PHYSICAL REVIEW D 99, 074026 (2019)
Heavy baryon-antibaryon molecules in effective field theory
† Jun-Xu Lu,1,2 Li-Sheng Geng,1,* and Manuel Pavon Valderrama1, 1School of Physics and Nuclear Energy Engineering, International Research Center for Nuclei and Particles in the Cosmos and Beijing Key Laboratory of Advanced Nuclear Materials and Physics, Beihang University, Beijing 100191, China 2Institut de Physique Nucl´eaire, CNRS-IN2P3, Université Paris-Sud, Universit´e Paris-Saclay, F-91406 Orsay Cedex, France
(Received 6 July 2017; revised manuscript received 24 February 2019; published 26 April 2019)
We discuss the effective field theory description of bound states composed of a heavy baryon and antibaryon. This framework is a variation of the ones already developed for heavy meson-antimeson states to describe the Xð3872Þ or the Zc and Zb resonances. We consider the case of heavy baryons for which the light quark pair is in S-wave and we explore how heavy quark spin symmetry constrains the heavy baryon- antibaryon potential. The one pion exchange potential mediates the low energy dynamics of this system. We determine the relative importance of pion exchanges, in particular the tensor force. We find that in Σ Σ¯ Σ Σ¯ Σ Σ¯ general pion exchanges are probably nonperturbative for the Q Q, Q Q, and Q Q systems, while for Ξ0 Ξ¯ 0 Ξ Ξ¯ 0 Ξ Ξ¯ the Q Q, Q Q, and Q Q cases they are perturbative. If we assume that the contact-range couplings of the effective field theory are saturated by the exchange of vector mesons, we can estimate for which quantum numbers it is more probable to find a heavy baryonium state. The most probable candidates to Λ Λ¯ Σ Σ¯ Σ Σ¯ Σ Σ¯ Λ Σ¯ Λ Σ¯ form bound states are the isoscalar Q Q, Q Q, Q Q, and Q Q and the isovector Q Q and Q Q systems, both in the hidden charm and hidden bottom sectors. Their doubly charmed and doubly bottom Λ Λ Λ Σð Þ Σð ÞΣð Þ counterparts ( Q Q, Q Q , Q Q ) are also good candidates for binding.
DOI: 10.1103/PhysRevD.99.074026
þ þ I. INTRODUCTION Pcð4380Þ and Pcð4450Þ pentaquark states [11], which ¯ Σ ¯ Σ ¯ Σ ¯ Λ 2590 — might contain D c, D c, D c, and even D cð Þ Heavy hadron molecules bound states composed of – heavy hadrons—are a type of exotic hadron. The theoreti- molecular components [12 18]. In the hidden bottom sector we have the Zbð10610Þ and Zbð10650Þ [19,20], cal basis for their existence is robust: in analogy with the ¯ ¯ nuclear forces that bind the nucleons, heavy hadrons can which might be BB , B B molecules [21,22].Ifwe 2317 exchange light mesons, generating exchange forces that consider the open charm sector, the Ds0ð Þ and 2460 might be strong enough to bind them [1–5]. The discovery Ds1ð Þ mesons [23,24] were discovered before the 3872 of the Xð3872Þ more than a decade ago [6] probably Xð Þ and have been theorized to have a large – provides the most paradigmatic candidate for a molecular DK=D K molecular component [25 28]. state. The Xð3872Þ turns out not to be alone: a series of We expect molecular states to be relatively narrow for similarly puzzling hidden charm (hidden bottom) states that states happening above the open charm threshold. For the do not fit in the charmonium (bottomonium) spectrum have moment the masses of the experimentally discovered states been found in different experiments since then. They are have reached the heavy meson-meson and heavy meson- 4 1 4 3 ¯ usually referred to as XYZ states and a few are particularly baryon threshold (3.7 and . = . GeV for DD and Λ ¯ Σ ¯ good candidates for molecular states. In the hidden charm cD= cD; respectively), but barely the heavy baryon- Λ Λ¯ Λ Σ¯ sector we have the Zcð3900Þ and Zcð4020Þ [7,8], which are baryon threshold (4.5, 4.7, and 4.9 GeV for c c, c c, ¯ ¯ ¯ suspected to be DD , D D molecules [9,10], and the and ΣcΣc, respectively). A narrow resonance near the heavy baryon-baryon threshold would be an excellent candidate *[email protected] for a heavy baryon-antibaryon bound state. Though these † [email protected] states have not been found yet, it is fairly straightforward to extend the available descriptions of heavy meson-antimeson Published by the American Physical Society under the terms of molecules to them and explore the relevant dynamics the Creative Commons Attribution 4.0 International license. behind these states. In a few instances it might be possible Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, to predict the location of heavy baryonium states, with the 3 ¯ and DOI. Funded by SCOAP . Λcð2590ÞΣc systems being an illustrative example [29].
2470-0010=2019=99(7)=074026(39) 074026-1 Published by the American Physical Society LU, GENG, and VALDERRAMA PHYS. REV. D 99, 074026 (2019)
Heavy hadron-antihadron molecules are among the most II. EFFECTIVE FIELD THEORY FOR HEAVY interesting theoretical objects of hadronic physics. Owing BARYON MOLECULES to their heavy-light quark content, they are simultaneously EFTs are generic and systematic descriptions of low subjected to isospin, SU(3)-flavor, chiral and heavy quark energy processes. They can be applied to physical systems symmetry, a high degree of symmetry that can translate into in which there is a distinct separation of scales, but where – a fairly regular spectrum [10,21,30 35]. This spectrum will the underlying high energy theory for that system is not be fully realized in nature: unless these states are unknown or unsolvable. Hadronic molecules are a good shallow they will be a mixture of molecule, charmonium candidate for the EFT treatment: the separation among the and other exotic components. Yet these potential regular- hadrons forming a hadronic molecule is expected to be larger ities in the molecular spectrum can be successfully than the size of the hadrons. When the hadrons are close to exploited to uncover the nature of a few of the XYZ states. each other they overlap and the ensuing description in terms ’ ’ The most clear example probably is the Zc s and Zb s of quantum chromodynamics (QCD) is unsolvable. But this resonances, which seem to be related by different realiza- is not the case when the hadrons are far away, in which case tions of heavy quark symmetry [10]. their interactions can be described in terms of well-known Heavy hadron molecules possess another interesting physics such as pion exchanges. In the following we will quality: they show a separation of scales. On the one hand present a brief introduction to the application of the EFT we have the size of the hadrons, which is of the order of framework to heavy baryon-antibaryon systems. 0.5 fm, while on the other we have the size of the bound state, which should be bigger than the individual hadrons A. The effective field theory expansion within it. As a consequence heavy hadron molecules are amenable to an effective field theory (EFT) treatment, EFTs rely on the existence of a separation of scales, where all quantities can be expressed as an expansion of a where a distinction is made between low and high energy light over a heavy energy scale. EFT descriptions of heavy physics and their respective characteristic momentum hadron molecules have been exploited successfully in the scales Msoft and Mhard, which are sometimes called the past specifically in systems composed of heavy mesons soft (or light) and hard (or heavy) scales. The separation of and antimesons [31,36–38]. In this manuscript we extend scales can be used to express physical quantities at low the heavy hadron EFT formulated in Ref. [32] and put to energies as expansions in terms of the small parameter use in Refs. [10,33,34] in the case of the heavy baryon- Msoft=Mhard. If we consider a system of heavy baryons for antibaryon molecules. As commented, these types of concreteness, there are two possible EFT expansions molecules might very well be discovered in the next depending on which type of low energy symmetry we few years. The purpose of this work is to explore the are considering: symmetry constraints and the kind of EFT that is to be (i) heavy quark spin symmetry (HQSS) and expected in these systems, rather than to make concrete (ii) chiral symmetry. Λ ∼ 200 predictions of the possible location of these states. Yet we For HQSS the soft and hard scales are QCD MeV ∼ will speculate a bit about this later issue on the basis of the and the mass of the heavy quark mQ, which is either mc 1 5 ∼ 4 5 relative strength of the long-range pion exchange and the . GeV or mb . GeV. For chiral symmetry we call the saturation of the EFT low energy constants by σ, ρ, ω and soft and hard scales Q and M, where if there are no baryons ϕ meson exchange. we have Q ∼ mπ ∼ q ∼ 100–200 MeV (with mπ the pion The manuscript is structured as follows. In Sec. II we mass and q the momenta of the pions) and M ∼ 2πfπ∼ make a brief introduction to the EFT formalism. In Sec. III 1 GeV. If there are baryons, Q includes the soft momenta we present the leading order EFT potential for heavy of the baryons, while practical calculations in the two- baryon-antibaryon states, which consists of a series of baryon sector suggest a more conservative value of M ∼ contact four-baryon vertices plus the time-honored one 0.5–1.0 GeV for the hard scale. We advance that the scale Q pion exchange potential. In Sec. IV we explore the question can contain more than the pion mass and the momenta of the of whether pions are perturbative or not for this type of pions and baryons, as we will discuss in Sec. II B. hadron molecule. In Sec. V we discuss the possible power Heavy baryons are nonrelativistic at the soft scales of countings to describe molecular states. In Sec. VI we either of the two previous symmetries. This implies that speculate about which heavy baryon-antibaryon molecules their interactions can be described in terms of an effective might be more probable. Finally in Sec. VII we present our potential VEFT, which admits the double expansion conclusions. In Appendix A we present the complete X Λ μ ν derivation of the one pion exchange potential, in ˆ ðμ;νÞ QCD Q VEFT ¼ V ; ð1Þ Appendix B we briefly explain the one eta and one kaon μ;ν mQ M exchange potential, and in Appendix C we derive the heavy quark symmetry constraints for the four-baryon contact where the indexes μ and ν indicate the order in the heavy vertices. quark and chiral expansion, respectively, with μ ≥ 0 and
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ν ≥−1 (this second point will be explained in Sec. II B). order of the LO contribution to the potential is ν ¼ 0.For The HQSS expansion converges remarkably faster than the the contact-range potential, this LO contribution is a chiral expansion, owing to the sizes of the soft and hard momentum- and energy-independent interaction, scales involved in each of these expansions. For this reason from now on we will work in the m → ∞ limit and ignore ⃗0 ð0Þ ⃗ ð0Þ Q hp jVC jpi¼C ; ð6Þ any HQSS breaking effect. With this in mind, the expansion of the EFT potential simplifies to with Cð0Þ a coupling, which will depend on the quantum
ν numbers of the two-body system under consideration and Xmax ν þ1 ν Q max will involve spin and isospin operators. The LO piece of the V ¼ Vð Þ þ O ; ð2Þ EFT M finite-range potential is given by one pion exchange (OPE), ν¼ν0 which we write (again schematically) as which converges for Q
ν subleading interactions, in particular the contact-range Xmax ν þ1 ðνÞ Q max ones, involve new free parameters which require additional V ¼ V þ O ; ð5Þ F F M data to be determined. These additional data are not ν ν0 F ¼ ð Þ expected to be experimentally available in the near future. where the power counting of V and V can differ: we C F B. Bound states and power counting introduce ν0ðCÞ and ν0ðFÞ to indicate that the expansion may begin at different orders. Power counting is not unique. NDA is simply the most For concreteness we will temporarily consider that obvious choice for building a power counting, but not the (chiral) power counting is given by naive dimensional only one. In particular, the existence of bound states analysis (NDA). We warn that NDA is incompatible with requires modifications to the power counting [39–42],as the existence of bound states, but we will address this we will illustrate below. Bound states are solutions of a problem later. Within NDA the power counting of a contribution to the potential is determined by the powers 1Modulo numerical factors, which for nonrelativistic scattering of the heavy baryon momenta, pion momenta, and pion will be multiples of 4π. As we are focusing on the scaling only, masses included in a particular contribution. In NDA the we do not include these factors explicitly.
