New spectrum of negative-parity doubly charmed baryons: Possibility of two quasistable states

The discovery of $\Xi_{cc}^{++}$ by the LHCb Collaboration triggers predictions of more doubly charmed baryons. By taking into account both the $P$-wave excitations between the two charm quarks and the scattering of light pseudoscalar mesons off the ground state doubly charmed baryons, a set of negative-parity spin-1/2 doubly charmed baryons 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 order are connected with those describing light pseudoscalar mesons scattering off charmed mesons, which have been well determined from lattice calculations and experimental data. Our calculations result in a spectrum richer than that of heavy mesons. We find two very narrow $J^P=1/2^-$ $\Omega_{cc}^P$, which very likely decay into $\Omega_{cc}\pi^0$ breaking isospin symmetry. In the isospin-1/2 $\Xi_{cc}^P$ sector, three states are predicted to exist below 4.2~GeV with the lowest one being narrow and the other two rather broad. We suggest to search for the $\Xi_{cc}^{P}$ states in the $\Xi_{cc}^{++}\pi^-$ mode. Searching for them and their analogues are helpful to establish the hadron spectrum.


I. INTRODUCTION
One of the most challenging problems in fundamental physics is to understand how the strong interaction, formulated in terms of quantum chromodynamics (QCD), organizes its spectrum observed as hadrons. The phenomenological constituent quark model achieved a great success in describing the majority of the hadron spectrum especially in the heavy quark sector until 2003 when a few hadrons were discovered with unexpected properties. Since then many hadronic resonances beyond the conventional quark model were discovered, and various exotic or non-exotic models were proposed to explain the internal structure of these resonances.
The new hadrons discovered in 2003 include the scalar and axial-vector charm-strange mesons D * s0 (2317) and D s1 (2460) [1,2]. Their masses are far below the quark model predictions [4] for the lowest charm-strange mesons with the corresponding quantum numbers. The subsequence observations of broad charm-nonstrange resonances D * 0 (2400) and D 1 (2430) [3] brought more puzzles. Thanks to the recent developments in lattice QCD calculations of heavy-mesonlight-meson systems [5][6][7][8][9][10][11], to the precise experimental data of B − → D + π − π − [12], and to the theoretical analysis of these lattice and experimental data in the framework of unitarized chiral perturbation theory [5,[13][14][15][16], a consistent picture which can explain all the puzzles in these positive parity charmed mesons has emerged [14]. In this picture, the D * s0 (2317) and D s1 (2460) are mainly DK and D * K bound states [17][18][19][20][21][22], respectively, and there are two nonstrange 0 + states and two 1 + states with isospin I = 1/2 in the ranges of the D * 0 (2400) and D 1 (2430) masses, respectively. According to the heavy quark flavor symmetry, all of these states have their corresponding counterparts in the bottom meson spec-trum. These low-lying positive-parity heavy mesons owe their existence to hadron-hadron interactions. This scenario needs to be checked against experimental and lattice results in other related processes.
The recent discovery of doubly charmed baryon Ξ ++ cc with a mass of (3621.40±0.78) MeV in Λ + c K − π + π + final states by the LHCb Collaboration [23] opens new opportunities in further testing the scenario mentioned above: First, this finding suggests the potential of discovering more low-lying doubly charmed baryons at the LHC in the near future, and thus one needs to have a solid theoretical basis for the corresponding spectrum. Second, one would expect the positive-parity heavy mesons to have analogous counterparts as negative-parity doubly-heavy baryons, since the scattering of the pseudo-Nambu-Goldstone bosons (NGBs) of the spontaneous breaking of chiral symmetry in QCD (π, K and η) off heavy sources in universal at leading order. Moreover, employing an approximate symmetry of QCD even subleading terms can be fixed as detailed below.
