Heavy Exotic Molecules with Charm and Bottom

We revisit the formation of pion-mediated heavy-light exotic molecules with both charm and bottom and their chiral partners under the general strictures of both heavy-quark and chiral symmetry. The chiral exotic partners with good parity formed using the $(0^+, 1^+)$ multiplet are about twice more bound than their primary exotic partners formed using the $(0^-,1^-)$ multiplet. The chiral couplings across the multiplets $(0^\pm, 1^\pm)$ cause the chiral exotic partners to unbind, and the primary exotic molecules to be about twice more bound, for $J\leq 1$. Our multi-channel coupling results show that only the charm isosinglet exotic molecules with $J^{PC}=1^{++}$ binds, which we identify as the reported neutral $X(3872)$. Also, the bottom isotriplet exotic with $J^{PC}=1^{+-}$ binds, which we identify as a mixture of the reported charged exotics $Z^+_b(10610)$ and $Z^+_b(10650)$. The bound isosinglet with $J^{PC}=1^{++}$ is suggested as a possible neutral $X_b(10532)$ not yet reported.


I. INTRODUCTION
A decade ago, both the BaBar collaboration [1] and the CLEOII collaboration [2] have reported narrow peaks in the D + s π 0 (2317 MeV) and the D * + s π 0 (2460 GeV) channels as expected from general chiral symmetry arguments [3,4]. In QCD the light quark sector (u, d, s) is dominated by the spontaneous breaking of chiral symmetry, while the heavy quark sector (c, b, t) is characterized by heavy-quark symmetry [5]. The combination of both symmetries led to the conclusion that the heavylight doublet (0 − , 1 − ) = (D, D * ) has a chiral partner (0 + , 1 + ) = (D,D * ) that is about one consituent mass heavier [3,4].
Recently, the Belle collaboration [6] and the BESIII collaboration [7] have reported the observations of multiquark exotics. A major provider for these exotics is Υ(10860) and its ideal location near the thresholds for BB * π (10744) and B * B * π (10790) decays. The smallness of the available phase space in the hadronic decay of Υ(10860) calls for a compound with a long life-time, perhaps in a molecular configuration with heavy meson constituents. Several heavy exotic molecules with quantum numbers uncommensurate with the excited states of charmonia and bottomia have been reported, such as the neutral X(3872) and the charged Z c (3900) ± and Z b (10610) ± . More of these exotics are expected to be unravelled by the DO collaboration at Fermilab [8], and the LHCb collaboration at Cern [9].
Theoretical arguments have predicted the occurence of some of these exotics as molecular bound states mediated by one-pion exchange much like deuterons or deusons [10,11]. A number of molecular estimates regarding the occurence of doubly heavy exotic mesons with both charm and bottom content were suggested by * Electronic address: yizhuang.liu@stonybrook.edu † Electronic address: ismail.zahed@stonybrook.edu many [11][12][13][14][15][16]. Non-molecular heavy exotics were also discussed using constituent quark models [17], heavy solitonic baryons [18,19], instantons [20] and QCD sum rules [21]. The molecular mechanism favors the formation of shallow bound states near treshold, while the non-molecular mechanism suggests the exitence of deeply bound states. The currently reported exotics by the various experimental collaborations are in support of the molecular configurations. The purpose of this paper is to revisit the formation of heavy-light molecules under the general strictures of chiral and heavy quark symmetry, including the mixing between the heavy doublets and their chiral partners which was partially considered in [11][12][13][14][15]. In leading order, chiral symmetry fixes the intra-and cross-multiplet couplings. In particular, bound moleculesDD with charm andBB with bottom may form through channel mixing, despite the absence of a direct pion coupling by parity. The P-wave inter-multiplet mixing in the (0 − , 1 − ) is enhanced by the almost degeneracy of the constituents by heavy-quark symmetry, while the S-wave cross-multiplet mixing in the (0 ± , 1 ± ) is still substantial due to the closeness of the constituents by chiral symmetry. The latter prevents the formation of dual chiral molecules such asDD with charm andBB with bottom, as we will show. Throughout, the coupling to the low-lying resonances in the continuum with more model assumptions will be ignored for simplicity. Also interactions mediated by shorter range massive vectors and axials will be mostly cutoff through the use of a core cutoff in the pion mediated potential of 1 GeV. Only the channels with total angular momenta J ≤ 1 will be discussed.
The organization of the paper is as follows: In section 2 we briefly derive the essential contruct for doubly charmed exotic molecules using the strictures of chiral and heavy quark symmetries and explicit the coupled channel problem for the lowest bound states. We also show how the same coupled channel problem carries to the chiral partners. In section 3, we extend our analysis to the doubly bottom exotic molecules and their chiral partners. Our conclusions are given in section 4.