074026-3 LU, GENG, and VALDERRAMA PHYS. REV. D 99, 074026 (2019) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Λ 2μΔ dynamical equation, such as Schrödinger or Lippmann- CC ¼ CC; ð13Þ Schwinger, and require nonperturbative physics. We can μ Δ see this from the Lippmann-Schwinger equation as applied with the reduced mass of the system and CC the energy to bound states, difference of the transition. Coupled channel effects can be argued to be suppressed by a factor of Ψ Ψ j Bi¼G0Vj Bi; ð9Þ Q 2 Ψ Λ ; ð14Þ where j Bi is the bound state wave function and G0 ¼ CC 1=ðE − H0Þ the resolvent operator, with H0 the free Λ Σ Σ Hamiltonian. When G0 appears in loops it is counted as Q: with CC the coupled channel scale. For the c c family Λ 400 564 Z of systems, we have that CC ¼ = MeV depending 3⃗ 1 on the transition, while for the Σ Σ case we have d l ∼ μ b b 3 2 Q; ð10Þ Λ ¼ 350=495 MeV. This scale is softer than M,but ð2πÞ E − l CC 2μ not much softer: we can effectively ignore the coupled channels at the price of reducing the range of applicability μ where refers to the reduced mass. Generating a bound of the EFT. state requires the iteration of the potential, which in terms That is, there are two choices for constructing the EFT in of power counting is only consistent if this case: (i) consider the coupled channel effects to be subleading (at the price of reducing the convergence radius O O O … ðVÞ¼ ðVG0VÞ¼ ðVG0VG0VÞ¼ ; ð11Þ of the EFT) or (ii) include them at leading order. Here we will opt for the first option, owing to its simplicity. Be that −1 from which V ∼ Q is required. To explain why the as it may, most of the results of this manuscript can be potential is counted this way we have to revisit the easily extended to the coupled channel case. estimations of the size of the couplings Cð0Þ and Fð0Þ contained in Eq. (8). If any of these two couplings contains D. Kaon/eta exchanges and SU(3) symmetry a light scale If we want to preserve SU(3) flavor symmetry, the 1 1 exchange of kaons and eta mesons should be treated on Cð−1Þ ∼ and=or Fð−1Þ ∼ ; ð12Þ MQ MQ equal footing as the exchange of pions, at least in principle. But the masses of the kaon and the eta meson are of the the LO potential will be promoted from Q0 to Q−1, order of 0.5 GeV, which is comparable to the hard scale M. allowing for the existence of bound states.2 The light We have two choices: (i) ignore kaon and eta exchanges or momentum scale that appears in Cð−1Þ can be identified (ii) include them explicitly. with the inverse scattering length of the two-body system In the first option, the contributions from kaon and eta −1 exchange are implicitly included in the contact-range [39–42], while the light scale in Fð Þ is related with the potential. There is a disadvantage, though: the contact- strength of the OPE potential [45]. We stress that it is range potential breaks SU(3)-flavor symmetry in this case. enough to promote one of the two couplings Cð0Þ and Fð0Þ 0 −1 We expect the size of this breaking to be parametrically from Q to Q , where for a more detailed discussion we small for heavy baryon-baryon and heavy baryon-anti- refer to Sec. V. Σ Σ Σ Σ¯ Ξ0 Ξ0 Ξ0 Ξ¯ 0 baryon systems of the type Q Q, Q Q, Q Q, Q Q, etc., i.e., systems containing only one species of baryon. C. Coupled channels Besides the pion, this type of system only exchanges eta Now we consider the power counting of coupled channel mesons, where their coupling to the heavy baryons is effects. Heavy baryons can come in HQSS multiplets considerably weaker than that of the pions. Regarding the which are degenerate in the heavy quark limit, e.g., the kaons, they are relevant for heavy baryon-baryon systems Σ Σ Σ Σ that involve different species: Σ Ξ0 , Ξ0 Ω , etc. But if we c and the c in the charm sector or the b and the b in the Q Q Q Q bottom sector. If we take the Σc and the Σc heavy baryons consider instead a heavy baryon-antibaryon system, the as an example, the two-heavy-baryon system can have exchange of a single kaon implies a transition between two- transitions of the type ΣcΣc → ΣcΣc, ΣcΣc → ΣcΣc, etc. In baryon states with different thresholds, which involves 0 ¯ ¯ 0 EFT, these transitions have a characteristic momentum coupled channel effects. For example in the ΞcΣc → ΩcΞc scale, transition mediated by the exchange of a kaon/antikaon, the Δ 240 energy gap is CC ¼ MeV and the coupled channel Λ 790 2We mention in passing that the promotion can also be scale is CC ¼ MeV, which is certainly hard. understood in terms of the anomalous dimension of the coupling For these reasons we expect the exchange of kaon and C0, i.e., of its scaling with respect to the cutoff [43,44]. eta mesons to have a small impact on the description of
074026-4 HEAVY BARYON-ANTIBARYON MOLECULES IN EFFECTIVE … PHYS. REV. D 99, 074026 (2019) heavy baryon-antibaryon systems in general. This suggests the discussion, the notation will be complex. We overview that ignoring explicit kaon and eta exchange, which the most used notation in this section in Table I. amounts to including them implicitly in the contact-range couplings, is not likely to generate a sizable breakdown of A. The heavy baryon superfields SU(3) flavor in the contact-range potential. The implicit Heavy baryons have the structure inclusion of eta and kaon effects in the contact-range couplings will change their values from the ones expected jQðqqÞi; ð16Þ from exact SU(3)-flavor symmetry, but this change will probably be numerically smaller than the usual 20% where Q is the heavy quark and qq the light quark pair, uncertainty associated with SU(3)-flavor symmetry rela- which is in S-wave. The light quarks can couple their spin tions. This point is supported by an analysis of the strength of the eta and kaon exchanges in Appendix B, where we also present the kaon and eta exchange potentials in case TABLE I. List of the most used symbols in Sec. III. one wants to include them explicitly in the EFT. Symbol Meaning
B3¯ Antitriplet heavy baryon field III. THE LEADING ORDER POTENTIAL 1 B6 Sextet heavy baryon field, ground state (J ¼ 2) 3 In this section we write down the heavy baryon-anti- B6 Sextet heavy baryon field, excited state (J ¼ 2) baryon potential at LO within the EFT expansion. This M3¯ Antitriplet heavy baryon mass potential can contain a contact- and a finite-range piece, 1 M6 Sextet heavy baryon mass, ground state (J ¼ 2) 3 0 0 0 M6 Sextet heavy baryon mass, excited state (J ¼ 2) Vð Þ ¼ Vð Þ þ Vð Þ; ð15Þ EFT C F T Antitriplet heavy baryon superfield S Sextet heavy baryon superfield where we are assuming NDA for the purpose of fixing the notation and simplifying the discussion. If there are bound C (i) C-parity states, the actual power counting of the heavy baryon- (ii) Coupling of the momentum- and antibaryon system will differ from NDA; see Sec. II B.But energy-independent contact interaction we will address this problem later in Secs. IV and V. G G-parity The LO contact-range potential is a momentum- and ðMÞ Antitriplet-antitriplet (A), antitriplet-sextet (B), AS energy-independent potential in momentum space (or a L and sextet-sextet (C) contact interactions ðMÞ ðMÞ M refers to the SU(3)-flavor representation, Dirac delta in coordinate space). The LO finite-range BS D, BS D 3 L L potential is the OPE potential. For the contact-range SL to total light quark spin, and CðMÞ D and E to whether it is a direct or component, we cannot determine if it is perturbative or SL not a priori without resorting to experimental or phenom- exchange term ⃗ i 1 enological input. For the finite-range components, i.e., the Ii Generic isospin operator for vertex ¼ ,2 pion exchanges, the situation is different and we can, in of the two-body potential ⃗ fact, determine if they are perturbative; see Sec. IV. The ti Isospin-0 to isospin-1 transition matrices τ⃗ 1 discussion about the possible power countings that arise i Isospin-2 Pauli matrices ⃗ Isospin-1 matrices depending on which pieces of the EFT potential are Ti ⃗ 1 perturbative and nonperturbative will be presented later ai Generic spin operator for vertex i ¼ ,2 in Sec. V. of the two-body potential 1 This section is organized as follows. We begin by σ⃗i Spin-2 Pauli matrices ⃗ ⃗þ Spin-1 to spin-3 transition matrices explaining the details of how the heavy baryons are Si, Si 2 2 organized in superfields that are well behaved according ⃗ 3 Σi Spin-2 angular momentum matrices to HQSS in Sec. III A. Next we will consider the C- and G- C12 a⃗1 a⃗2 parity properties of the heavy baryon-antibaryon system in Spin-spin operator ( · ) S12 Tensor operator (3a⃗1 · rˆa⃗2 · rˆ − a⃗1 · a⃗2) Sec. III B. After this, we will first introduce the general 2Sþ1 form of the contact-range potential and the constraints LJ Spectroscopic notation for partial waves imposed on it by heavy quark spin symmetry, SU(2)- with S the spin, isospin, and SU(3)-flavor symmetry in Sec. III C. Last, we L the orbital angular momentum, and J the total angular momentum will present the general form of the OPE potential and its partial wave projection in Sec. III D. Owing to the scope of C12 Matrix elements of the spin-spin operator in the partial wave basis S12 Matrix elements of the tensor operator in the 3 3 Regarding the exchange of the other SUð Þ Nambu- partial wave basis Goldstone bosons, we refer to the discussion in Sec. II D.
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P 1þ to SL ¼ 0, 1. If the light spin is SL ¼ 0,wehaveaJ ¼ 2 For a more complete account of the heavy baryon fields and heavy baryon: superfields, we refer to Appendix A.
B3¯ Q qq : 17 ¼j ð ÞSL¼0i ð Þ B. C- and G-parity We are considering heavy baryon-antibaryon states. If a This type of heavy baryon belongs to the 3¯ representation state is electrically neutral and does not have strangeness, of the SUð3Þ flavor group. If the light spin is SL ¼ 1,we P 1þ P 3þ then C-parity will be a well-defined quantum number. If the have instead a J ¼ 2 or a J ¼ 2 baryon, P heavy baryon and antibaryon have identical SL and J , i.e., if they have the structure B6 Q qq 1 2; 18 ¼j ð ÞSL¼1ijJ¼ = ð Þ ¯ ¯ ¯ B¯ B¯ ; B6B6 ; B B ; 24 j 3 3i j i j 6 6i ð Þ B Q qq 3 2; 19 6 ¼j ð ÞSL¼1ijJ¼ = ð Þ then the C-parity of the system is which belong to the 6 representation of SUð3Þ. If the heavy quark within the heavy baryons is a charm C ¼ð−1ÞLþS; ð25Þ ¯ quark, Q ¼ c, the flavor components of the B3 field are 4 0 1 with L and S the orbital angular momentum and spin. 0 Examples of this type of heavy baryon-antibaryon system Ξc B C ΛþΛ− Σ0Σ¯ 0 Σ þΣ − þ are c c , c c, and c c . B3¯ ¼ @ −Ξc A; ð20Þ P If the light quark spin SL and spin-parity J of the heavy Λþ c baryon and antibaryon are not identical, we first have to choose a C-parity convention, for instance, where we follow the convention of Cho [46]. For the B6 ¯ field, the flavor components are CjB3¯ i¼þjB3¯ i; ð26Þ 0 1 1 1 ¯ Σþþ pffiffi Σþ pffiffi Ξþ0 CjB6i¼þjB6i; ð27Þ B c 2 c 2 c C B 1 0 1 0 C B pffiffi Σþ Σ pffiffi Ξ 0 C − ¯ B6 ¼ @ 2 c c 2 c A: ð21Þ CjB6i¼ jB6i; ð28Þ 1 1 0 0 pffiffi Ξþ0 pffiffi Ξ 0 Ω 2 c 2 c c where there is a relative minus sign for the C-parity transformation of the spin-3 fields with respect to the 2 For the B6 baryons we have exactly the same components 1 spin-2 fields. With this convention we define the states as for B6, but with a star to indicate that they are spin-3=2 baryons. Depending on the case, it can be practical to ¯ 1 ¯ ¯ jB3¯ B6ðηÞi ¼ pffiffiffi ½jB3¯ B6iþηjB6B3¯ i ; ð29Þ simply consider the SUð2Þ-isospin structure rather than the 2 complete SUð3Þ-flavor one. 1 The fields B3¯ , B6,andB6 can be organized into the ¯ ¯ ¯ jB3¯ B ðηÞi ¼ pffiffiffi ½jB3¯ B iþηjB B3¯ i ; ð30Þ superfields T and S, which have good transformation proper- 6 2 6 6 ties under the rotation of the heavy quark. For nonrelativistic 1 ¯ ¯ ¯ heavy baryons we write the superfields as [46] jB6B ðηÞi ¼ pffiffiffi ½jB6B iþηjB B6i ; ð31Þ 6 2 6 6 1 ⃗ ⃗ T ¼ B3¯ ; S ¼ pffiffiffi σ⃗B6 þ B6; ð22Þ 3 where η ¼ 1, for which the C-parity is η −1 LþS where the letters T and S stands for (anti-)triplet and sextet. C ¼ ð Þ ; ð32Þ The definition of the nonrelativistic superfield T is redundant where L (S) is the total orbital angular momentum (spin) of (it acts merely as a second name for B3¯ ), but we include it for 5 3 2 the heavy baryon-antibaryon pair. Examples of this type of completeness. Notice that we have written the spin- = ΛþΣ− Ξ00Ξ¯ 0 Ξ0þΞ − ⃗ molecule include c c , c c, and c c . heavy baryon field as a vector: B6. The reason is that this is a 3 2 Rarita-Schwinger field, where the spin- = nature of this 4 field is taken into account by coupling a spatial vector with a This comes from multiplying the intrinsic C-parity of a 3 2 fermion-antifermion system with the symmetry factors of Dirac spinor and then projecting to the spin- = channel with exchanging the particles, i.e., C ¼ð−1Þ × ð−1ÞL × ð−1ÞSþ1. ⃗ 5 the condition σ⃗· B6 ¼ 0. Under rotations of the heavy quark The C-parity is the product of the intrinsic C-parity and spin, the superfields behave as the symmetry factor of exchanging the particles, which now includes a contribution from η, i.e., C ¼ðþηÞ × ð−1Þ × ð−1ÞL × −1 Sþ1 −η −1 −1 L −1 S iϵ⃗·σ⃗ ⃗ iϵ⃗·σ⃗ ⃗ ð Þ for B3¯ B6 and C ¼ð Þ × ð Þ × ð Þ × ð Þ for T → e2 Q T; S → e2 Q S: ð23Þ B3¯ B6=B6B6.
