A tetraquark or not a tetraquark: A holography inspired stringy hadron (HISH) perspective
AA tetraquark or not a tetraquark?A holography inspired stringy hadron (HISH)perspective
Jacob Sonnenschein [email protected]
Dorin Weissman [email protected]
The Raymond and Beverly Sackler School of Physics and Astronomy , Tel Aviv University, Ramat Aviv 69978, Israel
July 10, 2018
Abstract
We suggest to use the state Y (4630), which decays predominantly to Λ c Λ c , as awindow to the landscape of tetraquarks. We propose a simple criterion to decide whethera state is a stringy exotic hadron - a tetraquark - or a “molecule”. If it is the formerit should be on a (modified) Regge trajectory. We present the predictions of the massand width of the higher excited states on the Y (4630) trajectory. We argue that thereshould exist an analogous Y b state that decays to Λ b Λ b and describe its trajectory. Weconjecture also a similar trajectory for tetraquarks containing strange quarks, and themodified Regge trajectories can in fact be predicted for any resonances found decayingto a baryon-antibaryon pair. En route to the results regarding tetraquarks, we also makesome additional predictions on higher excited charmonium states. We briefly discuss thezoo of exotic stringy hadrons and in particular we sketch all the possibilities of tetraquarkstates. a r X i v : . [ h e p - ph ] J un ontents c ¯ c mesons . . . . . . . . . . . . . . . . . . . . 187.1.2 The Y (4630) resonance: an overview . . . . . . . . . . . . . . . . . . . 187.1.3 Deciphering the Y (4630) state . . . . . . . . . . . . . . . . . . . . . . 207.1.4 Regge trajectory of the Y (4630) . . . . . . . . . . . . . . . . . . . . . 217.2 Tetraquarks in the bottomonium sector . . . . . . . . . . . . . . . . . . . . . 227.3 Lighter tetraquarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 As early as the early days of the quark model and the “old duality”, the question of whethernature exhibits exotic hadronic states on top of the mesons, baryons, and glueballs was raised[1], and it has remained unresolved in fact until these days. There has been indirect evidencefor the existence of tetraquarks, for instance, it was pointed early on that duality requires theexistence of tetraquarks as intermediate states in baryon-antibaryon scattering [1, 2]. How-ever, there is no consensual direct observation of a tetraquark state in the hadronic spectrum.In recent years various experimental observations reignited this subject (for reviews see forinstance [3,4]). In particular a remarkable proliferation of exotic charmonium-like resonanceshas been discovered, some of which, like the charged charmonium-like Z c resonances, cannotbe interpreted as mesons. Up to date there is no theoretical consensus regarding all the newlydiscovered states and several conflicting descriptions have been proposed [4].In this note we consider exotic hadrons in the context of the holography inspired stringyhadron (HISH) model [5]. Previously we have addressed by using this model the spectra ofmesons [6], baryons [7], and glueballs [8]. We found that a significant part of the knownhadrons fall into modified Regge trajectories (MRT) associated with the model. The present1ork deals with the possibility to have exotic hadronic resonances in this model. We sketchpossible arrangements constructed from the basic building blocks, which are open strings,massive particles (quarks) at the endpoints of the open strings, baryonic vertices, diquarksattached to baryonic vertices, and closed strings. We discuss first configurations withoutquarks where all the strings end on baryonic vertices (BVs) and anti-BVs. These includesthe cube, the planar polygon of any even order, tiling the whole plane with hexagons, andmore. We also describe setups that include quarks and diquarks in addition to the vertices.A special class of HISH exotics is that of a baryonic vertex with a diquark attached toit connected by a string to a antibaryonic vertex with an antidiquark. This forms whatwe refer to as a stringy tetraquark . For five quark flavors, there is a total of 225 possibletetraquarks, which can be grouped into 15 symmetric, 110 semisymmetric, and 100 asym-metric tetraquarks.Classically the stringy tetraquark is described by a string that may rotate or not, andwith massive endpoints. This is the same as the string that is associated with a mesonor a baryon. The rotating classical strings with massive endpoints have states which aremembers of modified Regge trajectories with the same slope as those of the mesons andbaryons. Quantum mechanically there are differences between the original Regge trajectorieswhen there are no massive endpoints and the modified trajectories when the strings end onmassive particles [5]. We will argue that in addition there is a difference between the quantumtrajectory of a meson or a baryon and that of a tetraquark, following the same reasoning asfor the difference between mesons and baryons. In the later case there is a difference betweenthe two intercepts which is in charge of the difference between the masses of a baryon anda meson with the same orbital angular momentum and endpoint masses. We will suggest ascenario for which it is plausible that the difference between the intercept of a tetraquarkand a meson will be twice the difference between a baryon and meson.The most natural decay mode of the tetraquark configuration, at least for high spinexcited states, where the description of the tetraquark as a diquark and antidiquark shouldhold, is similar to that of meson or a baryon. It involves the breaking of the string into twostrings, one connecting the BV to a quark and the other an antiquark to an anti-BV (seefigure 8). This means that the natural decay mode of the stringy tetraquark, provided itis sufficiently massive, is to a baryon-antibaryon pair, as a BV and an anti-BV are alreadyparts of the tetraquark. The computation of the decay width of such a process is in fact verysimilar to that of a meson decaying into two mesons [9].As was mentioned above an important question about the peculiar states that have beenobserved recently is whether they are genuine exotic hadrons or “molecules” [10], i.e. boundstates of non-exotic hadrons. We propose a simple criterion to help decide whether a state isa genuine stringy exotic hadron or a molecule. If it is the former it should be on a (modified)Regge trajectory while for the latter there is no apparent reason to have such a trajectory. InQCD terms, this is the difference between the color interaction of the diquark and antidiquark(equivalent to an antiquark and a quark) and the mechanism, probably pion exchange [10],holding a bound state of hadrons together. It should be emphasized however that the stringydescription is expected to be valid for excited states, when long strings are formed between thediquark and antidiquark forming the tetraquark. It is not obvious that the lower tetraquarkstates (the ones that we are more likely to see in experiment) should be on trajectories, butexperience has shown us that this holds for both mesons and baryons, even those made up of2eavy quarks [6, 7]. The notion that the tetraquarks have Regge behavior is not new, datingback to the 1970s [11, 12], but now, with the wealth of new data, it is time to reexamine it.Out of the zoo of special states we suggest to use the state Y (4630), which decays pre-dominantly to Λ c Λ c , as a window to the landscape of tetraquarks. The baryon-antibaryondecay signals it out as a state that can match our description of a stringy tetraquark. Wepropose to use our simple criterion to decide whether this state is a genuine stringy exotichadron or a molecule. We present the predictions of the mass and width of the higher excitedstates of the Y (4630) trajectory. We compute the trajectories associated with M and theangular momentum J , as well as the trajectories associated with M and the excitation level n . In this way we also make some predictions of yet undiscovered higher excited charmoniumstates.We argue that there should exist an analogous Y b state in the bottomonium sector thatdecays to Λ b Λ b , which should also belong to a trajectory. Whereas such a state can followalso from a non-stringy picture, the existence of a (modified) trajectory would be clearly anindication of a stringy nature. We also propose a similar trajectory for tetraquarks containing s ¯ s and we would like to encourage an experimental search for such tetraquarks decaying toΛΛ. A priori such objects made out of light quarks are also not excluded, and in fact, anyresonance which may be found to decay to any pair of baryon and antibaryon - such as Λ c Σ c ,Λ c Λ b , Λ b Λ, or many other options - could be a tetraquark with an appropriate trajectoryfollowing from its stringy structure.The paper is organized as follows. In section 2 we briefly review the basic concepts andfeatures of the HISH model. In particular we explain the two main ingredients of the model,the string endpoint mass and the baryonic vertex. We also describe the basic string config-urations that constitute