Hydrogen Bond of QCD in Doubly Heavy Baryons and Tetraquarks

PHYSICAL REVIEW D 100, 074002 (2019)

Hydrogen bond of QCD in doubly heavy baryons and tetraquarks

† ‡ Luciano Maiani ,1,2,* Antonio D. Polosa,2, and Veronica Riquer1,2, 1T. D. Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China 2Dipartimento di Fisica and INFN, Sapienza Universit`a di Roma, Piazzale Aldo Moro 2, I-00185 Roma, Italy

(Received 8 August 2019; published 4 October 2019)

In this paper we present in greater detail previous work on the Born-Oppenheimer approximation to treat the hydrogen bond of QCD, and add a similar treatment of doubly heavy baryons. Doubly heavy exotic resonances X and Z can be described as color molecules of two-quark lumps, the þ analogue of the H2 molecule, and doubly heavy baryons as the analog of the H2 ion, except that the two heavy quarks attract each other. We compare our results with constituent quark model and lattice QCD calculations and find further evidence in support of this upgraded picture of compact tetraquarks and baryons.

DOI: 10.1103/PhysRevD.100.074002

I. INTRODUCTION Section II describes the Born-Oppenheimer approxima- Systems with heavy and light particles allow for an tion applied to a QCD double heavy hadron and gives the approximate treatment where the light and heavy degrees of two-body color couplings derived from the restriction that freedom are studied separately and solved one after the other. the hadron is an overall color singlet. Section III recalls the This is the Born-Oppenheimer approximation (BO), intro- salient features of the constituent quark model and gives duced in nonrelativistic quantum mechanics for molecules quark masses and hyperfine couplings derived from the S and crystals, where electrons coexist with the much heavier mass spectra of the -wave mesons and baryons. Section IV nuclei. We have recently reconsidered this method for the introduces the string tension for confined systems and QCD interactions of multiquark hadrons containing heavy discusses extensions beyond charmonium. (charm or bottom) and light (up and down) quarks [1], Sections V, VI, and VII illustrate the main calculations following earlier work in [2,3], and, for lattice calculations, and results for doubly heavy baryons, hidden heavy in [4]. flavor tetraquarks, and doubly heavy flavored tetraquarks, In this paper, based on our previous communication [1], respectively. we consider tetraquarks in terms of color molecules: lumps Results are summarized in Sec. VIII and conclusions of two-quark colored atoms (orbitals) held together by given in Sec. IX. Technical details are expanded in three color forces and treated in the BO approximation. The appendixes. variety of bound states described here identifies a new way of looking at multiquark hadrons, as formed by the QCD II. BORN-OPPENHEIMER APPROXIMATION analog of the hydrogen bond of molecular physics. WITH QCD CONSTITUENT QUARKS We restrict to doubly heavy-light systems, namely the We consider doubly heavy systems with open or hidden doubly heavy baryons, qQQ, not considered in [1], the heavy flavor, and discuss the application of the BO hidden flavor tetraquarks QQq¯ q¯ (see [5–8] for a review), approximation along the lines used for the treatment of and QQq¯ q¯ systems [9–13]. the hydrogen molecule (see [14,15]). The plan of the paper is the following. We denote coordinates and mass of the heavy quarks by xA, xB, and M and those of the light quarks by x1, x2, and m. * Coordinate symbols here include spin and color quantum [email protected] † [email protected] numbers, to be discussed later. ‡ veronica.riquer@cern.ch The Hamiltonian of the whole system is

Published by the American Physical Society under the terms of 1 X 1 X H P2 p2 the Creative Commons Attribution 4.0 International license. ¼ 2M i þ 2m i Further distribution of this work must maintain attribution to heavy light ’ the author(s) and the published article s title, journal citation, V x ; x V x ; x ; x ; x : and DOI. Funded by SCOAP3. þ ð A BÞþ Ið A B 1 2Þ ð1Þ

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We have separated the heavy quark interaction VðxA; xBÞ, 1=a Λ ¼ ; ð9Þ e.g., their Coulombic QCD interaction, from the general 1=b interactions involving light-heavy and light-light quarks. We start by solving the eigenvalue equation for the light where a and b are the lengths over which f or ψ show an particles for fixed values of the coordinates of the heavy appreciable variation. ones The length a is simply the radius of the orbitals, which we determine by minimizing the Schrödinger functional of X p2 i the light quark. As will be discussed below, we typically VI xA; xB; x1; x2 fα Eα xA; xB fα; 2 2m þ ð Þ ¼ ð Þ ð Þ 1=a A ∼ 0 3 a ∼ 0 7 light find ¼ . GeV, i.e., . fm. The length b has to be formed from the dimensional where quantities over which the Born-Oppenheimer equation (6) depends. In the case of double heavy baryons and hidden heavy flavor tetraquarks, Secs. V and VI, Eq. (6) depends fα ¼ fαðxA; xB; x1; x2Þ; ð3Þ on 1=M,ona, and on the string tension k, which has dimensions of GeV2. and focus on the lowest eigenvalue and eigenfunction, A quantity b with dimensions of length can be formed as which, dropping the subscript for simplicity of notation, we denote by E and f. Next, we look for solutions of the −1=4 eigenvalue equation of the complete Hamiltonian (1) of b ¼ðMkAÞ : ð10Þ the form Therefore Ψ ¼ ψðxA; xBÞfðxA; xB; x1; x2Þ: ð4Þ Λ ¼ A3=4ðkMÞ−1=4; ð11Þ When applying the Hamiltonian (1) to Ψ one encounters terms of the kind which is 0.57 for charm and 0.43 for beauty, using 2 ∂ k ¼ 0.15 GeV and the constituent masses of charm and P Ψ ψ x ; x i f x ; x ; x ; x A;B ¼ ð A BÞ ∂x ð A B 1 2Þ beauty from the tables in the next section. A;B We note in Sec. VII that the Born-Oppenheimer potential ∂ for double heavy tetraquarks does not depend on the string þ i ψðxA; xBÞ fðxA; xB; x1; x2Þ: ð5Þ ∂xA;B tension, which is screened by gluons for color octet orbitals. In this case, we get The Born-Oppenheimer approximation consists of neglecting the first with respect to the second term in all such b ¼ðMAÞ−1=2 ð12Þ instances so that, after factorizing f, we obtain the Schrödinger equation of the heavy particles, and X P2 i 1=2 þ V ðxA; xBÞ ψ ¼ Eψ ð6Þ A 2M BO Λ ; 13 heavy ¼ M ð Þ with the Born-Oppenheimer potential given by giving 0.42 for charm and 0.24 for beauty. In the following, V V x ; x V x ; x E x ; x : for convenience we shall include quark masses in BO,but BOð A BÞ¼ ð A BÞþ ð A BÞ ð7Þ it is worth noticing that the error we are estimating is the error on the binding energies, which turn out to be around For QED in molecular physics, the parameter which 100 MeV or smaller in absolute value. So, the errors regulates the validity of the approximation is estimated corresponding to (11) and (13) may be in the order of in [14] to be 20–50 MeV. We comment now about color. Treating heavy quark and/ m 1=4 ϵ ¼ : ð8Þ or antiquark as external sources implies specifying their M combined SUð3Þc representation. Restriction to an overall color singlet fixes completely the color composition of the We apply the same method to our case as follows. constituents. The ratio of the first (neglected) to the second (retained) Recall that the color coupling between any pair of term in (5) is given approximately by particles in color representation R is given by

