On the existence of near threshold exotic hadrons containing two heavy quarks

Transcription

1 On the existence of near threshold exotic hadrons containing two heavy quarks Yiming Cai and Thomas Cohen Maryland Center for Fundamental Physics and the Department of Physics, University of Maryland, College Park, MD, USA Dated: January 18, 019 This paper focuses on tetraquark containing two heavy quarks. There are well-established model-independent arguments for the existence of deeply bound exotic q q tetraquark in the formal limit of arbitrarily large heavy quark masses. However, these previous arguments did not address the question whether parametrically narrow tetraquark states exist close to the threshold for break up into two heavy mesons. Here we present a model-independent argument that establishes the existence of parametrically narrow tetraquark states that are parametrically close to threshold. The argument here is based on Born-Oppenheimer and semi-classical considerations. their quantum numbers cannot be made as quark-antiquark or three-quark color-singlet states. For example, there is no way for an isospin meson to be constructed as a constituent quark-antiquark state; such sates must have an isospin less than or equal to unity. Such NE states, if observed in nature must be exotic independently of how one interprets their structure. Cryptoexotics are hadrons whose quantum numbers can be constructed as quark-antiquark or three quark states in a constituent quark model, but for which there is a substantial phenomenological reason to believe that they are not so constructed. A discussion of CE hadrons is complicated by the fact that cryptoexotics can mix with ordinary hadrons and accordingly the identification of CE hadrons has been beset by ambiguities. From the perspective of theory, one interesting class of results has been the demonstration of limits of CD in which clearly discernible quantum-number exotic hadrons must exist. For example, in the large number of colors N c limit, it can be shown that narrow hadronic resonance with the quantum numbers of hybrid mesons such as J P C = 1 + must exist[4]; their widths can be shown to scale with N c as 1/N c, which indicates self-consistently these hadrons are narrow at large N c thus clearly identifiable as distinct resonances. Demonstrations of this sort show clearly, that the structure of nonabelian gauge theories with quarks at the very least allows for some types of exotic hadrons. As such, this kind of demonstration lends some support to the view that CD may have exotic hadronic states of particular types. However, since the demonstrations are based on limits they are not definitive in describing the physical world. For example, while that large N c limit and the 1/N c expansion gives significant insight into the real world such as the OZI rule for typical hadronic interactions [5] and some semi-quantitative predictive power for some observables such as relationships between nucleon and observables that arise due to a contracted SUN f symmetry at large N c [6 8], it also fails to describe other phenomenon even crudely such as the large OZI violating effects associated with the anomaly that is responsible for the large η mass[9, 10]. It is not clear a priori whether the prediction of discernible NE hybrid mesons is in the class of observables that the large N c limit gets qualitatively right. There has been significant interest in exotic mesons over the decades. Until relatively recently, however, much of the interarxiv: v1 [hep-ph] 16 Jan 019 I. INTRODUCTION One of the central goals of hadronic physics is to understand the rich structure of hadron spectroscopy in terms of the underlying theory CD. Exotic hadrons offer an critical window into this physics. This paper focuses on exotic hadrons containing two heavy quarks. The goal is to demonstrate given a few plausible technical assumptions that in the formal heavy quark limit, there must exist parametrically narrow exotic hadrons parametrically close to the threshold for two heavy hadrons. While this result is not directly related to experimental candidates for exotics containing a c c pair near threshold, it may give some insight into them. The simplicity of the naive constituent quark model in which mesons are described as a quark-antiquark pair in a color singlet and baryons are modeled as a three-quark color singlet state[1] makes it a natural picture by which many hadronic physicist visualize hadrons. The model has proven to be a useful guide to understanding much but by no means all hadronic physics. Historically, the constituent quark model played a critical role in the development of quantum chromodynamics CD[], which is strongly believed to be the correct theory of the strong interactions. Ironically, despite this history, the relationship of the naive constituent quark model to CD remains quite unclear. In this context, the study of so-called exotic hadrons hadrons that cannot be described in the simplest constituent quark model in terms of quark-antiquark and three quark states are important. Such states include hybrid mesons which have valence gluons as well as a constituent quark-antiquark pair, glueballs and multiquark hadrons eg. tetraquarks or pentaquarks. Such states act to focus on physics contained in CD but not the simplest constituent quark model and hence give significant insight into the nature of CD. There are two basic types of exotics: quantum-number exotic NE hadrons and cryptoexotic CE hadrons[3]. uantum-number exotic hadrons are ones that by virtue of yiming@umd.edu cohen@physics.umd.edu

