New spectrum of negative-parity doubly charmed baryons: Possibility of two quasistable states
Mao-Jun Yan, Xiao-Hai Liu, Sergi Gonzàlez-Solís, Feng-Kun Guo, Christoph Hanhart, Ulf-G. Meißner, Bing-Song Zou
NNew spectrum of negative-parity doubly charmed baryons:Possibility of two quasistable states
Mao-Jun Yan , , Xiao-Hai Liu , , ∗ Sergi Gonz`alez-Sol´ıs , Feng-Kun Guo , , † Christoph Hanhart , Ulf-G. Meißner , , and Bing-Song Zou , CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Institute for Advanced Simulation, Institut f¨ur Kernphysik and J¨ulich Centerfor Hadron Physics, Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany Helmholtz-Institut f¨ur Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universit¨at Bonn, D-53115 Bonn, Germany (Dated: November 16, 2018)The discovery of Ξ ++ cc by the LHCb Collaboration triggers predictions of more doubly charmed baryons. Bytaking into account both the P -wave excitations between the two charm quarks and the scattering of light pseu-doscalar mesons off the ground state doubly charmed baryons, a set of negative-parity spin-1/2 doubly charmedbaryons are predicted already from a unitarized version of leading order chiral perturbation theory. Moreover,employing heavy antiquark-diquark symmetry the relevant low-energy constants in the next-to-leading orderare connected with those describing light pseudoscalar mesons scattering off charmed mesons, which have beenwell determined from lattice calculations and experimental data. Our calculations result in a spectrum richerthan that of heavy mesons. We find two very narrow J P = 1 / − Ω Pcc , which very likely decay into Ω cc π breaking isospin symmetry. In the isospin-1/2 Ξ Pcc sector, three states are predicted to exist below 4.2 GeV withthe lowest one being narrow and the other two rather broad. We suggest to search for the Ξ Pcc states in the Ξ ++ cc π − mode. Searching for them and their analogues are helpful to establish the hadron spectrum. PACS numbers:
One of the most challenging problems in fundamentalphysics is to understand how the strong interaction, formu-lated in terms of quantum chromodynamics (QCD), organizesits spectrum observed as hadrons. The phenomenologicalconstituent quark model achieved a great success in describingthe majority of the hadron spectrum especially in the heavyquark sector until 2003 when a few hadrons were discoveredwith unexpected properties. Since then many hadronic reso-nances beyond the conventional quark model were discovered.The new hadrons discovered in 2003 include the scalarand axial-vector charm-strange mesons D ∗ s (2317) and D s (2460) [1, 2]. Their masses are far below the quarkmodel predictions [3]. The subsequent observations of broadcharm-nonstrange resonances D ∗ (2400) and D (2430) [4]brought more puzzles. Thanks to the recent developmentsin lattice QCD calculations of heavy-meson–light-meson sys-tems [5–11], to the precise experimental data of B − → D + π − π − [12], and to the theoretical analysis of these lat-tice and experimental data in the framework of unitarized chi-ral perturbation theory [5, 13–16], a consistent picture whichcan explain all the puzzles in these positive-parity charmedmesons has emerged [14]. In this picture, the D ∗ s (2317) and D s (2460) are mainly DK and D ∗ K bound states [17–22], respectively, and there are two nonstrange + states andtwo + states with isospin I = 1 / in the ranges of the D ∗ (2400) and D (2430) masses, respectively. According tothe heavy quark flavor symmetry, all of these states have theircorresponding counterparts in the bottom meson spectrum.These low-lying positive-parity heavy mesons owe their ex- ∗ Electronic address: [email protected] † Electronic address: [email protected] istence to hadron-hadron interactions. This scenario needs tobe checked against experimental and lattice