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For a heavy baryon-antibaryon state that is not electri- where SL is the total light quark spin of the heavy baryon- ’ cally neutral but has no strangeness and belongs to the same antibaryon system. The dSL s are coefficients that depend SUð2Þ isospin representation as a neutral state, C-parity is on the heavy and light quark decomposition of the specific not a well-defined quantum number but there exists an heavy baryon-antibaryon molecule; see Appendix C for extension that includes isospin. This extension is G-parity details. [47], which can be defined as follows: The contact-range couplings of the TT¯ , ST¯ ,and SS¯ molecules are independent and we will use a π G ¼ Cei I2 ; ð33Þ different notation for each case: A, B,andC respec- tively. For the TT¯ system we write the contact-range that is, a C-parity transformation combined with a rotation potential as in isospin space.6 For an electrically charged state, the G- parity is well defined and its eigenvalues are ¯ ð0Þ ¯ hTTjVC jTTi¼A0; ð37Þ G ¼ Cð−1ÞI; ð34Þ where we are already taking into account that the total 0 where I is the isospin of the electrically charged state light quark spin is always SL ¼ (we also ignore the coefficient because there is actually no decomposition). and C is the C-parity of the electrically neutral component ¯ of the isospin multiplet. For example, if we consider For the ST system we write ΣþþΣ¯ 0, its isospin is I ¼ 2 and the neutral component c c ¯ ð0Þ ¯ of its isospin multiplet is a linear combination of hSTjVC jSTi¼d1DB1D; ð38Þ 0 ¯ 0 þ − þþ −− þþ ¯ 0 ΣcΣc, Σc Σc , and Σc Σc : the G-parity of Σc Σc is then −1 I −1 LþS ¯ ð0Þ ¯ G ¼ð Þ C ¼ð Þ . hSTjVC jTSi¼d1EB1E; ð39Þ 1 C. The contact-range potential where the total light spin is always SL ¼ ,butwhere we have to make the distinction between a diagonal and The LO contact-range potential takes the generic form nondiagonal potential. For the SS¯ system we have 0 hp0jVð Þjpi¼C; ð35Þ C ¯ ð0Þ ¯ hSSjVC jSSi¼d0C0 þ d1C1 þ d2C2; ð40Þ with C a coupling constant and where p (p0) is the center- of-mass momentum of the incoming (outgoing) heavy where the total light spin is SL ¼ 0,1,2.Welistthe baryon-antibaryon pair. In principle, there should be one contact-range potential for heavy baryon-antibaryon independent coupling C for each JPC quantum number and molecules with well-defined C-parity in Table II,which type of heavy baryon-antibaryon molecule. But the contact- also applies by extension to the molecules with well- range potential is constrained by HQSS and SUð3Þ-flavor defined G-parity. symmetry, which greatly reduces the number of possible If the molecules do not have well-defined C- or G-parity couplings. We first consider the HQSS structure of the (i.e., molecules with strangeness), the form of the contact- contact-range potential and then the SUð3Þ flavor one. range potential depends on the particular case. For the family of molecules 1. HQSS structure ¯ ¯ ¯ The application of HQSS to the heavy baryon-antibaryon jB3¯ B3¯ i; jB6B6i; jB6B6ið41Þ system implies that the contact-range coupling does not ¯ ¯ 0 ¯ depend on the heavy quark spin, only on the light quark (e.g., ΛcΞc, ΣcΞc, ΣcΞc), the contact-range couplings spin. This means that the coupling C can be decomposed in are exactly as shown in Table II for the case in which terms of light quark components, C-parity is well defined. For the molecules involving X different types of heavy hadrons, the contact-range poten- tials are defined in coupled channels. If we consider the C ¼ dSL CSL ; ð36Þ bases
6Notice that the G-parity transformation is sometimes defined ¯ ¯ B1 ¼fjB3¯ B6i; jB6B3¯ ig; ð42Þ as G ¼ Ce−iπI2 with a minus sign. For particles with integer isospin, this is equivalent to the definition with a plus sign, þiπI2 ¯ ¯ G ¼ Ce . For particles with half-integer isospin, each of B2 ¼fjB3¯ B6i; jB6B3¯ ig; ð43Þ these conventions generates antiparticle states that differ by a sign. This has no observable consequence, as it amounts to a B ¯ ¯ global redefinition of the amplitudes by a phase. 3 ¼fjB6B6i; jB6B6ig; ð44Þ
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¯ 0 ¯ TABLE II. HQSS decomposition of the contact-range cou- Examples of bases 1, 2, and 3 are the ΞcΣc − ΞcΛc, plings for heavy baryon-antibaryon molecules with well-defined ¯ ¯ 0 ¯ ¯ ΞcΣc − ΞcΛc, and ΞcΣc − ΞcΣc systems. C-parity. A, B, and C refer to the couplings of the TT¯ , ST¯ , and SS¯ systems, respectively, with the subscript indicating the light quark spin decomposition. For the cases where C-parity is not well 2. SU(3)-flavor structure defined, we refer to the explanations in the main text. Besides HQSS, heavy baryon-antibaryon systems also have SU(3)-flavor symmetry. In SU(3) flavor, the T and S System JP=JPC V (HQSS) C heavy baryons belong to the antitriplet and sextet repre- ¯ 0−þ B3¯ B3¯ A0 sentation (3¯ and 6), respectively. For the TT¯ , the HQSS ¯ 1−− B3¯ B3¯ A0 coupling is further divided into the SU(3)-flavor represen- ¯ −þ − 3¯ ⊗ 3 1 ⊕ 8 B3¯ B6 0 B1D B1E tations ¼ , a singlet and an octet. That is, there ¯ −− B3¯ B6 0 B1D þ B1E are two independent SU(3)-flavor contact interactions ¯ −þ 1 B3¯ B6 1 B1 − B1 D 3 E 1 8 ¯ 1−− 1 → ð Þ ð Þ B3¯ B6 B1D þ 3 B1E A0 fA0 ;A0 g: ð51Þ ¯ 1−þ 1 B3¯ B6 B1D þ 3 B1E For the ST¯ case, we have 6 ⊗ 3 ¼ 8 ⊕ 10: ¯ 1−− − 1 B3¯ B6 B1D 3 B1E ¯ 2−þ − 8 10 B3¯ B6 B1D B1E → ð Þ ð Þ B1D fB1D;B1D g; ð52Þ ¯ −− B3¯ B6 2 B1D þ B1E ¯ 0−þ 1 2 → ð8Þ ð10Þ B6B6 3 C0 þ 3 C1 B1E fB1E ;B1E g: ð53Þ ¯ 1−− 1 6 20 B6B6 27 C0 þ 27 C1 þ 27 C2 ¯ 6 ⊗ 6¯ 1 ⊕ 8 ⊕ 27 ¯ −þ Finally for the SS case we have ¼ : B6B6 1 C1 ¯ 1−− 16 6 5 B6B6 27 C0 þ 27 C1 þ 27 C2 ð1Þ ð8Þ ð27Þ − 1 2 C → fC ;C ;C g: ð54Þ ¯ 2 þ SL SL SL SL B6B6 3 C1 þ 3 C2 ¯ 2−− B6B6 C2 The decomposition for a specific molecule can be found in ¯ 0−þ 2 1 B6B6 3 C0 þ 3 C1 Table III, which has been obtained from the SU(3) Clebsch- ¯ 1−− 10 15 2 3¯ ⊗ 3 6 ⊗ 3 6 ⊗ 6¯ B6B6 27 C0 þ 27 C1 þ 27 C2 Gordan coefficients for the , , and of ¯ 2−þ 2 1 Ref. [48]. Notice that we are not explicitly considering the B6B6 3 C1 þ 3 C2 ¯ −− SU(2)-isospin structure as it is a subgroup of SU(3) flavor. B6B6 3 C2
TABLE III. SU(3)-flavor decomposition of the contact-range we end up with the following contact-range potentials: couplings depending on the type heavy baryon-antibaryon molecule. Notice that the HQSS structure of the couplings is − − B1D B1E independent of the SU(3) one, which is why we do not show the VB ð0 Þ¼ ; ð45Þ 1 S −B1E B1D light spin indices for the couplings. For the heavy baryons we show the decomposition for the lightest member of the HQSS 1 multiplet only. B1 B1 1− D 3 E VB1 ð Þ¼ 1 ; ð46Þ 3 B1E B1D System Type Isospin VC Λ Λ¯ TT¯ 0 1 Að1Þ 2 Að8Þ 1 c c 3 þ 3 B1 − B1 Ξ Ξ¯ ¯ 2 ð1Þ 1 ð8Þ − D 3 E c c TT 0 3 A þ 3 A VB ð1 Þ¼ ; ð47Þ 2 1 Ξ Ξ¯ ¯ 1 ð8Þ − 3 B1E B1D c c TT A Ξ0 Ξ¯ ST¯ 0 Bð8Þ − c c − B1D B1E Ξ0 Ξ¯ ¯ 1 ð8Þ 2 ð10Þ 2 c c ST 1 3 B þ 3 B VB2 ð Þ¼ ; ð48Þ − Σ Λ¯ ¯ 2 ð8Þ 1 ð10Þ B1E B1D c c ST 1 3 B þ 3 B ! Ω Ω¯ ¯ 1 ð1Þ 8 ð8Þ 3 ð27Þ 16C0þ33C1þ5C2 16C0−21C1þ5C2 c c SS 0 6 C þ 15 C þ 10 C 1− 54 54 Ξ0 Ξ¯ 0 SS¯ 0 1 Cð1Þ 1 Cð8Þ 3 Cð27Þ VB3 ð Þ¼ ; ð49Þ c c 3 þ 15 þ 5 16C0−21C1þ5C2 16C0þ33C1þ5C2 Σ Σ¯ ¯ 1 ð1Þ 2 ð8Þ 1 ð27Þ 54 54 c c SS 0 2 C þ 5 C þ 10 C 0 ¯ 0 ¯ 1 ð8Þ 4 ð27Þ ! ΞcΞc SS 1 C þ C 1 5 1 1 5 5 C1 C2 C1 − C2 Σ Σ¯ ¯ 4 ð8Þ 1 ð27Þ − 6 þ 6 6 6 c c SS 1 5 C þ 5 C VB ð2 Þ¼ : ð50Þ 3 1 1 1 5 Σ Σ¯ SS¯ 2 Cð27Þ 6 C1 − 6 C2 6 C1 þ 6 C2 c c
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TABLE IV. Numerical, isospin, and spin factors associated with each vertex in the hST¯ jVjTS¯i and hSS¯jVjSS¯i heavy baryon- antibaryon potential. The arrows are used to indicate the final ⃗ baryon state in the vertex. The symbols τ⃗i and Ti represent the Pauli matrices (in isospin space) and the isospin I ¼ 1 matrices, respectively, while ti is a special isospin matrix for connecting the isoscalar Λc with the isovector Σc and the pion. The symbols σ⃗i ⃗ and Σi are the Pauli matrices and the spin S ¼ 3=2 matrices, while ⃗ Si is a 2 × 4 matrix that is used for the transitions involving a spin-1=2 and spin-3=2 baryon. Notice that this table can also be used to compute the heavy baryon-baryon potential. FIG. 1. Diagrams for OPE potential between two heavy R ¯ ⃗ ⃗ Vertex i Ri Ii ai hadrons. On the left we show the TS potential and on the right qffiffi qffiffi the SS potential, where T (S) is the heavy baryon with light spin Λ → Σ ⃗ σ⃗ c c 2 − 2 ti i 0 1 3 3 SL ¼ (SL ¼ ). pffiffiffi pffiffiffi Λ → Σ 2 2 ⃗ ⃗† c c ti Si qffiffi qffiffi Ξ → Ξ0 1 σ⃗ Finally, we remind the reader that the SU(3)-flavor c c 2 2 2 τ⃗i i 3 − 3 pffiffiffi pffiffiffi structure of the contact-range potential can be broken if Ξ → Ξ 2 2 1 τ⃗ ⃗† the finite-range potential is not SU(3)-flavor symmetric. c c 2 i Si Σ → Σ 2 − 2 ⃗ σ⃗ Whether this happens depends on two factors. The first is c c 3 3 Ti i 1 1 Σ → Σ pffiffi pffiffi ⃗ ⃗ the particular power counting we are using and the order we c c 3 3 Ti Si 1 1 † are considering within the EFT expansion, e.g., if the Σ → Σ pffiffi pffiffi ⃗ ⃗ c c 3 3 Ti Si contact-range interaction is leading, but the exchange of Σ → Σ 2 2 ⃗ ⃗ c c − T Σ pions, kaons, and etas is subleading, the violations of SU(3)- 3 3 i i Ξ0 → Ξ0 2 2 1 σ⃗ flavor symmetry if we ignore kaon and eta exchanges will be c c 3 − 3 2 τ⃗i i 1 1 1 Ξ → Ξ0 pffiffi pffiffi τ⃗ ⃗ subleading. The second factor is that kaon and eta exchanges c c 3 3 2 i Si 1 1 1 † are parametrically small, as was explained in Sec. II D, and a Ξ0 → Ξ pffiffi pffiffi τ⃗ ⃗ c c 3 3 2 i Si more detailed derivation can be found in Appendix B. Ξ → Ξ 2 − 2 1 τ⃗ ⃗ c c 3 3 2 i Σi
D. The one pion exchange potential The OPE potential in momentum space reads be described solely in terms of contact interactions at lowest order. ¯ ð0Þ 0 ¯ 0 hT1T2jV jT T i¼0; ð55Þ Regarding the isospin structure of the OPE potential, we F 1 2 ⃗ ⃗ have that I1 and I2 are the isospin matrices corresponding 2 to vertex 1 and 2. If we have a heavy baryon with isospin ð0Þ g3 a⃗1 · q⃗a⃗2 · q⃗ τ⃗ ¯ 0 ¯ 0 − ¯ ⃗ ⃗ ⃗ i hT1S2jVF jS1T2i¼ R1R2 2 I1 · I2 2 2 ; ð56Þ 1=2 at vertex i, we can simply make the substitution Ii ¼ 2, 2fπ q þ μπ where τ⃗are the Pauli matrices. If the heavy baryon at vertex 1 2 i has isospin 1, we use the J ¼ angular momentum 0 g a⃗1 · q⃗a⃗2 · q⃗ ¯ ð Þ 0 ¯ 0 − ¯ 2 ⃗ ⃗ ⃗ hS1S2jVF jS1S2i¼ R1R2 2 I1 · I2 2 2 ; ð57Þ matrices in isospin space, for which we use the notation Ti; 2fπ q þ μπ ⃗ ⃗ i.e., we make the substitution Ii ¼ Ti. The exact isospin factor for each type of vertex can be found in Table IV. where we have chosen the specific notation above to cover Regarding the spin structure, we note that the spin all the possible combinations. The subscripts 1 and 2 are operators a1 and a2 depend on which is the initial and used to denote the vertices 1 and 2 in the diagrams of Fig. 1. ¯ final spin of the heavy baryons at vertex 1 and 2. If the In the equation above, R1 and R2 are numerical factors initial and final heavy baryons at vertex 1(2) have spin 1=2, which depend on the transition we are considering; see we have a⃗1 ¼ σ⃗1 (a⃗2 ¼ σ⃗2). If the initial and final heavy Table IV (the bar indicates an antibaryon to antibaryon 3 2 ⃗ Σ⃗ ⃗ ⃗ ⃗ ⃗ baryons at vertex 1 (2) have spin = ,wehavea1 ¼ 1 transition). I1 and I2 are isospin matrices, while a1 and a2 ⃗ ⃗ (a⃗2 Σ2), where Σ are the S 3=2 spin matrices. If the spin matrices. For the couplings we have that g2 is the axial ¼ ¼ initial and final heavy baryons at vertex 1 (2) switch from coupling for the SL ¼ 1 heavy baryon, g3 the coupling 1 2 3 2 ⃗ ⃗ involved in T → Sπ transitions, and fπ the pion decay spin = to spin = (or vice versa), then a1 ¼ S1 ⃗ ⃗ constant. μπ is the effective pion mass for the vertices (a⃗2 ¼ S2), where S are special 2 × 4 spin matrices that involved in the particular channel considered. Finally we describe the transition from a different initial to final spin notice that OPE vanishes for the TT¯ molecules, which can (see their definition in Appendix A). As with isospin, the
074026-9 LU, GENG, and VALDERRAMA PHYS. REV. D 99, 074026 (2019) exact spin matrix to use in each type of vertex can be ¯ ð0Þ ⃗ 0 ¯ 0 hT1S2jVF ðrÞjS1T2i checked in Table IV. 2 ¯ ⃗ ⃗ g3 3 ¯ ⃗ ⃗ Regarding the axial couplings g2 and g3, we notice that ¼ −R1R2I1 · I2 C12δ ðr⃗ÞþR1R2I1 · I2½C12W ðrÞ þþ þ þ 6 2 C the value of g3 can be extracted from the Σc → Λc π fπ decay measured by Belle in Ref. [49], yielding g3 ¼ þ S12ðrˆÞWTðrÞ ; ð60Þ 0 973þ0.019 . −0.042 [50] (notice that the previous reference originally uses the convention of Yan et al. [51] to define ¯ ð0Þ ⃗ 0 ¯ 0 hS1S2jVF ðrÞjS1S2i the axial couplings, instead of the one by Cho [46,52] that 2 we employ here, and we have consequently adapted the − ¯ ⃗ ⃗ g2 δ3 ⃗ ¯ ⃗ ⃗ ¼ R1R2I1 · I2 2 C12 ðrÞþR1R2I1 · I2½C12WCðrÞ numbers of Ref. [50] to our convention). In contrast, g2 is 6fπ experimentally unavailable, but on the basis of quark model ˆ pffiffiffi þ S12ðrÞWTðrÞ ; ð61Þ relations one can estimate it to be g2 ¼ − 2g3 ¼ −1.38. If we consider the values of g2 and g3 from the lattice where C12 and S12 are the spin-spin and tensor operators, QCD calculation of Ref. [53], we obtain instead g2 ¼ defined as −0.84 0.20 and g3 ¼ 0.71 0.13, where it is important to mention that they are calculated in the mQ ¼ ∞ limit C12 ¼ a⃗1 · a⃗2; ð62Þ (notice again that our convention for g2 differs from the definition used in Ref. [53] by a sign, which has been taken S12ðrˆÞ¼3a⃗1 · rˆa⃗2 · rˆ − a⃗1 · a⃗2: ð63Þ into account). Had we applied the quark model relations to the lattice QCD value of g3, we would have obtained The OPE potential contains a Dirac-delta contribution g2 ¼ −1.00, which is considerably larger than the lattice which can be reabsorbed into the contact-range potential QCD determination but yet within its error bar. if one wishes to. The spin-spin and tensor pieces of the For the effective pion mass, we have that if the particles potential WC and WT can be written as in the vertex 1 and 2 have the same mass, then μπ ¼ mπ.On 2μ3 −μπ r the other hand if they have different masses (e.g., S1 ¼ B, gi π e 0 0 Δ WC ¼ 2 ; ð64Þ S1 ¼ B ) and the mass splitting is given by , then we have 24πfπ μπr μ2 2 − Δ2 that π ¼ mπ (a relation that assumes heavy, non- relativistic baryons). 2μ3 −μπ r 3 3 gi π e 1 Finally we notice that we can also compute the heavy WT ¼ 2 þ þ 2 ; ð65Þ 24πfπ μπr μπr ðμπrÞ baryon-baryon potential by making the change μ ¯ where gi ¼ g2 or g3 depending on the case and π is the R1R2 → R1R2 ð58Þ effective pion mass for the channel under consideration. The spin-spin piece of the OPE potential is often referred to in Eqs. (56) and (57) and checking the proper values in as “central” OPE, a naming convention that often appears Table IV. in nuclear physics for historical reasons and which per- The most explicit way to construct the potential for one meates the notation WC and WT. Central is used in particular channel is to make use of Table IV, where all the opposition to tensor to convey the idea that the central factors are listed. For instance if we are considering the piece carries no orbital angular momentum (while the Ξ Σ¯ → Ξ0 Σ¯ Ξ → c c c c potential, we can see that it contains a c tensor piece carries two units of orbital angular momen- Ξ0 transition in vertex 1 and a Σ¯ → Σ¯ transition in vertex “ ” c c pc ffiffiffi tum). The term central OPE is indeed convenient and we 1 3 τ⃗ 2 ⃗ 2. If we use Table IV we find R1 ¼ffiffiffi = , I1 ¼ 1= , a1 ¼ will use it in what follows (instead of the more accurate ⃗ ¯ p ⃗ ⃗† S1 for vertex 1 and R2 ¼ 1= 3, I2 ¼ T1, a⃗2 ¼ S2 for spin-spin OPE). We notice that the OPE potential also vertex 2. Putting the pieces together, the potential reads contains a contact-range contribution, which is mostly harmless: it can be reabsorbed into the EFT contact-range 2 ⃗ ⃗ ⃗† 0 1 g τ⃗1 · T2 S1 · q⃗S · q⃗ contribution to the potential by a redefinition of the Ξ Σ¯ ð Þ Ξ0 Σ¯ − 3 2 h c cjVF j c ci¼ 2 2 2 ; ð59Þ couplings. Hence it can be simply ignored. 3 2fπ 2 q þ μπ
