Review Heavy Tetraquarks in the Relativistic Quark Model
Rudolf N. Faustov 1, Vladimir O. Galkin 1,* and Elena M. Savchenko 2
1 Federal Research Center “Computer Science and Control”, Institute of Cybernetics and Informatics in Education, Russian Academy of Sciences, Vavilov Street 40, 119333 Moscow, Russia; [email protected] 2 Faculty of Physics, M.V. Lomonosov Moscow State University, 119991 Moscow, Russia; [email protected] * Correspondence: [email protected]
Abstract: We give a review of the calculations of the masses of tetraquarks with two and four heavy quarks in the framework of the relativistic quark model based on the quasipotential approach and QCD. The diquark-antidiquark picture of heavy tetraquarks is used. The quasipotentials of the quark-quark and diquark-antidiquark interactions are constructed similarly to the previous consideration of mesons and baryons. Diquarks are considered in the colour triplet state. It is assumed that the diquark and antidiquark interact in the tetraquark as a whole and the internal structure of the diquarks is taken into account by the calculated form factor of the diquark-gluon interaction. All parameters of the model are kept fixed from our previous calculations of meson and baryon properties. A detailed comparison of the obtained predictions for heavy tetraquark masses with available experimental data is given. Many candidates for tetraquarks are found. It is argued that the structures in the di-J/ψ mass spectrum observed recently by the LHCb collaboration can be interpreted as ccc¯c¯ tetraquarks.
Citation: Faustov, R.N.; Galkin, V.O.; Keywords: tetraquark spectroscopy; diquark; relativistic quark model Savchenko, E.M. Heavy Tetraquarks in the Relativistic Quark Model. Universe 2021, 7, 94. https://doi.org/ 10.3390/universe7040094 1. Introduction
Academic Editors: David Blaschke, The possibility of the existence of exotic multiquark hadrons, with the content of the Vitaly Bornyakov, Konstantin valence quarks and antiquarks being different from a quark–antiquark pair for mesons, Klimenko, Alexey Luchinsky, Roman and three quarks for baryons, had been considered since the early days of the quark model. Rogalyov and Roman Zhokhov However, the absence of convincing experimental evidence for such multiquark state made investigation of marginal interest for several decades. The situation dramatically Received: 2 March 2021 changed in the last two decades. This subject became a hot topic after the first explicit Accepted: 6 April 2021 experimental evidence of the existence of hadrons, with compositions different from the Published: 8 April 2021 usual qq¯ for mesons and qqq for baryons, became available (for recent reviews, see [1–6] and references therein). Candidates for both the exotic tetraquark qqq¯q¯ and pentaquark Publisher’s Note: MDPI stays neutral qqqqq¯ states were found. However, in the literature there is no consensus about the with regard to jurisdictional claims in composition of these states [1–6]. For example, significantly different interpretations for published maps and institutional affil- the qqq¯q¯ candidates were proposed: molecules composed from two mesons loosely bound iations. by the meson exchange, compact tetraquarks composed from a diquark and antidiquark bound by strong forces, hadroquarkonia composed of a heavy quarkonium embedded in a light meson, kinematic cusps, etc. The discrimination between different approaches is a very complicated experimental task. Copyright: © 2021 by the authors. The simplest multiquark system is a tetraquark, composed of two quarks and two Licensee MDPI, Basel, Switzerland. antiquarks. Heavy tetraquarks are of particular interest, since the presence of a heavy This article is an open access article quark increases the binding energy of the bound system and, as a result, the possibility distributed under the terms and that such tetraquarks will have masses below the thresholds for decays to mesons with conditions of the Creative Commons open heavy flavour. In this case, the strong decays, which proceed through the quark and Attribution (CC BY) license (https:// antiquark rearrangements, are kinematically forbidden, and the corresponding tetraquarks creativecommons.org/licenses/by/ can decay only weakly or electromagnetically, and thus have a tiny decay width. If the 4.0/).