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α s Double beauty tetraquarks: bb in 3¯.—The lowest energy VCðrÞ¼λq q ðRÞ ; 1 2 r state corresponds to bb in spin one and light antiquarks in 1 spin and isospin zero. The tetraquark state λq q ðRÞ¼ ½C2ðRÞ − C2ðq1Þ − C2ðq2Þ; ð14Þ 1 2 2 T ¼jðbbÞ3¯ ; ðq¯ q¯Þ3i1 ð19Þ where q1;2 are the irreducible representations of the particles can be Fierz transformed into in the pair and C2 are the quadratic Casimir operators. ¯ rffiffiffi rffiffiffi We note the results: C2ð1Þ¼0; C2ðRÞ¼C2ðRÞ; 1 2 C 3 4=3 C 6 10=3 C 8 3 T qb¯ ; qb¯ − qb¯ ; qb¯ 2ð Þ¼ ; 2ð Þ¼ ; 2ð Þ¼ . ¼ 3jð Þ1 ð Þ1i1 3jð Þ8 ð Þ8i1 ð20Þ If the pair q1q2 in the tetraquark TðqiqjqkqlÞ is in a superposition of two SUð3Þc representations with ampli- with all attractive couplings tudes a and b, we use 2 1 λbb ¼ λq¯ q¯ ¼ − αS; λbq¯ ¼ − αS: ð21Þ T a q q … b q q … ; 3 3 ¼ jð 1 2ÞR1 i1 þ jð 1 2ÞR2 i1 λ a2λ R b2λ R : Double beauty tetraquarks: bb in 6.—We start from q1q2 ¼ q1q2 ð 1Þþ q1q2 ð 2Þ ð15Þ T bb ; q¯ q¯ ; The different cases are as follows. ¼jð Þ6 ð Þ6¯ i1 ð22Þ Doubly charmed baryon cc in 3¯ — : . In a color singlet a case also considered in [13]. We find 3¯ baryon, all pairs are in color , and the color couplings are rffiffiffi rffiffiffi distributed according to 2 1 T qb¯ ; qb¯ qb¯ ; qb¯ ; ¼ 3jð Þ1 ð Þ1i1 þ 3jð Þ8 ð Þ8i1 ð23Þ λcc ¼ λcq ¼ −2=3: ð16Þ therefore Hidden flavor tetraquarks.—Color of the heavy particles 1 5 λ λ α ; λ q − α : can be either 1 or 8. In the first case, the interaction between bb ¼ q¯ q¯ ¼þ3 S bq¯ ¼ 6 S ð24Þ QQ¯ and qq¯ pairs goes via the exchange of color singlets. H We are in a situation dominated by nuclearlike forces, The situation is again analogous to the 2 molecule, with eventually leading to the formation of the hadrocharmo- two identical, repelling light particles. nium envisaged in [16]. We shall not consider QQ¯ in color singlet any further. III. QUARK MASSES AND HYPERFINE QQ¯ in 8.—Suppressing coordinates COUPLINGS FROM MESONS AND BARYONS The constituent quark model, in its simplest incarnation, ¯ A A T ¼ðQλ QÞðq¯λ qÞð17Þ describes the masses of mesons and baryons as due to the masses of the quarks in the hadron, Mi, with hyperfine with the sum over A ¼ 1; …; 8 understood. If we restrict to interactions added. The Hamiltonian is one-gluon exchange, Eq. (17) determines the interactions H ¼ H þ H ; between different pairs. Xmass hf Both QQ¯ and qq¯ are in color octet, and we read their H ¼ Mi; coupling from Eq. (14). The couplings of the other pairs are mass Xi found using the Fierz rearrangement formulas for SUð3Þc to H 2κ s s ; bring the desired pair in the same quark bilinear (see hf ¼ ijð i · jÞ ð25Þ i

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TABLE I. Constituent quark masses (MeV) from S-wave gluons, unlike photons in QED, may screen color charges mesons and baryons (see [6,8])(q ¼ u, d). by lowering the dimension of the color representation. The simplest case is color 6 charge. Since 6 ⊗ 8 ⊃ 3¯ the string Quark flavors q s c b tension strength of a pair 6 ⊗ 6¯ → 1 is reduced from 10=3 Quark mass (MeV) from mesons 308 484 1667 5005 to 4=3, i.e., the string tension of 3 ⊗ 3¯ → 1 [19]. Quark mass (MeV) from baryons 362 540 1710 5044 Screening by an arbitrary number of gluons reduces Casimir scaling of string tension to the simplest triality scaling. One can see this through the following steps: A reasonable hypothesis, advanced in [17], is that the (1) Recall that a generic SUð3Þc charge is represented difference of quark masses derived from mesons and b by a tensor ta [a; b; … ¼ 1, 2, 3 are SUð3Þc baryons is due to the different pattern of QCD interactions indices], with n upper and m lower fully sym- in systems with two or three constituents, which should be metrized indices, vanishing under the contraction apparent even in the lowest order, one-gluon exchange of an upper and a lower index [20]. Exchanging n approximation. We shall follow this hypothesis. Since the and m gives the complex conjugate representation, basic ingredient of the BO approximation are two-body which has the same Casimir, so that we may assume orbitals, we feel the natural choice is to take quark masses n ≥ m; t ¼ðn − mÞ mod 3 ¼ 0, 1, 2 is the triality of from the meson spectrum and leave to the QCD interactions the representation. between orbitals the task to compensate for the difference t tbAa (2) Saturating the tensor with gluon fields: a b , of quark masses from mesons with those derived from we reduce to tensors which have only n − m upper baryons in the naive constituent quark model. indices. (3) 8 ⊗ 80 ⊃ 10; 10 as can be seen from the simple IV. QUARK INTERACTION composition of two (different) octets: AND STRING TENSION G A d A0 eϵ ∈ 10: fabcg ¼½ð Það Þb cdeabc symmetrized ð28Þ The prototype of nonrelativistic quark interaction is the t G so-called Cornell potential [18] introduced in connection (4) Saturating with fabcg, we reduce the upper indices with the charmonium spectrum, to those of the lowest triality representations, namely ð1;t¼ 0Þ; ð3;t¼ 1Þ; ð6;t¼ 2Þ, equivalent to1 ¯ 4 αS ð3;t¼ −1Þ. VðrÞ¼− þ kr þ V0 ¼ VCðrÞþV ðrÞþV0: ð26Þ 3 r conf The upshot is that, for conjugate charges combined to a singlet, we have only two possibilities for the string The potential refers to the case of a heavy color triplet pair, ¯ tension: QQ, in an overall color singlet state. V0 is determined from (i) t ¼ 0 charges, e.g., 8 ⊗ 8 → 1,havek ¼ 0 and are the mass spectrum. We shall generalize (26) to several not confined. different cases. (ii) t ≠ 0 charges have k ¼ kð3Þ, equal to the charmo- The first term in (26) is obtained in the one-gluon nium string tension. exchange approximation by (14). It is generalized to any For other cases, e.g., qQ, we adopt (27) and write pair of colored particles in a color representation R α 3 λ by Eq. (15). S j q1q2 j Vq q ðrÞ¼λq q þ kr þ V0; ð29Þ The second term in (26), which dominates over the 1 2 1 2 r 4 Coulomb force at large separations, arises from quark confinement. In the simplest picture, confinement is due where k is the string tension taken from the charmonium to the condensation of Coulomb lines of force into a string spectrum. that joins the quark and the antiquark. The linearly rising In the numerical applications, we take αS and k at the term in (26) describes the force transmitted by the charmonium scale from the lattice calculation in [21]: string tension. In this picture, it is natural to assume that 2 the string tension, embodied by the coefficient k, scales αSð2McÞ¼0.30;k¼ 0.15 GeV : ð30Þ with the Coulomb coefficient At the Bc meson and bottomonium mass scales we take the k ∝ λ : q1q2 j q1q2 j ð27Þ same string tension and run αS with the two loops beta function, to get For color charges combined in an overall color singlet, the assumption leads to k ∝ C2ðqÞ, whence the name of αSðMc þ MbÞ¼0.24; αSð2MbÞ¼0.21: ð31Þ Casimir scaling; see [19] for an extensive discussion. Casimir scaling would give a string tension that increases 1 n−m with the dimensionality of color charges. However, QCD Indeed, t ¼ðn − mÞ mod 3 ≡ ðn − mÞ − 3b 3 c.