For a doubly-heavy baryon, the distance between the two heavy quarks QQ may be estimated as r d ∼ 1/(m Q v Q ), with v Q the heavy quark velocity. For doubly charmed baryons one finds m c v c ∼ 800 MeV [32]. On the other hand the distance of the light quark to the QQ-pair may be written as r q ∼ 1/Λ QCD , with Λ QCD ∼ 250 MeV the scale of nonperturbative QCD. Thus one may expand in r d /r q ∼ 0.3. To leading order in this expansion the QQ diquark appears as a point-like color antitriplet source, similar to a heavy antiquark, and this leads to an approximate heavy antiquarkdiquark symmetry (HADS) [24]. This approximate symmetry allows one to predict doubly heavy tetraquarks based on input from heavy mesons as well as doubly and singly heavy baryons [25,26] (see also Ref. [27]) and, more relevant to our work, to relate doubly heavy baryons to those of singly-heavy mesons [24,[28][29][30][31][32][33][34][35][36]. Therefore, one can construct a chiral effective field theory (EFT) describing the NGBs scattering off doubly charmed baryons (or antibaryons). The low-energy constants (LECs) in such a theory can be connected with those in the EFT describing NGBs scattering off anticharmed (or charmed) mesons. The latter has been extensively studied [5,14,16,19,21,[37][38][39][40][41][42][43]. In particular, the LECs in the next-to-leading-order (NLO) chiral Lagrangian have been fixed by fitting the lattice QCD results of several charmedmeson-light-meson S-wave scattering lengths [5], and the unitarized amplitudes using these inputs have been shown to be in a remarkable agreement with lattice QCD energy levels [10] in the center-of-mass frame for the S-wave coupled channels Dπ, Dη and D sK [13], to be consistent with the lattice energy levels [11] for the S-wave D ( * ) K [15], and to describe well the precise LHCb measurements [12] of the Dπ angular moments for the decay B − → D + π − π − [14]. As will be discussed in this Letter, the masses and decay widths of a set of negative-parity doubly heavy baryons can then be predicted. In fact, the existence of doubly-charmed baryons analogous to the D * s0 (2317) has been proposed in Ref. [44], and was recently studied by considering only the leading order (LO) potentials [45] or potentials via light vector meson exchange [46]. In this Letter, in addition to using the NLO potentials, we notice that the P -wave excitations between the two heavy quarks have to be taken into account as dynamical degrees of freedom. As a result, a distinct spectrum of the J P = 1/2 − Ω P cc baryons is predicted: In the most likely region of the parameter space for an unknown coupling constant, there are two quasi-stable 1/2 − Ω P cc baryons, both of which are below the Ξ ccK threshold and thus the only allowed strong decay mode, the Ω cc π 0 final state, breaks isospin symmetry. We also predict three 1/2 − Ξ * cc baryons below 4.2 GeV.

II. THEORETICAL FRAMEWORK
We consider the S-wave interactions between the lightest pseudoscalar mesons and the J P = 1/2 + ground state doubly charmed baryons in the energy region around the corresponding thresholds. We are interested in the sectors with (strangeness, isospin) (S, I) = (−1, 0) and (S, I) = (0, 1/2), which have Ξ ccK , Ω cc η and Ξ cc π, Ξ cc η, Ω cc K, respectively, as the relevant two-body coupled channels. The coupledchannel scattering amplitudes are collected in a T -matrix fulfilling unitarity, which can be written as [47][48][49][50][51] where s is center-of-mass energy squared. G(s) is a diagonal matrix with the nonvanishing elements G ii (s) = G(s, M ψcc,i , M φ,i ) being the scalar one-loop function in the i th channel depending on the corresponding doubly charmed baryon and light meson masses M ψcc,i and M φ,i . The loop function carries the unitary cut, and is calculated using a oncesubtracted dispersion relation with the subtraction constant a(µ) with µ an energy scale [49] serves as a regulator of the ultraviolet divergences. The matrix V(s) stands for the S-wave projection of the potentials. It is split into two parts V(s) = V c (s) + V s (s). V c (s) represents the contact terms derived from the chiral Lagrangian up to