Universe 2021, 7, 94. https://doi.org/10.3390/universe7040094 https://www.mdpi.com/journal/universe Universe 2021, 7, 94 2 of 21
predicted tetraquarks have masses slightly (a few MeV) above these thresholds, then they can be also observed as resonances. The excited tetraquark states could also be narrow, notwithstanding the large phase space, since their decays will be suppressed, either by the centrifugal barrier between quarks and antiquarks, by the nodes of the wave function of radially excited states, or both. In Table1, we collect experimental data on hidden-charm mesons with exotic prop- erties [7–11]. We use the XYZ naming scheme, where X are neutral exotic charmonium- like states, observed in hadronic decays, Y are neutral exotic charmonium-like states with JPC = 1−−, observed in e+e− collisions, and Z are charged (isospin triplet I = 1) charmonium-like states. The later ones are explicitly exotic, since they could not simply be cc¯ states and, in order to have a nonzero charge, these states should at least contain additional light quark and antiquark. The experimentally determined quantum numbers JPC, masses M, total decay widths Γ, observation channels and the names of the exper- iments where they were first observed are given in Table1[ 9–11]. To determine the quantum numbers of X and Z states, a rather complicated angular analysis was necessary, while those of Y states, which coincided with the quantum numbers of the photon, are determined by the observation channel. Note that the X(4140), X(4274), X(4500), X(4700) and X(4740) states were observed as resonances in the J/ψφ mass spectrum, thus they should contain the strange quark and strange antiquark instead of the u and d quarks and antiquarks. Very recently, the BESIII Collaboration [8] reported the first candidate for the charged charmonium-like state with the open strangeness Zcs(3885). It is important to point out that most of these exotic states have masses close to the thresholds of the open and/or hidden flavor meson production.
Table 1. Experimental data on hidden-charm exotic mesons.
State JPC M (MeV) Γ (MeV) Observed in Experiment X(3872) 1++ 3871.69 ± 0.17 <1.2 B± → K±π+π− J/ψ Belle ± +− + − + − Zc(3900) 1 3888.4 ± 2.5 28.3 ± 2.5 e e → π π J/ψ BESIII ?? +27 + − X(3940) ? 3942 ± 9 37−17 e e → J/ψX Belle − + +1.8 +5.3 + − + − ∗0 ∗ 0 Zcs(3985) 1 3982.5−2.6 ± 2.1 12.8−4.4 ± 3.0 e e → K (Ds D + Ds D ) BESIII ± ?− + − + − Zc(4020) ? 4024.1 ± 1.9 13 ± 5 e e → π π hc(1P) BESIII ± ?+ +20 +21+47 ¯0 − + Zc(4050) ? 4051 ± 14−41 82−17−22 B → K π χc1(1P) Belle ± ?− + − + − Zc(4055) ? 4054 ± 3 ± 1 45 ± 11 ± 6 e e → ψ(2S)π π Belle ± ?? +18 +60 0 + − Zc(4100) ? 4096 ± 20−22 152 ± 58−35 B → K π ηc LHCb ++ +8 + + X(4140) 1 4146.8 ± 2.4 22−7 B → φJ/ψK CDF, LHCb ± +− +31+17 +70 ¯0 − + Zc(4200) 1 4196−29−13 370 ± 70−132 B → K π J/ψ Belle −− + − Y(4230) 1 4218.7 ± 2.8 44 ± 9 e e → ωχc0 BESIII ± −− +45 +108 0 + − Zc(4240) 0 4239 ± 18−10 220 ± 47−74 B → K π ψ(2S) LHCb ± ?+ +44+180 +54+316 ¯0 − + Zc(4250) ? 4248−29−35 177−39−61 B → K π χc1(1P) Belle −− + − + − Y(4260) 1 4230 ± 8 55 ± 19 e e → γISRπ π J/ψ BaBar ++ +8 + + X(4274) 1 4274−6 49 ± 12 B → J/ψφK CDF, LHCb −− + − + − Y(4360) 1 4368 ± 13 96 ± 7 e e → γISRπ π ψ(2S) Belle −− +16 + − + − Y(4390) 1 4392 ± 7 140−21 e e → π π hc BESIII ± +− +15 ± Zc(4430) 1 4478−18 181 ± 31 B → Kπ ψ(2S) Belle ++ +12 +21 + + X(4500) 0 4506 ± 11−15 92 ± 21−20 B → J/ψφK LHCb −− +8+5 +40+10 + − + − Y(4630) 1 4634−7−8 92−24−21 e e → Λc Λc Belle −− + − + − Y(4660) 1 4633 ± 7 64 ± 9 e e → γISRπ π ψ(2S) Belle ++ +14 +42 + + X(4700) 0 4704 ± 10−24 120 ± 31−33 B → J/ψφK LHCb ?+ + − X(4740) ? 4741 ± 6 ± 6 53 ± 15 ± 11 Bs → J/ψφπ π LHCb X(6900) ??+ 6905 ± 11 ± 7 80 ± 19 ± 33 pp → J/ψJ/ψX LHCb Universe 2021, 7, 94 3 of 21
