Pion-assisted charmed dibaryon candidate
aa r X i v : . [ nu c l - t h ] J u l Pion-assisted charmed dibaryon candidate
A. Gal, ∗ H. Garcilazo, † A. Valcarce, ‡ and T. Fern´andez-Caram´es § Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel Escuela Superior de F´ısica y Matem´aticasInstituto Polit´ecnico Nacional, Edificio 9, 07738 M´exico D.F., Mexico Departamento de F´ısica Fundamental,Universidad de Salamanca, E-37008 Salamanca, Spain (Dated: October 24, 2018)
Abstract
The Λ c (2286) N system is studied in a chiral constituent quark model and the resulting s -waveinteraction is used in separable form within three-body models of the π Λ c N system with quantumnumbers ( C, I, J P ) = (+1 , , + ). Separable interactions are also used for the dominant p -wavepion-baryon channels dominated by the ∆(1232) and Σ c (2520) resonances. Faddeev equations withrelativistic kinematics are solved on the real axis to search for bound states and in the complex planeto search for three-body resonances. Some of the models considered generate a very narrow boundstate, requiring isospin violation for its strong decay. Other models lead to a narrow resonance(Γ < ∼ . c (2455) N threshold. This would be the lowest-lying C = +1 dibaryon, with mass estimated as ≈ ±
15 MeV.
PACS numbers: 12.39.Pn, 13.75.Gx, 14.20.Lq, 11.80.JyKeywords: Quark models, pion-baryon interactions, charmed baryons, Faddeev equations ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: valcarce@fis.usal.es § Electronic address: [email protected] . INTRODUCTION Several pion-assisted dibaryon candidates of the type πBB ′ , with a p -wave pion interact-ing with baryons B and B ′ that interact in s waves, were suggested in Ref. [1]. Consideredin detail was the π Λ N system in the channel ( I, J P ) = ( , + ) which is dominated by config-urations where the p -wave πN ( I, J P ) = ( ,
32 + ) ∆(1232) resonance is coupled to an s -waveΛ, and the p -wave π Λ (
I, J P ) = (1 ,
32 + ) Σ(1385) resonance is coupled to an s -wave nu-cleon. The choice ( I, J P ) = ( , + ) ensures that the spins and isospins of the three hadronsare parallel, with the two baryons necessarily in a S state, leading to maximal attrac-tion since all spin and isospin recoupling coefficients in this channel are equal to one. The( I, J P ) = ( , + ) π Λ N – π Σ N coupled-channel system was studied subsequently [2, 3], con-cluding that it resonates some 10-20 MeV below the π Σ N threshold [3]. Other pion-assisteddibaryon candidates suggested in [1] include π Ξ N , π ΞΛ and π Λ c N . In the present work weapply the same formalism [3] to study the charmed π Λ c N system where one replaces the
12 +g . s . Λ(1116) baryon by the
12 +g . s . Λ c (2286) charmed baryon, and the
32 +
Σ(1385) resonanceby the
32 + Σ c (2520) charmed resonance. An interesting aspect of this π Λ c N system is thata bound state, if occurring, will decay only by isospin-violating interactions since the lowestisospin-conserving decay channel Σ c (2455) N lies ≈
30 MeV above the π Λ c N threshold.To formulate and solve a π Λ c N three-body model one needs to specify the input pair-wise interactions. Whereas the construction of p -wave separable interactions describing thepion-baryon ∆(1232) and Σ c (2520) resonances is straightforward, the construction of thenecessary s -wave separable interaction describing the Λ c (2286) N system requires specialattention. In the present exploratory study we neglect its coupling to the Σ c (2455) N sys-tem, reporting briefly on a straightforward application of the chiral constituent quark model(CCQM) within the charm sector [4, 5]. This model, tuned by fitting to the baryon andmeson spectra as well as to the N N interaction, provides predictions for charm C = +1two-hadron systems that will become testable in due course. For an extensive review of theCCQM, see Ref. [6].The paper is organized as follows. The input pion-baryon phenomenological interactionsare discussed in Sect. II, and the input Λ c N CCQM interactions are discussed in Sect. III.Results of three-body calculations using Faddeev equations with relativistic kinematics aregiven and discussed in Sect. IV, with conclusions drawn in the last Sect. V.2