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E 2M M H V : V. THE DOUBLY CHARMED BARYON 0 ¼ c þ q þh imin þ 0 ð39Þ þ The baryon Ξcc ¼ qcc is analogous to the H ion [15], 2 Perturbation theory.—To first order in the perturbation except that the two heavy quarks attract each other, (33), the BO potential is given by Eq. (16). The interaction Hamiltonian is 2 1 V r − α E ΔE r ; BOð ABÞ¼ S þ 0 þ ð ABÞ ð40Þ 2 1 3 rAB HI ¼ − αS 3 jxA − xBj ΔE r f H f ð ABÞ¼h j pertj i 2 1 1 2α 1 − αS þ : ð32Þ S 3 x − x x − x ¼ − ½I1ðrABÞþI2ðrABÞ: ð41Þ j Aj j Bj 3 1 þ S cq c x We consider the orbital made by , with located in A. I1;2 are functions of rAB defined in terms of ψ and ϕ: The perturbation Hamiltonian that remains is Z 3 2 1 2 1 I1ðrABÞ¼ d ξjψðξÞj ; ð42Þ H − α : ξ − x pert ¼ S ð33Þ j Bj 3 jx − xBj Z 1 cq — I r d3ξψ ξ ϕ ξ ; The orbital. As potential, we take the Coulombic 2ð ABÞ¼ ð Þ ð Þ ξ − x ð43Þ interaction from (32) and a linear term with the string j Bj tension rescaled according to Eq. (29), where the vector ξ originates from A, taken in the origin, 2 α 1 and jxBj¼rAB. V − S kr V : S; I cq ¼ 3 r þ 2 þ 0 ð34Þ Analytic expressions for 1;2 are given in [15] for the hydrogen wave functions. We evaluate them numerically We assume a radial wave function RðrÞ of the form for the orbitals corresponding to the potential (34). Boundary condition for rAB → 0.—The perturbation 3=2 Hamiltonian (33) embodies the interaction of the light A −Ar RðrÞ¼pffiffiffiffiffiffi e ð35Þ quark when the other charm quark is far from the orbital. If 4π ¯ we let rAB vanish, the charm pair reduces to a single 3 q and determine A by minimizing the Schrödinger functional source generating the same interaction that would see inside a qc¯ charmed meson. This is in essence the heavy 1 – R r ; − Δ Vcq − V0 R r quark-diquark symmetry (see [22 24]). ð ð Þ ð 2Mq þ Þ ð ÞÞ H A : The symmetry means that E0 þ ΔEðrABÞ, when we h ð Þi ¼ R r ;R r ð36Þ ð ð Þ ð ÞÞ subtract Mc from it and let rAB → 0, has to reproduce the spin independent mass of a D¯ meson, which, by We take quark masses from the meson spectrum, Table I, αS definition, is Mc Mq. In formulas k A 0 32 ; H 0 48 þ and from (30). We find ¼ . GeV h imin ¼ . GeV. We consider as unperturbed ground state the symmetric E ΔE 0 − M M M V H ΔE 0 superposition of the two orbitals with q either attached to 0 þ ð Þ c ¼ c þ q þ 0 þh imin þ ð Þ c x ψ x c x ð AÞ, which we denote by ð Þ, or attached to ð BÞ, ¼ Mc þ Mq: ð44Þ denoted by ϕðxÞ: The condition determines the value of the a priori ψðxÞþϕðxÞ Rðjx − xAjÞ þ Rðjx − xBjÞ fðxÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð37Þ unknown V0, 2ð1 þ SÞ 2ð1 þ SÞ V H ΔE 0 0 0 þh imin þ ð Þ¼ ð45Þ The denominator in (37) is needed to normalize fðxÞ, and it arises because ψ and ϕ are not orthogonal (see and Appendix A) with the overlap S defined as (ψ and ϕ real) 2 1 Z V r − α ΔE r C; BOð ABÞ¼ 3 S r þ ð ABÞþ 3 AB SðrABÞ¼ d ξψðξÞϕðξÞð38Þ C ¼ 2Mc þ Mq − ΔEð0Þ: ð46Þ and rAB ¼jxA − xBj. Numerically, we find from (41) The energy corresponding to fðxÞ is given by the quark constituent masses plus the energy of the orbital ΔEð0Þ¼−65 MeV: ð47Þ