2 2 2 0 where μπ ¼mπ −Δ , with Δ ≃ ðmðΞcÞ − mðΞcÞÞ ≃ ðmðΣcÞ − 2. Partial wave projection of the OPE potential Σ ≃ 70 μ ∼ 120 mð cÞÞ MeV and π MeV. The other cases can We consider now the projection of the coordinate space be obtained analogously. potential into the partial wave basis. For that we work with baryon-antibaryon states with well-defined total angular 1. The OPE potential in coordinate space momentum and parity JP. If the total strangeness of the If we Fourier-transform the potential into coordinate baryon-antibaryon state is zero, we will consider states with space we obtain well-defined C-parity (for neutral systems) or G-parity (if
074026-10 HEAVY BARYON-ANTIBARYON MOLECULES IN EFFECTIVE … PHYS. REV. D 99, 074026 (2019) the system is not electrically neutral). Also, we will only irreducible diagrams enter at order Q2 in the chiral consider states that contain an S-wave, as they are the more expansion. likely to form a bound state. If we use the spectroscopic 2Sþ1 notation LJ to denote the partial waves, we have the A. The central potential following combinations: The perturbative nature of the central piece of OPE can ¯ − 1 be determined from the comparison of tree-level versus jB B6ð0 Þi¼fS0g; ð66Þ Q once-iterated central OPE, i.e., V and VG0V in operator form. This type of comparison was made in Ref. [54] in the ¯ 1− 3 3 jBQB6ð Þi¼fS1; D1g; ð67Þ context of nucleon-nucleon scattering.7 Here we are merely adapting it to the particular case of the heavy baryon- ¯ 1− 3 3 5 jBQB6ð Þi¼fS1; D1; D1g; ð68Þ antibaryon system. The ratio of iterated vs tree-level OPE can be expressed as a ratio of scales, ¯ − 3 5 5 5 jBQB6ð2 Þi¼fD2; S2; D2; G2g; ð69Þ hpjVG0Vjpi Q ¯ 0− 1 5 ¼ ; ð75Þ jB6B6ð Þi ¼ f S0; D0g; ð70Þ hpjVjpi ΛC
¯ − 3 3 7 7 where Q is a light scale (either the external momentum p or jB6B6ð1 Þi¼fS1; D1; D1; G1g; ð71Þ the pion mass mπ) and ΛC is a scale that characterizes ¯ − 1 5 5 5 central OPE. The evaluation of this ratio at p ¼ 0 leaves the jB6B6ð2 Þi¼fD2; S2; D2; G2g; ð72Þ pion mass as the only light scale left, in which case we ¯ − 3 3 7 7 7 7 obtain the following value for the central scale: jB6B6ð3 Þi¼fD3; G3; S3; D3; G3; I3g; ð73Þ 1 24π 2 Λ fπ with BQ ¼ B3¯ or B6. The calculation of the matrix elements C ¼ ¯ 2 ; ð76Þ jστj μjR1R2jg is in general straightforward, where we refer to Appendix A i for the details. The result of these calculations is that the with μ the reduced mass of the system, σ and τ the C12 and S12ðrˆÞ operators can be expressed as matrices, evaluation of the spin and isospin operators corresponding which we denote with the C12 and S12 notation. With this in to the particular case under consideration and where R1 and 0 ¯ mind for r> we write the OPE potential as R2 can be found in Table IV. For the charm and bottom sectors the values are, respectively, ð0Þ ¯ ⃗ ⃗ V ðrÞ¼R1R2I1 · I2½C12W ðrÞþS12W ðrÞ ; ð74Þ F C T 1060 Λ MeV CðQ ¼ cÞ¼ στ ¯ 2 ; ð77Þ where the dimension of the matrices is set by the number of j jjR1R2jgi partial waves. The explicit matrices that apply in each case can be found in Appendix A, where it is also explained how 450 MeV Λ Q b ; Cð ¼ Þ¼ στ ¯ 2 ð78Þ they are calculated. j jjR1R2jgi
IV. HOW TO COUNT THE ONE PION which depend on the value of the couplings g2 and g3. EXCHANGE POTENTIAL The discussion about the values of the axial couplings, and in particular g2, is important because it can change the The LO EFT heavy baryon-antibaryon potential can in ¯ ¯ value of ΛC by a large factor. For the TS and SS molecules, principle contain a contact- and a finite-range piece, where the value of ΛC in the charm sector is the latter is the well-known OPE potential. While there is þ100 no a priori way to determine if the contact-range potential ¯ 1120 MeV ΛTS ∼ −40 C ðQ ¼ cÞ ¯ ; ð79Þ is perturbative, this is not the case for the OPE potential jστjjR1R2j where there exists a series of theoretical developments to ¯ 560 − 1500 MeV evaluate its strength. In this section we will check how these ΛSS ∼ ð Þ C ðQ ¼ cÞ ¯ ; ð80Þ ideas apply to the central and tensor pieces of the OPE jστjjR1R2j potential. Before starting the discussion it is important to stress that we make a very explicit distinction ΛSS¯ where C can change almost by a factor of 3 owing to the between iterated OPE (or reducible multipion exchange) uncertainty of g2 (notice that instead of a number with an and irreducible multipion exchanges. The former is merely the outcome of iterating the EFT potential within the 7Recently a more sophisticated method for determining the Schrödinger or Lippmann-Schwinger equations while the perturbativeness of OPE has been developed in Ref. [55] for latter is a genuine contribution to the EFT potential, though peripheral waves with L ≥ 1. Unfortunately it has not been a subleading one: the lowest order two pion exchange extended yet to S-waves.
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Λ Λ TABLE V. The central scale C (in units of MeV) for a few C varies considerably as a consequence of the uncertainty selected heavy baryon-antibaryon molecules. For momenta above of the axial coupling g2. In particular if the absolute value of Λ this scale, p> C, the central force becomes nonperturbative. the axial coupling jg2j is on the high end, i.e., the value The scale Λ is inversely proportional to στ, reaching its C g2 ¼ −1.38 deduced from the quark model, central minimum for the channels with the lowest total spin and isospin OPE will be important for certain molecules in the hidden and growing quickly for other configurations. The selection charm sector. In the bottom sector the situation is more includes the heavy baryon-antibaryon states for which clear: central OPE will be nonperturbative for the afore- Λ < 0.5 GeV, most of which are of the Σð ÞΣ¯ ð Þ type. In C Q Q mentioned Σ Σ¯ and Σ Σ¯ molecules. Finally for having Ξð0= ÞΞ¯ ð0= Þ Q Q Q Q addition we include the Q Q molecule for which the a comparison with a well-known state, we mention that central force is strongest and the antitriplet-sextet molecules. the central scale for OPE in the two-nucleon system The uncertainty in the antitriplet-sextet case comes from the axial is Λ ≃ 590 MeV. 0 973þ0.019 C coupling, which is taken to be g3 ¼ . −0.042 in the charm sector [50] and g3 ¼ 0.71 0.13 in the bottom one [53]. For the sextet-sextet case in the charm sector, the axial coupling is not B. The tensor potential ∼ 0 84–1 38 well determined and we take it in the range jg2j . . , The tensor piece of the OPE potential requires a more while in the bottom sector we have g2 ¼ −0.84 0.20 [53]. For involved analysis. A direct comparison of V and VG0V is comparison, in the deuteron channel of the two-nucleon system, not possible. The reason is that the iteration of the tensor the central scale is Λ ≃ 590 MeV. C piece of OPE diverges; see, for instance, Refs. [32,55] for
Channel I Sign ΛC Channel I Sign ΛC a detailed explanation. Thus we must resort to a method that does not involve the direct evaluation of the iterated Ξ Ξ¯ 0 ð0− Þ 0 ∓ 720þ70 Ξ Ξ¯ 0 ð0− Þ 0 ∓ 580þ290 c c −30 b b −170 tensor OPE. Λ Σ¯ 0− 580þ50 Λ Σ¯ 0− 450þ220 c cð Þ 1 −20 b bð Þ 1 −130 The type of power-law behavior of the tensor OPE Ξ0 Ξ¯ 0 0−þ − – Ξ0 Ξ¯ 0 0−þ − 620þ450 c cð Þ 0 530 1420 b bð Þ 0 −210 potential is analogous to the behavior of a few physical ¯ −þ − – ¯ −þ − þ170 ΣcΣcð0 Þ 0 210 560 ΣbΣbð0 Þ 0 240−80 systems studied in atomic physics. The potential between two dipoles is of the 1=r3 type, just like the tensor force. Ξ Ξ¯ 0 1−þ 0 − 580–1560 Ξ Ξ¯ 0 1−þ 0 − 680þ490 c cð Þ b bð Þ −230 The failure of standard perturbation theory for these Σ Σ¯ 1−þ 0 − 230–610 Σ Σ¯ 1−þ 0 − 260þ190 c cð Þ b bð Þ −90 systems is well known in atomic physics, where alternative Σ Σ¯ 1−− − – Σ Σ¯ 1−− − 320þ230 c cð Þ 0 280 750 b bð Þ 0 −110 techniques have been developed to deal with this type of Σ Σ¯ 2−− – Σ Σ¯ 2−− 320þ230 c cð Þ 0 þ 280 750 b bð Þ 0 þ −110 potential. The work of Cavagnero [56] explains that the Ξ Ξ¯ ð0−þÞ 0 − 410–1100 Ξ Ξ¯ ð0−þÞ 0 − 490þ350 divergences of perturbation theory in these types of systems c c b b −170 is similar to the role of secular perturbations in classical Σ Σ¯ ð0−þÞ 0 − 160–440 Σ Σ¯ ð0−þÞ 0 − 190þ140 c c b b −70 mechanics, i.e., a type of perturbation that is small at short Σ Σ¯ 1−− 0 − 220–590 Σ Σ¯ 1−− 0 − 260þ190 c cð Þ b bð Þ −90 timescales but ends up diverging at large timescales. The Σ Σ¯ 3−− 0 270–740 Σ Σ¯ 3−− 0 320þ230 8 c cð Þ þ b bð Þ þ −110 solution is to redefine (or, loosely speaking, renormalize ) some quantity in order to obtain a finite result again. For the perturbative series of the 1=r3 potentials, the quantity we error, we have simply indicated a range of possible values). renormalize is the angular momentum. The zeroth order In the bottom sector it is instead more advisable to use the term in the perturbative expansion of the wave function lattice QCD determination for g2 and g3, leading to is now
þ440 ¯ 900 MeV ð0Þ Jνðka3ÞðkrÞ ΛTS ∼ −260 Ψ ðr; kÞ¼ pffiffiffi ; ð83Þ C ðQ ¼ bÞ ¯ ; ð81Þ l jστjjR1R2j r
440 660þ Jlþ1=2ðkrÞ ¯ −220 MeV instead of the standard pffiffi , where Jν z refers to the ΛSS Q b ∼ : r ð Þ C ð ¼ Þ στ ¯ ð82Þ j jjR1R2j Bessel function of order ν. In the expression above, ν is the renormalized angular momentum, which happens to be a The previous values have to be combined with the function of the momentum k and a length scale a3 that is στ ¯ j jjR1R2j factor. The maximum value of this factor related to the strength of the potential (it will be defined happens for the channels with lowest spin and isospin. later). The secular series is built not only by adding higher In Table V we list the scale Λ for a few representative C order terms but also by making νðxÞ depend on κ ¼ ka3.If heavy baryon-antibaryon molecules. In general ΛC ∼ 0 5 . GeV (if not harder) in most cases, which means that 8 we expect central OPE to be perturbative. The exceptions We simply adopt the terminology in use in the field of atomic 0−þ Σ Σ¯ 0−þ 1−− Σ Σ¯ physics for these redefinitions in secular perturbation theory, are the isoscalar Q Q and , Q Q molecules, though it does not exactly correspond with the standard meaning at least in the bottom sector. In the charm sector the scale of renormalization.
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0 L2 1 … 1 we switch off the potential and take ka3 ¼ ,wehave j ¼ diagðl1ðl1 þ Þ; ;lNðlN þ ÞÞ: ð86Þ 1 νð0Þ¼l þ 2 and we recover the free wave function. For small enough values of κ we expect νðκÞ to be expansible in The reduced wave function uk;j is an N × N matrix, where powers of x, i.e., to be perturbative. By reexpanding the column j represents a solution that behaves as a free wave secular series and the renormalized angular momentum we with angular momentum lj when we take a3 → 0. The can recover the original perturbative series. However, the solution of the Schrödinger equation is a linear combina- interesting feature of the series above is that we can tion of the functions ξ and η: κ ν κ determine the values of for which ð Þ is analytic. X When νðκÞ is not analytic, it does not admit a power series uk;jðrÞ¼ ½αl ξl ðr; kÞþβl ηl ðr; kÞ ; ð87Þ in κ anymore. This in turn means that there is no way to j j j j fljg rearrange the secular series into the standard perturbative series, leading to its failure. where we sum over the possible values of the angular 1 3 For the =r potential, which is equivalent to the tensor momenta and with ξ and η N-component vectors that can be 1 force for distances mπr< , the secular series has been written as sums of Bessel functions: studied in detail by Gao [57] for the uncoupled channel case. Birse [45] extended the previous techniques for the X∞ pffiffiffi coupled channel case and particularized them for the ξ ðr; kÞ¼ b ðν Þ rJ ν ðkrÞ; ð88Þ lj m lj mþ lj nucleon-nucleon system. In a previous publication by m¼−∞ one of the authors [32], the analysis of Birse was applied X∞ ffiffiffi to the heavy meson-antimeson system. In this work we m p η ðr; kÞ¼ ð−1Þ b− ðν Þ rJ− −ν ðkrÞ; ð89Þ lj m lj m lj extend it to the heavy baryon-antibaryon system. m¼−∞ We will consider the tensor force in the limit mπ → 0, for which the OPE potential can be written as ν ’ κ ν ν κ where the lj s are functions of ¼ ka3, i.e., lj ¼ lj ð Þ. The ν ’s are the renormalized angular momenta that we a3 lj 2μVðrÞ¼ S ; ð84Þ r3 j previously introduced in Eq. (83). In turn the expansion of the reduced wave functions ξ and η in Eqs. (88) and (89) is where the potential is a matrix in the coupled channel space simply the extension of Eq. (83) to arbitrary orders.9 We ξ η and Sj is the tensor operator (in matrix form, as written in have N different solutions for and that we have labeled κ 0 Sec. III D), with j referring to the total angular momentum. with the subscript lj to indicate that for ¼ they behave We have that μ is the reduced mass of the heavy baryon- as a free wave of angular momentum lj. The recursive antibaryon system and a3 is the length scale that determines b ν relation from which one can compute mð lj Þ can be found the strength of the tensor piece of the potential. The in Ref. [45] but is of no concern if we are only interested in potential in this limit is amenable to the secular perturbative ν ’ κ 0 b ν 0 the li s. For ¼ , only the mð lj Þ coefficient for m ¼ series developed in Refs. [32,45,57]. The corrections survives. mπ stemming from the finite value of were considered in The renormalized angular momenta ν ¼ ν ðκÞ (with Ref. [32] and will be discussed later on in this section. li li κ ¼ ka3) can be calculated as follows. First we define the following N × N matrix: C. The renormalized angular momentum Now we explain how the secular perturbation theory κ2 F ν κ ≡ f ν − R ν − R −ν looks like and most importantly, how to calculate the jð ; Þ jð Þ ν ½ 1ð Þ 1ð Þ ; ð90Þ renormalized angular momenta ν. We begin by writing the reduced Schrödinger equation in coupled channels (the which depends on two other matrices, fjðνÞ and R1ðνÞ; uncoupled channel can be found in Ref. [32]) for the fjðνÞ is a diagonal matrix defined as 3 particular case of a pure 1=r potential, f ν 1 2 1 2 jð Þ 2 2 2 ¼ diag ν − l1 þ ;…;ν − l þ ; ð91Þ a3 L 2ν 2 N 2 −u00 þ S þ j u ðrÞ¼k2u ðrÞ; ð85Þ k;j j r3 r2 k;j k;j while R1ðνÞ is given by the recursive relation where we are considering N angular momentum channels. → 0 R ν f ν − κ2S R ν S −1 Notice that we have taken the chiral limit mπ , which nð Þ¼½ jðn þ Þ j nþ1ð Þ j ; ð92Þ means that only the tensor piece of the OPE potential survives; see Eq. (84). In the equation above, Sj is the 9We note that Eq. (83) is written in terms of the standard wave tensor operator matrix, while Lj is a diagonal matrix 2 function, while Eqs. (88) and (89) use the reduced wave function representing the angular momentum operator L⃗: instead.