I. PION-BARYON p -WAVE INTERACTIONS Following the discussion of the π Λ N system in Ref. [1], the dominant two-body in-teractions in the π Λ c N system are the p -wave πN ( I, J P ) = ( ,
32 + ) ∆(1232) and π Λ c ( I, J P ) = (1 ,
32 + ) Σ c (2520) channels, and the s -wave Λ c N interaction in the I = , S chan-nel. In this section we describe the appropriate separable-interaction meson-baryon models,assigning particle indices 1,2,3 to charmed-hyperons, nucleon and pion, respectively. A. The πN subsystem The Lippmann-Schwinger equation for the pion-nucleon interaction is given by [7]: t ( p , p ′ ; ω ) = V ( p , p ′ ) + Z ∞ p ′′ dp ′′ × V ( p , p ′′ ) 1 ω − p m N + p ′′ − p m π + p ′′ + i ǫ t ( p ′′ , p ′ ; ω ) , (1)so that using a separable potential V ( p , p ′ ) = γ g ( p ) g ( p ′ ) , (2)one gets t ( p , p ′ ; ω ) = g ( p ) τ ( ω ) g ( p ′ ) , (3)where [ τ ( ω )] − = 1 γ − Z ∞ p dp g ( p ) ω − p m N + p − p m π + p + i ǫ . (4)A fit to the P phase shift and scattering volume using the form factor g ( p ) = p [exp( − p /β ) + Cp exp( − p /α )] , (5)with parameters given in Table I, was shown and discussed in Ref. [7]. Listed in the tableare also the r.m.s. radii of the form factors g ( p ) in momentum space and ˜ g ( r ) in coordinatespace, where ˜ g ( ~r ) = ˆ r ˜ g ( r ) is the Fourier transform of the p -wave form factor g ( ~p ) = ˆ pg ( p ),given by ˜ g ( r ) ∼ Z j ( pr ) g ( p ) p dp, (6)with j the spherical Bessel function for ℓ = 1. As elaborated in Ref. [7], ˜ g ( r ) is not positivedefinite, which may result in negative values of < r > . A spatial-size substitute for √ < r >
3s provided then by r ( πN )0 , the first zero of ˜ g ( r ). Both values of √ < r > and r listed inTable I are seen to be close to each other, but this need not necessarily be the case for othersubsystems, as will become evident in the next subsection. TABLE I: Fitted parameters of the πN separable p -wave interaction (2) with form factor g ( p )defined by Eq. (5). Listed also are values of its r.m.s. momentum p < p > g (in fm − ), and r.m.s.radius p < r > ˜ g and zero r ( πN )0 (both in fm) of the coordinate-space form factor ˜ g ( r ). γ (fm ) α (fm − ) β (fm − ) C (fm ) p < p > g p < r > ˜ g r ( πN )0 − The pion-nucleon amplitude in the three-body system with a Λ c as spectator is given by t ( p , p ′ ; W , q ) = g ( p ) τ ( W , q ) g ( p ′ ) , (7)where W is the invariant mass of the three-body system, q is the relative momentumbetween the spectator and the c.m. of the πN subsystem and[ τ ( W , q )] − = 1 γ − Z ∞ p dp g ( p ) W − r(cid:16)p m N + p + p m π + p (cid:17) + q − q m c + q + i ǫ . (8) B. The π Λ c subsystem Here, the separable potential V ( p , p ′ ) = γ g ( p ) g ( p ′ ) , (9)is used with the form factor g ( p ) = p (1 + Ap ) exp( − p /β ) , (10)where the three parameters γ , β and A were fitted to the two pieces of data available,namely, the position and width of the Σ c (2520) resonance [8]. A family of such fittedparameters is given in Table II. Scanning over A between 0 and 1 gave unrealistically smallpositive values of < r > ˜ g associated with the form factor g , decreasing rapidly with A and becoming negative for A exceeding 0.2. Our alternative choice of r for a size parameter4ives values monotonically increasing from 1 to 1.5 upon increasing A = 0. Anticipating r ( π Λ c )0 to somewhat exceed r ( πN )0 = 1 .