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The cq orbital is confined.—The interactions embodied TABLE II. S-wave mesons and baryons: spin-spin interactions in Eq. (46) originate from one gluon exchange and vanish at of the lightest quarks with the heavier flavors [6,8]. For the hf large separations. However, the orbital cq and the external c couplings of cc¯, cc, and similar ones for b quarks see text. ¯ quark carry 3 and 3 colors combined to a color singlet and ¯ ¯ Mesons ðqq¯Þ1 ðqs¯Þ1 ðqc¯Þ1 ðsc¯Þ1 ðqbÞ1 ðcc¯Þ1 ðbbÞ1 are confined. To take this into account we add a linearly κ rising term to the BO potential in (46), determined by the (MeV) 318 200 70 72 23 56 30 qq qs qc sc qb cc bb k Baryons ð Þ3¯ ð Þ3¯ ð Þ3¯ ð Þ3¯ ð Þ3¯ ð Þ3 ð Þ3 string tension of charmonium (see Sec. IV) and the onset a b R κ (MeV) 98 59 15 50 2.5 28 15 point, 0. The complete Born-Oppenheimer potential reads κMES Ratio κ 3.2 3.4 4.7 1.6 9.2 BAR a V r V r V r ; 0.5κ½ðcc¯Þ1. totð Þ¼ BOð Þþ conf ð Þ ð48Þ b ¯ 0.5κ½ðbbÞ1. V r k r − R θ r − R conf ð Þ¼ × ð 0Þ × ð 0Þð49Þ to be compared with the LHCb value [26] −1 with R0 ≥ 2A . For orientation, we start with R0 ∼ 8 −1 ∼ 1 6 c M Ξ 3621 2 0 7 : GeV . fm, where we may assume that sees ð ccÞExpt ¼ . . MeV ð54Þ the orbital as a point source and study the results for different values of R0. We do not attempt to give an overall theoretical error to The Schrödinger equation for the charm pair with the result in Sec. (53), which cannot be less than 30 MeV. V r V r − C potential ð Þ¼ totð Þ is solved numerically [25]. It is interesting to compare our calculation with the −1 Results are reported in Fig. 1.ForR0 ¼ 8 GeV , we plot calculation presented in [17]. These authors obtain the cc¯ VðrÞ, the radial wave function χðrÞ, and the lowest binding energy from charmonium using quark masses from eigenvalue E ¼ −0.041 GeV. The average distance of the meson spectrum (particle names denote their masses in the charm pair is ∼0.9 fm. The eigenvalue has an appreci- MeV) able dependence from R0. We find 1 Bcc¯ ¼ ðηc þ 3J=ψÞ − 2Mc ¼ −271; ð55Þ þ17 −1 4 E ¼ −41−7 MeV for R0 ¼ 8 2 GeV : ð50Þ where the first term is charmonium mass subtracted of its The contribution of hyperfine interactions to the J ¼ 1=2þ hyperfine interaction. The cc binding energy is obtained by Ξcc is multiplication of the color factor 1=2, and the result is used 1 as the binding energy of the cc quarks in Ξcc,tobe H Ξ −2κ κ −16 cc hf ð ccÞ¼ qc þ 2 cc ¼ MeV ð51Þ subtracted from quark mass derived from the baryon spectrum. Adding hyperfine interactions, they obtain [17] with the numerical value from Table II. Finally Ξcc ¼ 3628 12: ð56Þ 1 M Ξ 2M M − ΔE 0 E − 2κ κ ð ccÞTh ¼ c þ q ð Þþ qc þ 2 cc ð52Þ The consistency of results derived by two alternative routes with themselves and with the experimental value is leading to worth noticing. Ωcc.—Replacing the light quark mass with the strange þ17 quark mass in (36) and inserting the appropriate hyperfine MðΞccÞ ¼ 3655−7 MeV ð53Þ Th couplings, we obtain the mass of the strange-doubly charmed baryon, ½scc, denoted by Ωcc. (a) (b) Mass of cb and bb baryons.—With similar methods we 0 may compute M½Ξcb, M½Ξcb (see Appendix C), and M½Ξbb. Comparisons.—Our results are summarized in Table III, fourth column, and compared to the results in Refs. [17,27], reported in the fifth column. We differ for bc and bb by 50 and 150 MeV, which perhaps points to a significant FIG. 1. Born-Oppenheimer potential þ confinement in the discrepancy. (a) qcc and (b) scc baryons. Eigenfunction χðrÞ¼rRðrÞ and Predictions of the masses of doubly heavy baryons, based eigenvalue E in the fundamental state are shown. Here and in on different methods, have appeared earlier in the literature the following, on the y axes energies are in GeV and χ in [30–42]. Numerical values are summarized in [17] and arbitrary units. spread in a typical range of 100–200 MeVaround our values.

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TABLE III. Our results on doubly heavy baryon masses, fourth Unlike the H2 case, light particles are not identical and the column, compared to quark model and lattice QCD results, fifth unperturbed ground state is nondegenerate. and sixth columns. E − Δ E represents the correction to the The energy of fð1; 2Þ is given by constituent quark mass formula, with quark masses taken from meson spectrum. E 2 M M H V : 0 ¼ ð c þ q þh imin þ 0Þ ð60Þ −ΔEð0Þ EM½ΞQQ [17,27] [28,29] Perturbation theory.—The perturbation Hamiltonian of Ξ 65 −41 3628 12 cc þ 3656 3634(20) this case is Ωcc þ75 −44 3769 3692 16 3712(11)(12) Ξ 50 −37 6920 13 cb þ 6961 6945(22)(14) 7 1 1 Ξ0 þ50 −37 6993 6935 12 6966(23)(14) H − α cb pert ¼ 6 S x − x þ x − x Ξbb þ44 −53 10311 10162 12 j 1 Bj j 2 Aj 1 1 þ αS : ð61Þ 6 jx1 − x2j The results of recent lattice QCD calculations [28,29] are H reported in the last column. Reference [29] reviews the To first order in pert, Eq. (61), the BO potential is results of today’s available lattice calculations for doubly 1 1 heavy baryons. V r α E ΔE r ; BOð ABÞ¼þ S þ 0 þ ð ABÞ ð62Þ Experimental results are eagerly awaited. 6 rAB

where rAB ¼jxA − xBj. VI. HIDDEN HEAVY FLAVOR ΔE f H f ¼h j pertj i evaluates to We consider the hidden heavy flavor case, specializing to 7 1 the hidden charm and following closely the approach to the ΔE − α 2I r α I r ¼ 6 S 1ð ABÞþ6 S 4ð ABÞð63Þ H2 molecule in [15] (see Appendix A). With cc¯ and qq¯ taken in color 8 representation, Eq. (17), in terms of the function I1, Eq. (42), and we describe the unperturbed state as the product of two Z orbitals, bound states of one heavy and one light particle 3 3 2 2 1 around xA or xB, and treat the interactions not included in I4ðrABÞ¼ d ξd ηjψðξÞj jϕðηÞj ; ð64Þ jξ − ηj the orbitals as perturbations. cq c¯ q¯ cq¯ Two subcases are allowed: (i) and or (ii) where the vector ξ originates from A, taken in the origin, and cq¯ . x r cq — and j Bj¼ AB. The orbital. We take the Coulombic interaction cq orbitals are confined.—The orbitals cq and c¯ q¯ carry given by λcq in (18) and rescale the string tension from the 2 nonvanishing color and are confined. Similar to Sec. V,we charmonium one, according to Eq. (29): thus add a linearly rising term to the BO potential in (63), determined by a string tension kT and the onset point R0. 1 α 1 V − S kr V : The complete Born-Oppenheimer potential reads cq ¼ 3 r þ 4 þ 0 ð57Þ V r V r V r ; ð Þ¼ BOð Þþ confð Þ As the previous case, we assume an exponential form for V r k r − R θ r − R : radial wave function RðrÞ, confð Þ¼ T × ð 0Þ × ð 0Þ ð65Þ