the mesons, baryons, and glueballs. In section 3 we discuss possibleexotic stringy configurations. We mention several classes of them, for instance configurationswith no quarks or diquarks, formed of only baryonic and antibaryonic vertices. We also spec-ify the classes of exotic tetraquark configurations. In section 4 we describe the symmetric,semisymmetric and asymmetric tetraquark configurations. We then describe in section 5 themodified Regge trajectories of tetraquarks. In particular we discuss the differences betweenthe intercept of mesons and baryons and we conjecture about the intercept of the exotictetraquarks. Section 6 is devoted to the decay processes of the HISH tetra quarks. We writedown the width of the decay of a tetraquark which has in fact the same form as the width ofthe decay of a meson and a baryon.Readers interested in the phenomenological aspects of our work and predictions of newexotic states will find them in section 7, where we make the comparison between the HISHtetraquarks and hadron phenomenology. We start with the tetraquark in the charmoniumsector. We show that all of the known charmonium states - i.e. the non-exotic c ¯ c mesons -fit nicely on Regge trajectories, and extract the parameters of the Regge slope and endpointmass of the c quark. We review the Y (4630) resonance which is our main candidate tobe a tetraquark. We describe the various options to understand this state and we writedown its modified Regge trajectory. We then describe the possibility that there are suchtetraquarks also in the bottomonium sector and in the light quark sector, and offer sometentative predictions for these states. In section 8 we summarize and mention several openquestions. 3 A brief review of the HISH model
The holographic duality is an equivalence between certain bulk string theories and boundaryfield theories. The original duality was between the N = 4 SYM theory and string theoryin AdS × S . Obviously the N = 4 theory is not the right framework to describe hadronsthat resemble those found in nature. Instead we need a stringy dual of a four dimensionalgauge dynamical system which is non-supersymmetric, non-conformal, and confining. Thetwo main requirements on the desired string background is that it admits confining Wilsonlines, and that it is dual to a boundary that includes a matter sector, which is invariant undera chiral flavor symmetry that is spontaneously broken. There are by now several ways to geta string background which is dual to a confining boundary field theory. For a review paperand a list of relevant references see for instance [13].Practically most of the applications of holography of both conformal and non-conformalbackgrounds are based on relating bulk fields (not strings) and operators on the dual boundaryfield theory. This is based on the usual limit of α (cid:48) → α (cid:48) , is not very large. In gaugedynamical terms the IR region is characterized by λ = g N c of order one and not a very largeone.It is well known that in holography there is a wide sector of hadronic physical observableswhich cannot be faithfully described by bulk fields but rather require dual stringy phenomena.This is the case for a Wilson, ’t Hooft, or Polyakov line.In the holography inspired stringy hadron (HISH) model we argue that in fact also thespectra, decays and other properties of hadrons - mesons, baryons, glueballs and exotics -should be recast only by holographic stringy configurations and not by fields. The majorargument against describing the hadron spectra in terms of fluctuations of fields, like bulkfields or modes on probe flavor branes, is that they generically do not admit Regge trajectoriesin the spectra, namely, the (almost) linear relation between M and the angular momentum J . Moreover, for top-down models with the assignment of mesons as fluctuations of flavorbranes one can get mesons only with J = 0 or J = 1. Higher J will have to be related tostrings, but then there is a big gap of order λ (or some fractional power of λ depending onthe model) between the low and high J mesons. Similarly the attempts to get the observedlinearity between M and the excitation number n are problematic whereas for strings it isan obvious property.The construction of the HISH model is based on the following steps. (i) Analyzing stringconfigurations in holographic string models that correspond to hadrons, (ii) devising a tran-sition from the holographic regime of large N c and large λ to the real world that bypassestaking the limits of N c and λ expansions, (iii) proposing a model of stringy hadrons in flatfour dimensions that is inspired by the corresponding holographic strings, (iv) confrontingthe outcome of the models with the experimental data (as was done in [6–8]).Confining holographic models are characterized by a “wall” that truncates in one way oranother the range of the holographic radial direction. A common feature to all the holographicstringy hadrons is that there is a segment of the string that stretches along a constant radialcoordinate in the vicinity of the “wall”, as in figure 3. For a stringy glueball it is the wholefolded closed string that rotates there and for an open string it is part of the string, the4orizontal segment, that connects with vertical segments either to the boundary for a Wilsonline or to flavor branes for a meson or baryon. The fact that the classical solutions of theflatly rotating strings reside at fixed radial direction is a main rationale behind the mapbetween rotating strings in curved spacetime and rotating strings in flat spacetime. A keyingredient of the map is the “string endpoint mass”, m sep , that provides in the flat spacetimedescription the dual of the vertical string segments. It is important to note that this massparameter is neither the QCD mass parameter nor that of the constituent quark mass, andthat the m sep parameter is not an exact map of a vertical segment. The endpoint mass isobtained as an approximation that is more accurate the longer the horizontal string, whenthe vertical segments are most pronounced (for shorter strings the configuration will resemblea more curved “U” shape).The stringy picture of mesons has been thoroughly investigated in the past (see [14] andreferences therein). It turns out that also in recent years there were several attempts todescribe hadrons in terms of strings. Papers on the subject that have certain overlap withour approach are [15–18]. Another approach to the stringy nature of QCD is the approachof low-energy effective theory on long strings. This approach is similar to the approachpresented in this paper. A recent review of the subject can be found in [19].The HISH model describes hadrons in terms of the following basic ingredients. • Open strings which are characterized by a tension T (or a slope α (cid:48) = πT ). The openstring generically has a given energy (mass) and angular momentum associated with itsrotation. The latter gets contribution from the classical configuration and in additionthere is also a quantum contribution (the intercept a ). Any such open string can be inits ground state or in an excited state. • Massive particles - or “quarks” - attached to the ends of the open strings which canhave four different values m u/d , m s , m c , m b . The latter are determined by fitting theexperimental spectra of hadrons. These particles naturally contribute to the energyand angular momentum of the hadron of which they are part. • “Baryonic vertices” (BV) which are connected to a net number of three ( N c ) strings.One particularly important configuration would involve a single long string and two veryshort ones. Since the endpoints of the two short strings are one next to the other wecan consider them as forming a diquark. We emphasize that unlike other models whichhave quarks and diquarks as elementary particles (such as [20]), in the HISH modelthe diquark is always attached to a baryonic vertex. A BV can be also connected to acombination of quarks and antibaryonic vertices as will be discussed in the next section. • Closed strings which have an effective tension that is twice the tension of the openstring, leading to a slope that is halved, α (cid:48) closed = α (cid:48) open . They can have non-trivialangular momentum by taking a configuration of a folded rotating string. The excitationnumber of a closed string is necessarily even, and so is the angular momentum (on theleading Regge trajectory). Their intercept, the quantum contribution to the angularmomentum, is also twice that of an open string ( a closed = 2 a open ). There is no a priori reason to impose m u = m d in our model, but, for all our purposes the two masses canbe taken to be the same. The difference between them is smaller than the degree to which the masses can bedetermined from the phenomenology. • A single open string attached to two massive endpoints corresponds to a meson. • A single string that connects on one hand to a quark and on the other hand a baryonicvertex with a diquark attached to it is the HISH description of a baryon. • A single closed string is a glueball.