2 est the experimental and phenomenological side has focused on rather ambiguous situations in which various states are argued to be cryptoexotic hadrons. The reason for this focus is that there were comparatively few cases were there was strong evidence for quantum number exotic hadrons. Perhaps cleanest case for quantum number exotic hadrons are putative states with the quantum numbers of hybrid mesons. The best candidates [11] are for I G J P C = mesons: the π , π and π However, even in these cases there are experimental issues and issues of interpretation that make it hard to know unambiguously that the observations correspond to hadronic resonances. The situation has changed over the past 15 years. During this period there has been a renaissance in hadronic physics, driven by the discovery of the so-called X, Y and Z mesons and subsequently the P baryons[1, 13]. The various X, Y and Z resonances from their masses and decays generically look like some form of c c state. However, they are not expected in the standard charmonium picture, and thus might be identified as cryptoexotic hadrons. Similarly the P baryons, which also look like they contain a c c pair are, by definition, CE hadrons. However, in a very real sense, some of these states are more than merely cryptoexotic. Note that, in the realm of hadronic physics, CD dynamics suppresses the creation and annihilation of heavy quark pairs[14]. Indeed, this is a prerequisite for the viability of NRCD, which is a powerful tool for describing quarkonium states [15]. Thus to good approximation one can think of CD dynamics inside of hadrons as preserving the number of charm quarks and charm antiquarks separately. To the extent that this is sensible, then the number of heavy quarks and the number of heavy quarks and heavy anti-quarks in a state act as conserved quantum numbers. A critical feature of many of the putative exotics containing a charm-anticharm pair is the fact that many are quite close to the threshold for breakup into heavy mesons with one containing a charm quark and the other charm antiquark. One important question is the extent to which near-threshold exotic mesons containing heavy quarks is a generic feature of CD. While the current experimental interest is in exotics containing a pair, this paper will focus on exotics containing two heavy quarks as it simplifies the analysis. Moreover, the focus is on the formal limit where the quark masses become heavy as this allows a systematic analysis. It is, of course worth recalling that while there is strong evidence for stronginteraction stable non-exotic doubly-charmed baryons[16], as of now the are no known examples of doubly-charmed or doubly-bottomed tetraquarks. However, it is also worth noting that this does not mean that these states do not exist in CD, but could indicate that they are very difficult to make in typical laboratory experiments. In this paper, it is shown in a model-independent way that Exotic tetraquarks of the type must exist as parametrically narrow resonances in the formal heavy quark limit in at least some spin-flavor channels. For sufficiently heavy quarks, multiple parametrically narrow exotic hadrons will exist with fixed quantum numbers. As the heavy quark masses increase the number of such states grows without bound. In channels with exotic hadrons, exotics exist that are below, but parametrically close, to the threshold for decay into two non-exotic hadrons each of which contains a heavy quark. For simplicity the analysis done here will assume that the light quarks in the state are either up or down rather than strange and that isospin is an exact symmetry. However, these assumptions can be relaxed with no fundamental change in the argument. Only strong interaction effects will be considered in this work. Electro-weak interactions will be excluded and thus decays of heavy quarks into lighter ones will be ignored. One subject that has received considerable attention is the extent to which various X, Y and Z states are really tetraquarks as opposed to di-meson molecules. The perspective of this paper is that question of whether these states are tetraquarks or molecules is not well posed and it is not useful to make the distinction. Thus for example, even the description of the X387, probably the most thoroughly studied of these states is still controversial. Despite significant effort in describing the X387 as primarily D 0 D 0 molecular[17] because of its exceedingly closeness to D 0 D 0 threshold, arguments exist for questioning this interpretation and suggesting a large c c component[1, 17] Much of the reason for the controversy is that the way one decomposes a quantum state into constituent parts depends on the choice of basis and choices of basis are not physical. Moreover even with a fixed basis there will be mixing between tetraquark and molecular configurations. Whether hadrons are regarded as molecules or true tetraquarks appears to depend on how one attempts to model the hadron in simplified treatments. Significantly, even as a matter of principle, there is no known approach starting with the CD Lagrangian that quantifies the extent to which a given hadronic resonance can be regarded as a molecule. It is noteworthy that hadrons that have been identified as good candidates for molecules such as the X387 are typically near the threshold for a channel in which the hadrons break up into two lighter hadrons. It may seem natural to identify these as molecules since the quantum states are dominated by regions in which the two lighter hadrons are distinct and far from each other. To make this notion precise, consider a quantum system in which there is a parameter that controls the masses of two self-bound subsystems and also of a potential stable bound state of the two. This parameter can be varied continuously moving the system from a stable bound state to a resonant scattering state of the two subsystems. Starting from a stable bound state and varying the parameter towards the threshold for break up, the wave function of the system is such that probability that the system will consist of the two nearly unmodified subsystems separated by a distance greater than any arbitrary constant R will diverge as the threshold is approached. In other words, as the threshold is approached the system appears to become arbitrarily well-defined as a bound state of the two subsystems with each subsystem maintaining