results in otherrelated processes, in order to reveal the proper paradigm ofexcited heavy hadrons.The recent discovery of the doubly charmed baryon Ξ ++ cc with a mass of (3621 . ± . MeV in Λ + c K − π + π + finalstates by the LHCb Collaboration [23] opens new opportu-nities: First, this finding suggests the potential of discover-ing more low-lying doubly charmed baryons in the near fu-ture, and thus one needs to have a solid theoretical basis forthe corresponding spectrum. Second, one would expect thepositive-parity heavy mesons to have analogous counterpartsas negative-parity doubly-heavy baryons, since the scatteringof the pseudo-Nambu–Goldstone bosons (NGBs) ( π , K and η ) off heavy sources is universal at leading order (LO). More-over, employing an approximate symmetry of QCD even sub-leading terms can be fixed as detailed below.For a doubly heavy baryon, the distance between the twoheavy quarks QQ may be estimated as r d ∼ / ( m Q v Q ) , with v Q the heavy quark velocity. For an S -wave charm diquarkone finds m c v c ∼ MeV [24]. On the other hand the dis-tance of the light quark to the QQ pair is r q ∼ / Λ QCD , with Λ QCD ∼ MeV the scale of nonperturbative QCD. Thusone may expand in r d /r q ∼ . . To LO in this expansion the S -wave QQ diquark appears as a point-like color antitripletsource, similar to a heavy antiquark, and this leads to an ap-proximate heavy antiquark-diquark symmetry (HADS) [25].Diquarks with higher partial waves are spatially much moreextended, and such an approximation is not expected to workfor them. This approximate symmetry allows one to predictdoubly-heavy tetraquarks based on input from heavy mesonsas well as doubly and singly heavy baryons [26–28] and,more relevant to our work, to relate doubly heavy baryons a r X i v : . [ h e p - ph ] N ov to singly heavy mesons [24, 25, 29–36]. Therefore, one canconstruct a chiral effective field theory (EFT) describing theNGBs scattering off the ground state (positive-parity) dou-bly charmed baryons. The low-energy constants (LECs) insuch a theory can be connected with those in the EFT de-scribing NGBs scattering off ground state (negative-parity)anticharmed mesons. The latter has been extensively stud-ied [5, 14, 16, 19, 21, 37–43]. In particular, the LECs inthe next-to-leading-order (NLO) chiral Lagrangian have beenfixed by fitting to the lattice QCD results of several charmed-meson–light-meson S -wave scattering lengths [5], and theunitarized amplitudes using these inputs have been shown tobe in a remarkable agreement with lattice QCD energy lev-els [10] in the center-of-mass frame for the S -wave coupledchannels Dπ, Dη and D s ¯ K [13], to be consistent with thelattice energy levels [11] for the S -wave D ( ∗ ) K [15], andto describe well the precise LHCb measurements [12] of the Dπ angular moments for the decay B − → D + π − π − [14].The predicted lowest positive-parity bottom-strange mesonmasses [13] also agree nicely with the lattice QCD results [9].The existence of doubly charmed baryons analogous to the D ∗ s (2317) has been proposed in Ref. [44], and was recentlystudied by considering potentials at LO [45] or via light vec-tor meson exchange [46]. In this Letter, in addition to usingthe NLO potentials, we notice that the P -wave excitations be-tween the two heavy quarks have to be taken into account asdynamical degrees of freedom, leading to a distinct spectrumof novel states.We consider the S -wave interactions between NGBs andthe J P = 1 / + ground state doubly charmed baryonsin the energy region around the corresponding thresholds.We are interested in the sectors with (strangeness, isospin) ( S, I ) = ( − , and ( S, I ) = (0 , / , which have ψ cc φ =Ξ cc ¯ K, Ω cc η and Ξ cc π, Ξ cc η, Ω cc K , respectively, as the rele-vant two-body coupled channels. The coupled channel scat-tering amplitudes are