In addition to the basic hadron configurations with which we ended the previous section,there is a very wide range of other options to combine the elementary building blocks of theHISH model.To decide which of them are stable, one has to show that they are solutions of the quantumequations of motion. One can check the equations of the HISH model [5] or better, checkthe underlying holographic string equations. The physical properties that enter the formerequations are the string tension, the endpoint masses, and the angular momentum (or thecorresponding centrifugal force). There are also electromagnetic forces between the endpointcharges, although the impact of those is typically much smaller. These are features of theclassical configurations. Quantum mechanically there is a Casimir force effect which is relatedto the intercept. For instance the η meson has J = L = S = 0 and it is the negative intercept( a ∼ − .
23) that corresponds to a repulsive Casimir force that balances the tension. Inaddition as will be discussed in section 6 the main quantum instability associated with thebreaking mechanism of the string. We will not perform the stability analysis for all thepossible HISH states (but just refer to the decays of tetraquarks in section 6) . Moreover, wedo not intend to survey all the possible stringy configurations since we will be interested inthis work only in those that correspond to tetraquarks. Nevertheless, we sketch here severalclasses of possible configurations.Configurations that involve open strings are characterized by n B , n ¯ B , n q , and n ¯ q , whichare the numbers of baryonic vertices, antibaryonic vertices, quarks, and antiquarks respec-tively. They must obey the relation13 ( n B − n ¯ B ) − ( n q − n ¯ q ) = 0 (1) Exotic configurations without quarks:
For the case of no quarks or antiquarks, namelywith only baryonic vertices and antibaryonic vertices, we necessarily have n B = n ¯ B . In thiscase we can have a configuration similar to a closed string with three strings connecting theBV and the anti-BV. See (a) in figure (1).Another class of possible configurations consists of any planar polygon with an evennumber n B + n ¯ B vertices such that from each baryonic vertex there is one line going to aneighboring antibaryonic vertex, and two lines going to the other anti-BV on the other side,as is depicted in figure 1 (b). We can have also have a three dimensional configuration suchas the cube, where each vertex has three strings connected to it. This is (c) in figure 1. Inaddition we can tile the whole plane with hexagons, as seen in figure 2.6igure 1: Configurations of BVs and anti-BVs. (a) Three strings connecting one BV andone anti-BV. (b) A planar polygon. (c) A cube.Let us emphasize again that we do not aim here at finding all possible configurations andnot to check their stability. We merely demonstrate several potential possibilities, and we dothis only for the case of N c = 3. The number of strings can be further generalized to giveeven more possibilities. Exotic hadrons with quarks:
Next we can discuss the case of quarks and antiquarks ontop of the BVs and anti-BVs.The basic configuration of a baryonic vertex is that there are three strings attached toit. As was mentioned above, baryons are in fact characterized by a string tension, or a slopewhich is the same as for mesons, and hence one concludes that there is only one long stringattached to the BV, with two short strings connected it as a diquark.Generically, the single string can be connected either to another quark or to an antibary-onic vertex to which an antidiquark is connected. The first possibility is the baryon, whileand the second one is what we will refer to as the tetraquark. In figure 3 we draw for exam-ple a tetraquark including c and ¯ c quarks. We distinguish between a baryonic vertex and anantibaryonic vertex by the orientation of the strings attached to it. We take that strings arecoming into the BV and are coming out of the anti-BV.In fact the condition for the neutrality of Abelian gauge field that resides on the wrapped D -brane that constitutes the baryonic vertex [21], can be obeyed also if there are N c + k strings and k anti-strings for the general case of N c [22]. In particular we can have a BV withan attached diquark and single string whose end is a quark and, in addition, k strings thatend on a quark and k strings with the opposite orientation that end on antiquarks. As suchwe get a configuration which can be considered as pentaquarks (for k = 1) or “heptaquarks”etc. for higher k . In figure 4 we draw the holographic picture of a pentaquark that has thesame quark content of a Λ c and a K meson. It is easy to notice that these exotic hadronsmay have electric charges that cannot be carried by an ordinary hadron, in particular theremay be a pentaquark state with electric charge 3.Based on the experience with ordinary baryons that are based on a single string it is7igure 2: Tiling the whole plane with hexagons of BVs and anti-BVs. The pattern can berepeated to infinity.Figure 3: An example of a holographic tetraquark based on a diquark of c and u and anantidiquark of ¯ c and ¯ u c baryon (a) and a pentaquark (b) with an addedquark and an antiquark.Figure 5: An example of a holographic tetraquark built with two baryonic and two antibary-onic vertices.probable that the additional pairs of string-antistring will connect the BV to the nearby flavorbranes with very short strings like for the diquarks. One can similarly attach quark-antiquarkpairs to the tetraquark configuration of above thus yielding hexaquarks, octoquarks, and soforth.Instead of a single BV or a pair of one BV and one anti-BV as was just described one canalso have configurations where n B baryonic vertices are connected to n ¯ B = n B antibaryonicvertices. See for instance a configuration with n B = n ¯ B = 2 drawn in figure 5. In this casethe exotic configuration can carry a charge of − −
1, 0, 1 or 2.In these configurations each BV is connected to two anti-BVs and to a quark, and theanti-BVs similarly connect each to two BVs and an antiquarks.One can also imagine the configurations described above where the pairs of strings withopposite orientations, are not connected to flavor branes in the holographic picture or to aquark and antiquark in the HISH picture, but rather merge one into the other thus formingclosed strings.This is just a partial list of possible hypothetical hadronic configurations. In a future9ublication we will discuss these exotics in more detail.
From the zoo of potential exotic multiquark states discussed in the previous section we nowwant to focus on the tetraquarks. The configuration of a BV attached to a diquark connectedby a string to an anti-BV attached to an antidiquark, which is referred to as tetraquark, canbe classified by distinguishing between the following possibilities. • Symmetric tetraquarks where the antidiquark is built from the antiquarks associatedwith those found in the diquark. For instance, in figure (3) the diquark cu , and theantidiquark cu . • Semisymmetric tetraquarks in which there is one pair of quark and antiquark of thesame flavor and one pair which include a quark and an antiquark of different flavors,for instance ( cu )( cs ). • Asymmetric tetraquarks, where both pairs are of different flavor.