3 3 its structure which seems to be what one expects in a molecular picture. However, the preceding argument can be misleading. Suppose, one asks whether the ρ meson is really a quarkantiquark state or a two-pion state. On the theoretical level, one can control whether the ρ is above or below the two π threshold by varying the mass of the up and quarks taking m u = m d though out and in practice this can be done in lattice studies[18, 19]. The preceding argument appears to imply that as the threshold is approached from the bound state side, the system becomes more and more unambiguously a molecule of two pions. However, this violates the intuitive sense that a ρ meson is naturally described as a quarkantiquark state and that this description does not change as one approaches the threshold. While the kinematics change dramatically at the threshold, the underlying dynamics does not: it is the interaction of quarks at short distances that determines the existence of the ρ regardless of whether it is a bound state or a resonance. There is no reason to think as one nears the threshold, the essence of the ρ meson ceases to be of a quark-antiquark nature. Indeed, if one considers CD near the limit of a number of colors N c [0], the ρ meson is well described as quark-antiquark state regardless of whether the quark masses put it near threshold. Given this, the mere fact of a state being near threshold clearly does not automatically make it a molecule. The upshot of these considerations is that there appears to be considerable theoretical ambiguity into whether a particular observed state should be regarded as having a molecular or tetraquark character. Accordingly, without prejudice to their true character, all of these hadrons will be referred in this work as tetraquarks since all of them have at least two quarks and two antiquarks. However, a key point is that in the extreme heavy quark limit, some of these will necessarily be near threshold in a parametric sense. These near-threshold tetraquarks are a principal focus of this paper. To understand why the present focus is on tetraquarks, it is important to note two differences between the case and the case. The first is the possible annihilation of the pair and the second is the possibility of a rearrangement in which the can form a color-singlet quarkonium state leaving behind an ordinary meson. As discussed previously, the annihilation of pairs is suppressed in the heavy quark regime. However, the second issue is potentially serious. Similarly, rearrangement issues are potentially important for pentaquarks of either or varieties. As it happens, the formalism developed in this work for tetraquarks does not account for possible rearrangement effects that occur in either pentaquarks or tetraquarks. It is plausible that the formalism developed here could be modified to account for rearrangement and the conclusions drawn here might also apply to these cases. This possibility will be addressed in future work. The principal result of this paper is the existence of parametrically narrow near-threshold exotics parametrically close to threshold in the heavy quark limit. It is useful to recall that there are well-established and compelling arguments[1 3] for the existence of exotic q q tetraquarks where q and q are light quarks with possibly distinct flavors. This paper borrows heavily from one of these arguments[]. It is also worth recalling, however, that previous arguments concerning the existence of exotic hadrons containing pairs [] did not address the issue of whether such systems in the extreme heavy quark limit were guaranteed to have exotic states parametrically close to the threshold for dissociation into two ordinary hadrons each containing a heavy quark. It will be shown in this paper that this is indeed the case. Before undertaking this analysis, a word about scales: we are interested in how observables scale with the heavy quark mass. It is natural to express this as a dimensionless ratio with a non-heavy hadronic scale. We typically denote this scale as and take it to be a number of order unity times CD. However, part of the analysis involves long-ranged forces characterized by a length scale of 1/m π and the dimensionless ratio of m π /m enters the problem. In studies of the light sector one often considers there to be a scale separation between m π and. Indeed such a separation is the entire basis for chiral perturbation theory[4]. However, for the present purpose we are really interested in asymptotic scaling as the heavy quark mass grows. For the purposes of this analysis, it is convenient to consider m π to be of order but simply numerically small. The point is that the mass of the pion is very insensitive to the mass of the heavy quarks; accordingly in studies of the scaling with m, it acts like other light scales. This paper begins with a brief review of the known salient features of tetraquarks containing a pair. The next section focuses on an analysis that shows in a simple treatment based on the Schrödinger equation as the heavy quark mass increases, multiple exotic states exist as bound states for each channel containing exotics and that some of these states are parametrically close to the threshold for decay into two nonexotic heavy mesons. Following this it will be shown that the full CD Hamiltonian in the sector with exotic quantum numbers for the full theory can be decomposed to a piece describable via the Schrödinger equation used to deduce the existence of near threshold tetraquarks and a remainder, and that the effect of the remainder is render the exotic bound states of the Schrödinger equation into parametrically narrow resonances. The conclusion then is that parametrically narrow exotic tetraquarks states parametrically close to the break up threshold exist in the heavy quark limit. Following this is a discussion section that explores the challenges posed by rearrangement effects and possible phenomenological implications of the results. II. ON THE EXISTENCE OF TETRAUARKS WITH TWO HEAVY UARKS It is useful to review compelling arguments that exotic hadrons with the quantum numbers of tetraquarks and containing a pair must exist as strong-interaction stable particles. Note that because these contain a pair, these are all, of necessity quantum-number exotics. There are actually two distinct arguments for the existence of strong-interaction stable q q tetraquarks. The first one