II. CHARMED EXOTICS MOLECULES
The low energy effective action of heavy-light mesons interacting with pions is constrained by both chiral and heavy quark symmetry. In short, the leading part of the heavy-light Lagrangian for the charmed multiplet (0 − , 1 − ) with pions reads [3,5] with ∆m i = m i − m C of the order of a quark constituent mass. The molecular exotics of the type DD * and alike, follows from (1) through one-pion exchange. The nonrelativistic character of the molecules yield naturally to a Hamiltonian description. For all available 2-body channels, the pertinent matrix entries for the interaction are readily found in the form with the isospin factor The spin polarizations of D * and its conjugateD * are referred to as v and v * respectively. Here V (r) is the regulated one-pion exchange using the standard monopole form factor by analogy with the pion-nucleon form factor [22]. Denoting by D 00 ( r) the wave function of the molecular scalar, byȲ 0ī ( r) and Y i0 ( r) the wavefunctions of the molecular vectors, and by T ij ( r) the wavefunction of the molecular tensors, we can rewrite (2) as The explicit reduction of the molecular wavefunctions will be detailed below, for all channels with J ≤ 1.
The one-pion mediated interaction is defined with a core cutoff Λ m π [11,22] Once inserted in (4) it contributes a scalar and a tensor through which are shown in Fig. 1 for g H = 0.6 [3,4] and Λ = 1 GeV in units of Λ. The strength of the regulated onepion exchange potential increases with increasing cutoff Λ. The dependence of the results on the choice of core cutoff Λ is the major uncertainty of the molecular analysis to follow. The tensor contribution in (6) is at the origin of the notorious D-wave admixing in the deuteron state [22], and is distinctly different from the gluonic based exchanges in heavy quarkonia [17].

B.
(0 + , 1 + ) chiral partners and their mixing The leading part of the heavy-light chiral doublers Lagrangian for the charmed (0 + , 1 + ) multiplet with pions reads [3] with again ∆mĩ = mĩ − m C of the order of a quark constituent mass. The (0 + , 1 + ) multiplet mixes with the (0 − , 1 − ) by chiral symmetry. The leading part of the interaction in the chirally mixed parity channels reads [3,4] δL C. J=0 channels To analyze the coupled molecular ground states, we present the analysis for the J = 0 coupled channels. We first discuss the mixing in the (0 − , 1 − ) multiplet, followed by the mixing in the (0 + , 1 + ) chiral mirror multiplet, and finally the cross mixing between the (0 ± , 1 ± ) multiplets. The pertinent 0 P C channels with their spectroscopic S L J assignments are We have added the primes to track the different contributions in the numerical results below. Here, T SL,JM ij refers to the tensor spherical harmonics with spin S, orbital angular momentum L, and total angular momentum J and projection J z = M . As all JM = 00, we have omitted them in (9) for convenience. Also Y L 0i ≡ Y L,JM i refers to the vector spherical harmonics with orbital angular momentum L, total angular momentum J with J z = M . The explicit form of the properly normalized tensor and vector spherical harmonics in this case, are readily obtained as Here, we note that T 00 ij , T 11 ij , T 22 ij carries explicitly charge conjugation C = +. However,Ȳ 1 0k , Y 1 k0 carry C = ±. It is straightforward to project onto states of good C and rewrite the interactions to follow in this basis, but for J = 0 it is not needed, as only the C = + combination is seen not to vanish. It will not be the case for J = 1 as we will discuss below. The even-parity channels T 0 , T 2 , D 0 mix , and the odd-parity channels T 1 , Y 1 ,Ȳ 1 mix.
The mixing part of the interaction in the J P C = 0 ++ channel is We note that while the one-pion mediated DD → DD intraction in (11) vanishes by parity, the cross interactions DD → D * D * and D * D → DD * do not. As a result bound states through mixing DD → D * D * → DD could and will form in the same order. The corresponding mass shifts and kinetic terms are and dr 2 (rφ), m 12 = m 1 + m 2 and the corresponding reduced masses are The empirical masses are m D ± = 1.870 GeV, m D 0 = 1.865 GeV and m D * ± = 2.010 GeV, m D * 0 = 2.007 GeV. Below, we will use the averages over the isotriplets for m 1,2,3 . Specifically, m 1 = 1.005 GeV, m 2 = 0.934 GeV and m 3 = 0.968 GeV. Here m π = 137 MeV and f π = 93 MeV, with g H = 0.6 [3,4].
