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Hydrogen Bond of QCD in Doubly Heavy Baryons and Tetraquarks

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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. ‡ [email protected] 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 2470-0010=2019=100(7)=074002(14) 074002-1 Published by the American Physical Society MAIANI, POLOSA, and RIQUER PHYS. REV. D 100, 074002 (2019) 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 neglect- ing 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 074002-2 HYDROGEN BOND OF QCD IN DOUBLY HEAVY BARYONS AND … PHYS. REV. D 100, 074002 (2019) 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.
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