4 4 is that tetraquark of the q q is expected to exist in the heavy quark mass limit m as strong-interaction stable states, simply because two heavy quarks Coulomb potential is attractive in a 3 color configuration[1, ]. In the extreme heavy quark limit, this will lead to a possible deeply bound diquark configuration, with a binding energy of order αsm which for very heavy quarks is much large than typical hadronic scales of CD times a factor of order unity, and a characteristic size of order M α s 1 where α s is taken at the scale of the inverse size. This means that in the extreme heavy quark limit, there will exist a tightly bound diquark of small size in the 3 color configuration. Because it is heavy, tightly bound and small in dynamics on the hadronic scale it will act essentially as a nearly point-like source of a color Coulomb field. In terms of hadronic physics a single heavy quark in the extreme heavy quark limit and viewed from the rest frame of the hadron acts simply as a static source of a color Coulomb field its spin becomes irrelevant as M and the quark s recoil is suppressed. A tightly-bound diquark with small size will also act as a static source of a color Coulomb field but with its color in the 3 representation. Thus it acts dynamically in the same way as a heavy antiquark. A recent lattice simulation also supports this theoretical prediction[5]. This doubly-heavy-diquark-antiquark DHDA duality means that so long as the system is well below the excitation energy of the heavy diquark which is formally order αsm and thus well above CD every hadronic state in the system with two heavy quarks has an analog state for the system with a heavy antiquark strictly speaking with a set of states with nearly degenerate masses which differ by the spin of the heavy antiquark with the excitation spectrum of the doubly-heavy system identical to the spectrum of the system containing a single heavy antiquark up to corrections associated with the size of the diquark, virtual excitations of the diquark and finite mass corrections. This DHDA duality implies that since antibaryons containing heavy antiquarks exist as strong-interaction stable states in CD, then so to do q q tetraquarks, at least in the extreme heavy quark limit. Thus DHDA duality implies the existence of doubly heavy exotic mesons; moreover the excitation spectra of these two systems will coincide as M. Unfortunately, the charm mass is significantly too small for the DHDA to apply and that the bottom mass is at best quite marginal[1, ]. Thus, this approach is not viable as the basis of a quantitative effective field theory approach for the description of possible doubly-charmed tetraquarks in the real world. However, on a theoretical level the argument remains valid for the case of extreme masses and minimally shows that the structure of a class of gauge theories that includes CD does not exclude tetraquarks containing two heavy quarks. Unfortunately, by construction this argument yields deeply bound tetraquarks and provides little insight into near threshold states. There is another argument[1, ] which also predicts strong-interaction stable q q tetaquarks; this argument can be generalized to q q systems unfortunately with a critical caveat. The argument treats strong-interaction stable tetraquarks of the q q type as a bound state problem of two interacting heavy mesons, each one containing a heavy quark. At first blush, this may seem as though it is valid only if the molecule picture is correct. In fact, however, it holds regardless of the underlying structure of the tetraquark state. The argument depends on a theoretical framework which was implicit in Refs.[1, ] but which plays a critical and somewhat subtle role in the current context. Accordingly, it will be discussed in some detail here before the argument is given in detail. The idea is to focus on the interaction between two mesons each containing a heavy quark. If the processes under consideration restrict the motion of the heavy mesons so that no additional light particles are created in the dynamics and also ensure that the heavy mesons remain distinct, that they move non-relativistically and that they remain at large distances from each other, then the system can be described accurately by a Schrödinger equation. Consider now what happens in a system containing two subsystems whose relative dynamics is describable by Schrödinger equation if some parameter of the problem, λ, is adjustable and this allow for the mass of the two subsystems to increase to arbitrarily large size and may or may not affect the potential. For simplicity, let us first focus on an s-wave channel without mixing with purely non-relativistic kinematics. The time-independent Schrödinger equation for the n th bound state, given in terms of the radial coordinate is then given by 1 µλ + V r; λ u dr n r; λ = E n λur; λ 1 where µ m 1 m /m 1 + m is the reduced mass and the λ dependence of all quantities is made explicit and u is normalized so that dr ur = 1. It is useful to multiply both sides by µλ: + µλv r; λ u dr n r; λ = µλe n λu n r; λ. Thus, increasing µ plays the same role in determining the existence of normalizable eigenstates as increasing V, since only the product µv matters. Thus, it is natural if V is attractive then at large enough µ, the system should have bound states. To make this more concrete, suppose that the potential has the property that it remains finite and well-defined in the large λ limit: lim V r; λ = V 0r, 3 λ with at least some region in r for which the V 0 r < 0 i.e. the potential has an attractive region, while at the same time the mass of the two subsystems and hence the reduced mass diverge with λ: µλ lim = µ 4 λ λ

5 5 where µ is a finite constant. In this situation, it is trivial to see that there must be bound states. The easiest way to see this is by a variational argument: Consider a normalized trial wave function that is centered around some point r 0 with V 0 r 0 < 0 and with a width which goes to zero with increasing λ but does so more slowly than λ 1/, for example: u trial r = 3λ λ 1 4 r r 0 r 0 Θ 1 λ 1 4 r r 0 r 0 5 8r0 where Θ is a Heaviside step function; and choose r 0. The expectation value of the kinetic energy in the trial wave function in Eq. 5 is 3/8λ 1/ µr 0 which goes to zero as λ. On the other hand. lim λ u trial r = δr r 0, so that lim λ u trial V u trial = V 0 r 0. Thus, u trial H u trial = V r 0 < 0. 