36 fm (Table I), because the pionic Λ c → Σ c p -waveexcitation energy of 231.5 MeV is smaller than the corresponding excitation energy 293 MeVfor N → ∆, we consider the last two rows in Table II as the most physically acceptable fits.For further discussion of form-factor sizes, see Ref. [7]. TABLE II: Fitted parameters of the π Λ c separable p -wave interaction defined by Eqs. (9) and (10),for chosen values of the parameter A . Listed also are values of the r.m.s. momentum p < p > g (in fm − ), the r.m.s. radius p < r > ˜ g and zero r ( π Λ c )0 (both in fm) of the Fourier transform ˜ g ( r ). A (fm ) γ (fm ) β (fm − ) p < p > g p < r > ˜ g r ( π Λ c )0 − − − − − − − − III. THE Λ c N SUBSYSTEM
There is no experimental data on the Λ c N subsystem that one may rely upon to fit aseparable potential form. Therefore, and as a guide, we have generated local potentials inthe I = , S channel from the recent application of the CCQM to the charmed mesonsector [5]. A brief description of the essential properties required in this model to provideinteraction output for the Λ c N system follows. A. Extension of the CCQM to the charm sector
Baryons are described in the CCQM as clusters of three interacting massive con-stituent quarks, with the light-quark ( u, d ) mass generated by the spontaneously broken5 U (2) L ⊗ SU (2) R chiral symmetry of the QCD Lagrangian. Hence, light quarks interactnonperturbatively via Nambu-Goldstone boson-exchange potentials V χ ( ~r ij ) = V OSE ( ~r ij ) + V OPE ( ~r ij ) , (11)given in obvious notation by V OSE ( ~r ij ) = − g π Λ Λ − m σ m σ (cid:20) Y ( m σ r ij ) − Λ m σ Y (Λ r ij ) (cid:21) , (12) V OPE ( ~r ij ) = g π m π m i m j Λ Λ − m π m π ( (cid:20) Y ( m π r ij ) − Λ m π Y (Λ r ij ) (cid:21) ~σ i · ~σ j + (cid:20) H ( m π r ij ) − Λ m π H (Λ r ij ) (cid:21) S ij ) ~τ i · ~τ j , (13)where g / π is the chiral coupling constant, Y ( x ) is the Yukawa function, Y ( x ) = e − x /x ,and H ( x ) = (1 + 3 /x + 3 /x ) Y ( x ) is associated with the quark-quark tensor operator S ij = 3 ( ~σ i · ˆ r ij )( ~σ j · ˆ r ij ) − ~σ i · ~σ j . The values used for the mass, coupling-constant andcut-off parameters are listed in Table 2 of [5]. In the case of the heavy charmed quark c , forwhich chiral symmetry is explicitly broken, no boson-exchange is operative in its interactionswith the other quarks.Perturbative effects within QCD are accounted for by the one-gluon-exchange (OGE)potential V OGE ( ~r ij ) = α s ~λ c i · ~λ c j (cid:20) r ij − (cid:18) m i + 12 m j + 2 ~σ i · ~σ j m i m j (cid:19) e − r ij /r r r ij − S ij m q r ij (cid:21) , (14)where λ c are the SU (3) color matrices, r is a flavor-dependent regularization that scaleswith the reduced mass of the interacting pair, and α s is the QCD scale-dependent couplingconstant which assumes values of α s ∼ .
54 for light-quark pairs and α s ∼ .
43 for uc and dc pairs [4].Finally, to fully simulate QCD one needs to incorporate confinement. While negligiblefor hadron-hadron interactions, lattice calculations suggest that the confinement potentialis screened upon increasing the interquark distance [9], V CON ( ~r ij ) = [ − a c (1 − e − µ c r ij )] ~λ ci · ~λ cj , (15)with a scale given by a c = 230 MeV and a screening mass identified here with the pion mass: µ c = m π . 6he CCQM yields a good description of meson [4] and baryon spectra [10]. Furthermore,by applying resonating-group methods it enables one to derive baryon-baryon ( BB ) poten-tials and, in particular, to reproduce the main features of the N N interaction [6]. Thus,the B n B m → B k B l local transition potential V B n B m ( L S T ) → B k B l ( L ′ S ′ T ) ( R ) is derived within aBorn-Oppenheimer approximation as V B n B m ( L S T ) → B k B l ( L ′ S ′ T ) ( R ) = ξ L ′ S ′ TL S T ( R ) − ξ L ′ S ′ TL S T ( ∞ ) , (16)where ξ L ′ S ′ TL S T ( R ) = D Ψ L ′ S ′ TB k B l ( ~R ) | P i 518 fm for the ( u, d ) light quarks, here denoted n , and b c = 0 . c quark.However, whereas the value adopted for b n was deduced long ago by fitting to the N N phaseshifts and the deuteron binding energy [11] (see also the discussion in Ref. [12]), the value b c = 0 . BB data. It was argued in Ref. [13] thata considerably smaller value of b c , in fact b c ≈ . b q scaleswith quark-mass as b q ∼ m − / q for harmonic-oscillator quark potential. For CCQM quarkmasses m n = 313 MeV and m c = 1752 MeV, the widely adopted value b n = 0 . 