Let us start with a description the configurations described above in holography. Similar tofigure 3 where the diquark includes a c quark, one can have also configurations that includeinstead a diquark with any other flavor quarks, as shown in figure 6.Notice that these configurations are symmetric in the sense that for every quark connectedto the baryonic vertex there is an antiquark of the same flavor connected to the antibaryonicvertex. It thus obvious that these symmetric tetraquark configurations are flavorless andcarry zero electric charge. In the lower part of figure 6 we draw the corresponding picturesof each of the tetraquarks in the HISH setup. In fact, we drew only the cases where one ofquarks that build the diquarks is a light u or d quarks but in fact we can also have othercombinations like those with a c and s diquark. Altogether there are 15 different symmetrictetraquarks for 15 unique types of diquarks composed of quarks of five flavors (as the diquark( ij ) is identical to the diquark ( ji )). Another class of tetraquark is the semisymmetric one. In this class of tetraquarks only onepair in matched and the other is not. Thus the flavor content of these exotic hadrons isthe same as of mesons that carry non-trivial flavor. We have 5 possibilities for a matchedpair and 20 for the unmatched pair, so altogether there are 100 possible semisymmetrictetraquarks. For example in figure 7 the left figure may describe a semisymmetric tetraquarkwith a matched pair of u and u , with a b and a c as the unmatched pair.Unlike the symmetric tetraquarks which are always chargeless, semisymmetric tetraquarkscan carry a charge of +1, 0, or +1. 10igure 6: Symmetric holographic tetraquarks and their corresponding HISH pictures. Thediquark include a b , s and u/d quarks for the left, middle and right templates correspondingly. We could have any pair of quark forming the diquark and any two antiquarks forming theantidiquarks thus altogether there are a priori 15 ×
15 = 225 possibilities of tetraquarks. Outof the 225 tetraquarks, we have 15 that are symmetric, 100 semisymmetric, and thus 110are asymmetric ones. The asymmetric tetraquarks can carry a charge of − − ± bu )( cd ), an example of an asymmetric tetraquark. Following the holographic description, the four dimensional flat spacetime HISH descriptionof a tetraquark takes the form of a single string with massive endpoints, which in addition tomass carry certain flavor and charge. It is thus clear that similarly to the HISH configurationsthat describe mesons and baryons, the tetraquarks also admit modified Regge trajectories(MRT). As was analyzed in [6] the modification of the linear Regge trajectories is due to themassive particles at the endpoints of the string. The endpoint masses are given, in the mapping from holographic to flat spacetime, by thelength of the vertical segments of the string. For a diquark, there is a priori a contributionto the endpoint mass from the baryonic vertex. However, in [7] it was found that the BV Additional modification is caused by the electric charges of the endpoints. This modification is lesssignificant than that due to the masses so we will ignore it in what follows. b quark and their corresponding HISH pictures.and the short strings completing the diquark do not make a significant contribution to theendpoint mass, at least not in the case where one or both of the quarks in the diquark is light,i.e. a d or u quark. The mass of the BV also depends on its height in the radial coordinate,in a way that will vary between different models, and in the case of a diquark composedof a heavy quark and a light one, it is energetically favorable for the BV to remain on thelower flavor brane, where it is less massive. The details will be model dependent. With thesetwo points in mind, but in particularly the phenomenological result of [7], we can take theendpoint mass of a cu diquark, for example, to be simply the mass of the c quark.The existence of the baryonic vertex in the holographic picture may affect the interceptof the HISH in the quantum case, as will be discussed in section 5.2. The trajectories are given in terms of the energy and the angular momentum of the stringwith massive endpoints. In the holographic picture the masses of the endpoints come fromthe vertical segments of the string and the BV in the case of a diquark, but from a flatspacetime perspective we do not see the details, only the masses themselves. Classically thetrajectories are given by E = (cid:88) i =1 , m i β i arcsin( β i ) + (cid:113) − β i − β i , (2)12 = (cid:88) i =1 , πα (cid:48) m i β i (1 − β i ) (cid:18) arcsin( β i ) + β i (cid:113) − β i (cid:19) , (3)where β i = ωl i is the velocity of an endpoint, and l i is the length of each of the arms of thestring, namely the segments from the center of mass, around which the string rotates, to theendpoints. The velocities are related by the boundary condition Tω = m β − β = m β − β . (4)In the low mass limit where the endpoints move at a speed close to the speed of light, β i →
1, we have an expansion in ( m/E ): J = α (cid:48) E × (cid:32) − (cid:88) i =1 (cid:32) √ π (cid:16) m i E (cid:17) / + 2 √ π √ (cid:16) m i E (cid:17) / + · · · (cid:33)(cid:33) , (5)from which we can easily see that the linear Regge behavior is restored in the limit m i → √ E .The high mass limit holds when( E − m − m ) / ( m + m ) (cid:28) . (6)We take the limit β i →
0, and the expansion is J = 4 π √ α (cid:48) (cid:114) m m m m ( E − m − m ) / + 7 √ π √ α (cid:48) m − m m + m m m (cid:112) ( m + m ) ( E − m − m ) / + · · · . (7) The classical trajectories are uplifted to the quantum trajectories when quantum fluctuationsare included. For a linear Regge trajectory of a string with no massive endpoints the passagefrom the classical to the quantum picture is simply via the replacement J cl = α (cid:48) E cl → J + n = α (cid:48) E + a (8)where n are the eigenvalues of the world sheet Hamiltonian, w n = n , of the n -th excitationlevel, and a is the intercept, a = D − (cid:80) n w n . In the last expression, D is the numberof dimensions in the target spacetime. In fact this expression holds only for the criticaldimension and for the case of non-critical dimensions it is corrected by the Liouville term.As was explained in detail in [5], for the MRT the story is much more complicated in variousrespects.Firstly, the eigenvalues w n are affected by the masses at the ends of the string, w n = w n ( n, T Lm ) (cid:54) = n . Moreover [18], the eigenvalues are different for the fluctuations transverseto the plane of rotation and for the one along the direction of the string (which is not there13t all in the massless case). Correspondingly, the intercept a is modified. For the staticnon-rotating string this was derived in [23].Another complication is that since the string associated with the tetraquark resides in fourdimensions, i.e. a non-critical dimension, one has to include the Liouville or the Polchinski-Strominger term for consistent quantization [24]. This term diverges for the massless caseand has to be regularized and renormalized. This was discussed for the massless case in [17]and was touched upon for the massive case in [5].The passage from the classical to the quantum trajectory that we have just presented isthe one required for the stringy meson. The question is whether the same treatment appliesalso to the exotic tetraquark configuration, or whether there are special features associatedwith it. On the one hand it seems that from the point of view of the fluctuations of thestring there is no difference between the mesonic, baryonic, and the tetraquark string, sincein all these three cases it is a string between massive endpoints. However, that is not thewhole story. In fact just by comparing the mesonic and baryonic trajectories we observe, bycomparison to the experimental data, that although the tension and the endpoint masses arevery similar, there are several differences.Obviously the angular momenta along the baryon trajectories are half integers whereasfor the mesons they are integers. This follows of course from the contribution of the spindegrees of freedom to the total angular momentum. The role of the spin in the HISH has notyet been worked out. In the holographic picture the quark is in fact the vertical (along theholographic radial direction) segment of the string and its endpoint is on the flavor brane.Correspondingly the spin associated with each quark can be related to the string along theradial direction or to its endpoint. The former is the case if the vertical segment is a fermionicstring. In this case one needs to explain how to connect the horizontal bosonic string to thevertical fermionic spin-carrying one. The latter option is that the connection point on theflavor brane is a fermion. In principle one can attribute a spin S = to the baryonic vertex,however that will not explain the spin values of the various mesons. So we will assume herethat there is a spin S = to each connection of a string and a flavor brane.To compare the intercept of hadrons with different spin, like baryon and a meson, it isuseful to define a different intercept as follows. Let us start with the linear case, which wecan write as J = L ± S = α (cid:48) E + a → L = α (cid:48) E + ˜ a (9)by defining ˜ a ≡ a ∓ S (10)For the MRT of strings with massive endpoints we still define ˜ a in the same way and relateit to the trajectory of L , the orbital angular momentum.When comparing the intercepts of mesons and baryons we find that they are differenteven when they are composed of the same type of quarks. In particular we find that thereis a difference between ˜ a of the lightest baryon composed of u and d quarks and the lightestmeson (which is not a Goldstone boson).As was mentioned before in both cases the intercept is a measure of quantum fluctuationsof a straight string between massive particles on its ends. We must conclude that the fluc-tuations of a system that includes a string and an endpoint built from a BV and a diquarkis different from that with an ordinary quark as an endpoint. There are two possible reasons14or this difference: (i) the quantum fluctuations of the BV, or (ii) a change of the boundarycondition for the string by the BV which changes the eigenvalues its fluctuations.If we denote by ˜ a m and ˜ a b the intercepts of a meson and a baryon of the same flavorstructure, the the difference of the intercepts is ∆˜ a = ˜ a b − ˜ a m . If the cause of the differenceis the quantum fluctuations of the BV then we anticipate that˜ a t ≡ ˜ a m + 2∆˜ a = ˜ a b + ∆˜ a . (11)But the intercept of the tetraquark will not be of this form if it is due to the change of theboundary conditions of the string. In any case, since we do not know how to determine thisdifference from string theory (recall that the BV is a wrapped D p brane over a c p cycle), wewill use the experimental data to determine ˜ a b , ˜ a m , and ˜ a t .Another difference between the meson and baryon trajectories is the fact that for baryonsthere is a different intercept for even and odd orbital angular momentum states. This happensfor baryons composed of light quarks, namely the N and ∆ baryons, and not for thosebuilt from heavy quarks. Where this effect occurs we see two parallel trajectories, with thetrajectory of the odd states above that of the even states, L = (cid:40) αM + a e , L even αM + a o , L odd . (12)For the N trajectory a e − a o ≈ .