collected in a T -matrix fulfilling unitar-ity, which can be written as [47–51] T ( s ) = [1 − V ( s ) G ( s )] − V ( s ) , (1)where s is center-of-mass energy squared. G ( s ) is a di-agonal matrix with the nonvanishing elements G ii ( s ) = G ( s, M ψ cc ,i , M φ,i ) being the scalar one-loop function in the i th channel depending on the corresponding doubly charmedbaryon and light meson masses M ψ cc ,i and M φ,i . The loopfunction carries the unitary cut, and is calculated using a once-subtracted dispersion relation with the subtraction constant a ( µ ) , where µ is an energy scale, [49] serves as a regula-tor of the ultraviolet divergence. The matrix V ( s ) stands forthe S -wave projection of the potentials. It is split into twoparts V ( s ) = V c ( s ) + V s ( s ) . V c ( s ) represents the contactterms derived from the chiral Lagrangian up to NLO takinga similar form as that for the charmed mesons [5, 38, 52]with the charmed meson fields replaced by those of thedoubly charmed baryons. The HADS relates the involvedLECs ( c , ,..., ) to those in the charmed meson Lagrangian( h , ,..., ), as can be easily worked out with the superfield for- malism of Refs. [24, 35]: c i = h i M D , i = 0 , , , , (2)where c = c + c ¯ M ψ cc and c = c + c ¯ M ψ cc . Here, ¯ M D and ¯ M ψ cc are the averaged masses of the ground statecharmed mesons and doubly charmed baryons, respectively.For recent studies of doubly charmed baryons in chiral pertur-bation theory, we refer to Refs. [53–55]. Furthermore, V s ( s ) contains s -channel doubly charmed-baryon exchange poten-tials as discussed below.The lowest excitations of doubly charmed baryons are dueto the P -wave excitation inside the cc diquark. Since the po-tential inside the color antitriplet cc diquark is believed to behalf of that between the c and ¯ c in a charmonium, one expectsthat the P -wave excitation energy is roughly half of that forcharmonia [35], i.e. , M ψ Pcc − M ψ cc (cid:39) ( M h c − M J/ψ ) / MeV, where ψ Pcc denotes the doubly charmed baryonswith a P -wave diquark excitation. This value is similar to thatcalculated in quark models, see, e.g., Refs. [56–58]. With theexcitation energy being of O ( M π ) , the ψ Pcc baryons have to beincluded explicitly as dynamical degrees of freedom. There-fore, for a proper description of the low-energy ψ cc φ interac-tions, we need the S -wave coupling [24, 35] L P = λ ¯ ψ Pcc γ µ u µ ψ cc + h.c., (3)where ψ Pcc = (Ξ P ++ cc , Ξ P + cc , Ω P + cc ) T represents the dou-bly charmed baryons with a P -wave cc diquark, and u µ = −√ ∂ µ φ/F + O ( φ ) is the axial current. Here, F de-notes the pion decay constant in the chiral limit, and φ = (cid:80) i =1 λ i φ i / √ , with λ i the Gell-Mann matrices, collects theSU(3) NGB octet. Fermi statistics fixes the total spin ofthe cc diquark in the ground state ψ cc and in the ψ Pcc to be and 0, respectively. Thus, the transition ψ Pcc → ψ cc φ needs a flip of the charm quark spin, breaking heavy quarkspin symmetry, and the dimensionless coupling constant λ should be λ = O (Λ QCD /m c ) (cid:28) . The tree-level ampli-tude for ψ icc ( p ) φ i ( p ) → ψ fcc ( p ) φ f ( p ) from exchanging a ψ Pcc reads V s = 2 λ F C ( s ) ¯ u f ( p , σ f ) /p /P − ˚ M ψ Pcc /p u i ( p , σ i ) , (4)where σ i ( σ f ) indicates the polarization of the initial (final)state baryon, P = p + p = p + p , and the coupled channelcoefficients C ( s ) are given in matrix form as (cid:32) − √ − √ (cid:33) , and
23 12 √ √ √