518 fm impliesthat b c = b n ( m n /m c ) / = 0 . 219 fm. This strong dependence on the constituent quark massby far overshadows the weak flavor dependence of the harmonic-oscillator 1¯ hω excitationenergy, of order hundreds of MeV, in mesons and in baryons. B. The CCQM I = , S Λ c N interaction Adopting the CCQM s -wave potentials for the Λ c N interacting pair, we show on the l.h.s.of Fig. 1 three such potentials for the I = , S channel, using three different values of thecharmed quark oscillator parameter b c within the six-quark wave function: b c = 0 . b c = 0 . b c = 0 . b c = 0 . IG. 1: Left: CCQM Λ c N local potentials in the S channel for several values of the charmedquark harmonic oscillator size parameter b c . Right: corresponding Λ c N S phase shifts, in solid(dashed) lines, produced by the CCQM local (separable) potentials. The smaller b c is, the higheris the maximum of the phase shift δ . repulsive core and the model with b c = 0 . c N bound state. The S phase shiftsproduced by these potentials are shown by the solid lines on the r.h.s. of the figure wherethe change of sign of the phase shift for the model with b c = 0 . N N or Λ N , whilefor the models with b c = 0 . b c = 0 . V ( p , p ′ ) = − g a ( p ) g a ( p ′ ) + g r ( p ) g r ( p ′ ) , (18)so that the corresponding two-body t -matrix is given by t ( p , p ′ ; ω ) = − X α = a,r X β = a,r g α ( p ) τ αβ ( ω ) g β ( p ′ ) , (19)8here τ ar ( ω ) = τ ra ( ω ) = G ar ( ω )[1 + G aa ( ω )][1 − G rr ( ω )] + [ G ar ( ω )] , (20) τ aa ( ω ) = 1 − G rr ( ω )[1 + G aa ( ω )][1 − G rr ( ω )] + [ G ar ( ω )] , (21) τ rr ( ω ) = − G aa ( ω )[1 + G aa ( ω )][1 − G rr ( ω )] + [ G ar ( ω )] , (22)with G αβ ( ω ) given by G αβ ( ω ) = Z ∞ p dp g α ( p ) g β ( p ) ω − p p + m N − q p + m c + iǫ . (23)The form factors g β ( p ) are chosen to be of the Yamaguchi form g β ( p ) = √ γ β p + α β ( β = a, r ) , (24)and the parameters of these models are given in Table III together with values of the as-sociated scattering lengths and effective ranges. The relatively small size of the scatteringlengths a Λ c N clearly indicates that the S Λ c N system is far from binding on its own. TABLE III: Parameters of the S Λ c N separable potential models Eqs. (18), (24). b c (fm) γ a (fm ) α a (fm − ) γ r (fm ) α r (fm − ) a (fm) r (fm)0.2 1.8915 1.7672 2.6210 2.1523 − − − IV. RESULTS AND DISCUSSION Solutions of the Faddeev equations corresponding to bound states and resonance polesin the ( I, J P ) = ( , + ) channel of the π Λ c N three-body system were found applying searchprocedures described in Refs. [1–3]. A single bound state or resonance was established forany combination of each one of the π Λ c interaction models specified in Table II and eachone of the Λ c N interaction models specified in Table III, as well as for the case when there isno Λ c N interaction. The resulting bound-state and resonance energies are given in Table IVwith respect to the π Λ c N threshold mass E th ≈ ABLE IV: Energy eigenvalue of the ( I, J P ) = ( , + ) π Λ c N state (in MeV with respect to the π Λ c N threshold) for the eight models of the π Λ c interaction characterized by the parameter A andthe three models of the Λ c N interaction b c = 0 . b c = 0 . 