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which can be accurately solved with between 20 and 30 TABLE VI. Reduced critical momenta κc for the different S- 0 ¯ iterations [45] (that is, one takes RN ¼ 0 for large enough wave heavy baryon-antibaryon systems B B. – N, e.g., 20 30, and solves the recursion relation back- κ ν ν κ Channel c wards). Once we have the matrix Fj, we obtain li ¼ li ð Þ ¯ − ¯ − by finding the zeros of B3B6ð0 Þ=B6B6ð0 Þ … ¯ − ¯ − B3B6ð1 Þ=B6B6ð1 Þ 0.6835 F ν κ 0 det ð jð ; ÞÞ ¼ : ð93Þ ¯ − B3B6ð1 Þ 1.412 ¯ 2− This equation admits N solutions, one for each value of the B3B6ð Þ 1.934 ¯ −þ angular momentum. For κ ¼ 0 these solutions behave as B6B6ð1 Þ 0.8533 ¯ −− B6B6ð1 Þ 0.7264 1 ¯ −þ ν κ 0 B6B6ð2 Þ 0.5784 li ð ¼ Þ¼li þ ; ð94Þ 2 ¯ −− B6B6ð2 Þ 0.6998 1 … κ ν κ ¯ 0−þ 0.6378 with i ¼ ; ;N.As increases li ð Þ moves slowly B6B6ð Þ ν ¯ 1−− downwards. Once we reach li ¼ li at the critical value B6B6ð Þ 0.6674 κ κ ν ¯ 2−þ ¼ c, we have that li splits into the complex conjugate B6B6ð Þ 0.6833 solutions ν κ l iρ κ . This is a nonanalyticity ¯ −− li ð Þ¼ i li ð Þ B6B6ð3 Þ 0.5922 ν κ which marks the point above which li ð Þ cannot be expressed as a perturbative series. This in turn defines κ κ ν κ ¯ c, the critical value of for which there is a li ð Þ that the B6B6 system we will be computing the critical values of κ ν becomes nonanalytic in . Usually the first lj to split is the the matrix one that corresponds to the smallest angular momentum 3 and also the one that determines the breakdown of the S SD − SE j ¼ j j : ð97Þ perturbative series. 4 For computing the critical momenta we need first the D. Critical momenta matrix elements of the tensor operator in the channel under ¯ ¯ ¯ The critical κ values for which the convergence criterion consideration. For the B3¯ B3¯ , B6B6, and B6B6 molecules c fails are listed in Table VI for the different possible heavy this is trivial: first we take the tensor force matrix Sj, which can be found in Eq. (138) and Eqs. (A147)–(A150) of baryon-antibaryon states that contain an S-wave. The Appendix A, and then we plug this matrix into Eq. (92), previous values have been obtained under the assumption that the effective pion mass can be taken to be zero. The from which finally we solve Eq. (93) to obtain κc. For the ¯ effect of finite pion mass was considered in Ref. [32], B3B molecules the procedure is the same, with the tensor 6 where it was found that it increases the range of momenta force matrices defined in Eqs. (A167) and (A168). The ¯ where the tensor part of OPE is perturbative by the B6B 6 molecules are the most complicated because they following factor, contain both a direct and exchange tensor operators, which ¯ → ¯ ¯ → ¯ mediate the B6B6 B6B6 and B6B6 B6B6 potential, þmπ Rc κcðmπÞ¼κce ; ð98Þ respectively. In addition the effective pion masses are different for the direct and exchange tensor operators. where Rc is the radius below which we do not expect the Here we ignore this effect: in the present calculation we OPE potential to be valid. The value of this radius is rather are making the approximation that HQSS is exact and ambiguous. In Ref. [32] the estimation Rc ¼ 0.5–0.8 fm therefore there is no mass splitting between the B6 and B6 was proposed, yielding heavy baryons. However, the length scale a3 is different for the direct and exchange operators, i.e., κcðmπÞ ≃ 1.5κc: ð99Þ
D E a3 D a3 E Higher values might be more appropriate indeed, but here 2μV ¯ ðrÞ¼ S þ S : ð95Þ B6B6 r3 j r3 j we will stick to this value. To obtain the critical momenta we multiply κcðmπÞ by It happens that both scales are proportional to each other, the relation kc ¼ κcðmπÞ=ja3j, where a3 is the tensor length → 0 3 scale. If we match the mπr limit of the OPE potential E − D to the 1=r3 form we have used to derive κ , we find that a3 ¼ 4 a3 ; ð96Þ c μ 2 as can be checked by inspecting the coupled-channel form ¯ τ gi ja3j¼jR1R2 j 2 ; ð100Þ of the potential in Eq. (A106) from Appendix A. Thus in 4πfπ
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TABLE VII. The tensor scale ΛT (in units of MeV) for a series of heavy baryon-antibaryon molecules. For momenta above this scale, p>ΛT , the tensor force becomes nonperturbative. The tensor scale depends on the pion mass: ΛT ð0Þ and ΛT ðmπÞ represent the value in the chiral limit and physical pion mass, respectively. The values of ΛT are shown for the antitriplet-sextet and sextet-sextet molecules. In Σð ÞΣ¯ ð Þ the latter case we concentrate on the isoscalar Q Q molecules, for which pion exchanges are stronger (for the isovector case, the value Λ Ξð0= ÞΞ¯ ð0= Þ of T is twice that of the isoscalar case). For the Q Q molecules we only show the channel in which the tensor force is strongest. For comparison purposes, in the deuteron channel of the two-nucleon system the tensor scale is ΛT ð0Þ¼66 MeV and ΛT ðmπÞ¼99 MeV, respectively.
Channel I ΛT ð0Þ ΛT ðmπÞ Channel I ΛT ð0Þ ΛT ðmπÞ Ξ Ξ¯ 0 1− 246þ23 368þ34 Ξ Ξ¯ 0 1− 200þ100 300þ150 c cð Þ 0 −10 −14 b bð Þ 0 −60 −90 ¯ − þ18 þ27 ¯ − þ80 þ110 ΛcΣcð1 Þ 1 196−7 295−11 ΛbΣbð1 Þ 1 150−40 230−70 Ξ0 Ξ¯ 0 1−− – – Ξ0 Ξ¯ 0 1−− 210þ160 320þ230 c cð Þ 0 180 490 270 740 b bð Þ 0 −70 −110 ¯ −− – – ¯ −− þ60 þ90 ΣcΣcð1 Þ 070190 110 290 ΣbΣbð1 Þ 0 80−30 120−40 Ξ Ξ¯ 0 2−þ – – Ξ Ξ¯ 0 2−þ 180þ130 270þ200 c cð Þ 0 150 410 230 610 b bð Þ 0 −60 −90 Σ Σ¯ 1−þ – – Σ Σ¯ 1−þ 100þ80 150þ110 c cð Þ 090240 130 350 b bð Þ 0 −30 −50 Σ Σ¯ 1−− – – Σ Σ¯ 1−− 90þ60 130þ90 c cð Þ 070200 110 300 b bð Þ 0 −30 −40 Σ Σ¯ 2−þ – – Σ Σ¯ 2−þ 70þ50 100þ80 c cð Þ 060160 90 240 b bð Þ 0 −20 −30 Σ Σ¯ 2−− – – Σ Σ¯ 2−− 80þ60 130þ90 c cð Þ 070190 110 290 b bð Þ 0 −30 −50 Ξ Ξ¯ 0 3−− – – Ξ Ξ¯ 0 3−− 190þ130 280þ200 c cð Þ 0 160 420 230 630 b bð Þ 0 −70 −100 Σ Σ¯ 0−þ – – Σ Σ¯ 0−þ 80þ50 110þ80 c cð Þ 060170 100 260 b bð Þ 0 −30 −40 Σ Σ¯ 1−− – – Σ Σ¯ 1−− 80þ60 120þ90 c cð Þ 070180 100 270 b bð Þ 0 −30 −40 Σ Σ¯ 2−þ – – Σ Σ¯ 2−þ 80þ60 120þ90 c cð Þ 070190 100 280 b bð Þ 0 −30 −40 Σ Σ¯ 3−− – – Σ Σ¯ 3−− 70þ50 100þ80 c cð Þ 060160 90 240 b bð Þ 0 −20 −40
τ ⃗ ⃗ þ80 where ¼ T1 · T2 and gi ¼ g2, g3 depending on whether ¯ 100 MeV ¯ ¯ ΛSS ≃ κ −40 we are considering the SS or the TS potential. The factors T ðmπ;Q¼ bÞ cðmπÞ ¯ τ ; ð105Þ ¯ jR1R2 j R1 and R2 and the proper isospin operator to use can be ¯ checked in Table IV. For the B6B6 case we will use the where we have used the lattice QCD values of g2 and g3 factors corresponding to the direct channels, i.e., B6 → B6 [53]. These scales look rather soft at first sight but the → ¯ and B6 B6, in agreement with the convention that we factors R1, R2, and τ will increase the values of ΛT have used in Eq. (97) for writing their tensor matrices. considerably in most cases. A few representative values From the previous, we can define the tensor scale as of ΛT are compiled in Table VII both for the chiral limit and the physical pion mass. In general we find that tensor OPE is 2 κðmπÞ κðmπÞ 4πfπ Ξ Ξ¯ 0 Λ considerably stronger than central OPE. For the Q Q and TðmπÞ¼ ¼ ¯ 2 ; ð101Þ ja3j R1R2τ μg ¯ j j i ΛQΣQ molecules we find that ΛT is markedly softer than ΛC, and in the bottom sector the tensor force probably requires a which is useful because it allows a direct comparison with nonperturbative treatment. For the isoscalar Σ Σ¯ , Σ Σ¯ , the central scale Λ that we defined in Eq. (76).Ifwe Q Q Q Q C Σ Σ¯ molecules the tensor scale Λ is moderately soft, particularize for the TS¯ and SS¯ in the charm sector, Q Q T particularly in the bottom sector. We notice that the same þ18 ¯ 186 MeV comments are also valid in the two-nucleon system, in which ΛTS ≃ κ −8 Λ 0 66 Λ T ðmπ;Q¼ cÞ cðmπÞ ¯ ; ð102Þ Tð Þ¼ MeV in the chiral limit [45] and ðmπÞ¼ jR1R2τj 99 MeV for the physical pion mass.
¯ 90 − 250 MeV ΛSS ≃ κ 3 ðmπ;Q¼ cÞ cðmπÞ ¯ ; ð103Þ jR1R2τj V. POWER COUNTING FOR HEAVY BARYON MOLECULES while for the bottom sector we obtain In this section we discuss the different possible power
þ80 counting rules for the heavy baryon-antibaryon states. We ¯ 140 MeV ΛTS ≃ κ −40 are interested in the case where there are bound states. This T ðmπ;Q¼ bÞ cðmπÞ ¯ ; ð104Þ jR1R2τj excludes NDA, for which
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TABLE VIII. Possible power countings for the heavy baryon-antibaryon system: NDA and the three scenarios (a), (b), (c) that we consider in Sec. V. Q refers to the soft scales in each power counting, which include the momenta p of the heavy baryons and pions, the pffiffiffiffiffiffiffiffiffiffi mass mπ of the pions, the binding momentum 2μB2 of a heavy baryon-antibaryon bound state (with μ the reduced mass and B2 the binding energy) and the central and tensor scales ΛC and ΛT defined in Eqs. (76) and (101) and listed in Tables Vand VII for a series of heavy baryon-antibaryon systems. The LO and NLO columns indicate the counting Qν of the leading order and first subleading correction. In the VLO and VNLO columns we write the contributions to the EFT potential in each case. C and Dðp2 þ p02Þ refer to a contact-range potential without derivatives and with two derivatives of the heavy baryon field, respectively. VOPE is the OPE potential, VOPEðCÞ its central piece, and VOPEðTÞ its tensor piece. Finally VTPE refers to the two pion exchange potential (i.e., irreducible diagrams containing two pions), which we have not considered in this work.