NLO taking a similar form as that for the charmed mesons [5,38,52] with the charmed meson fields replaced by those of the doubly charmed baryons. The HADS relates the involved LECs (c 0,1,...,5 ) to those in the charmed meson Lagrangian (h 0,1,...,5 ), as can be easily worked with the superfield formalism of Ref. [32,35]. For recent studies of doubly charmed baryons in chiral perturbation theory, we refer to Refs. [53][54][55]. Furthermore, V s (s) contains s-channel doubly-baryon exchange potentials as discussed below. The lowest excitations of the doubly charmed baryons are due to the P -wave excitation inside the cc diquark. Since the potential inside the color antitriplet cc diquark is believed to be half of that between the c andc in a charmonium, one expects that the P -wave excitation energy is roughly half of that for charmonia [35], i.e., M ψ P cc − M ψcc (M hc − M J/ψ )/2 = 214 MeV, where ψ P cc denotes the doubly charmed baryons with a P -wave diquark excitation. This value is similar to that calculated in quark models, see, e.g., Ref. [56]. With the excitation energy being O(M π ), the ψ P cc baryons have to be included explicitly as dynamical degrees of freedom. Therefore, for a proper description of the low-energy ψ cc φ interactions, we need the S-wave coupling [32,35] where ψ P cc = (Ξ P ++ cc , Ξ P + cc , Ω P + cc ) T represents the doubly charmed baryons with a P -wave cc diquark, and u µ = is the axial current. Here, F 0 denotes the pion decay constant in the chiral limit, and φ = 2, with λ i the Gell-Mann matrices, collects the SU(3) NGB octet. Fermi statistics fixes the total spin of the cc diquark in the ground state ψ cc and in the ψ P cc to be 1 and 0, respectively. Thus, the transition ψ P cc → ψ cc φ needs a flip of the charm quark spin, breaking heavy quark spin symmetry (HQSS), and the dimensionless coupling constant where σ i (σ f ) indicates the polarization of the initial (final) state baryon, P = p 1 + p 2 , and the coupled-channel coefficients C (s) are given in matrix form as for (S, I) = (−1, 0) and (S, I) = (0, 1/2), respectively. The S-wave projection of V s gives the elements of the matrix V s (s). It is worthwhile to notice that, analogous to the charmed meson case [57,58], the u-channel exchange of doubly charmed baryons gives a negligible contribution to the Swave ψ cc φ scattering, as checked in Ref. [45].

III. POLES IN THE COUPLED-CHANNEL DYNAMICS
Combining the doubly charmed baryons and anticharmed mesons into a single superfield and then expanding the pertinent effective Lagrangians following Refs. [32,35], one finds the HADS relations for the LECs where and c 5 = c 5M 2 ψcc . Here,M D andM ψcc are the averaged masses of the ground state charmed mesons and doubly charmed baryons, respectively. The values of the h i have already been fixed from fitting to the lattice results for several charmed-meson-NGB scattering lengths at a few pion masses [5], which lead to the prediction of 2317 +18 −28 MeV for the mass of the D * s0 (2317). This set of parameters has been shown [13] to be able to postdict the I = 1/2 coupledchannel (Dπ, Dη, D sK ) finite-volume energy levels calculated by the Hadron Spectrum Collaboration at a pion mass of about 391 MeV [10], and to yield a remarkable description [14] of the precise data of the D + π − angular moments for the process B − → D + π − π − [12]. Using the matching prescription in Refs. [21,59], the subtraction constant a(µ) in the charmed meson sector [5] is translated to the doubly charmed sector as a ψccφ (1 GeV) = −2.79 +0.04 −0.05 . As input for the hadron masses we take the isospin averaged values 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 3725 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 bare mass of Ξ P cc , i.e.,M Ξ P cc = 3838 MeV, corresponding to the P -wave diquark excitation energy being 217 MeV (a value of 225 MeV was used in Ref. [35]). And we useM Ω P cc M Ωcc + 217 MeV 3942 MeV.M is used to emphasize that these values are the bare masses for the 1/2 − states without the ψ cc φ dressing, to be distinguished from the pole masses from the coupled-channel dynamics in the following.
