by LHCb Collaboration.Considering the possible channels in Table 1, we find that ¯ D ∗ Λ c , J/ψp and ¯ D Λ c are the threedominant decay channels. The dependance of total decay widths of the two P c states and branchingfractions of these three channels on cutoff Λ and Λ are shown in Ref. [15]. Choosing Λ = 1 . = 2 . Table 2:
Partial widths of P c (4380) as a ¯ D Σ ∗ c molecule and a ¯ D ∗ Σ c molecule with J P = − , and P c (4450) as a¯ D ∗ Σ c molecule with J P = − and
52 + , into different final states with Λ = 1 . = 2 . Mode Widths (MeV) P c (4380) P c (4450)¯ D Σ ∗ c ( − ) ¯ D ∗ Σ c ( − ) ¯ D ∗ Σ c ( − ) ¯ D ∗ Σ c (
52 + )¯ D ∗ Λ c J/ψp D Λ c πN χ c p η c p ρN ωp D Σ c D Σ ∗ c - - 8.9 0.5¯ D Λ c π P c (4380) being a ¯ D Σ ∗ c molecule will have larger width than being a ¯ D ∗ Σ c molecule, and the width of P c (4450) having spin-parity
52 + is several times smaller than havingspin-parity − when supposing it is a ¯ D ∗ Σ c molecule. Compared with the widths reported by theLHCb Collaboration, the P c (4380) in the J P = − ¯ D Σ ∗ c picture and P c (4450) in the J P =
52 + ¯ D ∗ Σ c picture in more favorable.The cross sections of these reactions through s- and u- channel exchange of two P c states with γp and πp incident and ¯ D ∗ Λ c , J/ψp and ¯ D Λ c outgoing respectively are calculated in Ref. [15]. Theproductions for ¯ D ∗ Λ c and ¯ D Λ c final states are much larger than the J/ψp final state. It is helpfulto figure out the nature of these two pentaquarklike P c states to seek for them in the ¯ D ∗ Λ c and ¯ D Λ c production in the future experiments. 3 cknowledgment We thank Bing-Song Zou, Feng-Kun Guo and Ju-Jun Xie for useful suggestions and the effects on thiswork. This project is supported by NSFC under Grant No. 11261130311 (CRC110 cofunded by DFGand NSFC) and Grant No. 11647601, and by the Thousand Talents Plan for Young Professionals,and by the CAS Key Research Program of Frontier Sciences under Grant No. QYZDB-SSW-SYS013.
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Decay behaviors of the P cc