62 1 √ (5)for ( S, I ) = ( − , and ( S, I ) = (0 , / , respectively.The S -wave projection of V s gives the elements of the ma-trix V s ( s ) . It is worthwhile to notice that, analogous to thecharmed meson case [59, 60], the u -channel exchange of dou-bly charmed baryons gives a negligible contribution to the S -wave ψ cc φ scattering, as checked in Ref. [45]. Γ [ M e V ] Ω cc P , H Ω cc P , L M [ M e V ] λ BC λ RC M Ξ cc + M K _ M Ω cc P ∘ λ FIG. 1: The widths (upper panel) and the masses of the two low-est / − Ω Pcc states (lower panel) depending on the value of λ withisospin symmetry imposed. The green, orange and red bands corre-spond to the cases of bound state, virtual states and resonance, re-spectively. The bands are obtained by taking into account uncertain-ties of the subtraction constant and the LECs determined in Ref. [5]. The values of LECs are fixed from Eq. (2). The values ofthe h i have already been fixed from fitting to the lattice resultsfor several charmed-meson–NGB scattering lengths at a fewpion masses [5], which lead to the prediction of +18 − MeVfor the mass of the D ∗ s (2317) . Using the matching prescrip-tion in Refs. [21, 61], the subtraction constant a ( µ ) in thecharmed meson sector [5] is translated to the doubly charmedsector as a ψ cc φ (1 GeV ) = − . +0 . − . .As input for the hadron masses we take the isospin averagedvalues for all the mesons involved and use 3621.4 MeV [23]for the Ξ cc . For the ground state Ω cc we use a mass of MeV fixed by requiring M Ω + cc − M Ξ + cc = M D + s − M D + from HADS [33]. The quark model prediction from Ref. [56],which correctly predicted the Ξ cc mass, is used as the baremass of Ξ Pcc , i.e., ˚ M Ξ Pcc = 3838
MeV, corresponding to the P -wave diquark excitation energy being 217 MeV. And weuse ˚ M Ω Pcc (cid:39) M Ω cc + 217 MeV (cid:39)
MeV. The symbol ˚ M is used to emphasize that these values are the bare masses forthe / − states without the ψ cc φ dressing, to be distinguishedfrom the pole masses from the coupled channel dynamics inthe following. The only free parameter is the coupling λ inthe s -channel potential.The masses and widths of the low-lying / − doublycharmed baryons can be obtained by searching for poles ofthe coupled channel T -matrix with the corresponding quan-tum numbers. Depending on the channels and parameters,there can be real bound state poles in the first Riemann sheetof the complex energy plane, and/or poles in the second Rie-mann sheet (corresponding to a virtual state if the pole is realand below threshold, and a resonance if the pole is complex).The position of a real pole gives the mass of a physical state,and for a resonance, the pole is denoted as M − i Γ / with M the mass and Γ the width.We first focus on the sector with ( S, I ) = ( − , and λ = 0 . Then, in addition to the Ω Pcc with a P -wave cc ex-citation, one finds a pole below the Ξ cc ¯ K threshold from the Γ [ k e V ] ( a ) Ω cc P , H λ Γ [ k e V ] ( b ) Ω cc P , L FIG. 2: Isospin symmetry-breaking decay width of the higher Ω P,Hcc (a) and the lower Ω P,Lcc (b). Ξ cc ¯ K – Ω cc η coupled channel dynamics at about 4.07 GeV,analogous to the D ∗ s (2317) . The pole couples dominantly to Ξ cc ¯ K . As long as λ takes a nonvanishing value, as it should,the two states will mix with each other. It is expected that thestate from the P -wave diquark excitation gets pushed downand the dynamically generated state is pushed up (denotedby Ω P,Lcc and Ω P,Hcc , respectively). When λ is larger than acritical value λ BC , the higher pole Ω P,Hcc will change froma bound state to a virtual state. Increasing λ further, Ω P,Hcc will become a resonance with the critical value denoted by λ RC , see Fig. 1. Such a behavior for an S -wave pole has al-ready been observed in the study of the quark mass depen-dence of the lightest scalar meson f (500) [62] and of thescalar charmed mesons [38, 60]. The mass of Ω P,Lcc decreasesmonotonically. As already discussed, the natural value for λ should be O (Λ QCD /m c ) = O (0 . . From Fig. 1, one seesthat if λ (cid:46) . , both / − Ω Pcc states are below the Ξ cc ¯ K threshold. In this case, the only allowed strong decay mode is Ω cc π which breaks isospin symmetry. Therefore, both