The only free parameter is the HQSS breaking coupling λ in the s-channel potential, and it is expected to be of O(Λ QCD /m c ) 1. The masses and widths of the low-lying 1/2 − doubly charmed baryons can be obtained by searching for poles of the coupled-channel T -matrix with the corresponding quantum numbers. Depending on the channels and parameters, there can be real bound state poles in the first Riemann sheet of the complex energy plane, and/or poles in the second Riemann sheet (corresponding to a virtual state if the pole is real and 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/2 with M the mass and Γ the width.
We first focus on the sector with (S, I) = (−1, 0) and λ = 0. Then, in addition to the Ω P cc with a P -wave cc excitation, one finds a pole below the Ξ ccK threshold from the Ξ ccK -Ω cc η coupled-channel dynamics at about 4.07 GeV,  [56]. The higher horizontal dotted line represents the ΞccK threshold, and the two vertical lines represent λBC and λRC defined in the text. For the higher Ω P,H cc , the green, orange and red bands correspond to the cases of bound state, virtual states and resonance, respectively. Note that the line for the virtual state formally continues also for lower values of λ as indicated by the dashed continuation, however, it looses its impact on observables. The bands are obtained by taking into account uncertainties of the subtraction constant and the LECs determined in Ref. [5].
analogous to the D * s0 (2317). The pole couples dominantly to Ξ ccK . As long as λ takes a nonvanishing value, as it should, the two states will mix with each other. It is expected that the state from the P -wave diquark excitation gets pushed down and the dynamically generated state is pushed up (denoted by Ω P,L cc and Ω P,H cc , respectively). When λ is larger than a critical value λ BC , the higher pole Ω P,H cc will change from a bound state to a virtual state. Increasing λ further, Ω P,H cc will become a resonance with the critical value denoted by λ RC . Such a behavior for an S-wave pole has already observed in the study of the quark mass dependence of the lightest scalar meson f 0 (500) [60] and of the scalar charmed mesons [38,58]. The trajectory of the higher pole as a function of λ shown in Fig. 1 displays exactly such a behavior. The mass of the lower state decreases monotonously. As already discussed, λ should be much smaller than 1, and its natural value is O(Λ QCD /m c ) = O(0.2). From Fig. 1, one sees that if λ 0.45, both 1/2 − Ω P cc states are below the Ξ ccK threshold. In this case, the only allowed strong decay mode is Ω cc π 0 which breaks isospin symmetry. Therefore, both states are expected to be very narrow.
For an S-wave bound state with a small binding energy, the so-called compositeness [61][62][63][64][65][66]  around 55-80% of Ξ ccK when it is below the Ξ ccK threshold. If we use different values for the so far unobserved doubly charmed baryons, numerical results will change. However, the general mixing picture shown in Fig. 1 remains. For instance, the critical value λ BC changes to 0.40 if we increaseM Ω P cc by 40 MeV and keep all the other masses fixed. This is consistent with the expectation that the closerM Ω P cc to the dynamically generated pole the stronger the mixing and thus the smaller λ BC .