−1 A3=2 For orientation, we choose R0 ¼ 10 GeV , greater than RðrÞ¼pffiffiffiffiffiffi e−Ar; ð58Þ 2A−1 ∼ 7 4 −1 4π . GeV , where the two orbitals start to separate. In principle, R0 should be considered a free parameter, to be fixed on the phenomenology of the tetraquark, as we and determine A by minimizing the Schrödinger functional discuss below. (36) for the potential (57), with quark masses from the As for kT, we note that the tetraquark T ¼jðcc¯ Þ8ðqq¯ Þ8i1 meson spectrum, Table I, and parameters of the potential can be written as from (30). We find A ¼ 0.27 GeV; hHi ¼ 0.30 GeV. min rffiffiffi rffiffiffi The wave function of the two noninteracting orbitals is 2 1 T cq c¯ q¯ − cq c¯ q¯ : ¼ 3jð Þ3¯ ð Þ3i1 3jð Þ6ð Þ6¯ i1 ð66Þ fð1; 2Þ¼ψð1Þϕð2Þ¼Rðjx1 − xAjÞRðjx2 − xBjÞ: ð59Þ At large distances the diquark-antidiquark system is a ¯ ¯ 2In our previous analysis [1], string tension 1=4k was superposition of 3 ⊗ 3 → 1 and 6 ⊗ 6 → 1. Equation (66) considered as an alternative possibility to string tension k. and the hypothesis of strict Casimir scaling of kT would give

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(a) (b) The cq¯ orbital.—One obtains the new orbital by replacing −1=3αS → −7=6αS in Eq. (57) and string tension 7 k cq¯ k: ð Þ¼8 ð68Þ A 0 40 ; H 0 66 Correspondingly ¼ . GeV h imin ¼ . GeV. The perturbation Hamiltonian appropriate to this case is FIG. 2. (a) Dominant cq¯ and cq¯ attraction þ confinement; qq¯ 1=6α ∼ (b) dominant repulsion þ confinement, letting þ S 1 1 1 1 1 0 05 → 3 3 χ r rR r H − α α . . in Eq. (18). Eigenfunction ð Þ¼ ð Þ and eigen- pert ¼ S þ þ S value E of the tetraquark in the fundamental state are shown. 3 jx1 − xBj jx2 − xAj 6 jx1 − x2j Diquarks are separated by a potential barrier, and there are two ð69Þ different lengths: Rqc ∼ 0.4 fm and the total radius R ∼ 1.5 fm, as in [43]. and 1 1 V α E ΔE r BO ¼þ S þ 0 þ ð ABÞð70Þ 2 1 C2ð6Þ 6 rAB kT ¼ þ k ¼ 1.5k: ð67Þ 3 3 C2ð3Þ with

However, as discussed in [19] and in Sec. IV, gluon screening 1 1 ¯ ΔE ¼ − αS2I1 þ αSI4: ð71Þ gives the 6 diquark a component over the 3 bringing kT closer 3 6 to k. For simplicity, we adopt kT ¼ k. The potential VðrÞ computed on the basis of Eq. (65) is The tetraquark state is given in Fig. 2(a). Also reported are the wave function and rffiffiffi 8 1 the eigenvalue obtained by solving numerically the radial T ¼ jðcq¯ Þ1ðqc¯ Þ1i1 − pffiffiffi jðcq¯ Þ8ðqc¯ Þ8i1: ð72Þ Schrödinger equation [25]. 9 9 As is customary for a confined system such as charmo- x − x nia, we fix V0 to reproduce the mass of the tetraquark, so At large j A Bj the lowest energy state (two color the eigenvalue is not interesting. However, the eigenfunc- singlet mesons) has to prevail, as concluded in Sec. IV on tion gives us information on the internal configuration of the basis of the triality scaling due to gluon screening of the tetraquark. In Fig. 2(a), with one-gluon exchange octet charges. Therefore there is no confining potential to couplings, a configuration with c close to c¯ and the light be added to the BO potential in (70). r → ∞ — r → ∞ quarks around is obtained, much like the quarkonium Boundary condition for AB . For AB , V → H V adjoint meson described in [2]. BO h imin þ 0. Including constituent quark masses, r ∞ E 2 M M Figure 2(b) is obtained by increasing the repulsion in the the energy of the state at AB ¼ is ∞ ¼ ð c þ q þ H V qq¯ interaction associated with the function I4, letting h imin þ 0Þ and it must coincide with the mass of a pair 1=6αS ¼ 0.05 → 3.3. The corresponding cc¯ wave function of noninteracting charmed mesons, with spin-spin inter- clearly displays the separation of the diquark from the action subtracted. Therefore we impose antidiquark suggested in [43] and further considered H V 0: in [44]. h imin þ 0 ¼ ð73Þ The presence of a barrier that c has to overcome to reach c¯, apparent in Fig. 2(b), explains the suppression of the A minimum of the BO potential is not guaranteed. If there J=ψ þ ρ=ω decay modes of Xð3872Þ, otherwise favored is such a minimum, as in Fig. 3(a), it would correspond to a by phase space with respect to the DD modes. With 2 −3 the parameters in Fig. 2(b), we find jRð0Þj ¼ 1.6 × 10 (a) (b) with respect to jRð0Þj2 ¼ 1.9 × 10−2 with the perturbative parameters of Fig. 2(a). The tetraquark picture of Xð3872Þ and the related Zð3900Þ and Zð4020Þ have been originally formulated in terms of pure 3¯ ⊗ 3 diquark-antidiquark states [6,7,43]. The 6 ⊗ 6¯ component in (66) results in the opposite sign of qq¯ cq c¯ q¯ the hyperfine interactions vs the dominant and FIG. 3. Born-Oppenheimer potential VðrÞ vs RAB for cq¯ one, and it could be the reason why Xð3872Þ is lighter orbitals. Unit length: GeV−1 ∼ 0.2 fm. (a) Using the perturbative than Zð3900Þ. parameters; (b) with repulsion enhanced.

074002-8 HYDROGEN BOND OF QCD IN DOUBLY HEAVY BARYONS AND … PHYS. REV. D 100, 074002 (2019) configuration similar to the quarkonium adjoint meson in Fig. 2(a). If repulsion is increased above the perturbative value, e.g., changing þ1=6αS ∼ 0.05 to a coupling ≥ 1 in analogy with Fig. 2(b), the BO potential has no minimum at all, Fig. 3(b).

VII. DOUBLE BEAUTY TETRAQUARKS FIG. 4. Left panel: BO potential, eigenfunction, and eigenvalue ðbbÞ3¯ q¯ q¯ tetraquark. Right panel: Same for ðccÞ3¯ q¯ q¯. We consider bb tetraquarks, analyzing in turn the two options for the total color of the bb pair. tight bb diquark, of the kind expected in the constituent bb in 3¯. We recall from Sec. II that the lowest energy quark model [10,11,13]. state corresponds to bb in spin one and light antiquarks The BO potential in the origin is Coulomb-like, and it in spin and isospin zero. The tetraquark state is T ¼ tends to zero, for large rAB, due to (73). The (negative) jðbbÞ3¯ ; ðq¯ q¯Þ3i1, whence one derives the attractive color eigenvalue E of the Schrödinger equation is the binding couplings reported in (21) and energy associated with the BO potential. The masses of the bb ; q¯ q¯ B 1 lowest tetraquark with ð ÞS¼1 ð ÞS¼0, and of the k bq¯ k: ð Þ¼4 ð74Þ mesons are