6 Since the energy of the true ground state of the system needs to be below that of the trial wave function, it follows that the ground state has negative energy i.e. that a bound state exists, provided that there is not an additional open channel with lower energy than the variational state. A key point in this analysis is that while the description in terms of a simple onedegree of freedom Schrödinger equation need not be valid for the entire Hilbert space, one can choose a trial wave-function in the domain where the description is valid, and provided one has a trial state with negative expectation value of energy, the system must have a bound state, even if it is outside the domain of validity of the Schödinger dynamics. Thus, if there is a regime in which a Schrödinger description is valid and which has a region with an attractive potential, a bound state must exist at large λ assuming that there is no open channel with lower energy. An analogous argument holds for any partial wave and to systems with mixing of degrees of freedom such as spin and orbital, or distinct flavors and to systems with relativistic kinematics. Unfortunately there is a caveat to this logic. If an open channel with lower energy does exist, the argument does not hold. Let us now consider what this argument tells us about the existence of heavy exotics of the q q type viewed as the interaction between two heavy mesons. The previous argument should hold, since the lowest threshold for fixed quantum numbers will be for break into two heavy hadrons. The long distance part of the strong-interaction potential in the heavy quark limit is well known. It is given by one-pion exchange. The precise form that the long-distance potential takes depends on heavy quark symmetry, in particular the fact the spin of the light decouples from the dynamics at large heavy quark mass. One can use heavy-hadron chiral perturbation theory HHχPT [6] to fix this form as was done in Ref. []. However, the result does not depend on the light quarks being in a regime where chiral perturbation theory applies. Rather the key point is simply that the pion is the lightest hadrons and remains so as the heavy quark masses get heavy and hence dominate the potential at long distances. In describing the potential in the limit of heavy quark masses, it is important to recall that the pseudoscalar and vector heavy mesons form a multiplet that becomes degenerate in the heavy quark limit. We denote these two generically as H and H. This structure follows because the spin of the heavy quark becomes irrelevant in each meson. It is clear that because the pion is a pseudoscalar, it couples to an angular momentum and because the spin of heavy quark is irrelevant it couples to the internal angular moment of the light degrees of freedom in the heavy meson which for simplicity we refer to as spin of the light quark. Thus the generic form of the long-range potential in momentum space is given by[] Ṽ long q = I 1 I 4g f π s l1 q s l q q m π where the tilde indicates the potential in the heavy quark limit, I is the isospin, s l is the spin of the light quark in the heavy meson, g is the coupling constant for H H π coupling evaluated at q=0 and f π 93 MeV is the pion decay constant; the quantities g, f π and m π are all understood to be at their heavy-quark limit. Many possible spin-isospin channels exist for this system and the one pion exchange potential depends on the channel. Bose symmetry between the two heavy mesons constrains the possible quantum numbers: 1 = 1 I+S l+s +L+1 where L is the orbital angular momentum, I the total isospin, S l is the total spin of the light degrees of freedom, and S is the total spin of the heavy quark. For simplicity and concreteness, this paper focuses on states with I = 1 which by construction are quantum number exotic and S equal zero. Recall that S can be fixed at the outset and is conserved in the heavy quark limit. Our initial focus is on states with J = 0 wherej = L + S l + S is the total angular momentum, positive parity and heavy quark spin, S, of zero. Such states automatically have S l = 0. To see why, note that for states with I = 1 and S = 0 the constraint in Eq. 8 becomes 1 = 1 Sl+L. To the extent that the state is composed of two heavy mesons each of which is a multiplet of a pseudoscalar and a vector, each heavy meson has the spin of the light degrees of freedom equal to 1/ so that the only possible values of S l for the total system are 0 and 1. Since S = 0, the total angular momentum is J = L + S l and the parity is positive so 1 L = 0 and L is even. Since J = 0 and S l is either 0 or 1, the only possible values of L are 0 and 1 but only L = 0 is even and thus L = 0. J = 0 with positive parity is only possible then with S l = 0. In fact it is quite simple to construct such states in the basis of the two heavy mesons. Such states are L = 0 spatial states with mesons in the configuration H 1 H + H 1 H where H is the pseudoscalar meson and H is the vector. S l = 0 channels are simple to analyze as the long-range potential is central and does not mix orbital partial waves. The potential in configuration space is obtained by Fourier transforming Eq. 7. For S l = 0 states this is given by a simple Yukawa potential: Ṽ long r = II g m π e mπr 4 πfπ r 7 8 9

6 6 where I is the total isospin of the state. An obvious and significant point is that for states with I = 1 the potential is negative. Given the argument above, this is sufficient to demonstrate that bound tetraquarks of the q q exist in the heavy quark limit; this argument holds independent of the argument based on the DHDA duality. Strictly speaking, the argument as given only applies to the particular quantum number I = 1, J = 0, S = 0, positive parity however, it should be clear that analogous arguments can be given for other channels that are attractive at long distance. It should also be clear at this point that this argument does not go through for the case of tetraquarks with content: in those cases there are open thresholds below the variational state, for example in the channel with a pion plus a J/ψ. Of course, if there is a dynamical reason this open channel is weakly coupled to the dynamics under consideration it is plausible that arguments analogous to the ones here might imply narrow resonant states. Let us return to the case for the next step in the analysis. III. ON THE EXISTENCE OF MULTIPLE TETRAUARKS WITH FIXED UANTUM NUMBERS INCLUDING NEAR THRESHOLD STATES One key point that was not stressed in refs. [1] and [] is that for channels with given spin, parity exotic channel with two heavy quarks, a Schrödinger equation with a potential whose long-range part is given in Eq. 9 