Power counting Q LO VLO NLO VNLO
0 2 2 2 NDA p, mπ C, V 0 ffiffiffiffiffiffiffiffiffiffi Q OPE Q Dðp þ p Þ, VTPE p −1 0 2 02 (a) p, mπ, 2μB2 Q C Q Dðp þ p Þ, VOPE Λ −1 0 (b) p, mπ, C Q VOPEðCÞ Q C, VOPEðTÞ Λ −1 3=2 2 02 (c) p, mπ, T Q C, VOPE Q Dðp þ p Þ
LO ⃗ ∼ 0 LO ∼ 0 VC ðqÞ Q ;VF Q ; ð106Þ short-range nature. Within the EFT language this amounts to the promotion of the contact-range potential from Q0 to as this counting leads to purely perturbative heavy baryon- Q−1. Within this power counting, the leading order antibaryon interactions. The existence of bound states (LO ≡ Q−1 in this case) potential will be composed of requires that at least one of the components of the potential contact terms, while the next-to-leading order potential −1 is promoted to Q ; see the discussion in Sec. II B for (NLO ≡ Q0) will contain the OPE potential plus a few details. There are different choices depending on which additional contact interactions: piece of the interaction is promoted. We will consider three −1 scenarios: VLO ¼ Vð Þ; ð107Þ (a) promotion of the contact terms, C
(b) promotion of central OPE, and NLO ð0Þ (c) promotion of tensor OPE. V ¼ VC þ VOPE: ð108Þ Each scenario represents a different binding mechanism: (a) short range, We do not have to promote all the possible contact (b) long range, and interactions that we obtain from the heavy-light spin (c) a combination of both, decomposition: in general a subset of it will be enough. where we notice that (c) is not obvious but a consequence There is one important detail with this counting. If we of the technicalities of power counting, as we will explain. consider the S-wave contact-range interactions in EFT, they We present an overview of these scenarios in Table VIII. admit the momentum expansion But we stress that the discussion here will be theoretical: in 0 02 2 … the absence of experimental data it is not particularly useful hp jVCjpi¼C þ Dðp þ p Þþ ; ð109Þ to consider the subleading orders of the EFT expansion. The exploration in this section provides information about where the dots denote couplings involving more derivatives the theoretical uncertainties that are to be expected from a of the baryon fields. Here we use C and D as a generic LO calculation in each scenario, notation for the couplings of a contact-range potential with (a) (Q=M), no derivatives (C) or with two derivatives (D). The naive (b) (Q=M), expectation for the scaling of the C and D couplings is 5 2 (c) ðQ=MÞ , 1 1 ∼ ∼ where we will explain in detail how we obtain these C 2 ;D4 : ð110Þ uncertainties and also what is the general form of the first M M subleading corrections, which can actually be found in But if we promote the coupling C to LO, the coupling D Table VIII. For a more in-depth discussion of the power must also be promoted [39–41]: counting of heavy meson-antimeson in particular and two- body systems in general we refer the reader to Refs. [32,44]. 1 1 C ∼ ;D∼ : ð111Þ MQ M2Q2 A. Counting with perturbative pions The first possibility—scenario (a)—is that the binding As a consequence the ordering of the contact-range mechanism for heavy baryon-antibaryon molecules is of a potential will be
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0 LO hp jVC jpi¼C; ð112Þ C. Counting with nonperturbative tensor OPE The third possibility—scenario (c)—arises when tensor 0 NLO 02 2 hp jVC jpi¼Dðp þ p Þ: ð113Þ OPE is nonperturbative. This is the most involved of the three power countings considered. Tensor OPE is a singular That is, the NLO potential will contain a contact-range potential, which means that it diverges as fast as (or faster 2 interaction with two derivatives on the baryon fields. As a than) 1=r for r → 0. Singular potentials in general lead to consequence if we promote a particular C coupling to Q−1, nontrivial consequences in EFT [58–63]. The tensor force the corresponding derivative coupling with D will be is not only singular, but also attractive for the case at hand: promoted to Q0. We notice that in this work we have for an S-wave heavy baryon-antibaryon state that mixes with a D-wave, there is always a configuration for which not explicitly considered a contact-range potential with 10 derivatives. The take-home message is that in this scenario the tensor force is attractive. For attractive singular the theoretical uncertainty of the calculations is Q=M potentials short-range physics is enhanced: the nonpertur- because the first correction to a LO calculation is sup- bative treatment of attractive singular potentials requires the pressed by one order in the EFT expansion. inclusion of a contact-range interaction at LO [60,61]. The application of these ideas for a heavy baryon- antibaryon S-wave molecule implies that a nonperturbative B. Counting with nonperturbative central OPE tensor force requires a nonperturbative contact potential. As The second possibility is that the binding mechanism a consequence, the LO potential will be depends also on the attraction provided by the central OPE. We can distinguish two cases: (i) the binding depends on LO −1 V ¼ VOPE þ VC ; ð119Þ central OPE alone, i.e., scenario (b), and (ii) the binding depends on the interplay of the contact terms and central ð−1Þ 11 with VC the lowest order contact-range potential. OPE, i.e., scenario (a þ b). The counting of the contacts will be modified as — — In the first case scenario (a) we have a relatively follows [44,45]: simple power counting in which 1 1 LO C ∼ ;D∼ ; ð120Þ V ¼ VOPEðCÞ; ð114Þ MQ M7=2Q1=2 where OPEðCÞ means the central piece of OPE. The or equivalently we can write NLO potential contains the tensor OPE and the contact interactions 0 ð−1Þ hp jVC jpi¼C; ð121Þ NLO ð0Þ V ¼ VOPEðTÞ þ V ; ð115Þ C 0 ð3=2Þ 02 2 hp jVC jpi¼Dðp þ p Þ; ð122Þ where where the contacts with derivatives get promoted by half an 0 ð0Þ order. Thus the theoretical uncertainty of a LO calculation hp jV jpi¼C0: ð116Þ C is ðQ=MÞ5=2 (the first subleading correction, a derivative 5=2 Contacts with 2n derivatives on the baryon fields will enter contact interaction, enters Q orders after LO), which is 2 considerably better than for the other scenarios. at order Q n. In this scenario the relative uncertainty of a The previous analysis is a simplification though: tensor LO calculation is Q=M because the first correction to the forces mix channels with different orbital angular momen- EFT potential enters at NLO. tum, which might lead to complications in certain cases (in The second case—scenario (a þ b)—is identical to the particular the power counting of contact interactions mix- power counting of scenario (a) except for the fact that we ing partial waves). We have not addressed these problems include OPE in the LO: here: they depend on the particular system under consid- LO LO eration and the aim of the present discussion is to provide V ¼ VC þ VOPEðCÞ; ð117Þ
10 NLO NLO This can be seen by inspecting the partial wave projection of V ¼ VC þ VOPEðTÞ; ð118Þ the tensor operator S12, which can be found in Appendix A.It happens that for these matrices there is always at least one LO positive and one negative eigenvalue. where VOPEðTÞ is the tensor piece of OPE, while VC and 11 NLO We have simply included the full OPE potential in LO VC are the contact-range potentials of Eqs. (112) and because the addition of central OPE does not further modify the (113). The uncertainty of the LO calculation is Q=M. power counting induced by tensor OPE.
074026-17 LU, GENG, and VALDERRAMA PHYS. REV. D 99, 074026 (2019) an overview of power counting rather than a detailed others). Meanwhile the authors of Ref. [68] used the OPE account. potential instead to make qualitative predictions about 1 probable hadronic molecules, including a possible I ¼ 2, VI. PREDICTING HEAVY BARYON MOLECULES P 3− ¯ J ¼ 2 D Σc molecule. Finally EFT and EFT-inspired In this section we investigate the question of whether we works have explained the properties of shallow molecular can predict heavy baryon molecules. The answer to this states solely on the basis of short-range interactions question depends on which is the binding mechanism. If the [31,36,69], without making explicit assumptions about binding mechanism is of a short-range nature, the pre- the binding mechanism. diction of bound states will rely on phenomenology. Within In this section we will examine the short- and long-range the EFT framework this is illustrated by the fact that the binding mechanisms for heavy baryon-antibaryon mole- contact-range couplings are free parameters. If there is no cules. The most obvious short-range mechanism is the preexisting experimental information about the heavy saturation of the EFT contact-range couplings from scalar baryon-antibaryon system, we will have to determine the and vector meson exchange, while the most important long- contact-range couplings by matching to a phenomenologi- range mechanism is the OPE potential. Now we will cal model. Conversely if the binding mechanism is of a explain these binding mechanisms in detail. long-range nature, the prediction of bound states is possible Λ Σ¯ within EFT. Examples are the c1 c [29] and DDs0=D Ds1 A. Short-range binding dynamics [64] systems, which interact via a long-range Yukawa potential that is strong enough to bind. This is not the First we explore the short-range dynamics, in particular standard situation though and more often than not we will scalar and vector meson exchange. For taking this effect need phenomenological input. into account we saturate the contact-range couplings of the At this point it is interesting to notice the relation EFT with the exchange of a meson with mass mS of the ∼ between power counting and the predictability of heavy order of the hard scale of the EFT (mS M); see Ref. [70] baryon-antibaryon molecules. In Sec. V we proposed three for a detailed exposition of this idea. For this we expand the ⃗2 ≪ power counting scenarios: (a), (b), and (c). Scenario exchange potential VS for momenta q mS and match it (a) corresponds to a short-range binding mechanism, which with the expansion of the EFT contact-range potential requires phenomenological input. Scenario (b) corresponds ⃗ ⃗2 … to a long-range binding mechanism, which allows for EFT VSðqÞ¼V0 þ V2q þ ; ð123Þ predictions. Finally scenario (c) is a mixture of short- and ⃗ ⃗2 … long-range binding, which in a few cases will lead to VCðqÞ¼C þ Dq þ ; ð124Þ predictions. The heavy baryon-antibaryon system belongs to scenario (a) or (c) depending on the particular state and from which we arrive to quantum numbers considered. Λ ∼ ∼ ⃗ 0 Theoretical studies of hadronic molecules have attributed Cð mSÞ V0 ¼ VSðjqj¼ Þ; ð125Þ the binding mechanism to either short- or long-range Λ Λ ∼ causes. In the pioneering work of Voloshin and Okun where is the cutoff. Notice that we take mS: this is [1] it is the exchange of light mesons (π, σ, ρ, and ω) which because the saturation hypothesis is only expected to work generates heavy hadron molecules, i.e., a mixture of short- if the cutoff is of the order of the mass of the exchange and long-range physics. Early speculations [2–5] often meson [70]. For a Yukawa-like meson exchange potential predicted binding from the OPE potential (long-range 2 physics) alone. It is notable to mention that Ericson and ⃗ gS VSðqÞ¼⃗2 2 ; ð126Þ Karl [4] indicated that hadronic molecules should be q þ mS possible in the charm sector and that Törnqvist [5] predicted the existence of an isoscalar 1þþ D D¯ bound the saturated contact-range coupling is proportional to state. The experimental discovery of the Xð3872Þ a decade 2 after [6] suggests that these theoretical speculations were g C ∝ S ; ð127Þ on the right track. At this point we find it interesting to 2 mS notice that a molecular Xð3872Þ also arises naturally from short-range physics [65]. Before the discovery of the where the proportionality constant will depend on the Pcð4450Þ by the LHCb [11], which is suspected (but not details of the regularization process. This argument is ¯ confirmed) to be a D Σc molecule [14,16–18], there were independent of the nature of the exchanged meson; it only theoretical predictions of its existence too. The authors of matters that the mass of this meson is of the order of the Refs. [66,67] used contact-range interactions derived from hard scale. vector meson exchange saturation to make quantitative Next we calculate the scalar and vector meson exchange 1 P 3− ¯ predictions of an I ¼ 2, J ¼ 2 D Σc molecule (among contribution to the saturation of the EFT coupling. We
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TABLE IX. Contact-range coupling from saturation of vector We continue with the vector meson exchange potential, 2 2 and scalar meson exchange in units proportional to gρ=mV and for which the starting point is the heavy baryon-vector- 2 ≃ 2 9 ≃ 3 4 gσqq=mσ,withgρ . and gσqq . . Scalar and vector meson meson Lagrangian for the SSV and TTV vertices (where V 3 Σ Ξ exchange saturations of the spin-2 sextet heavy baryons ( Q, Q, represents the vector meson). If we consider interactions Ω 1 Σ Ξ Ω Q) are identical to their spin-2 partners ( Q, Q, Q) and are not with no derivatives, which allow for saturation of the lowest listed independently. The contributions to vector meson exchange order EFT couplings, we can write the following are also listed separately as CV ¼ Cρ þ Cω þ Cϕ. Though not Lagrangians: listed, the saturation for the heavy baryon-baryon system can be obtained from the heavy baryon-antibaryon case by changing the L λ ϵ ϵ ¯ l μ i TTV ¼ T ikl jkmTQvμðV ÞjTQm; ð129Þ sign of the ω and ϕ contributions. For comparison purposes we ¯ ¯ Σ also include the NN (two-nucleon), D D, and D c systems, L λ ¯ μ i νjk SSV ¼ SSQνikvμðV ÞjSQ ; ð130Þ which are related to the deuteron, the Xð3872Þ, and the Pcð4450Þ, respectively. where the latin indices indicate the sum over the SU(3) 12 System Isospin CS CV Cρ Cω Cϕ components. The vector meson nonet field is given by NN 0 −9 þ6 −3 þ9 0 0 1 ρ0 ω0 −9 10 1 9 pffiffi pffiffi ρþ K þ NN 1 þ þ þ 0 B 2 þ 2 C ¯ B 0 0 C DD 0 −1 −4 −3 −1 0 B − ρffiffi ωffiffi 0 C V ¼ @ ρ − p þ p K A; ð131Þ DD¯ 1 −1 0 þ1 −1 0 2 2 − ¯ 0 ϕ Σ ¯ 1 −2 −2 −4 2 K K cD 2 þ 0 Σ D¯ 3 −2 þ4 þ2 þ2 0 c 2 where the Lorentz index μ is implicitly understood. The ¯ −4 −4 −4 ΛQΛQ 0 0 0 vector meson exchange contribution to the potential can be ¯ −1 −6 −3 −1 −2 ΞQΞQ 0 worked out along the lines of Appendix A. A few ¯ −1 −2 1 −1 −2 representative examples are ΞQΞQ 1 þ Ξ0 Ξ¯ −1 −6 −3 −1 −2 2 Q Q 0 λ τ⃗ τ⃗ − 3 Ξ Ξ¯ Ξ Ξ¯ T ð 1 · 2 Þ Ξ0 Ξ¯ 1 −1 −2 þ1 −1 −2 h Q QjVj Q Qi¼ 2 2 ; ð132Þ Q Q q⃗ þ m 2 ¯ −4 −4 −4 V ΣcΛQ 1 0 0 ¯ λ λ τ⃗1 · τ⃗2 − 3 Ω Ω 00−8 00−8 Ξ0 Ξ¯ Ξ0 Ξ¯ S T ð Þ Q Q h QjVj Qi¼ 2 2 ; ð133Þ ¯ −1 −6 −3 −1 −2 Q Q q⃗ þ m 4 ΞQΞQ 0 V ¯ −1 −2 1 −1 −2 ΞQΞQ 1 þ 2 λ ðτ⃗1 · τ⃗2 − 3Þ Σ Σ¯ 0 −4 −12 −8 −4 0 Ξ0 Ξ¯ 0 V Ξ0 Ξ¯ 0 S ; Q Q h Q Qj j Q Qi¼⃗2 2 8 ð134Þ ¯ −4 −8 −4 −4 q þ mV ΣQΣQ 1 0 Σ Σ¯ −4 4 −4 Q Q 2 0 þ 0 λ2 ⃗ ⃗ − 1 Σ Σ¯ Σ Σ¯ S ðT1 · T2 Þ h Q QjVj Q Qi¼⃗2 2 2 ; ð135Þ q þ mV begin by considering scalar meson exchange. The sigma meson exchange potential is which have been calculated in the SU(3) limit. We have taken mρ ¼ mω ¼ mϕ ¼ mV with mV the vector meson
2 mass. Notice that we do not have to write explicitly the gσ potential for the sextet spin-3=2 heavy baryons: the vector Vσðq⃗Þ¼− ; ð128Þ ⃗2 2 Σ Ξ Ω q þ mσ meson potentials for the Q, Q, and Q are identical to the Σ Ξ0 Ω λ λ ones for Q, Q, and Q. The couplings S and T are not with gσ the sigma coupling. We can determine gσ from the arbitrary: they can be determined from the universality of quark model, in which gσ is simply proportional to the the ρ coupling constant [72]. If we consider the ρ-meson number of u and d quarks in the hadron. Here we take exchange potential between two isospin-1=2 baryons the sigma-nucleon-nucleon coupling as input, which in the pffiffiffi 2 2 ∼ 10 2 gρ nonlinear sigma model [71] is gσNN ¼ MN=fπ . , ⃗ τ⃗ τ⃗ VρðqÞ¼ 2 2 1 · 2; ð136Þ where MN is the nucleon mass and fπ the pion decay q⃗ þ mρ 3 ∼ 3 4 Ξ Ξ0 constant. From this we have gσ ¼ gσNN= . for Q, Q, and Ξ and gσ ¼ 2gσ =3 ∼ 6.8 for Λ , Σ , and Σ . The Q NN Q Q Q 12Notice that we are not considering TSV vertices because they contributions of the scalar meson to the saturation of the involve derivatives and do not saturate the LO contact-range contact-range couplings are listed in Table IX. couplings.