Many theoretical interpretations of these states were suggested in the literature (for recent reviews, see [1–6] and references therein). The main interpretations are as follows. The conventional cc¯ states influenced by the open flavor thresholds. It is clear that such an interpretation is inapplicable, at least to the charged Z states. However, the cc¯ admixture may be present in some of the neutral states. Thus, exotic interpretations were proposed. They include: 1. Molecules, which are two heavy mesons (Qq¯)(Qq¯ ), loosely bound by meson ex- changes [3]; 2. Tetraquarks, which are four-quark QqQ¯ q¯ states, tightly bound by the color forces [12–14]; 3. Hybrids, which are QQ¯ -gluon states with excited gluonic degrees of freedom [15]; 4. Hadro-quarkonium, which are compact quarkonium states QQ¯ embedded in an excited light-quark matter [16]; 5. Kinematic or rescattering effects at corresponding thresholds [17]; 6. QQ¯ core plus molecular-like components [18]. In this review,we consider these exotic heavy mesons as heavy tetraquarks [12–14,19–21]. Our main assumption is as follows. Tetraquarks are composed from a diquark and antidiquark in color 3¯ and three configurations, which are bound by color forces. This assaumption reduces the very complicated four-body relativistic calculation to a more simple two-step, two-body calculation. First, a diquark d (antidiquark d¯) is considered as a qq0 (q¯q¯0) bound state (as in baryons). Note that only the color triplet configuration contributes, since there is a repulsion between quarks in a color sextet. Second, a tetraquark is considered as the dd¯0 bound state where constituents are assumed to interact as a whole. This means that there are no separate interactions between quarks, composing a diquark, and antiquarks, composing an antidiquark [14]. The resulting tetraquark has a typical hadronic size. We consider diquarks in the ground state only, as in the case of heavy baryons [22]. All excitations are assumed to be in the dd¯-bound system. A rich spectroscopy is predicted, since both radial and orbital excitations can occur between diquarks. However, the number of predicted excited states is significantly lower than in a pure four-body picture of a tetraquark. When one constructs a diquark, it is necessary to remember that it is a composite (qq0) system. Thus, a diquark is not a point-like object. Indeed, its interaction with gluons is smeared by the form factor, which can be expressed through the overlap integral of diquark wave functions. Aldditionally, the Pauli principle should be taken into account. For the ground state diquarks, it leads to the following restrictions. The (qq0) diquark, composed from quarks of different flavours, can have spins S = 0, 1 (scalar [q, q0], axial vector {q, q0} diquarks, while the (qq) diquarks, composed from quarks of the same flavour, can have only S = 1 (axial vector {q, q} diquark). The scalar S diquark is more tightly bound and has a smaller mass because of the larger attraction due to the spin–spin interaction. It is often called a “good” diquark and the heavier axial vector A diquark is called a “bad” diquark. It is important to emphasize that we treat both light and heavy quarks and diquarks fully relativistically, without application of the nonrelativistic (v/c) expansion. In this review, we consider the following tetraquarks: 1. Heavy tetraquarks (Qq)(Q¯ q¯0) with hidden charm and bottom [14,20,21]. The neutral X should be split into two states ([Qu][Q¯ u¯] and [Qd][Q¯ d¯]) with ∆M ∼ few MeV. The model predicts the existence of their charged partners X+ = [Qu][Q¯ d¯], − X = [Qd][Q¯ u¯] and the existence of tetraquarks with open Xsq¯ = [Qs][Q¯ q¯] and hidden Xss¯ = [Qs][Q¯ s¯] strangeness; 2. Doubly heavy tetraquarks (QQ0)(q¯q¯0) with open charm and bottom [19]. These tetraquarks are explicitly exotic, with heavy flavor number equal to 2. Their ob- servation would be a direct proof of the existence of multiquark states. The estimates of the production rates of such tetraquarks indicate that they could be produced and detected at present and future facilities. We considered the doubly heavy (QQ0)(q¯q¯0) tetraquark (Q, Q0 = b, c and q, q0 = u, d, s) as the bound system of the heavy diquark (QQ0) and light antidiquark (q¯q¯0); Universe 2021, 7, 94 4 of 21
3. Heavy tetraquarks (cq)(b¯q¯0) with open charm and bottom [19]. We considered heavy (cq)(b¯q¯0) tetraquark (q, q0 = u, d, s) as the bound system of the heavy–light diquark (cq) and heavy–light antidiquark (b¯q¯0); 4. QQQ¯ Q¯ tetraquarks composed from heavy (Q = c, b) quarks only [23]. The new structures in double-J/ψ spectrum have been very recently observed by the LHCb Collaboration in proton–proton collisions [10]. On the other hand, the absence of narrow structures in the Υ-pair production was reported by the LHCb [24] and CMS [25] Collaborations. We considered heavy (QQ0)(Q¯ Q¯ 0) tetraquark as the bound system of the doubly heavy diquark (QQ0) and doubly heavy antidiquark (Q¯ Q¯ 0).