supports a parametrically large number of distinct bound states for fixed quantum numbers including some parametrically close to threshold when treated in the limit that the heavy quark mass becomes large. Superficially this appears to imply that at very large heavy quark masses there are parametrically many distinct exotic hadrons with some of these close to threshold. This turns out to be correct, but the analysis is somewhat subtle: most of these these states, including all of the near threshold states exist not as bound states describable in terms of a two-body Schrödinger equation but as parametrically narrow resonances in a description with more asymptotic degrees of freedom. The subtle part of the analysis lies in demonstrating that the resonances are parametrically narrow. This issue will be addressed in the following section. A. Description based on the two-body Schrödinger equation The reason why the Schrödinger equation description implies a large number of bound states at large heavy quark mass is easily understood from the semi-classical limit. Semiclassical analysis is far simpler in contexts where there is only a single dynamical degree of freedom so that Bohr- Sommerfeld quantization and the WKB approximation can be invoked. For this reason we start with a description of I = 1, J = 0, S = 0, positive parity states which does not involve coupled channels and this has a single dynamical degree of freedom. States with other quantum numbers involve a slightly more complicated analysis which will be given in the next subsection but the results are qualitatively identical. For systems with a single dynamical degree of freedom such as states in the the I = 1, J = 0, S = 0, positive parity sector of a potential model between heavy mesons, standard semi-classical analysis implies that the number of bound states between energy E 0 and E 0 + E is given by E0+ E E NE 0, E 0 de dp dq δe Hp, q π dp dq ΘE0 + E Hp, qθhp, q E 0 = π 10 where H is the classical Hamiltonian and Θ is a Heaviside step function. The number of bound levels is approximately π 1 times the area of phase space with energies in the appropriate range. The approximation becomes increasingly accurate when NE 0, E becomes large. Suppose that the Hamiltonian is of the for = p µ + V q where µ is the reduced mass and V q is independent of µ. It is trivial to see that phase space area between E 0 and E 0 + E is directly proportional to µ and accordingly so is NE 0, E. Let us first apply this to the long-range dynamics for the I = 1, J = 0, S = 0, positive parity channel whose potential at large quark mass is given in Eq. 9 and whose kinetic is given by p /µ p /m. These quantum numbers guarantee that long distance dynamics depend on only a single radial degree of freedom. By construction, such a treatment is valid in the limit of that the heavy quark masses gets large and for dynamics restricted to a regime where no particles are created or destroyed. However, within this restriction this description should be accurate. Since this is a problem with one degree of freedom, the semiclassical expression for NE 0, E should hold at least for this long distance part. Let us denote N long E 0, E, as the contribution to integral in Eq. 10 coming from values of r that are well described by the long-distance potential. Provided that as N long E 0, E 1 in some energy range, it seems apparent that the system is in the semiclassical domain and, moreover, the total system should have more bound doubly heavy tetraquark states in that energy range than given by N long E 0, E since only part of the phase space is being considered and because this analysis was only applied to the s l = 0 channel while other channels can also have bound states. The reduced mass in the long-distance Hamiltonian is just the heavy meson mass over two and the heavy meson mass is given by m. Thus, as m, N long E 0, E diverges as m ; at large m, the total number of bound doubly heavy tetraquark states in any energy should grow as m. Consider, the implications of this for near-threshold states, which we will define as states that are below the threshold for separation into two heavy mesons by, a binding energy B which goes to zero as m. We show that there exist bound states for which B ɛ for any ɛ with 0 < ɛ < 1. 11

7 7 Thus, there are tetraquark states that are parametrically close to threshold. The semi-classical estimate for the number of bound doubly heavy pentaquark states, n B is then bounded by m Ṽ long r n B N long B, B dr 1 π r B where r B is defined implicitly through Ṽ long r B = B 13 that is r B is the distance at which the potential is equal to the binding energy. The second inequality in Eq. 1 comes from evaluating the integral only over the region greater than r B rather than the entire long distance region; this is done for simplicity of analysis. Finally, using the explicit form for Ṽ long, it is straightforward to prove that if r B > log then m π m Ṽ long r m B dr > π. m π r B 14 Moreover, since r B diverges as B goes to zero, for sufficiently small binding energy and recalling that our interest is in parametrically small binding energies the condition r B > log m π will be satisfied for small B. Putting this all together yields, m B n B > m B. 15 m π where we take m π to be of order of a characteristic hadronic scale. Equation 15 implies that if B is held fixed, n B the number of states with energy less than B grows with m at least as as m. Suppose that B is not held fixed; rather, that B is chosen to decrease parametrically with increasing m to ensure that the states under consideration are parametrically close to threshold at large m : B = b π m ɛ π ɛ 16 where b is a dimensionless numerical constant of order unity and 0 < ɛ 1. With the condition in Eq. 16, Eq. 15 yields n b > b m /m π ɛ/. In the limit of large m this implies that there are a large number of states satisfying Eq.16; i.e. a large number of near threshold states. Thus, at this level of analysis near threshold tetraquark states exist. Of course, for Eq. 15 to be valid the system needs to be in the semi-classical limit; however, this should be true if n b 1, as it is provided m is sufficient large and 0 < ɛ. While the particular analysis was done for channels with s l = 0, it should be clear that analogous results hold for other