For the (0 + , 1 + ) multiplet, the classification of all the states remains the same. The relation between the matrix elements in the (0 − , 1 − ) sector and the (0 + , 1 + ) sector (primed below) can be made explicit if we note the relations With this in mind, the matrix elements between the different tensor projections are related as follows As a result we haveṼ 0++ = V 0++ and The kinetic contributionsK 0++ andK 0++ follows from (13)(14) with the appropriate substitution for the reduced masses. We will use the empirical masses for the reported chargeless doublet (D * 0 , D 0 1 ) with mD = 2.400 GeV and mD * = 2.420 GeV, which translate tom 1 = 1.210 GeV, m 2 = 1.200 GeV andm 3 = 1.205 GeV. Hereg H = 0.6 follows from heavy quark symmetry. The mixed coupling between the (0 − , 1 − ) and (0 + , 1 + ) induces a scalar interaction typically of the form δV (r) ≈ ∆m 2 V (r) with ∆m/m 1 ≈ 0.4/1.2 = 1/3. In the relevant range shown in Fig. 1, it is about the same as V 1 (r) and will be retained. The corresponding one-pion mediated potential in the 0 ++ is and in the 0 −± channel, is Here the empirical mass splittings are The stationary coupled channel problem for the ground states in J P C = 0 ++ and J P C = 0 −− , follows from the To proceed further, we need to solve the coupled channels problem numerically with (22) in each sector.
The pertinent projections onto the higher J P C channels of the molecular wavefunctions in (4) require the use of both vector and higher tensor spherical harmonics [23,24]. For J = 1, we will use the explicit forms quoted in [24] with the S L J assignment completly specified. For the (1 ∓ , 0 ∓ ) multiplets, there are 4 different 1 P C sectors with the JM labels omitted for convenience. The normalized tensor harmonics are [24] T 01,1m The DD * channels with definite charge conjugation C = ± are explicitly We note that In terms of the previous J = 1 channels, the onepion mediated interaction in the (0 − , 1 − ) multiplet in the J P C = 1 −− channel takes the block form with the blocks defined as Similarly, the one-pion mediated interaction in the J P C = 1 +− channel has the following block structure with each block defined as In the remaining J P C = 1 −+ and J P C = 1 ++ the one-pion mediated interactions are respectively given by and The one-pion mediated interaction within the (0 + , 1 + ) multiplet follows the same construct as in the (0 − , 1 − ) multiplet using the transfer rules in (16)(17). The same interaction across the two chiral multiplets introduces also a diagonal mixing of the form TheDD states with good charge conjugation follows from (25) through the substitution ± → ∓ only on the right hand side. The total Hamiltonian in the (0 + , 1 + ) sector to diagonalize is H = K + V . Including the chiral multiplet, the total Hamiltonian across the (0 ± , 1 ± ) sectors to diagonalize is twice larger H = K + V + W.

J. Results for charm exotic molecules
In the upper plot of Fig. 2 we show the typical Φ i radial components of the bound isosinglet charm wavefunction with energy E = 3.867 GeV for a cutoff Λ = 1 GeV, as a function of the radial distance r also in units of 1 GeV. The chiral cross coupling between the (0 − , 1 − ) and (0 + , 1 + ) multiplets induces a very small mixing to the molecular wavefunction in the (0 − , 1 − ) multiplet as displayed in Fig. 2. In the lower chart of Fig. 2 we show the percentage content of the contributions to the same wavefunction, with the S L J assignments referring to the (0 − , 1 − ) multiplet, and the SL J assignments referring to the (0 + , 1 + ) multiplet. The mixing results in a stronger binding in this channel wich is mostly an isosinglet 1 S 3 contribution in the (1 − , 0 − ) multiplet with almost no Dwave admixture. This molecular state carries J P C = 1 ++ assignment, and from our S L J assignments in (24) it is chiefly an isosinglet DD * molecule. We identify this state with the reported isosinglet exotic X(3872). The cross chiral mixing causes the dual chiral partnersDD * state to unbind. .
In the upper plot of Fig. 3 we show the typical Φ i radial components of the unbound isotriplet charm wavefunction for a cutoff Λ = 1 GeV, as a function of the radial distance r also in units of 1 GeV. The multi-channel coupling in the channel with J P C = 1 +− , shows that the dominant wave is 3 S 1 which is composed of a resonating isosinglet DD * (3876) compound. The wave shows a weak visible attraction near the origin that is not enough to bind. We conclude that the reported Z c (3900) ± is at best a resonance in the continuum in our analysis. All other channels are unbound for charm exotic molecules with our cutoff of 1 GeV, in both the isosinglet and isotriplet configurations

III. BOTTOM EXOTIC MOLECULES
Doubly bottom exotic molecules follow the same construction as before, all potentials and inteactions remains of the the same form , with now the new mass parameters The results for the chirally mixed states for the bottom exotic states involving the pair multiplet (0 ± , 1 ± ) can be obtained using similar arguments to those used for charm with the same cutoff choice. Since the one-pion exchange interaction is three times stronger in the isosinglet channel than the isotriplet channel, a multitude of isosinglet bottom exotic states will be revealed, thanks also to the heavier bottom mass and thus smaller kinetic energy in comparison to the charm exotic states.