4, while for the ∆, a e − a o ≈
1. The source of this difference,and its magnitude, must depend on the spins involved. In the baryon we have an spin 1/2quark, and a diquark which can carry spin 0 or 1.In the tetraquark, we have a diquark and antidiquark, where each may carry spin. If thesource of the difference between the even and odd intercepts is the BV and the diquark, weshould expect a similar split also for the tetraquark trajectories built from light quarks. If,on the other hand, it occurs only when the total angular momentum is a half-integer, anddepends on an interaction between the half-integer spin of the quark and the integer spin ofthe diquark, then for the tetraquarks that have integer angular momentum we should nothave this phenomenon. The details of this phenomenon in the baryon spectrum have notbeen worked out yet, and a fuller understanding is needed before speculating further on itsexistence in the case of the tetraquark.
In a similar manner to the decay of mesons, baryons, and glueballs, the exotic hadrons canalso decay generically to a combination of exotic and ordinary hadrons.For low mass tetraquarks, the constituent quarks and antiquarks may be close enough toeach other to annihilate and thus decay, but for excited tetraquarks that are described bya diquark and antidiquark joined by a (long) string the dominant mechanism of decay is asplit of a string and a generation of a quark-antiquark pair at the two ends of the torn apartstring. In this way we get a BV connected to a quark and a diquark on one side of the splitstring - which gives a baryon, while on the other side we get an antibaryon.In figure 8 we draw a sketch of the holographic decay of the Y ( cd )( cd ) /Y ( cu )( cu ) tetraquarkinto a Λ c Λ c pair. Figure 9 shows the same process in the HISH description including variousother flavor combinations and their outcomes.15igure 8: The decay of a holographic tetraquark composed of a diquark of c and u and ananti diquark of c and u into Λ c Λ c .Figure 9: Different types of tetraquarks based on a diquark of c and u and an anti diquarkof ¯ c and ¯ u , and their decay products. 16t is clear from these figures that the stringy tetraquarks cannot decay through thismechanism of breaking the string into two mesons like D and D , but rather only to a baryon-antibaryon pair. On the other hand it is also clear that a stringy charmonium state decaysvia a string breaking and a creation of quark antiquark pair into a pair of charmed mesons, D and D .The decay width of a tetraquark into a baryon and antibaryon via this mechanism hasthe same structure as that of a meson and a baryon. It was computed in [9], where the decaywidth is found to be proportional to the length of the string l times the probability for paircreation, Γ ∼ le − mqsep T , (13)where m qsep is the mass of the quark created as an endpoint of the torn apart string. Thelength is a function of the string tension, the energy, and of m and m , the masses of thediquark and antidiquark at the endpoints of the original tetraquark. For long strings (orsmall m and m ) we have the expressionΓ ∼ (cid:18) EπT − m + m T (cid:19) e − mqsep T . (14)As was shown in [9] the decay probability is proportional to the product of the probabilityof the horizontal string to reach due to quantum fluctuations a flavor brane (the exponentialfactor) and the probability that the string splits (which is proportional to the string length). In the previous sections we have sketched several exotic hadron configurations in the HISHmodel. Obviously, the most interesting issue is to what extent have such states been observedor could be observed in future experiments. The main idea we would like to introduce hereis that tetraquarks should reside on modified Regge trajectories, and that this could animportant tool in their phenomenological study.In this section, we begin by reexamining the trajectories of the regular c ¯ c mesons, showingthat they fit on trajectories of rotating strings with massive endpoints. We then introduce the Y (4630), a charmonium-like resonance which is the main tetraquark candidate, and shouldhave a trajectory similar to those of the charmonia. We offer predictions for higher tetraquarkstates.Following the discussion of the charmonium sector we discuss the possibility of tetraquarkscontaining b ¯ b or s ¯ s . In those cases, there are no experimentally known candidates thatdecay to a baryon and antibaryon, but we offer some predictions for tetraquark trajectoriesanalogous to that of the charmonium-like Y (4630).All the fits in this section were done using the formulae obtained from the rotating stringwith massive endpoints, given by E = 2 m (cid:32) β arcsin( β ) + (cid:112) − β − β (cid:33) , (15) J + n − a = 2 πα (cid:48) m β (1 − β ) (cid:16) arcsin( β ) + β (cid:112) − β (cid:17) . (16)17hese are the same equations as those of eqs. 2 and 3 when we take the symmetric case oftwo equal endpoint masses. Note that J is shifted for the quantum case to J + n − a . In thisway we also do the radial trajectories of increasing excitation number n .We note again that, as explained in section 5, the mass (as a string endpoint) of a diquarkof a heavy quark and a light one is expected to be the same as the mass of the heavier quark.That is why, to give one example, we take m = m c ≈ . In recent years, many new charmonium-like resonances have been discovered, and conse-quently, many papers have been written on the subject. Some useful reviews written in thelast few years are [4, 25–27]. c ¯ c mesons In [6], we presented the fits to the trajectories of the J/ Ψ meson, including excited Ψ mesonsand the χ c states. We can expand on the analysis there by adding the trajectories of all ofthe other c ¯ c mesons, namely the trajectories of the different η c and h c states. We take a massof m sep = m c = 1490 MeV for the c quark at the string endpoints. The Regge slope is foundto be different for the different types of trajectories. For the orbital trajectories, the slopeis 0.86 GeV − , while for the radial trajectories, the slope is 0.59 GeV − . We present allthe c ¯ c trajectories in figure 10. A detailed table of the states used and the masses obtained,including predictions of higher charmonia, is in table 1.In sorting the c ¯ c mesons into trajectories, we do not stray from the accepted assignmentsof the different states. Our predictions are also consistent with those obtained by potentialmodels [28]. One omission on the part of our model is that we do not offer a way of predictingthe spin-orbit splitting measured for the χ cJ and higher spin Ψ mesons. Therefore in table 1we give a single predicted mass for these triplets of states.The trajectories of the hidden charm tetraquarks are expected to have the same values ofthe slopes and endpoint masses as the c ¯ c trajectories, so we will continue to use the valueslisted above in the following section. Y (4630) resonance: an overview Of particular interest to us is the state Y (4630). It was observed by Belle in 2008, in theprocess e + e − → γ Λ + c Λ − c , with a significance of 8 σ [29]. The parameters measured there forthe Y (4630) were J P C = 1 −− , M Y (4630) = 4634 +9 − , Γ Y (4630) = 92 +41 − . (17)As described in section 6, the decay to a baryon-antibaryon pair is the natural decaymode of a tetraquark. We reach this conclusion from a string theory point of view, but thispossibility was also explored in other contexts elsewhere [30–33]. Note that 0.86 GeV − is almost the same slope one gets for the light mesons, so, at least in the ( J, M )plane, we have a unified description of the trajectories of all mesons comprised of u , d , s , and c quarks. c ¯ c mesons. Left:
Trajectories in n of the η c (black), J/ Ψ (blue),and χ c (red). Right:
Orbital trajectories of the η c (black), J/ Ψ (blue), and Ψ(2 S ) (red).See table 1 for a full specification of the states used. J P C n State M (Exp.) M (Thry.) n J P C
State M (Exp.) M (Thry.)0 − + η c (1 S ) 2983.6 ± − + η c (1 S ) 2983.6 ± η c (2 S ) 3639.2 ± + − h c (1 P ) 3525.4 ± η c (3 S ) ? 4023 2 − + η c (1 D ) ? 38041 −− J/ Ψ(1 S ) 3096.9 ± −− J/ Ψ(1 S ) 3096.9 ± S ) 3686.1 ± ++ χ c (1 P ) 3414.8 ± S ) 4039 ± ++ χ c (1 P ) 3510.7 ± S ) 4421 ± ++ χ c (1 P ) 3556.2 ± S ) ? 4621 1 −− Ψ(1 D ) 3773.1 ± + − h c (1 P ) 3525.4 ± −− Ψ (1 D ) ? 38301 h c (2 P ) ? 3932 3 −− Ψ (1 D ) ?0 ++ χ c (1 P ) 3414.8 ± j ++[ a ] χ cj (1 F ) ? 40721 χ c (2 P ) ? 3854 1 0 − + η c (2 S ) 3639.2 ± ++ χ c (1 P ) 3510.7 ± + − h c (2 P ) ? 39111 χ c (2 P ) ? 3922 1 1 −− Ψ(2 S ) 3686.1 ± ++ χ c (1 P ) 3556.2 ± ++ χ c (2 P ) ?1 χ c (2 P ) 3927.2 ± ++ χ c (2 P ) ? 39412 χ c (3 P ) ? 4266 2 ++ χ c (2 P ) 3927.2 ± j −− [ b ] Ψ j (2 D ) ? 4169Table 1: The c ¯ c mesons sorted in modified Regge trajectories, including predictions forhigher charmonia. The columns on the left show the radial trajectories of increasing n , whileon the right we have trajectories in the (orbital) angular momentum. For each trajectory wecompared measured to experimental masses and offer one higher predicted state. The squarebrackets around some of the predicted masses indicate that the respective states were notused in the fits (and are therefore not drawn in figure 10), because we lack the measurementsfor the next states in their trajectories. Their masses are then used as input to predict theirexcited partners. [a] j = 2 , ,
4. [b] j = 1 , , Y (4630), there is another reported resonance, the Y (4660). It was observed byboth Belle [34] and BaBar [35], also has J P C = 1 −− , and its measured mass and width are M Y (4660) = 4665 ± , Γ Y (4660) = 53 ± . (18)The channel in which it was observed is e + e − → γπ + π − Ψ(2 S ).There are claims, and possibly even a consensus, that the two resonances Y (4630) and Y (4660) are really one and the same, so what we have is a single state with both decaychannels, Λ c Λ c and Ψ(2 S ) π + π − . If this is the case, then the Λ c ¯Λ c is still the dominant decay,so this does not interfere so much with the tetraquark picture [30, 37].Other works discussing the Y (4630) and Y (4660) include [38–40]. Y (4630) state Let us now examine the description of the Y (4630) state, the state that decays primarily intoΛ c Λ c , in terms of possible HISH pictures. There are three possible explanations: • A highly excited charmonium state. Specifically, it could be the state Ψ(5 S ), as seenin table 1 (the mass of the Ψ(5 S ) should be about 4620 MeV). Such a state shouldreside on a charmonium Regge trajectory as described in section 7.1.1 and should haveheavier higher excited states on the same trajectory, which should also decay to Λ c Λ c .The decay process involves in this case a creation of a baryonic vertex with a diquarkattached to it together with an antibaryonic vertex joined to an antidiquark. Unlikethe case of quark-antiquark pair creation where we can estimate in holography thedecay width as in [9], the process of creating a pair of baryonic vertices translates inholography to the creation of a wrapped Dp over a p -cycle and there is no simple wayto estimate its probability. • The second option is the tetraquark.In this case too there should be higher excitedtetraquark states residing on a trajectory with the Y(4630) state. The decay processinto a pair of Λ c Λ c is the one described in section 6. The width for such a decay is thesame as the decay width of a stringy meson into two stringy meson s via the breakingof the string and creation of a pair of endpoints. • There is also an option which does not take the form of an exotic hadron but rather amolecule of two hadrons. Since a molecule is not a single string with certain dressingson its endpoints and hence we do not expect in such a case to have a trajectory of suchstates. The simplest possibility is a molecule of Λ c and Λ c , which then simply breaksapart into the baryon antibaryon pair. The measured mass of the state Y (4630) is higherthan twice the Λ c mass by some 60 MeV, when the uncertainty in the measurementof the mass is of about 10 MeV. Had the molecule been bounded with some non-trivial binding energy, its mass should have been smaller than twice the mass of Λ c .In [41] the binding energy was evaluated, in a calculation where it arose from σ or ω exchange. Here there is no such binding energy involved. The Λ c ¯Λ c molecule isthen disfavored, but not entirely ruled out given its large measured width of 92 +41 − MeV. On the other hand, the Y (4630) and the Y (4660) resonance mentioned in the These are the average values given by the PDG [36]. S ) f (980), whichis at 4676 ±
20 MeV. According to [37], the interpretation of both measurements as asingle resonance with this molecular configuration is not inconsistent with experimentalresults, and in particular can reproduce the Λ c ¯Λ c decay width. In the stringy picture,the decay of the proposed Ψ(2 S ) f (980) molecule into Λ c ¯Λ c would again require thecreation of a BV-anti-BV pair, and its probability difficult to estimate.To summarize, the current data regarding the Y (4630) resonance is not enough to deter-mine its nature. This is why we propose to search for more states on its Regge trajectory.Its excited partners, which, if found, will serve as evidence for the tetraquark picture, andin particular could make it preferable to the molecular configuration, which does not exhibittrajectories. Y (4630)If the Y (4630) is indeed a tetraquark, it should be part of a Regge trajectory. Using thevalues of m sep and α (cid:48) of the c ¯ c trajectories, we can use the known mass of the state Y (4630)and extrapolate from it to higher (or lower) states along the trajectory. From the combinedfit of section 7.1.1, we have the values m c = 1490 MeV , (19)and α (cid:48) J = 0 .