2. Relativistic Diquark–Antidiquark Model of Heavy Tetraquarks For the calculation of the masses of tetraquarks, we use the relativistic quark model based on the quasipotential approach and the diquark–antidiquark picture of tetraquarks. First, we calculate the masses and wave functions (Ψd) of the light and heavy diquarks as the bound quark–quark states. Second, the masses of the tetraquarks and their wave functions (ΨT) are obtained for the bound diquark–antidiquark states. These wave functions are solutions of the Schrödinger-type quasipotential equations [14,19]
b2(M) p2 Z d3q − ( ) = ( ) ( ) Ψd,T p 3 V p, q; M Ψd,T q , (1) 2µR 2µR (2π)
with the on-mass-shell relative momentum squared given by
[M2 − (m + m )2][M2 − (m − m )2] b2(M) = 1 2 1 2 , (2) 4M2 and the relativistic reduced mass
E E M4 − (m2 − m2)2 = 1 2 = 1 2 µR 3 . (3) E1 + E2 4M
The on-mass-shell energies E1, E2 are defined as follows
M2 − m2 + m2 M2 − m2 + m2 E = 2 1 , E = 1 2 . (4) 1 2M 2 2M
The bound-state masses of a diquark or a tetraquark are M = E1 + E2 , where m1,2 are the masses of quarks (Q1 and Q2) which form the diquark or of the diquark (d) and antidiquark (d¯0), which form the heavy tetraquark (T), while p is their relative momentum. The quasipotential operator V(p, q; M) in Equation (1) is constructed with the help of the off-mass-shell scattering amplitude, projected onto the positive-energy states. We consider diquarks in a tetraquark, as in a baryon, to be in the color triplet state, since in the color sextet there is a repulsion between two quarks. The quark–quark (QQ0) interaction quasipotential is 1/2 of the quark–antiquark (QQ¯ 0) interaction and is given by [22]
V(p, q; M) = u¯1(p)u¯2(−p)V(p, q; M)u1(q)u2(−q), (5)
with 1 4 µ µ V(p, q; M) = α D (k)γ γν + VV (k)Γ (k)Γ (−k) + VS (k) . 2 3 s µν 1 2 conf 1 2;µ conf
Here, Dµν is the gluon propagator in the Coulomb gauge, u(p) are the Dirac spinors and αs is the running QCD coupling constant with freezing
2 4π αs(µ ) = , (6) 2 µ2 + M2 11 − n ln B 3 f Λ2 Universe 2021, 7, 94 5 of 21
where the√ scale µ is chosen to be equal to 2m1m2/(m1 + m2), the background mass is MB = 2.24 A = 0.95 GeV, and n f is the number of flavours [26]. The effective long-range vector vertex contains both the Dirac and Pauli terms [27]
iκ Γ (k) = γ + σ k˜ν, k˜ = (0, k), (7) µ µ 2m µν where κ is the long-range anomalous chromomagnetic moment. In the nonrelativistic limit, the vector and scalar confining potentials in configuration space have the form
V S Vconf(r) = (1 − ε)(Ar + B), Vconf(r) = ε(Ar + B), V S Vconf(r) = Vconf(r) + Vconf(r) = Ar + B, (8)
where ε is the mixing coefficient. Therefore, in the nonrelativistic limit, the QQ0 quasipo- tential reduces to NR 1 NR 1 4 αs V 0 (r) = V (r) = − + Ar + B , (9) QQ 2 QQ¯ 0 2 3 r reproducing the usual Cornel potential. Thus, our quasipotential can be viewed as its relativistic generalization. It contains both spin-independent and spin-dependent relativis- tic contributions. Constructing the diquark–antidiquark (dd¯0) quasipotential, we use the same assump- tions regarding the structure of the short- and long-range interactions. Taking into account the integer spin of a diquark in the color triplet state, the quasipotential is given by [14,19]
0 0 0 0 hd(P)|Jµ|d(Q)i 4 µν hd (P )|Jν|d (Q )i V(p, q; M) = √ αsD (k) √ 2 EdEd 3 2 Ed0 Ed0
∗ ∗ 0 h µ V S i 0 0 +ψd (P)ψd0 (P ) Jd;µ Jd0 Vconf(k) + Vconf(k) ψd(Q)ψd (Q ), (10)
where ψd(p) is the wave function of the diquark, ( 1 for a scalar diquark ψd(p) = (11) εd(p) for an axial-vector diquark
Here the four-vector (εd · p) (εd · p)p µ εd(p) = , εd + , εd(p)pµ = 0, (12) Md Md(Ed(p) + Md) q 2 2 is the polarization vector of the axial-vector diquark with momentum p, Ed(p) = p + Md, and εd(0) = (0, εd) is the polarization vector in the diquark rest frame. The effective long- range vector vertex of the diquark Jd;µ is given by
(P + Q)µ p for a scalar diquark, 2 EdEd Jd;µ = (13) (P + Q) iµ − µ + d Σν k˜ for an axial-vector diquark, p 2M µ ν 2 EdEd d
˜ ν where k = (0, k). Here, the antisymmetric tensor Σµ is defined by