5, and b c = 0 . V Λ c N = 0. A (fm ) b c = 0 . b c = 0 . b c = 0 . V Λ c N = 00.0 − − i0.45 63.1 − i1.91 − − − i0.18 54.8 − i1.16 − − − i0.21 55.2 − i1.22 − − − i0.28 57.0 − i1.53 − − − i0.38 59.1 − i1.67 − − i0.00 35.9 − i0.51 61.2 − i1.96 3.9 − i0.000.7 7.0 − i0.01 40.8 − i0.80 64.4 − i2.34 11.3 − i0.011.0 14.7 − i0.07 46.2 − i1.10 68.2 − i3.05 19.5 − i0.07 Bound-state solutions appear in several of the b c = 0 . c N interaction is switched off, whereas the models b c = 0 . b c = 0 . c N threshold (Re E < 27 MeV) theresonance states are quite narrow with widths less than 0.4 MeV. This does not apply tothe models with b c = 0 . b c = 0 . c (2455) N threshold and its width is therefore larger than indicated by the tabulated widths.As concluded in Sect. II B, only values of A > . are acceptable in considering theΛ c (2286) + π → Σ c (2520) p -wave form factor relative to the N (939) + π → ∆(1232) p -waveform factor. Combined with the more plausible Λ c (2286) N interaction model defined bychoosing b c = 0 . c N interaction, we concludefrom Table IV that the ( I, J P ) = ( , + ) state of the π Λ c N system resonates at energy upto about 20 MeV above threshold, that is at 3363 < ∼ √ s < ∼ I π Λ c N = ) → ( I Λ c N = ),and the lowest Σ c N channel is closed at the energy range expected for the resonance.10 . CONCLUSION It was demonstrated in this work by solving Faddeev equations that the ( I, J P ) = ( , + )state of the π Λ c N system is a strong candidate for a pion-assisted charmed dibaryon. Al-though the CCQM Λ c N interaction is not sufficiently strong to bind a Λ c N S pair (andthis holds even more so for a S pair in the CCQM), the p -wave pion attractive interactionsinduced by the J P = 32 + ∆(1232) and Σ c (2520) resonances manage to bind the π Λ c N three-body system or more likely to make it resonate. The prediction of this dibaryon candidateis robust in the sense that its existence depends little on the Λ c N spin-triplet s -wave inter-action, even if the precise energy of the resonance is not pinned down between thresholdat ≈ lowest lying charmed dibaryon, considerablybelow the mass ≈ DN N bound state with quantum num-bers I = , J P = 0 − that may be viewed also as a Λ c (2595) N bound state [14]. These twocharmed-dibaryon predictions, with ( I, J P ) = ( , + ) (ours) versus ( I, J P ) = ( , − ) [14],bear structural resemblance to the strange-dibaryon predictions of π Λ N ( I = , J P = 2 + )[1–3] versus K − pp ( I = , J P = 0 − ) that may also be viewed as a Λ(1405) N quasiboundstate [15]. None of these dibaryon candidates has been confirmed by experiment.Denoting the ( I, J P ) = ( , + ) π Λ c N dibaryon candidate by Y c , in analogy to the( I, J P ) = ( , + ) π Λ N dibaryon candidate Y [15], the following production reactions of Y c are feasible with proton and pion beams in the high-momentum hadron beam line exten-sion approved at J-PARC: p + p → Y +++ c + D − ֒ → Σ ++ c (2455) + p , (25) π + + d → Y +++ c + D − ֒ → Σ ++ c (2455) + p , (26) π − + d → Y + c + D − ֒ → Σ + / c (2455) + n/p . (27)The Y c dibaryon resonance may be looked for both within inclusive missing-mass measure-ments by focusing on the outgoing D − charmed meson, and in exclusive invariant-mass11easurements focusing on the outgoing Σ c (2455) N decay pair provided that Y c is locatedabove the Σ c (2455) N threshold. Acknowledgments The research of A.G. is partially supported by the HadronPhysics3 networks SPHEREand LEANNIS of the European FP7 initiative, the research of H.G. is supported in part byCOFAA-IPN (M´exico), and the research of A.V. and T.F.C. is supported by the SpanishMinisterio de Educaci´on y Ciencia and EU FEDER under Contract No. FPA2010-21750,and by the Spanish Consolider-Ingenio 2010 Program CPAN (CSD2007-00042). [1] A. Gal and H. Garcilazo, Phys. Rev. D , 014013 (2008).[2] H. Garcilazo and A. Gal, Phys. Rev. C , 055205 (2010).[3] H. Garcilazo and A. Gal, Nucl. Phys. A , 167 (2013).[4] J. Vijande, F. Fern´andez, and A. Valcarce, J. Phys. G , 481 (2005).[5] T. F. Caram´es and A. Valcarce, Phys. Rev. D , 094017 (2012).[6] A. Valcarce, H. Garcilazo, F. Fern´andez, and P. Gonz´alez, Rep. Prog. Phys. , 965 (2005).[7] A. Gal and H. Garcilazo, Nucl. Phys. A , 153 (2011).[8] J. Beringer et al. , Phys. Rev. D , 010001 (2012) [http://pdg.lbl.gov].[9] G. S. Bali, Phys. 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