86 GeV − , α (cid:48) n = 0 .
59 GeV − . (20)There are two different slopes, one for orbital trajectories and one for radial trajectories.The higher states are only predictions at this point, as measurements at the relevantmasses are scarce, but it is interesting to see if we can continue the trajectory backwards,that is, to check if the Y (4630) is an excited state of an already known resonance. Thelower state would be below the Λ c ¯Λ c threshold, and its decay would require the baryonic andantibaryonic vertices to annihilate. This could mean that such a state is very narrow, due toa OZI-like suppression mechanism of these decays. This mechanism was discussed recentlyin [42].In the angular momentum plane, we can have a scalar state with J P C = 0 ++ precedingthe Y (4630) in its trajectory. There is no currently known scalar resonance at the masspredicted by the Regge trajectory, of about 4480 MeV.In the ( n, M ) plane we may find the Y (4360), another vector resonance that is at theright mass for the Y (4630) to be its first excited state. Its mass and width are M Y (4360) = 4354 ± , Γ Y (4360) = 78 ± . (21)It was observed in decays to Ψ(2 S ) π + π − . This is the same decay channel in which the Y (4660) was observed. In fact, suggestions that Y (4630) or Y (4660) is an excited partner of Y (4360) are not uncommon [37].The problem with assigning the Y (4360) as the unexcited partner of the Y (4630) is inits width. The widths of the two states are roughly the same, even though the lower state,being beneath the Λ c ¯Λ c threshold, should be significantly narrower than the above thresholdstate. 21ere we also have to reconsider the Y (4660), as it was observed in the same channel asthe Y (4360). One scenario exists in which the Y (4630) and Y (4660) are the same resonance,that decays both to Λ c ¯Λ c and Ψ(2 S ) π + π − . It is a tetraquark, but there is a limited phasespace for Λ c ¯Λ c decays, and we see it decay to Ψ(2 S ) π + π − . The Y (4360) is also a tetraquark,but has no phase space to decay into baryons and decays instead to Ψ(2 S ) π + π − . However,we have to ask why the tetraquark would have such a large decay width for non-baryonicchannels. We will leave this scenario for now, and consider the higher excitations of Y (4630).Whether or not there is a lower, unexcited version of Y (4630), we still predict higher exci-tations of the stringy tetraquark. For the higher states, the Λ c ¯Λ c decays should be dominant,and their masses should be such that they fall on a Regge trajectory. The predictions are intable 2.The intercept of the Regge trajectories is calculated by matching the mass formula ofeqs. 15 and 16 to the mass of the Y (4630). The values obtained are a = − . a = − . a t = ˜ a + 2∆˜ a . (22)The value ˜ a is the intercept of the meson with the same flavor content, in this case thecharmonium. We take the intercept of the J/ Ψ trajectory, which is very small for both theradial and orbital trajectories and equals ≈ − .
1. Now ∆˜ a is the difference in the interceptwhen replacing a quark with a diquark at one of the string’s endpoints. We would like toevaluate it in a case where a c quark is involved. We could compare the J/ Ψ intercept to theintercept obtained from the mass of the doubly-charmed baryon Ξ c . From these we have ∆˜ a in the range -0.7 to -1.0, depending of the value of the slope. Another measurement wouldcompare the intercept of charmed mesons ( D ) and baryons (Λ c ). Then we get ∆˜ a = ( − . − . a t is as negative as we foundit to be.This reflects the large mass difference between the Y (4630) state and the c ¯ c ground statemass, and is perhaps an indication that the Y (4630) is a radially excited state and not thefirst state in its trajectory. If we assume that it has n = 2, for example, instead of n = 0,then the intercept of its radial trajectory is − .
0, and we get a better agreement with thetheory provided there are two more states in the trajectory beneath the Y (4630). In the spectroscopy of heavy quarkonia, one usually expects analogies to exist between the c ¯ c and b ¯ b spectra. Tetraquarks and other exotics should be no exception. Therefore, wepropose to search for an analogous state to the Y (4630), which we will call the Y b , thatdecays primarily to Λ b ¯Λ b . We will assume that, like the Y (4630), the Y b state is located alittle above the relevant baryon-antibaryon threshold of Λ b Λ b .The mass of the Λ b is 5619 . ± .
23 MeV. As an estimate - take a mass of M ( Y b ) ≈ M (Λ b ) + 40 MeV∆ M ( Y b ) ≈
40 MeV (23) To be exact, the replacement is of an antiquark with a diquark, or a quark with an antidiquark. Mass Width“-2” 4060 ±
50 Narrow“-1” 4360 ±
50 Narrow0 + − + − ±
50 150–2502 5100 ±
60 200–3003 5305 ±
60 220–3204 5500 ±
60 250–350 J P C
Mass Width0 ++ ±
40 Narrow1 −− + − + − ++ ±
40 150–2503 −− ±
40 180–2804 ++ ±
45 200–3005 −− ±
45 250–350Table 2: Trajectories of the Y (4630). Based on the experimental mass and width of the Y (4630) we extrapolate to higher excited states on the trajectory. Uncertainties are basedon both experimental errors and uncertainties in the fit parameters. The excited statesare expected to decay into Λ c Λ c . Some possible lower states, with masses below the Λ c Λ c threshold, are also included. Y (4360), observed to decay to Ψ(2 S ) π + π − , is a candidate forthe n = − Y b would also have its own Regge trajectory, so, based on the mass we choose for it, wecan predict the rest of the states that lie on the trajectory. The trajectories are calculatedusing the slopes of the Υ trajectories, which are α (cid:48) ≈ .