channels. The scenario consistent with this result appears to be that as m increases, the total number of bound doubly heavy state tetraquark states with fixed quantum numbers increases in a manner proportional to m. Thus if one were to follow the spectrum as m is increased, new bound tetraquark states will continue to appear at threshold. After a new state appears, it moves below threshold with increasing m, eventually leaving the regime that can be identified as being near threshold. However, before the state leaves the near-threshold regime new states enter at threshold ensuring that there are always some near threshold states. While the precise number of such states identified as being near threshold depends on the choice of the arbitrary parameters b and ɛ, some such states will exist for sufficiently large m. A useful way to quantify this behavior is to look at B min m the binding energy of the least bound state as a function of m. This will be zero right when a new bound state appears, it will then grow monotonically with m and reach a local maximum just before the next bound state appears and will then discontinuously drop to zero. Since there are formally bound states with a binding energy given parametrically as ɛ for any 0 < ɛ 1 and ɛ can be arbitrarily small, B min m must go to zero as M goes to infinity, at least as as M 1. However, it can go to zero er than this since the semi-classical analysis breaks down for states right at threshold. We have verified that the scenario does indeed play out in the expected way with various toy models. These are nonrelativistic potential models for two heavy meson with an s-wave potential that is designed to become a Yukawa function mimicking Ṽlongr at long distances while being nonsingular at short distances. For example, V r = 10m Y e 4m Y r +1/, 17 4m Y r + 1 where m Y is the mass in the Yukawa function i.e the analog of m π. As expected from the analysis above, for the toy models we considered, as the mass of the heavy meson increased with the potential held fixed, the number of bound states increased and the binding energy of the least bound state, satisfied Inequality 11. This behavior is illustrated in Fig.1 which is based on the potential in Eq. 17: The dots represent the binding energy of the least bound state for values of just prior to a new state entering at zero these are local maxima of the binding. For simplicity both the binding energy and the mass of the heavy meson are given in units of the Yukawa mass. Since our interest is in the asymptotic behavior at large, the region of plotted is for > 00m Y. The solid line is a representation of the curve B = / ; on a log-log plot it is a straight line with a slope of -1. The value of was chosen so that the line exactly reproduce the binding energy of the dot with smallest mass on the plot. As can be seen in the figure all of the binding energies with masses that are larger than the minimum one are below the solid line. Indeed, from the figure it is apparent that the binding energies of these point fall off with sightly er than m 1 H. Clearly Inequality 11 is satisfied for these special points. Moreover, since the binding energy of the least bound state grows monotonically between

8 8 the entry of a new bound state and the entry of the next one, Inequality 11 is satisfied throughout the high mass region plotted in the figure Note that our analysis implies that near threshold states will exist and that some states are parametrically no further from threshold by amount that scales arbitrarily close to m 1. While our analysis does not allow us to conclude that the scaling is as as m 1 or er, it is certainly consistent with this. The behavior seen in Fig. 1 is what one expects based on the semi-classical analysis which should be valid for large masses. Moreover, we have explored a number of other potential models that asymptote at large distances to a Yukawa form; they all have the same qualitative behavior, with parametrically weakly bound states. Between the semi-classical analysis and the behavior of the toy models it seems apparent that parametrically light bound states at large mass exist for all potential models that asymptote at long distances to a Yukawa form. From this, one might be tempted to conclude that CD also has numerous bound tetraquark states including some that are parametrically very weakly bound B /m. After all, the potential description this is valid in the heavy quark limit for dynamics restricted to having the heavy meson or equivalently the heavy mesons restricted to long distances and to processes in which additional particles are neither created nor destroyed. Unfortunately, such a conclusion is not justified. The reason is that the dynamics of CD for these states allows for the creation of additional particles i.e. the emission of mesons which is beyond the regime of validity of the potential model description. As will be discussed in Sect. IV, a remnant of these would-be bound states do survive in the heavy mass limit of CD but as parametrically narrow resonances rather than bound states. B 0.00 m Y B. Other quantum numbers FIG. 1. Binding energy in units of the Yukawa mass of the least bound state for a toy model with the potential in Eq. 17 as a function of the heavy meson mass in units of the Yukawa mass. The dots corresponds to values of just prior to the entrance of a new bound state. The solid line with a slope of -1 on this log-log plot represents B = with fit to the lowest mass point on the plot. m Y Before addressing the question of why the possibility of meson emission does not vitiate the argument of the previous subsection, it is useful to show that the argument can be generalized to attractive channels with quantum numbers other than I = 1, J = 0, S = 0, positive parity states. The difficulty with such channels is that the pion exchange includes a tensor force which mixes the L quantum number which appears to render the inapplicable the simple semi-classical formalism of the previous section which depends on there being a single dynamical degree of freedom. While there are, in fact, coupled channels for other quantum numbers, the kinematics of the problem allows one to choose a basis in which the channels decouple up to corrections that vanish in the heavy quark mass limit which is the limit of