074026-19 LU, GENG, and VALDERRAMA PHYS. REV. D 99, 074026 (2019) with mρ ¼ 770 MeV, the universality of the ρ coupling pairs listed in Eq. (141) are the strongest candidates ¯ implies that gρ ¼ mρ=2fπ ≃ 2.9. If we match to the to bind. The particular case of ΛcΛc has been recently potentials in Eqs. (132) to (135), we find studied in Ref. [73], leading to binding in agreement with pffiffiffi pffiffiffi our conclusions. λT ¼ 2gρ and λS ¼ 2 2gρ: ð137Þ If we consider the heavy baryon-baryon system instead, the contribution of the ω and ϕ mesons is repulsive and in The saturation of the EFT contact couplings by the vector general there is less attraction than in the heavy baryon- mesons is easy to obtain and can be found in Table IX.We antibaryon case. Yet for the following heavy baryon-baryon mention that it is possible to consider the contributions of system, the different vector mesons separately: ΣQΣQðI ¼ 0Þ; ð142Þ CV ¼ Cρ þ Cω þ Cϕ: ð138Þ there is more short-range attraction than in the Xð3872Þ and This form is interesting because it makes it easy to deduce the Pcð4450Þ. Further candidates for binding can be the strength of the heavy baryon-baryon short-range inferred from a comparison with the deuteron, for which interaction from the heavy baryon-antibaryon one. This the short-range interaction is repulsive. In Table IX we see merely involves changing the sign of the contributions from that for the nucleon-nucleon system there is a strong short- the negative G-parity mesons, the ω and the ϕ, yielding range repulsion from the exchange of the ω meson but also 0 − − CV ¼ Cρ Cω Cϕ. Though the vector meson saturations a strong attraction coming from the exchange of the σ of the heavy baryon-baryon system is not listed here, they meson. The existence of the deuteron indicates that can be obtained from Table IX where the Cρ, Cω, and Cϕ attraction wins in this case. This is not surprising if we contributions are listed. notice that mσ
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TABLE X. Critical radius Rc for which OPE is able to bind certain heavy baryon-antibaryon molecules, where I refers to the isospin of ¯ −þ þ0.23 the system. For the ΣQΣCð0 Þ, where there is no tensor force, the notation 0 indicates that there is only binding for Rc ≤ 0.23 fm if the coupling g2 lies on the high end of the lattice calculations. For comparison purposes, the critical radius for the deuteron and the þ Pc ð4450Þ pentaquark are 1.00 fm and 0.30–0.49 fm, respectively. If the system does not bind, we indicate it with the “…” notation. Channel I ¼ 0 I ¼ 1 I ¼ 2 Channel I ¼ 0 I ¼ 1 I ¼ 2 ¯ −þ ………¯ −þ þ0.23 …… ΣcΣcð0 Þ ΣbΣbð0 Þ 0 Σ Σ¯ 1−− – – – Σ Σ¯ 1−− 0 86þ0.30 0 49þ0.21 0 43þ0.21 c cð Þ 0.42 0.94 0.22 0.55 0.19 0.48 b bð Þ . −0.31 . −0.19 . −0.18 Σ Σ¯ 1−þ – – – Σ Σ¯ 1−þ 0 88þ0.38 0 46þ0.23 0 32þ0.13 c cð Þ 0.39 1.00 0.19 0.53 0.15 0.36 b bð Þ . −0.35 . −0.20 . −0.12 Σ Σ¯ 1−− 0.44–1.11 0.22–0.59 0.18–0.44 Σ Σ¯ 1−− 0 99þ0.40 0 52þ0.26 0 39þ0.16 c cð Þ b bð Þ . −0.39 . −0.22 . −0.15 Σ Σ¯ 2−þ – – – Σ Σ¯ 2−þ 0 91þ0.31 0 51þ0.21 0 56þ0.24 c cð Þ 0.46 1.01 0.24 0.59 0.25 0.64 b bð Þ . −0.32 . −0.21 . −0.22 Σ Σ¯ 2−− – – – Σ Σ¯ 2−− 0 77þ0.27 0 45þ0.18 0 48þ0.23 c cð Þ 0.39 0.85 0.21 0.50 0.20 0.55 b bð Þ . −0.27 . −0.17 . −0.21 Σ Σ¯ 0−þ – – – Σ Σ¯ 0−þ 1 20þ0.46 0 64þ0.32 0 36þ0.14 c cð Þ 0.56 1.36 0.27 0.75 0.18 0.41 b bð Þ . −0.46 . −0.27 . −0.30 Σ Σ¯ 1−− – – – Σ Σ¯ 1−− 1 09þ0.42 0 59þ0.29 0 37þ0.15 c cð Þ 0.52 1.24 0.26 0.68 0.18 0.42 b bð Þ . −0.41 . −0.25 . −0.14 Σ Σ¯ 2−þ – – – Σ Σ¯ 2−þ 0 91þ0.34 0 50þ0.23 0 43þ0.18 c cð Þ 0.44 1.02 0.22 0.57 0.20 0.50 b bð Þ . −0.34 . −0.20 . −0.17 Σ Σ¯ 3−− – – – Σ Σ¯ 3−− 0 82þ0.27 0 48þ0.20 0 60þ0.28 c cð Þ 0.45 0.91 0.25 0.55 0.28 0.69 b bð Þ . −0.28 . −0.18 . −0.25 system, to which the Xð3872Þ is suspected to belong, scalar consider that the system is likely to bind. By large enough and vector meson exchange saturation predicts exactly the we mean for instance that the critical radius is larger than same potential for DD¯ , D D¯ , DD¯ , and D D¯ irrespec- the size of the hadrons or the range of other contributions to tively of the spin and C-parity quantum numbers. That is, the hadron-hadron potential that have not been taken into the saturation argument leads to the prediction of six account (e.g., two pion exchange). isoscalar heavy meson-antimeson molecules. This is to It is important to notice that most heavy baryon-anti- be compared with only one obvious molecular candidate, baryon molecules bind if rc is sufficiently small because of the Xð3872Þ. Analogously, the application of this argument the tensor force. Thus the crucial factor is not whether there to the heavy meson-antibaryon molecules leads to the is a critical radius for which the molecule binds, but ¯ ¯ ¯ ¯ prediction of seven DΣc, DΣc, D Σc, and D Σc molecules whether the critical radius Rc is reasonable or not. The but only one experimental candidate, the Pcð4450Þ. This reason why the tensor force is able to bind in most cases is situation also happens in other theoretical approaches that because for S-wave molecules it behaves as an attractive derive heavy hadron interactions from vector meson singular potential; see the discussion in Sec. VC. This is saturation [65–67]. The probable conclusion is that we why it is important to consider whether the distance at are probably missing something in the resonance saturation which OPE binds is reasonable or not. Σð ÞΣ¯ ð Þ arguments we are using to derive the LO couplings. Be that We list the critical radii for the Q Q molecules in as it may, for the set of molecules in Eq. (141) the short- Σð ÞΣ¯ ð Þ Table X. We have chosen the Q Q system because this is range attraction is expected to be remarkably stronger than the case in which the OPE potential is expected to be in the Xð3872Þ and Pcð4450Þ. stronger owing to the higher isospin of the ΣQ’s. Besides, from scalar and vector meson exchange saturation we B. Long-range binding dynamics expect a very strong short-range attraction. The isoscalar The long-range dynamics of the heavy baryon-anti- molecules are the ones showing more attraction and higher baryon system is driven by OPE. We assess the relative critical radii, reaching in a few cases 1 fm. For the hidden strength of the OPE potential for each channel in the charm molecules the uncertainty is really big as the value of following way: first we modify the OPE potential by g2 is not experimentally known. To give a sense of scale we including a cutoff mention that for the deuteron the critical radius is 1.00 fm. ¯ For the Pcð4450Þ pentaquarklike state as a ΣcD molecule, ; θ − – VOPEðr rcÞ¼VOPEðrÞ ðr rcÞ; ð145Þ the critical radius is 0.30 0.49 fm (where the uncertainty is again a consequence of g2). In comparison, for the heavy where rc is the cutoff. Then we calculate the largest rc for meson-antimeson molecular candidates, the radii are 0.30, which OPE alone is able to bind a molecule. We call this 0.10, and 0.26 fm for the Xð3872Þ, Zcð3900Þ, and 10610 radius rc ¼ Rc the “critical radius.” Notice that we are in Zbð Þ, respectively. The rather small critical radii of fact assuming that (i) OPE is valid from infinity till the the X, the Zc, and the Zb suggest that these hadron critical radius and (ii) there is no short-range physics. If this molecules depend on the short-range attraction (instead critical radius turns out to be large enough, we will of OPE) to bind. For the deuteron the critical radius is
074026-21 LU, GENG, and VALDERRAMA PHYS. REV. D 99, 074026 (2019) significantly larger, indicating that OPE is an important contact-range potential and central OPE. Symmetries in the component of the binding mechanism. Lastly, the situation S-wave interaction are likely to translate into symmetries in þ ¯ for the Pcð4450Þ pentaquark seems to be in the middle. the spectrum. For the TT case, the LO EFT potential does From Table X it is apparent that for the heavy baryon- not depend on the total spin of the system: antibaryon system OPE can provide as much attraction as ¯ − ¯ ¯ − ¯ in the deuteron. If we combine this observation with what hTTjVsð0 ÞjTTi¼hTTjVsð1 ÞjTTi; ð146Þ we know about short-range physics according to Table IX, Σð ÞΣ¯ ð Þ where the subscript s is used to indicate S-wave. That is, the the conclusion is that there will be a rich Q Q molecular ¯ spectrum, particularly in the I ¼ 0, 1 configurations. TT heavy baryon molecules are expected to come in pairs. For the TS=S¯ T¯ molecules we have the following two VII. CONCLUSIONS relations:
¯ − ¯ ¯ − ¯ In this work we have presented a general EFT framework hTSjVsð0 ÞjTSi¼hTSjVsð2 ÞjTSi; ð147Þ for the heavy baryon-antibaryon system. EFTs exploit the ¯ − ¯ ¯ −∓ ¯ existence of a separation of scales to express the observable hTB6jVsð1 ÞjTB6i¼hTB6jVsð1 ÞjTB6i; ð148Þ quantities of a low energy system as a power series. In the case at hand the size of a hadron molecule is expected to be where in the second line we have explicitly indicated larger than the hadrons forming it. As a consequence this whether the sextet heavy baryon is in the ground or excited type of system is amenable to an EFT description. Also, state. The conclusion is again that TS¯ molecules appear in heavy hadron molecules are constrained by chiral, SU(2)- pairs. For SS¯ molecules the contacts have a far richer isospin, SU(3)-flavor, and HQSS symmetries. This degree structure, with only one obvious symmetry relation: of symmetry translates into a few interesting regularities in their spectrum. ¯ −− ¯ ¯ −− ¯ hSSjVsð2 ÞjSSi¼hSSjVsð3 ÞjSSi: ð149Þ EFT explains the heavy baryon-antibaryon interaction in terms of contact-range interactions and pion exchanges. Tensor OPE mixes partial waves and will induce deviations The relative importance of these two contributions changes from the previous relations, which will be moderate if the from system to system. In general the LO EFT description bound states are shallow. At this point it is worth noticing involves four-baryon contact-range interactions and pion the analogy with the heavy meson-antimeson case, where exchanges (OPE), but this depends on the molecule. Pion this type of twin structure also happens for (i) the 1þ− D D¯ exchanges are expected to be particularly important in the and 1þ− D D¯ and (ii) the 1þþ D D¯ and 2þþ D D¯ Σ Σ¯ Σ Σ¯ Σ Σ¯ isoscalar Q Q, Q Q, and Q Q molecules (but less molecules. The first of these relations explains why the Zc’s important for other configurations). In contrast, OPE and Z ’s resonances appear in pairs [31,74], while the latter Ξ Ξ¯ Λ Λ¯ b vanishes in the Q Q and Q Q molecules, which can predicts that the Xð3872Þ should have a 2þþ partner, the be described in terms of a contact theory at LO. For the Xð4012Þ [32,33]. In the heavy meson-antimeson system Ξ0 Ξ¯ 0 Ξ Ξ¯ 0 Ξ Ξ¯ Q Q, Q Q, and Q Q molecules, particularly in the there is a series of dynamical effects (besides the afore- hidden charm sector, OPE is probably a NLO effect. We mentioned tensor OPE) that might break these patterns, warn that the conclusions about the relevance of the OPE which include decays into nearby channels [75], coupled potential are only well established for the bottom sector. In channel dynamics [76], the existence of nearby quarkonia the charm sector, the value of the g2 axial coupling that [77], and annihilation [78]. Though they have not been appears in the Σc → Σcπ amplitude is not known exper- studied in the heavy baryon-antibaryon case, these effects imentally, and a determination either in a future experiment could be relevant. or in the lattice will be welcomed. Particle coupled Finally there is the important question of whether the channels, i.e., transitions in which a heavy baryon changes existence of heavy baryon molecules can be predicted. from the ground to the excited state (B6 → B6), are EFTs are generic frameworks that usually require preexist- subleading if the moleculesp areffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi not too tightly bound, ing experimental input to make predictions. The EFT i.e., for binding momenta γ ¼ M6jEBj ≤ 350–400 MeV. potential is composed of a long-range and short-range The previous findings regarding pion exchanges and piece. The short-range piece involves unknown couplings, coupled channels are analogous to the ones in the heavy which have to be determined from external information. In meson-antimeson molecules [32]. It is also worth mention- the absence of experimental data, there is the possibility of ing that right now the LO EFT is more than enough for the using phenomenological arguments to estimate the contact- description of heavy hadron-antihadron molecules, where range couplings. If we assume the saturation of these the scarcity of experimental data makes it superfluous to couplings from ρ-, ω-, ϕ-, and σ-meson exchange, the most calculate subleading orders. probable candidates for a heavy baryon-antibaryon bound ¯ ¯ ¯ ¯ The EFT potential is constrained by HQSS. This is state are the isoscalar ΛcΛc, ΣcΣc, ΣcΣc, and ΣcΣc particularly evident for S-wave interactions, such as the LO molecules, located at 4573, 4906, 4970, and 5035 MeV
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¯ ¯ A A and the isovector ΛcΣc and ΛcΣc molecules at 4740 component to its 4-momentum). The amplitudes 1 and 2 and 4805 MeV. If we consider the heavy baryon-baryon may refer to baryons or antibaryons indistinctively. In the system instead, saturation indicates that the isoscalar, following lines we will explain how to do the derivation in doubly-charmed ΣcΣc, ΣcΣc, and ΣcΣc molecules are detail. good candidates for binding, followed by their isovector counterparts, the isoscalar ΛcΛc and isovector ΛcΣc and 1. The heavy baryon field ΛcΣc systems. For the heavy baryon-antibaryon system we supplement the saturation argument with an estimate of Heavy baryons contain a heavy quark and two light the relative strength of the OPE potential, which we assess quarks; i.e., they have the structure jQqqi. The total spin of 0 by calculating the radius for which OPE would be able the light quark pair is SL ¼ , 1. If the light quark spin is 0 1 2 to bind the system by itself. This second argument also SL ¼ , we have an antitriplet spin- = heavy baryon, points to isoscalar molecules as the most likely to bind. It which we denote by the Dirac field might be possible to observe these heavy baryon-anti- Ψ baryon molecules in experiments such as LHCb and 3¯Q: ðA3Þ PANDA, which is expected to be particularly suited for 1 1 2 the precision study of hidden charm exotic states [79],or If the light quark spin is SL ¼ , we have spin- = and alternatively in the lattice. spin-3=2 heavy baryons which we denote by the Dirac and Rarita-Schwinger fields ACKNOWLEDGMENTS Ψ Ψ 6Q; 6Qμ: ðA4Þ M. P. V. thanks Johann Haidenbauer for discussions and the Institute de Physique Nucl´eaire d’Orsay, where part of Notice that the Rarita-Schwinger field contains a Lorentz this work was carried out, for its hospitality. We also thank index: it is the external product of a Minkowsky vector and Chu-Wen Xiao for his comments on the manuscript. 1 a Dirac spinor. The product contains a spurious spin-2 This work is partly supported by the National Natural component that can be removed with the condition Science Foundation of China under Grants No. 11375024, No. 11522539, and No. 11735003; the Fundamental γμΨ ¼ 0: ðA5Þ Research Funds for the Central Universities; and the 6Qμ Thousand Talents Plan for Young Professionals. Within heavy hadron EFT it is customary to use the fields TðvÞ and SμðvÞ instead, which have good transformation APPENDIX A: THE ONE PION EXCHANGE properties under rotations of the spin of the heavy quark POTENTIAL IN HEAVY HADRON CHIRAL (where v refers to the velocity of the heavy quark). For the PERTURBATION THEORY Ψ 3¯Q heavy baryon, the definition is The OPE potential is the LO piece of the finite-range 1 EFT potential. In this Appendix we explain how to þ =v Ψ TQðvÞ¼ 3¯Q; ðA6Þ compute it. The idea is to obtain nonrelativistic amplitudes 2 for processes involving an incoming and outgoing heavy 1 =v baryon and a pion, which we write as ¯ Ψ¯ þ TQðvÞ¼ 3¯Q 2 : ðA7Þ AðB → B0π; q⃗Þ; ðA1Þ The superfield TQðvÞ transforms as 0 ⃗ where BðB Þ are the initial/final baryon and q the momen- − ϵ⃗⃗ → i ·Sv tum of the pion if outgoing (if incoming we change the TQðvÞ e TQðvÞ; ðA8Þ momentum to −q⃗). If written in a suitable normalization ⃗ 2 2 these amplitudes can be combined to compute the OPE where Sv is related to SUð Þv, the SUð Þ spin group of the Ψ Ψ potential (or for that matter any one boson exchange heavy quark Q moving at velocity v. For the 6Q and 6Qμ potential) as follows: sextet heavy baryons, the definition is
A1ðq⃗ÞA2ð−q⃗Þ 1 1 =v 1 =v 0 0 ffiffiffi γ γ þ Ψ þ Ψ hB1B2jVjB1B2i¼ 2 2 ; ðA2Þ SQμðvÞ¼p ð μ þ vμÞ 5 6Q þ 6Qμ; ðA9Þ q⃗ þ μπ 3 2 2 where 1 and 2 refer to the pion vertices 1 and 2 and μπ is 1 1 1 ¯ ¯ þ =v ¯ þ =v S μðvÞ¼− pffiffiffi Ψ6 γ5ðγμ þ vμÞþΨ ; the effective pion mass for this particular transition (which Q 3 Q 2 6Qμ 2 is not necessarily the physical pion mass because
0 − ≠ 0 ðA10Þ mB1 mB1 and gives the pion a nonvanishing zeroth
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γ γ where μ, 5 are the Dirac matrices. The SQμðvÞ superfield Notice that we are interested in the OPE potential: the contains a Lorentz index that comes from the Rarita- isoscalar ΩQ cannot exchange a single pion and will not be Ψ Schwinger field 6Qμ. It obeys a constraint analogous to further considered here. The isoscalar ΛQ and the isospinor Ξ Ξ0 Eq. (A5): Q can only exchange pions in vertices involving a Q and ð Þ a Σ respectively; i.e., there is no ΞQΞQπ or ΛQΛQπ μ Q 0 0 v SQμðvÞ¼ and =vSQμðvÞ¼SQμðvÞ: ðA11Þ Ξ Ξð Þ π Λ Σð Þπ vertex but there are Q Q and Q Q vertices.