There is only one possible orbital, namely bq¯, but the 1 3 MðTÞ¼2ðMb þ MqÞþE þ κbb − κqq; ð80Þ unperturbed state now is the superposition of two states 2 2 with the roles of q¯ 1 and q¯ 2 interchanged, such as electrons H 3 in the 2 molecule (see Appendix A), M B M M − κ : ð Þ¼ b þ q 2 bq¯ ð81Þ ψð1Þϕð2Þþϕð1Þψð2Þ fð1; 2Þ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð75Þ 2ð1 þ S2Þ The hyperfine interactions are taken from Table II, and E ¼ −67 MeV is the eigenvalue shown in Fig. 4(a) α 2M 0 20 The denominator needed to normalize fð1; 2Þ includes the with sð bÞ¼ . . Q T → 2B γ overlap function S defined in (38). The value for the decay þ is then The perturbation Hamiltonian is 1 3 Qbb ¼ E þ κbb − κqq þ 3κbq¯ ¼ −138ð−156Þ MeV 1 1 1 2 1 2 2 H − α − α pert ¼ S þ S 3 jx1 − xBj jx2 − xAj 3 jx1 − x2j ð82Þ ð76Þ for the string tension (74) (in parentheses with string k and tension ). Results for Qcc;bc are reported in Table IV using the 2 1 values of αS in (31). V r 2 H V − α ΔE; BOð ABÞ¼ ðh imin þ 0Þ S þ ð77Þ Equation (82) underscores the result obtained by Eichten 3 rAB and Quigg [11] that the Q value goes to a negative constant ΔE f H f limit for MQ → ∞: Q −150 MeV O 1=MQ . where ¼h j pertj i evaluates to ¼ þ ð Þ Double beauty tetraquarks: bb in 6.—Color charges are 1 1 2 given in (24) and ΔE ¼ − αS2ðI1 þ SI2Þ − αSðI4 þ I6Þ : ð78Þ 1 þ S2 3 3 5 kðbq¯Þ¼ k: ð83Þ I1;2;4 were defined previously, whereas [15] 8 Z 3 3 1 The situation is entirely analogous to the H2 molecule, with I6ðrABÞ¼ d ξd ηψðξÞϕðξÞψðηÞϕðηÞ : ð79Þ jξ − ηj two identical, repelling light particles. For the orbital bq¯, A 0 43 H 0 72 we find ¼ . GeV and h imin ¼ . GeV. The BO bq¯ A 0 26 ; H 0 32 For the orbital we find ¼ . GeV h imin ¼ . GeV. potential with the one-gluon exchange parameters admits The BO potential, wave function, and eigenvalue for the a very shallow bound state with E ¼ −30 MeV, quantum PC ¯ bb q¯ q¯ ¯ J 0þþ bb pair in color 3 and the one-gluon exchange couplings numbers: ð Þ6;S¼0 and ð Þ6;S¼0;I¼1, ¼ , and are reported in Fig. 4. There is a bound tetraquark with a charges −2; −1,0.

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TABLE IV. Q values in MeV for decays into meson þ meson þ γ obtained with string tension 1=4k in Eq. (57),in parentheses with string tension k. Models in [10,11,13] are different elaborations of the constituent quark model we use throughout this paper, and more details are found in the original references. In the last column are the lattice QCD results [45–48].

QQ0u¯ d¯ This work [10] [11] [13] Lattice QCD ccu¯ d¯ þ7ð−10Þþ140 þ102 þ39 −23 11 [45] cbu¯ d¯ −60ð−74Þ ∼0 þ83 −108 þ8 23 [46] bbu¯ d¯ −138ð−156Þ −170 −121 −75 −143 34 [45] −143ð1Þð3Þ [47]−82 24 10 [48]

As shown in Fig. 5, the potential is so shallow as to raise inside the orbitals are treated as perturbations, to obtain the doubts whether a bound tetraquark will indeed result. We first order BO potential that goes into the Schrödinger register nonetheless the Q value for the decay T → BB¯.For equation of the heavy constituents. the string tension (83) we find The non-Abelian nature of QCD produces a number of peculiarities. Given that the hadron is a color singlet and 3 3 given the representation of the heavy constituents, one can Qbb E − κbb − κqq 3κbq¯ −131 −133 MeV; ¼ 2 2 þ ¼ ð Þ deduce, for each pair, the coefficient of the Coulomb-like ð84Þ interaction and the strength of the string tension. The pair forming an orbital, except for the case of the baryon, is in parentheses the result with string tension k. general in a superposition of color representations with the same triality, e.g., 3¯ and 6. Orbitals with nonvanishing triality have to be confined, and we add to the BO potential VIII. SUMMARY OF RESULTS the appropriate linearly rising potential. Triality zero The present paper gives an extensive discussion of orbitals are not confined, as discussed in Sec. IV and doubly heavy hadrons, baryons, and tetraquarks, within in [19], and the BO potential vanishes for large separation the BO approximation. The paper is an expansion of the of the heavy constituents. A feature of the QCD Cornell potential, Sec. IV, is that it shorter communication [1], with the discussion of doubly V heavy baryons added, a case where we can compare contains an additive constant 0 that in charmonium directly theory to experimental results [26]. physics is determined from one physical mass of the þ spectrum. We are able to determine V0 (i) in the baryon In analogy with the QED treatment [15] of the H2 ion H case from a boundary condition related to the heavy quark- (the analog of a doubly heavy baryon) and the 2 molecule – QQq¯ q¯ (analog of a doubly heavy tetraquark), we start our diquark symmetry [22 24], Sec. V, and (ii) in discussion from orbitals: two-body, heavy-light, quark- tetraquarks from the condition that, at infinity, the potential quark, or quark-antiquark lumps held together by the gives rise to a meson-meson* pair, Sec. VII. For these QCD Coulomb-like interaction plus a linear confining term reasons, we get in these two cases an absolute prediction of with the appropriate string tension. their mass, which can be compared with the experimental value in the case of Ξcc, and which allows us to judge the The wave functions of the orbitals, obtained from the bbq¯ q¯ two-body Schrödinger equation, are taken as zeroth order stability of against strong or electromagnetic decays into DB or DB þ γ. approximation of the light constituents wave function ¯ On the other hand, V0 remains undetermined for QQqq¯ inside the hadron. QCD Coulomb-like interactions with tetraquarks and orbitals with nonvanishing triality and the the other constituents of the light quarks or antiquarks hadron mass cannot be predicted, at least for the ground state. However, the QQ¯ wave function provides interesting information on the tetraquark internal structure, with significant phenomenological implications. We now summarize the results of Secs. V–VII Doubly heavy baryon.—Our results are summarized in M Ξ 3652þ17 Table III, fourth column. We find ð ccÞTh ¼ −7 MeV M Ξ to be compared with the LHCb value [26] ð ccÞExpt ¼ 3621.2 0.7 MeV. The difference is within the theoretical uncertainty of our approach [see Eq. (11)]. For the heavier baryons, our results differ from the results in Refs. [17,27] by 50 and 150 MeV for bc and bb baryons, respectively. FIG. 5. A shallow bound state might be present in the color 6 Recent lattice QCD results [28,29], where available, are channel. intermediate between us and [17,27] (see Table III).