interest here. The argument is very much in the spirit of the Born-Oppenheimer approximation[7] of molecular physics. Recall that in the heavy quark limit, the centre of mass of the two mesons are parametrically slowly moving. relative to each other. The potential that couples the various channels can be viewed as a matrix, with diagonal terms within each channel and off diagonal terms for the coupling between channels. This potential matrix is a function of r, the separation between the two heavy mesons. For any given r one can diagonalize this matrix to express the potential in terms of an eigenbasis; the eigenvalues give the value of the potential for the new channels and the eigenvectors give these new channels in terms of the old ones. However, since the mesons are moving slowly, one can regard the variation as being adiabatic. Thus, if the system is the lowest eigenstate of the potential for one value of r, it will tend to remain in the lowest eigenvalue for all values of r. This tendency should become perfect as m. Thus, for large m, the system acts like a single channel one where the channel is the lowest eigenvector of the potential matrix. To make this explicit, consider the form of the Schrödinger equation for a coupled channel potential problem with k chan-

9 9 nels: 1 µ r + V ψ = Eψ ψr and ψ 1 r ψ r. ψ k r V r with V 11 r V 1 r... V 1k r V 1 r V r... V k r.... V k1 r V k r... V kk r, 18 where µ is the reduced mass. The matrix V is constructed to include the centrifugal barriers. Note that V r has a magnitude of order and varies over a distance of order 1 ; while the kinetic term is controlled by µ m. One can introduce a k k matrix U r that diagonalizes V r; U r varies over a length scale of order 1. Defining ψr U rψr, allows one to reexpress the Schrödinger equation as H 0 + H 1 + H ψ = E ψ with H 0 = 1 µ r + Ṽ r, H 1 = 1 U r U r r µ and H = 1 U r U r. µ 19 By construction H 0 is diagonal. Moreover, since spatial derivatives for the semi-classical bound states are of order m, nominally H 0, H 1 3 m and H m. Thus, at large m, the H 0 term is dominant. One can treat it as the leading term and treat H 1 and H perturbatively. Superficially, it looks as though the leading corrections to this picture come from the H 1 term which is nominally of order 3 m. However, it is easy to show that the H 1 is entirely off-diagonal so that it contributes only at second order. Thus, the leading contributions of both H 1 and H to the energy are of order m. Since the characteristic energy spacing in the semi-classical limit is 3 m, these become negligible at large m and H 0 dominates. To the extent that one can neglect H 1 and H and treat H 0 as the dominant term, the diagonal nature of H 0 implies the lowest eigenvalue of V r acts as a potential for a problem with a single radial degree of freedom. Once this fact is recognized, one can immediately exploit the arguments of Subsection III A without further work for all quantum numbers with angular momentum of order unity and attractive interactions at long distances. There is one potential subtle issue that should be noted in these cases, however. In the case of a purely s-wave interactions, there is no long-range repulsion due to a centrifugal barrier. For L 0 this is not the case. Since the analysis involves the nature of the long-distance interaction, this might seem problematic. However, provided M 0 that J is parametrically of order unity i.e.l, then the orbital angular momentum, L, will be as well and the centrifugal term is negligible parametrically; its contribution to the energy is of order O M while the binding energy is O where 1 ɛ > 0. ɛ M 1 ɛ IV. A MORE COMPLETE DESCRIPTION As was noted in Subsection III A when the quark masses are heavy, states near the threshold for decay into two heavy mesons can emit mesons and thus potential models become problematic. Recall that in Sect. II, there were two distinct arguments for the existence of exotic tetraquarks, one was based on the fact that an attractive potential binds as the heavy quark mass goes to infinity, and the other on formation of a doubly heavy diquark and the doubly-heavy-diquark-antiquark duality. It is important to note that the DHD duality argument implies the existence of relatively deeply bound tetraquarks, with a binding of order αsm relative to the threshold for dissociation into two heavy mesons with α s evaluated at a scale of α s M. The existence of tetraquarks with a binding of order αsm implies that energy of these states are below that of the putative weakly bound tetraquarks discussed in the previous subsection by more than m π indeed by much more than m π. In fact, most of the bound states predicted by the semiclassical analysis have other states that are more than m π below them. Moreover there is no symmetry that prevents a decay of such a weakly bound tetraquark state via the emission of one or more pions. Given the totalitarian principle[8] of particle physics that which is not forbidden is mandatory such parametrically weakly bound tetraquark states will decay via pion emission: they are not stable states in CD. An analysis based on a potential model that does not allow for pion emission is wrong or, more precisely, it is incomplete. As an aside, it is worth noting that the potential-based argument of Sect. II that concluded independent of the argument based on diquarks and the DHD duality that a double heavy tetraquark must exist remains valid. If one went through the potential-based argument agnostic about whether a deeply bound doubly heavy tetraquark existed, one would be faced with two possibilities: A deeply bound tetraquark exists and invalidates the potential-based analysis but requires a tetraquark to exist. Alternatively, if such a deeply bound state did not exist, then the potential based argument is valid and it predicts that a bound tetraquark must exist in the heavy quark limit. In either case, a tetraquark exists. But the validity of the argument for the existence of a bound tetraquark is distinct from the question of whether the argument is legitimate in its prediction of parametrically weakly bound tetraquark states. While, the potential-based argument fails for stronginteraction-stable bound tetraquarks states near threshold and for other semi-classical tetraquark states, the argument may