An anomalously large isospin-breaking partial decay width Γ(D * s0 (2317) → D + s π 0 ) of about 100 keV [5,16,52,[67][68][69] can be taken as an evidence for the D * s0 (2317) to be mainly a DK molecule rather than a P -wave cs meson. This prediction will be checked at the PANDA experiment in the future [70]. The reason is that in the hadronic molecule case, the D * s0 (2317) strongly couples to DK so that the mass splittings between the charged and neutral kaons and D mesons play a dominant role in driving an isospin breaking decay width much larger than the one generated by π 0 -η mixing only. Similarly, once the 1/2 − Ω P cc states will be discovered, one expects their isospin breaking decay widths to be also important to reveal their nature. In order to calculate these tiny widths, one needs to work in the particle basis instead of the isospin basis. There are four channels: Ω + cc π 0 , Ξ ++ cc K − , Ξ + ccK 0 and Ω + cc η. The isospin-breaking decay width receives contributions from both the π 0 -η mixing and the isospin mass splittings of doubly charmed baryons and kaons. We take the central values of all the meson masses from the Review of Particle Physics [71], and M Ξ ++ cc − M Ξ + cc = (2.16 ± 0.20) MeV from a lattice QCD computation [72]. Note that due to the interference between the 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 kaon isospin splittings contribute in opposite directions, so that the isospin-breaking decay width of the Ω P,H cc should be smaller than that of the D * s0 (2317) when λ = 0. This expectation is confirmed by the explicit calculations as can be seen from Fig. 2. It is found that the lower Ω P,L cc gets a width of a few keV, while the isospin breaking width for the higher Ω P,H cc is larger than 30 keV. The error bands in Fig.2 come from the uncertainties of the subtraction constant, of the LECs and of M Ξ ++ cc − M Ξ + cc . Now let us turn to the sector with (S, I) = (0, 1/2) in the isospin symmetric limit. Three resonance poles are found in the complex energy plane. Their positions with different λ values are displayed in Fig. 3, whereM Ξ P cc = 3838 MeV [56] is used. As can be seen, the lowest pole Ξ P,1 cc originates from the quark model state, and it has a small width less than 40 MeV. The seeds of the two broad poles Ξ P,2 cc and Ξ P, 3 cc are the doubly charmed baryon counterparts of the two poles found in the coupled-channel Dπ, Dη and D sK scattering amplitudes [13,14,21,37] belonging to the SU(3) flavor triplet and anti-sextet, respectively. Analogously, Ξ P,2 cc and Ξ P,3 cc couple most strongly to the Ξ cc π and Ω cc K, respectively. Increasing λ will make M Ξ P,1 cc and M Ξ P,3 cc smaller, and push M Ξ P,2 cc to larger values. When λ is small, the masses of Ξ P,1 cc and Ξ P,2 cc are close, thus 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 broad bump. Depending on the interference from coupled channels, there might also be a dip. The only allowed strong decay channel for both Ξ P,1 cc and Ξ P,2 cc is Ξ cc π. The natural channel to search for them the Ξ ++ cc π − . Presumably, the values of λ and the bare masses will be first determined from measuring the masses and widths of the lowest Ξ P cc states. Then the rest of the spectrum can be predicted.

IV. SUMMARY
In summary, under the assumption that the heavy antiquarldiquark symmetry is realized in nature, we investigated the low-lying spectrum of the doubly charmed baryons with J P = 1/2 − by studying the S-wave ψ cc φ interactions in channels with (S, I) = (−1, 0) and (S, I) = (0, 1/2) using a unitarized coupled-channel approach based on chiral effective Lagrangians up to NLO. Heavy antiquark-diquark symmetry is used to relate the parameters to those in the charmed meson sector which have already been fixed and tested. The essential new point in this Letter is that in addition to the mesonbaryon channels, the P -wave cc diquark excitations have to be taken into account as dynamical degrees of freedom. As a result, the spectrum of 1/2 − doubly charmed baryons becomes richer than that known for positive-parity charmed and bottom mesons, and is also predicted to be different than predictions from quark models. The numerical results depend on inputs for the unobserved doubly baryon masses, of which rough estimates are known, and on one unknown coupling λ = O(Λ QCD /m c ) 1. When λ 0.45, which is likely, there exist two 1/2 − Ω P cc whose only strong decay mode is the isospin breaking Ω cc π 0 . Thus, both states should be very narrow. In the (S, I) = (0, 1/2) sector there are three 1/2 − Ξ P cc states below 4.2 GeV. The lowest one has a narrow width while the other two are rather broad. We suggest to search for the lower states in the Ξ ++ cc π − decay mode. It is expected that the 3/2 − doubly charmed baryons and the (1/2, 3/2) − doubly bottom baryons possess the same pattern.
Searching for these particles and their analogues in future experiments will be helpful to establish the proper paradigm for excited hadrons. Given that LHCb already observed the Ξ + cc , we expect to see more exciting results in the near future on doubly charmed baryons.