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QQ — V Overall, the general consistency of results derived by Double heavy tetraquarks: ð Þ6. The BO potential alternative routes with themselves and with the experimen- for bb has a repulsive behavior at the origin, and it vanishes tal value is very encouraging. Experimental results on at large separations with a very shallow minimum. heavier baryons will allow a more significant comparison The binding energy E ¼ −30 MeV is at the limit of our and are eagerly awaited. visibility. If it exists, the bound state would make a second Hidden charm tetraquark: cq orbitals.—The interaction bb tetraquark, possibly stable. Its existence needs con- between the light quarks, q and q¯ is repulsive. Combined firmation by lattice QCD calculations. with the existence of a raising confining potential between the orbitals, this leads to envisage two regimes, exemplified IX. CONCLUSIONS in Figs. 2(a) and 2(b). The BO approximation gives a new insight on the multi- For the low value of the repulsive coupling, 1=6αS ∼ þ quark hadron structure and provides new opportunities for 0.05, implied by one gluon exchange, the equilibrium theoretical progress in the field of exotic resonances. configuration obtains for c and c¯ relatively close to each The restriction to a perturbative treatment followed here other, in a quarkonium adjoint meson configuration [2,3]; is, at the moment, a necessity for any analytical approach. see Fig. 2(a). Nonetheless, the consistency of the results we have found Increasing the repulsion, orbitals are split apart and for doubly heavy baryons and doubly heavy tetraquarks equilibrium obtains for a diquark-antidiquark configuration, with lattice QCD calculations seems to show that the Fig. 2(b), with well separated diquarks. As an example, 1=6α ∼ 0 05 → 3 3 perturbative approach is sufficiently robust (as it was for letting þ S . . in Eq. (18),diquarksare the hydrogen ion and molecule) to provide useful, quanti- separated by a potential barrier and there are two different R ∼ 0 4 tative indications. lengths: the diquark radius qc . fm and the total radius A critical case, where nonperturbative calculations are R ∼ 1 5 qq¯ . fm. A dominant, nonperturbative repulsion plus called for, is in the QQq¯ q¯ tetraquarks. As we have shown confinement gives the dynamical basis to the emergence of here, the strength of qq¯ repulsion is the critical parameter the repulsive barrier between diquarks and antidiquarks to determine the internal configuration of the tetraquark, suggested in [43]. The need to tunnel under the barrier from a quarkonium adjoint meson to a diquark-antidiquark explains why decays into charmonia occur at a lower rate configuration. The latter configuration is indicated by the with respect to decays into open charm mesons, as observed pattern of decay modes of Xð3872Þ and is compatible with X Z in and resonances. Diquark-antidiquark separation the existence of charged partners of the Xð3872Þ not to be may also be the reason why charged partners of the X have observed in open charm decays but only in final states not (yet) been observed and there is an almost degenerate containing charmonia, X → ρJ=ψ. The B meson may X0 doublet of u;d neutral states [43,44]. have a smaller branching fraction than expected for decays Hidden charm tetraquark: cq¯ orbitals.—The BO poten- that involve the charged X, and this requires some dedi- tial goes to þ∞ at zero separation, due to cc¯ repulsion, and cated experimental effort to go beyond the bounds which it vanishes at infinity, due to the zero triality of orbitals. The have been set years ago. existence of a minimum is not guaranteed. The situation is Nonperturbative investigations along these lines should shown in Figs. 3(a) and 3(b). For the one gluon exchange be provided by lattice QCD, following the growing interest parameters, there is indeed one minimum, Fig. 3(a), and a shown for doubly heavy tetraquarks. second tetraquark, in the quarkonium adjoint meson configuration. ACKNOWLEDGMENTS If the qq¯ repulsion is increased, letting e.g., þ1=6αS ∼ We are grateful for hospitality by the T. D. Lee Institute 0.05 to a value > 1, there is no minimum, Fig. 3(b). The and Shanghai Jiao Tong University where this work was lack of a second resonance with the same features of, initiated. We acknowledge interesting discussions with but well separated from, Xð3872Þ would speak in favor A. Ali, A. Esposito, A. Francis, M. Karliner, R. Lebed, of Figs. 2(b) and 3(b) supporting the enhancement of qq¯ N. Mathur, A. Pilloni, and W. Wang. repulsion. Double heavy tetraquarks: ðQQÞ3¯ .—Our results for the APPENDIX A: QED ORBITALS Q value of the lowest ½bb tetraquark against decays into AND MOLECULES DB þ γ are shown in Table IV and found to compare well with previous estimates done with quark model, We review here the Born-Oppenheimer approximation Refs. [10,11,13], and, remarkably, with lattice QCD results for the hydrogen molecule and sketch the perturbative [45–48], where available. method starting from the hydrogen orbitals [15] which Given the error estimate following Eq. (13), we support provides the basis of our treatment of heavy-light tetra- the proposal that the lowest ½bb and perhaps ½bc tetraquarks in QCD. quarks may be stable against strong and electromagnetic The Hamiltonian of two protons in xA and xB and two decays [10,11]; see also [49,50]. electrons in x1 and x2 is

074002-11 MAIANI, POLOSA, and RIQUER PHYS. REV. D 100, 074002 (2019) X 2 X 2 P p 1 I1 to I6 as functions of rAB are defined as H i i α ¼ 2M þ 2m þ x − x A;B 1;2 j A Bj Z 3 2 1 1 1 I1 ¼ d xψðxÞ ; − α þ þð1 ↔ 2Þ jxB − xj jxA − x1j jxB − x2j Z 1 1 I d3xψ x ϕ x ; α H H H H ; 2 ¼ ð Þ ð Þ þ ¼ AB þ A;1 þ B;2 þ pert ðA1Þ jxA − xj jx1 − x2j Z 1 I d3xd3xψ x 2ϕ y 2 ; 4 ¼ ð Þ ð Þ r where Z 3 3 1 X P2 1 I6 ¼ d xd x½ψðxÞϕðxÞ½ψðyÞϕðyÞ ðA7Þ H i α ; r AB ¼ 2M þ x − x A;B j A Bj 2 r x − y p1 1 with ¼j j. Explicit expressions of the integrals are HA;1 ¼ − α ; given in [15]. 2m jxA − x1j The Born-Oppenheimer potential is HB;2 ¼ same with∶ A → B; 1 → 2; 1 1 1 1 H −α α : V r α − α2m ΔE r : pert ¼ þ þ ðA2Þ BOð ABÞ¼þ þ ð ABÞ ðA8Þ jxA − x2j jxB − x1j jx1 − x2j rAB ψ x We denote by ð Þ the lowest energy eigenfunction of þ The potential diverges to þ∞ for rAB → 0 and tends to HA;1 and by ϕðxÞ the similar eigenfunction of HB;2, both 2 −α m (the energy of two hydrogen atoms) for rAB → ∞.A being real functions. Since they belong to two different numerical evaluation of the previous formulas shows that Hamiltonians, ψðxÞ and ϕðxÞ are not orthogonal, and we the potential has one minimum for denote by S the overlap function Z r ∼ 1 5 αm −1 0 79 0 76 ; 3 min . ð Þ ¼ . Åð . ÅÞ SðrABÞ¼ d xψðxÞϕðxÞðA3Þ V ∼ 0 23E 3 1 4 4 ; ½ BOmin . H ¼ . eVð . eVÞ with rAB ¼jxA − xBj. ψ and ϕ are usually called the H H orbitals of the 2 molecule. Neglecting pert, there are which compare favorably with the experimental numbers two degenerate lowest energy eigenstates, namely given in parentheses. ψðx1Þϕðx2Þ and ψðx2Þϕðx1Þ, which may be combined in Computed along the same lines, the BO potential for the H the symmetric or antisymmetric combinations. When pert antisymmetric combination (and electrons in the triplet is turned on, the antisymmetric combination turns out to state) shows no minimum. have a higher energy and we restrict to the symmetric combination (ψ and ϕ normalized to unity) APPENDIX B: FIERZ IDENTITIES ψðx1Þϕðx2Þþψðx2Þϕðx1Þ SU 3 f0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA4Þ The basic Fierz identity, in ð Þc, reads 2ð1 þ S2Þ 1 1 δγ δδ δγ δδ λA γ λA δ ; with energy α β ¼ 3 β α þ 2 ð Þβð Þα ðB1Þ 2 E0 ¼ 2EH ¼ −α m; ðA5Þ where from we derive i.e., twice the hydrogen ground level. Electrons being fermions, the symmetric combination (A4) is associated γ δ γ δ 2 γ δ 1 A γ A δ δαδ − δ δα − δ δα λ λ ; B2 with electron spins in the singlet combination, S ¼ 0. β β ¼ 3 β þ 2 ð Þβð Þα ð Þ H To first order in pert we find [15]