A. Results for bottom exotic molecules
In Fig. 4 we show the behavior of the typical isosinglet bound state wavefunctions contributing in the J P C = 0 ++ channel with energy E = 10.509 GeV (Λ = 1 GeV) (upper-plots). The percentage content of the same wavefunction is displayed as a histogram with the appropriate parity labels in the lower display, with the S L J assignments referring to the (0 − , 1 − ) multiplet, and the SL J assignments referring to the (0 + , 1 + ) multiplet. From the assignments given in (9), we see that the 0 ++ mixed bound state is chiefly an isosinglet BB ( 1 S 0 ) molecule, with relatively small B * B * ( 1 S 0 ) and B * B * ( 5 D 0 ) admixtures.
In Fig. 5 we show the behavior of the typical isosinglet bound state wavefunctions contributing in the J P C = 0 −+ channel with energy E =10.555 GeV (Λ = 1 GeV) (upper-plots). The percentage content of the same wavefunction is displayed as a histogram with the appropriate parity labels in the lower display, with the S L J assignments referring to the (0 − , 1 − ) multiplet, and the SL J assignments referring to the (0 + , 1 + ) multiplet. From the assignments given in (9), we see that the 0 −+ mixed bound state is a mixed molecule with about equal admixture of BB * ( 3 P 0 ), B * B ( 3 P 0 ) and B * B * ( 3 P 0 ) molecules all from the (0 − , 1 − ) multiplet as those from the (0 + , 1 + ) are shown to decouple and unbind. The effect of the latters is to cause the formers to bind twice  more.
In Fig. 6 we show the behavior of the typical isosinglet bound wavefunctions contributing in the J P C = 1 ++ channel with energy E = 10.532 GeV (Λ = 1 GeV) (upper-plots). The percentage content of the same wavefunction is displayed as a histogram with the appropriate parity labels in the lower display, with the S L J assignments referring to the (0 − , 1 − ) multiplet, and the SL J assignments referring to the (0 + , 1 + ) multiplet. From the assignments given in (24), we see that the 1 ++ mixed bound state is chiefly a BB * ( 3 S 1 ), with small B * B * ( 5 D 1 ) and BB * ( 3 D 1 ) admixtures. We see again the decoupling of the molecular configurations with SL J assignments as they are found to unbind, leaving the S L J assignments twice more bound as per our calculation. A quick comparison between Fig. 2 and Fig. 6 shows that this neutral bottom molecular state is the mirror analogue of the neutral charm molecular state which we suggest as X b (10532).
In Fig. 7 we show the behavior of the typical isosinglet bound wavefunctions contributing in the J P C = 1 +− channel with energy E = 10.550 GeV (Λ = 1 GeV) (upper-plots). The percentage content of the same wavefunction is displayed as a histogram with the appropriate parity labels in the lower display, with the S L J assignments referring to the (0 − , 1 − ) multiplet, and the SL J assignments referring to the (0 + , 1 + ) multiplet. From the assignments given in (24), we see that the 1 +− mixed bound state are primarily B * B * ( 3 S 1 ) and BB * ( 3 S 1 ) molecules, with small B * B * ( 3 D 1 ) and BB * ( 3 D 1 ) molecular admixtures. The molecules are mostly from the (0 − , 1 − ) multiplet as those from the (0 + , 1 + ) are shown to decouple and unbind. Again, the effect of the latters is to cause the formers to bind twice more. In Fig. 8 we show the behavior of the typical isosinglet bound wavefunctions contributing in the J P C = 1 −− channel with energy E = 10.558 GeV (Λ = 1 GeV) (upper-plots). The percentage content of the same wavefunction is displayed as a histogram with the appropriate parity labels in the lower display, with the S L J assignments referring to the (0 − , 1 − ) multiplet only. From the assignments given in (24), we see that the 1 −− isosinglet bound state is mostly P-wave with equal admixture of B * B * ( 5 P 1 ), BB * ( 3 P 1 ) and BB ( 1 P 1 ) molecules. We note the clear repulsion of the P-waves near the origin. The molecules are mostly from the (0 − , 1 − ) multiplet as those from the (0 + , 1 + ) decouple and unbind. The effect of the latters is to cause the formers to bind twice more. This isosinglet molecular exotic is well below the reported Y b (10888).