64 GeV − in the ( J, M ) plane, and α (cid:48) ≈ .
46 GeV − in the ( n, M ) plane. The string endpoint mass of the b quark is m b = 4730MeV. The resulting masses are listed in table 3.Like before, we get a large negative intercept if we assume the above threshold state Y b isthe first state in the trajectory with n = 0. For the radial trajectory, this intercept is -6.1. So,we may also continue the radial trajectory backwards. The masses of the states that wouldpreceed the above threshold Y b tetraquark are also listed in table 3. We can see that theobserved resonance Y b (10890) is a potential match to be the “ n = −
2” state. This resonancewas observed to decay to π + π − Υ( nS ) and has a mass of 10888 . ± . Y (4360) from the charmonium sector, which decayed to π + π − Ψ(2 S ) andwas at the right mass to be an unexcited partner of the tetraquark candidate Y (4630). Ifthe Y (10890) is on the trajectory, then we also predict another resonance following it, whichbeing similarly below the baryon-antibaryon threshold should have similar properties, at amass of ≈ An important and longstanding open question is why there are no tetraquarks, or strongtetraquark candidates among the light mesons. In this note we will not attempt to answerthis question, even though among the 225 possible tetraquarks we counted section 4, manyare made up of u , d , and s quarks and their antiquarks only.Instead, suppose that at least s quarks are heavy enough to accommodate tetraquarks.Then we again predict a state that would decay to Λ ¯Λ, and a trajectory of its excited states.We then predict again a trajectory of states beginning with a near-threshold state. Themasses are in table 4. Values taken from the Regge trajectory fits of [6]. Mass“-2” 10870 ± ±
500 11280 ±
401 11460 ±
402 11640 ±
403 11810 ±
404 11980 ± J P C
Mass1 −− ± ++ ± −− ± ++ ± −− ± Y b , a tetraquark containing b ¯ b and decaying toΛ b ¯Λ b . The mass of the first state is taken near threshold, masses of higher states are on theRegge trajectories that follow from the ground state.For the predictions, we take a ground state mass of M ( Y s ) ≈ (2 M (Λ) + 40) ±
40 MeV (24)with the usual slope for light quark ( u , d , s ) hadrons of 0.9 GeV − in the ( J, M ) plane. Inthe ( n, M ) the slope is lower, and we take 0.8 GeV − . The mass of the s quark is taken tobe 220 MeV.The resulting intercept is -3.6. Again we repeat the exercise of extrapolating the trajectorybackwards, listing two preceding states. The vector resonance ρ (1570) might be a match. n Mass“-2” 1580 ± ±
400 2270 ±
401 2540 ±
402 2780 ±
403 3000 ±
404 3210 ± J P C
Mass1 −− ± ++ ± −− ± ++ ± −− ± Y s , a tetraquark containing s ¯ s and decaying toΛ ¯Λ. In this note we have used the HISH model to shed some light on the tetraquark controversy,and we used recent data on the spectra and decays of heavy hadrons as a further test of theHISH model. The main theme of this note is quite simple. Taking as a definition that ahadron is a string (or a connected collection of strings), then any exotic hadron, just as thenormal hadrons, has to admit a Regge trajectory behavior in its spectrum. Conversely, thisis not the case for a “molecule”, or bound state of strings. Thus, a simple way to verify that astate is indeed a multiquark exotic hadron is to look for its excited Regge partners. Of course,the absence of these excited partners on a trajectory cannot be considered conclusive evidence24etraquarks containing c ¯ c Tetraquarks containing b ¯ b Tetraquarks containing s ¯ s (decaying to Λ c Λ c ) (decaying to Λ b Λ b ) (decaying to ΛΛ) n J P C M Γ n J P C
M n J
P C M −− + − + − −− ±
40 0 1 −− ±
400 2 ++ ±
40 150–250 0 2 ++ ±
40 0 2 ++ ±
401 1 −− ±
50 150–250 1 1 −− ±
40 1 1 −− ±
400 3 −− ±
40 180–280 0 3 −− ±
40 0 3 −− ±
402 1 −− ±
60 200–300 2 1 −− ±
40 2 1 −− ± Y (4630) (here in bold).that a state is not a tetraquark. However, the existence of a trajectory would strongly implythat the state is a tetraquark.We suggest to apply this rule to the resonance Y(4630). Due to the fact that it pre-dominantly decays to a baryon and an antibaryon, it seems plausible from its HISH modeldescription that it is indeed a genuine stringy tetraquark, and, if we consider the Reggetrajectory, it is a possible gateway leading to further excited tetraquark states. We furtherpropose to look for a similar phenomena also for hadrons containing a b quark and a ¯ b , aswell as s and ¯ s , also decaying to baryon-antibaryon. We recap the predictions for thesetetraquarks in table 5.We note that while the natural way to search for molecules is by looking in the vicinityof different two hadron thresholds [10], a good way to identify tetraquarks is to look abovethe two baryon threshold. It is there that we can look for particles that decay into a baryon-antibaryon pair, the stringy tetraquark’s natural and distinctive mode of decay, but it is alsoa region where data is still quite scarce. And while the predictions above are for the Y (4630),and its immediate analogues Y b and Y s , it should be noted that there could be similar stateswith non-symmetric decay channels, Λ c Λ b for example. For resonances found in such channelswe will also expect trajectories, which can be easily predicted using the HISH model.There are several future directions along which this research project can be continued.Here we list some of them. • While the focus here was on the Y (4630) state, as a window to the exotic hadrons, thereare in fact there are plenty of other exotic resonances in the charmonium and bottomo-nium sectors, the plethora of X , Y and Z resonances, that are awaiting to be part ofthe HISH community. They are mostly lower mass states, below the baryon-antibaryonthreshold, but some of these resonances may turn out to be themselves tetraquarks.Our model easily describes excited states, with a single long string stretched betweendiquark and antidiquark, but one should also think about tetraquarks where all fourconstituent quarks and antiquarks are close together. The latter picture is the onethat might be more appropriate to already discovered states, which do not decay tobaryon-antibaryon (and without being narrow as a result). This is related to the dif-ficult theoretical question of the applicability of these stringy models to the hadron25pectrum. As we have pointed out, the lowest spin mesons and baryons belong onRegge trajectories, showing a stringy nature even when effective long string approxi-mations are expected to break down. One must ask then to what extent this is alsotrue of tetraquarks and other exotics. • As was described in section 3 hypothetically there could be 225 types of tetraquarks,which include symmetric, semisymmetric and asymmetric ones. In this note we dis-cussed those that are chargeless and hence decay to a baryon and its antiparticle butthere could be also those that carry charges of 1 or even 2. We intend in the future toprovide theoretical predictions about their masses and in certain cases also their width.In addition, there are more options to get tetraquarks on top of that built from one BVand one anti-BV. The configuration drawn in figure 5 with two BVs and two anti-BVsis one such example. • Moreover, it is easy to visualize higher order multiquark states like pentaquarks (seefigure 4) heptaquarks, and so on. In a similar manner to the description of the MRTof the tetraquarks we will analyze in the future also the trajectories of potential highermultiquarks. • In this note we addressed the issues of the spectra and decay modes of candidates ofexotic quarks. We have not discussed the question of production mechanisms of suchstates. This issue is obviously crucial for the future generation of exotic hadrons. Oneshould separately discuss the production processes in e + e − colliders and in hadroncolliders. • From the theoretical side of the story, it is essential to understand better the quantumcorrections of the trajectories, namely the intercept, and its generalization, in particularfor hadrons that include baryonic vertices. As was mentioned in section 5.2, that is infact also the understanding of the difference between meson and baryon trajectories. • So far we used the HISH model to describe genuine exotic hadrons but one shouldbe able to address in this framework also states which are molecules of hadrons bothmesons and baryons. Accomplishing this task will shed more light of how to distinguishbetween the molecule and the exotic hadrons and in addition may be useful for theunderstanding of molecules, the most famous of which is probably the deuterium.
Acknowledgments
We are grateful to O. Aharony, V. Kaplunovsky, M. Karliner, S. Nussinov, A. Soffer and S.Yankielowicz for insightful conversations. We want to thank O. Aharony for his remarks onthe manuscript. J.S would to thank Hai-Bo Li of the BESIII collaboration for hosting himin the IHEP Beijing where part of this work was done. This work was supported in part bya center of excellence supported by the Israel Science Foundation (grant number 1989/14),and by the US-Israel bi-national fund (BSF) grant number 2012383 and the Germany Israelbi-national fund GIF grant number I-244-303.7-2013.26 eferences [1] J. L. Rosner, Possibility of baryon - anti-baryon enhancements with unusual quantumnumbers, Phys. Rev. Lett. 21 (1968) 950–952. doi:10.1103/PhysRevLett.21.950 .[2] J. L. Rosner, The Classification and Decays of Resonant Particles, Phys. Rept. 11 (1974)189–326. doi:10.1016/0370-1573(74)90007-6 .[3] P. 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