statesare expected to be very narrow.For an S -wave bound state with a small binding energy,the so-called compositeness [63–68] measures the probabil-ity of finding the composite component in the wave functionof the physical state. Here, one can evaluate the Ξ cc ¯ K com-positeness in Ω P,Hcc by using − g cc ¯ K ∂G Ξ cc ¯ K /∂s at the poleof Ω P,Hcc , where g cc ¯ K is the residue of the T -matrix elementfor the elastic Ξ cc ¯ K channel. It is found that Ω P,Hcc containsaround 55%–80% of Ξ cc ¯ K when it is below the Ξ cc ¯ K thresh-old.If we use different values for the so far unobserved doublycharmed baryons, numerical results will change. However, thegeneral mixing picture shown in Fig. 1 remains. For instance,the critical value λ BC changes to 0.40 if we increase ˚ M Ω Pcc by40 MeV and keep all the other masses fixed. This is consistentwith the expectation that the closer ˚ M Ω Pcc to the dynamicallygenerated pole the stronger the mixing and thus the smaller λ BC .An anomalously large isospin-breaking partial decay width Γ( D ∗ s (2317) → D + s π ) of about 100 keV [5, 16, 52, 69–71]can be taken as an evidence for the D ∗ s (2317) to be mainlya DK molecule rather than a P -wave c ¯ s meson. This pre- ▲▲▲▲▲▲▲ ■■■ ■ ■ ■ ■ ●●●●●●● [ GeV ] Γ [ M e V ] ▲ Ξ cc P ,1 ■ Ξ cc P ,2 ● Ξ cc P ,3 λ = λ = λ = λ = λ = λ = FIG. 3: Trajectories of the three resonance poles in the ( S, I ) =(0 , / channel by changing the λ value. Central values of LECsand a ψ cc φ are adopted, and ˚ M Ξ Pcc = 3838
MeV [56] is used. diction will be checked at the PANDA experiment [72]. Sim-ilarly, once the / − Ω Pcc states will be discovered, one ex-pects their isospin-breaking decay widths to be also impor-tant to reveal their nature. The reason is that in the hadronicmolecule case, the isospin mass splittings of the constituenthadrons play a dominant role in driving an isospin-breakingdecay width much larger than the one generated by the π - η mixing only. In order to calculate these tiny widths, one needsto work in the particle basis instead of the isospin basis. Thereare four channels: Ω + cc π , Ξ ++ cc K − , Ξ + cc ¯ K , and Ω + cc η . Wetake the central values of all the meson masses from Ref. [73],and M Ξ ++ cc − M Ξ + cc = (2 . ± . MeV from a lattice QCDcomputation [74]. Note that due to the interference betweenthe electromagnetic and m d − m u contributions [33], M Ξ ++ cc is a bit larger than M Ξ + cc . This implies that the Ξ cc and kaonisospin splittings contribute in opposite directions, so that theisospin-breaking decay width of the Ω P,Hcc should be smallerthan that of the D ∗ s (2317) when λ = 0 . This expectation isconfirmed by the explicit calculations as shown in Fig. 2. Itis found that the lower Ω P,Lcc gets a width of a few keV, whilethe width for the higher Ω P,Hcc is larger than keV. The errorbands in Fig.2 come from the uncertainties of the subtractionconstant, of the LECs and of M Ξ ++ cc − M Ξ + cc .Now let us turn to the sector with ( S, I ) = (0 , / in theisospin symmetric limit. Three resonance poles are found inthe complex energy plane. Their positions with different λ values are displayed in Fig. 3, where ˚ M Ξ Pcc = 3838
MeV [56]is used. As can be seen, the lowest pole Ξ P, cc originates fromthe P -wave cc excitation, and it has a small width less than40 MeV. The seeds of the two broad poles Ξ P, cc and Ξ P, cc are the doubly charmed baryon counterparts of the two polesfound in the coupled channel Dπ , Dη and D s ¯ K scatteringamplitudes [13, 14, 21, 37] belonging to the SU(3) flavortriplet and antisextet, respectively. Analogously, Ξ P, cc and Ξ P, cc couple most strongly to Ξ cc π and Ω cc K , respectively.Increasing λ will make M Ξ P, cc and M Ξ P, cc smaller and push M Ξ P, cc to larger values. When λ is small, the masses of Ξ P, cc and Ξ P, cc are close. Therefore, in experiments