In Fig. 9 we show the behavior of the typical isotriplet bound wavefunctions contributing in the J P C = 1 +− channel with energy E = 10.592 GeV (Λ = 1 GeV) (upper-plots). The percentage content of the same wavefunction is displayed as a histogram with the appropriate parity labels in the lower display, with the S L J assignments referring to the (0 − , 1 − ) multiplet, and the SL  assignments referring to the (0 + , 1 + ) multiplet. From the assignments given in (24), we see that the 1 +− mixed isotriplet bound state is mostly an S-state made primarily of BB * ( 3 S 1 ) molecules with a small admixture of B * B * ( 3 S 1 ) molecules. The molecules are mostly from the (0 − , 1 − ) multiplet as those from the (0 + , 1 + ) again decouple and unbind. We identify this exotic molecule as a mixed state of the reported pair of isotriplet exotics Z + b (10610) and Z + b (10650).

IV. CONCLUSIONS
We have analyzed molecular states of doubly heavy mesons mediated by one-pion exchange for both the chiral parteners (0 ± , 1 ± ) as a coupled channel problem, for all the molecular configurations with J ≤ 1. Our results show that the binding energy is sensitive to the cutoff used for the one-pion exchange interaction which is substantial in the lowest partial waves. All other parameters are fixed by symmetry and data. Our results complement and extend those presented in [11][12][13][14][15][16] by taking into account the strictures of chiral and heavy quark symmetry, and by retaining most coupled channels between the (0 − , 1 − ) multiplet and its chiral partner (0 + , 1 + ). The key aspect of this coupling is to cause the molecules in the (0 − , 1 − ) multiplet to bind about twice more, and the molecules in the (0 + , 1 + ) multiplet to unbind.
For channel couplings with J ≤ 1, we have found that only the charm isosinglet exotic molecules with J P C = 1 ++ is strictly bound for a pion-exchange cutoff Λ = 1 GeV. This state is identified with the reported isosinglet exotic X(3872) which in our case is mostly an isosinglet DD * molecule in the 1 S 0 channel with no Dwave admixture. The attraction in the isotriplet channel with J P C = 1 +− is too weak to bind the DD * compound, suggesting that the reported isotriplet Z C (3900) ± is at best a near treshold resonance. All other J P C assignments with charm for both the isotriplet and isosinglet are unbound. The noteworthy absence in our analysis of the Y (4260), Y (4360) and Y (4660) may point to the possibility of their constituents made of excited (D 1 , D 2 ) heavy mesons and their chiral partners [3,25], which we have not considered.
In contrast, and for the same choice of the cutoff, we have identified several isosinglet bottom exotic molecules in the J P C = 0 ±+ , 1 +± , 1 −− channels which are mostly admixtures of the heavy-light mesons in the (0 − , 1 − ) multiplet. We have only found one isotriplet bottom exotic molecule with J P C = 1 +− which we have identified with the pair Z + b (10610) and Z + b (10650), which is a mixed state in our analysis. The isosinglet bottom exotic molecule with J P C = 1 ++ is a potential candidate for X b (10532), the bottom analogue of the charm exotic X(3872).
Our results show that the cross chiral mixing between the (0 ± , 1 ± ) multiplets while strong, does not generate new mixed molecules of the type DD * and alike, as suggested in [13]. Rather, it prevents the formation of dual chiral molecules of the typeDD * and alike, which would be otherwise possible. In the process, it provides for a stronger binding of the low lying molecules in the (0 − , 1 − ) multiplet in comparison to the results in [14]. Most noteworthy, is the appearance of a single bound isosinglet J P C = 1 ++ charm exotic molecule in our analysis, with also one single bound isotriplet bottom exotic molecule with J P C = 1 +− but several isosinglet bottom exotic molecular states with J P C = 0 ±+ , 1 +± . The latters may transmute to broad resonances by mixing to bottomia with similar quantum numbers.
Clearly higher values of J > 1 may also be considered using the same construct, but the molecular configurations maybe too large to bind, a point in support of their absence in the currently reported experiments. The recoupling of the current bound state problem to the open channels with charmonia and bottomia is also important to consider, but requires a more extensive analysis of the multi-channel scattering problem. Finally, the extension of the present analysis to D s and B s molecules using the heavier eta-exchange [26], as well as exotic baryonic molecules shoud be of interest in light of the ongoing experimental programs.