γ γ 4 γ 1 γ E ¼ E0 þ ΔEðrABÞ; δ δδ δ δδ δ δδ λA λA δ : α β þ β α ¼þ3 β α þ 2 ð Þβð Þα ðB3Þ ΔE f H f ¼h 0j pertj 0i α α β ¯ −2 I SI I I : q Q Qγq¯ δ ¼ 2 ½ ð 1 þ 2Þþ 4 þ 6 ðA6Þ Saturating with the products , we obtain the ð1 þ S Þ identities

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ðQq¯ ÞðqQ¯ Þ − ðQQ¯ Þðqq¯ Þ only one state for total spin J ¼ 1=2. In the case of cb, there A A are two states with J ¼ 1=2 and Scb ¼ 0, 1. It is customary ðQQ¯ Þðqq¯ Þ pffiffiffi ðQ¯ λ QÞðq¯λ qÞ −2 2 2 ffiffiffi to classify the states according to the spin of the lighter ¼ 3 þ p 4 2 quarks, namely × ðQq¯ ÞðqQ¯ ÞþðQQ¯ Þðqq¯ Þ Ξ qc ; b ; Ξ0 qc ; b ; QQ¯ qq¯ pffiffiffi Q¯ λAQ q¯λAq ½ cb0 ¼jð Þ0 i1=2 ½ cb0 ¼jð Þ1 i1=2 ðC1Þ 4 ð Þð Þ 2 2 ð Þðffiffiffi Þ ¼ 3 þ p 4 2 where the subscript 0 on brackets refers to states before mixing and the subscript 0,1 inside kets refers to the total and factors in the denominators are introduced to have 3 spin of the lighter pair. quadrilinear forms normalized to unity. The hyperfine Hamiltonian is In terms of normalized kets, we have H ¼ 2κqcðsq · scÞþ2κqbðsq · sbÞþ2κcbðs · sbÞ; ðC2Þ ¯ 1 ¯ hf c jðQqÞ3¯ ðQq¯Þ3i1 ¼ pffiffiffijðQQÞ1ðqq¯ Þ1i1 3 rffiffiffi and to compute the matrix elements we need to know what 2 is the spin if the qb and cb pairs in the states (C1); see − QQ¯ qq¯ ; 3jð Þ8ð Þ8i1 e.g., [8]. rffiffiffi An elementary calculation gives (we drop for simplicity ¯ 2 ¯ 1 ¯ the subscript cb) jðQqÞ6ðQq¯Þ6¯ i1 ¼ jðQQÞ1ðqq¯ Þ1i1 þpffiffiffijðQQÞ8ðqq¯ Þ8i1: 3 3 pffiffiffi 3 1 QQ¯ Ξ qb c qb c The combination with in pure octet is therefore 0 ¼ 2 j½ð Þ1 1=2iþ2 jð Þ0 i pffiffiffi ¯ 3 1 T ¼jðQQÞ8ðqq¯ Þ8i1 − cb u − cb u ; rffiffiffi ¼ 2 j½ð Þ1 1=2i 2 jð Þ0 i ffiffiffi 2 ¯ 1 ¯ p ¼ jðQqÞ¯ ðQ q¯Þ i1 − pffiffiffi jðQqÞ ðQ q¯Þ¯ i1 1 3 3 3 3 3 6 6 Ξ0 − qb c qb c 0 ¼ 2 j½ð Þ1 1=2iþ 2 jð Þ0 i pffiffiffi so that 1 3 cb q − cb q : C3 ¼þ2 j½ð Þ1 1=2i 2 jð Þ0 i ð Þ 2 2 1 1 1 λQq λ ¯ − αS − αS: B4 ¼ Q q¯ ¼ 3 3 þ 3 3 ¼ 3 ð Þ Scalar products ðsi · sjÞ commute with the total spin Sij, and we find Saturating (B2) and (B3) with the combination: α β ¯ 3 1 Q q Qγq¯ δ, we express the diquark-antidiquark states in Ξ s s Ξ − ; Ξ0 s s Ξ0 ; h 0j q · cj 0i¼ 2 h 0j q · cj 0i¼þ2 terms of the bilinears with the pairs Qq¯ and qQ¯ and finally express the latter in terms of the state T: and rffiffiffi 8 ¯ 1 ¯ hΞ0jsq · sbjΞ0i¼hΞ0jsc · sbjΞ0i¼0; T ¼ jðQqÞ1ðqQ¯ Þ1i1 − pffiffiffi jðQqÞ8ðqQ¯ Þ8i1 ðB5Þ 9 9 0 0 0 0 hΞ0jsq · sbjΞ0i¼hΞcjsc · sbjΞci¼−1; pffiffiffi and 3 Ξ0 H Ξ κ − κ : h 0j hfj 0i¼ 2 ð qb cbÞ 7 λQq¯ ¼ λqQ¯ ¼ − : 0 6 The mixing matrix in the (Ξ0; Ξ0) basis is 0 1 pffiffi 3 3 B − 2 κqc 2 ðκqb − κcbÞ C MðΞÞ¼@ pffiffi A: ðC4Þ APPENDIX C: MASS AND MIXING 3 κ − κ 1 κ − κ − κ 0 2 ð qb cbÞþ2 qc qb cb OF Ξcb AND Ξcb cc bb For identical or flavors, color antisymmetry and Numerically, we use Tables I and II. Noting that κij ∝ Fermi statistics require the pair to be in spin 1, and there is −1 ðMiMjÞ (see [8]), we take pffiffiffiffiffiffiffiffiffiffiffiffi 3For an expression of the form T ⊗ T0 with T and T0 matrices κbc ¼ κccκbb † † in colorP space, we require TrðTT Þ¼TrðT0T0 Þ¼1. If we have a A 0A −35; −2 9 Ξ sum A T ⊗ T ;A¼ 1; …;N, with eachffiffiffiffi term normalized to to obtain the eigenvalues: ð . Þ MeV and the cb p 0 unity, we divide by an additional factor N. and Ξcb masses reported in Table III.

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