where these particles can be produced, one would expect to see in the Ξ cc π invariant mass distribution a narrow peak on top of a broadbump. Depending on the interference from coupled channels,there might also be a dip. The only allowed strong decay chan-nel for both Ξ P, cc and Ξ P, cc is Ξ cc π . The natural channel tosearch for them is the Ξ ++ cc π − . Presumably, the values of λ and the bare masses will be first determined from measuringthe masses and widths of the lowest Ξ Pcc states. Then the restof the spectrum can be predicted.Note that in the results presented no corrections to the as-sumed HADS were included. Those corrections can lead tovariations of O ( r d /r q ) ∼ in the LECs of the NLO inter-actions. While this in principle can lead to moderate quanti-tative deviations from the predictions given above, these cor-rections should not change the overall picture that is domi-nated by the leading interactions, fixed completely by the chi-ral symmetry of QCD, and the interplay with the s -channelpoles.In summary, we investigated the low-lying spectrum of thedoubly charmed baryons with J P = 1 / − by studying the S -wave ψ cc φ interactions in channels with ( S, I ) = ( − , and ( S, I ) = (0 , / using a unitarized coupled channelapproach based on chiral effective Lagrangians up to NLO.The HADS is used to relate the NLO parameters to those inthe charmed meson sector which have already been fixed andtested. The essential new point in this paper is that, in additionto the meson-baryon channels, the P -wave cc diquark excita-tions have to be taken into account as dynamical degrees offreedom. As a result, the spectrum of / − doubly charmedbaryons becomes richer than that known for positive-paritycharmed and bottom mesons, and is also predicted to be dif-ferent than predictions from quark models. The numericalresults depend on inputs for the unobserved doubly baryonmasses, of which rough estimates are known, and on one un-known coupling λ = O (Λ QCD /m c ) (cid:28) . When λ (cid:46) . ,which is likely, there exist two / − Ω Pcc whose only strongdecay mode is the isospin breaking Ω cc π . Thus, both statesshould be very narrow. In the ( S, I ) = (0 , / sector thereare three / − Ξ Pcc states below 4.2 GeV. The lowest one hasa narrow width while the other two are rather broad. We sug-gest to search for the lower states in the Ξ ++ cc π − decay mode.It is expected that the / − doubly charmed baryons and the (1 / , / − doubly bottom and charm-bottom baryons pos-sess the same pattern.Searching for these particles and their analogues in futureexperiments will be helpful to establish the proper paradigmfor excited hadrons. Given that LHCb already observed the Ξ ++ cc , we expect to see more exciting results in the near futureon doubly charmed baryons. Acknowledgments
Helpful discussions with Andreas Wirzba and De-LiangYao are gratefully acknowledged. We thank Zhen-Wei Yangfor discussions on the experimental perspective, and thankPedro Fernandez-Soler for pointing out a typo in a previ-ous version. This work is supported by the National Nat-ural Science Foundation of China (NSFC) and DeutscheForschungsgemeinschaft (DFG) through funds provided tothe Sino–German Collaborative Research Center “Symme-tries and the Emergence of Structure in QCD” (NSFC GrantNo. 11621131001, DFG Grant No. TRR110), by the NSFC(Grant No. 11747601), by the Thousand Talents Plan forYoung Professionals, by the CAS Key Research Program of Frontier Sciences (Grant No. QYZDB-SSW-SYS013), bythe CAS Key Research Program (Grant No. XDPB09), bythe CAS President’s International Fellowship Initiative (PIFI)(Grant Nos. 2017PM0031 and 2018DM0034), by the CASCenter for Excellence in Particle Physics (CCEPP), and bythe VolkswagenStiftung (Grant No. 93562). [1] B. Aubert et al. 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