The S μðvÞ superfield transforms as Q 2. The heavy baryon chiral Lagrangian at LO
−iϵ⃗·S⃗ The interaction of heavy baryons and pions can be S μ v → e v S μ v : Q ð Þ Q ð Þ ðA12Þ written as [46,52] ⃗ In general we take the velocity parameter to be v ¼ð1; 0Þ. LTTπ ¼ 0; ðA17Þ We are interested in heavy baryon-antibaryon molecules; i.e., we need the antibaryon fields. Here it is important to ¯ i μ j kl ijk ¯ μ μ l L π ¼ g3½ϵ T ðA Þ S þ ϵ S ðA Þ T ; ðA18Þ notice that ST ijk Q l Qμ Qkl j Qi
¯ † γ L ϵ ¯ μ ν σ i λ jk TQðvÞ¼TQðvÞ 0; ðA13Þ SSπ ¼ ig2 μνσλSQikv ðA ÞjðS ÞQ ; ðA19Þ
¯ † γ SQμðvÞ¼SQμðvÞ 0; ðA14Þ where the latin indices i, j, k, l indicate either the SU(2)- isospin or the SU(3)-flavor components; g2, g3 are coupling are the operators for creating heavy baryons, which are constants; and ϵμνσλ is the four-dimensional Levi-Civit`a unrelated to the antibaryon fields. Heavy antibaryons symbol. In the equation above, Aμ is the pseudo Goldstone ¯ require the definition of new T ¯ ðvÞ, T ¯ ðvÞ, S ¯ ðvÞ, and boson field, ¯ Q Q Q SQ¯ ðvÞ fields. We will not need to define them explicitly i though. Instead we will use C- and G-parity transforma- Aμ ¼ ðξ†∂μξ − ξ∂μξ†Þ; ðA20Þ tions to deduce the interactions of the heavy antibaryons 2 with the pions. where ξ is defined as The heavy baryon fields have SU(2)-isospin and SU(3)- flavor structure. If we add SU(3)-flavor indices for the i M ξ ¼ efπ ; ðA21Þ heavy baryons with Q ¼ b and SL ¼ 0,wehave 0 1 with the matrix M Ξ− B b C 0 1 B 0 C π0 η Ψ3¯ ¼ −Ξ ðA15Þ pffiffi þ pffiffi πþ Kþ b @ b A B 2 6 C 0 Λ B − π0 η 0 C b B π − pffiffi pffiffi C M ¼ B 2 þ 6 K C; ðA22Þ @ qffiffi A − ¯ 0 − 2η while for Q ¼ b, SL ¼ 1 we have K K 3 0 1 1 0 1 0 Σþ pffiffi Σ pffiffi Ξ 0 which entails that we are taking the normalization B b 2 b 2 b C B 1 0 − 1 − C choice fπ ≃ 132 MeV. Ψ B pffiffi Σ Σ pffiffi Ξ 0 C μ 6¯b ¼ @ 2 b b 2 b A; ðA16Þ If we consider the SUð2Þ subgroup of SUð3Þ, the A field 1 0 1 − − pffiffi Ξ 0 pffiffi Ξ 0 Ω reduces to the following expansion in the pion field: 2 b 2 b b 1 1 3 μ − ∂μπ π π ∂μπ … where the corresponding expressions for the spin-2 heavy A ¼ þ 3 ½ ; ½ ; þ ; ðA23Þ fπ 6fπ Ψ 3 baryons 6c are identical. For the SUð Þ-flavor structure of the Q ¼ c charmed baryons we refer to Eqs. (20) and (21). where π refers to the SUð2Þ submatrix in the equation In the following we will mostly consider the SUð2Þ-isospin above (after removing the contribution from the η), i.e., structure: when we talk about the T we could either be Q π0 ! pffiffi πþ referring to ΛQ (isoscalar) or ΞQ (isospinor), while when 2 0 π ¼ : ðA24Þ S Σ Ξ − π0 we talk about Q it could be Q (isovector), Q (isospinor), π − pffiffi 2 or ΩQ (isoscalar).
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3. The nonrelativistic limit TQ ¼ B3¯ ; ðA34Þ
The potential is well defined in the nonrelativistic limit, † † where the heavy baryon fields reduce to TQ ¼ B3¯ ; ðA35Þ rffiffiffi pffiffiffiffiffiffiffiffiffi χ0 1 s ⃗ ⃗ Ψ3¯ → 2M3¯ ; ðA25Þ σ⃗ Q 0 SQ ¼ 3 B6 þ B6; ðA36Þ rffiffiffi pffiffiffiffiffiffiffiffiffi χ 1 s ⃗† † ⃗ † Ψ6 → 2M6 ; ðA26Þ S ¼ B σ⃗þ B : ðA37Þ Q 0 Q 3 6 6 pffiffiffiffiffiffiffiffiffi 0 χ⃗ The pion field Aμ reduces in the heavy baryon non- ⃗ s Ψ ¼fΨ ; Ψ6 g → 2M ; ; ðA27Þ 6Qμ 6Q0 Q 6 0 0 relativistic limit to 3 0 1 π where χ , χ are standard spinors, while χ⃗ χ 1; χ 2; χ 3 ⃗ − ∇⃗π⃗ O s s s ¼ð s s s Þ A ¼ þ 3 ; ðA38Þ is a vector in which each component is a spinor. The vector fπ fπ χ⃗s fulfills the condition where we ignore the zeroth component of Aμ because it ⃗ couples to the zeroth component of S μðvÞ, which vanishes σ⃗· χs ¼ 0; ðA28Þ Q in the nonrelativistic limit. From this we can rewrite the i.e., the nonrelativistic version of Eq. (A5), which ensures Lagrangian as 3 Ψ ¯ that 6Qμ is a genuine spin-2 field. We have that M3 is the i† ⃗ j ⃗kl ijk ⃗† ⃗ l L π ¼ −g3½ϵ T ðAÞ · S þ ϵ S · ðAÞ T ; ðA39Þ SL ¼ 0 heavy baryon mass, while M6, M6 are the SL ¼ 1 ST ijk Q l Q Qkl j Qi heavy baryon masses. In the heavy quark limit the SL ¼ 1 L − ⃗† ⃗ ⃗ baryon masses are identical: SSπ ¼ ig2Tr½SQ · ðA × SQÞ ; ðA40Þ
M6 ¼ M6 for mQ → ∞: ðA29Þ where the trace is over isospin space. Alternatively we can expand the Lagrangian in terms of the fields B3¯ , B6, and B6: However, this does not happen with the M3¯ mass, which g3 † g3 † remains different from M6 and M in the heavy quark limit. L ffiffiffi σ⃗ ∇⃗π ffiffiffi σ⃗ ∇⃗π 0 6 STπ ¼ p B3¯ · B6 þ p B6 · B6 Putting all the pieces together, in the nonrelativistic limit 3fπ 3fπ 0 g3 g3 the SL ¼ heavy field reduces to †∇⃗π ⃗ ⃗ † ∇⃗π þ B3¯ · B6 þ B6 · B3¯ ðA41Þ fπ fπ 1 B3¯ ffiffiffiffiffiffiffiffiffi T v → ; p2 Qð Þ ðA30Þ g2 † ⃗ g2 ⃗ † ⃗ ⃗ M3¯ 0 LSSπ ¼ i B6σ⃗· ð∇ π⃗×σÞB6 þ i B6 · ð∇ π⃗×B6Þ 3fπ fπ g2 † 1 † ffiffiffi ⃗ ⃗ ffiffiffiffiffiffiffiffiffi ¯ → 0 þ i p B6σ⃗· ð∇ π⃗×B6Þ p TQðvÞ B3¯ ; ðA31Þ 3 2M3¯ fπ gffiffiffi2 ⃗ † ∇⃗π⃗ σ⃗ 1 þ i p B6 · ð × ÞB6; ðA42Þ while the SL ¼ heavy fields read 3fπ qffiffi ! 1 1σ⃗ ⃗ where we have removed the isospin/flavor indices to make ffiffiffiffiffiffiffiffiffi ⃗ → 3 B6 þ B6 p SQðvÞ ; ðA32Þ the expressions shorter. 2M6 0 qffiffi 4. The spin and isospin factors 1 ¯ ffiffiffiffiffiffiffiffiffi ⃗ → 1σ⃗ † ⃗ † 0 p SQðvÞ 3 B6 þ B6 ; ðA33Þ Now we calculate the matrix elements of the different 2M6 vertices, which depend on a series of spin and isospin factors. In the charm sector (Q ¼ c), the relations between with B3¯ , B6, and B6 the nonrelativistic heavy baryon fields. The notation can be further simplified (i) by noticing that the isospin and particle basis are ¯ ¯ † † there is no difference between the TQ=SQ and TQ=SQ fields þ B3¯ ðΛ Þ¼Λ ; ðA43Þ in the nonrelativistic limit, (ii) by ignoring the antibaryon c c componentsffiffiffiffiffiffiffiffiffi and (iii)ffiffiffiffiffiffiffiffiffi by absorbing the normalization þ0 p p Ξc factors 2M3¯ and 2M6 in a field redefinition. In this B3¯ ðΞ Þ¼ ; ðA44Þ c Ξ00 case we end up with c
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0 a Ξþ † T 0 c Σ πaΣ ffiffiffi B6 Ξ ; A45 c c ¼ p ; ðA57Þ ð cÞ¼ 00 ð Þ 2 Ξc 0 1 1 where T are the J 1 angular momentum matrices in Σþþ pffiffi Σþ a ¼ c 2 c Ξ Σ Σ @ A isospin space. The isospin factors for the c and c baryons B6ð cÞ¼ 1 0 ; ðA46Þ pffiffi Σþ Σ Ξ Σ 2 c c are identical to those of the c and c baryons. Next we factor out the spin in terms of angular 0 momentum matrices or equivalent expressions. For the for the Λc, Ξc, Ξc, and Σc baryons, respectively. The 0 STπ vertices the factors are relations for the excited Ξc and Σc sextet baryons are identical to those of the Ξ0 and Σ baryons. Alternatively if c c †σ⃗ ⃗ σ⃗ ⃗ we consider isospin vectors we can write B3¯ · qB6 ¼ · q; ðA58Þ
Λþ 00 † ⃗ ⃗ ⃗ ⃗ c ¼j iI; ðA47Þ B3¯ q · B6 ¼ S · q; ðA59Þ 1 1 1 1 π Ξþ Ξ0 − while for the SS vertices we have f c ; cg¼ 2 ; 2 ; 2 ; 2 ; ðA48Þ I I † B6σ⃗· ðq⃗× σ⃗ÞB6 ¼ −i2σ⃗· q;⃗ ðA60Þ 1 1 1 1 Ξþ0 ; Ξ00 ; ; ; − ; A49 f c c g¼ 2 2 2 2 ð Þ 2 I I ⃗ † ⃗ ⃗ − Σ⃗ ⃗ B6 · ðq × B6Þ¼ i 3 · q; ðA61Þ Σþþ Σþ Σ0 1 1 1 0 1 −1 f c ; c ; cg¼fj ; iI; j ; iI; j ; iIg: ðA50Þ † ⃗ ⃗ B6σ⃗· ðq⃗× B6Þ¼−iS · q;⃗ ðA62Þ The isospin factors can be extracted by first expanding the isospin/flavor indices in the particle basis and later rein- ⃗ † ⃗† B · q⃗× σ⃗B6 −iS · q;⃗ A63 terpreting the result in terms of matrices in the isospin 6 ð Þ ¼ ð Þ space. We begin with the STπ Lagrangian, for which ⃗ 3 where σ⃗are the Pauli matrices, Σ are the J ¼ 2 angular † a a ⃗ Λcπ Σc ¼ t ; ðA51Þ momentum matrices, and S are 2 × 4 matrices that connect the spin- 1 and spin- 3 baryons. These S⃗matrices read a 2 2 † τ Ξ πaΞ0 ; A52 c c ¼ ð Þ 1 1 ! 2 pffiffi 0 − pffiffi 0 2 6 a a S1 ¼ 1 1 ; ðA64Þ where π is the pion field in the Cartesian basis, τ are the 0 pffiffi 0 − pffiffi 6 2 Pauli matrices, and ta are given by ! 0 1 iffiffi iffiffi 1 p 0 p 0 pffiffi 2 6 2 S2 ¼ ; ðA65Þ B C iffiffi iffiffi 1 B C 0 p 0 p t ¼ @ 0 A; ðA53Þ 6 2 1 − pffiffi 0 qffiffi 1 2 0 − 2 00 0 1 B 3 C S3 @ qffiffi A; A66 piffiffi ¼ ð Þ 2 2 B C 00− 3 0 2 B C t ¼ @ 0 A; ðA54Þ piffiffi 2 which are normalized as follows: 0 1 0 † 2δij − iϵijkσk B C SiSj ¼ 3 : ðA67Þ t3 ¼ @ 1 A: ðA55Þ 0 Now we define the nonrelativistic amplitudes as
In the SSπ case we have AðB → BπaÞ¼−ihBπajLjBi; ðA68Þ
a † τ 0 a 0 ffiffiffi with B ¼ B3¯ , B6, B . For the transitions involving Λ we Ξc π Ξc ¼ p ; ðA56Þ 6 c 2 2 have
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g3 π a ffiffiffi a G ¼ Cei I2 ; ðA83Þ AðΛc → Σcπ Þ¼p t σ⃗· q;⃗ ðA69Þ 3fπ with I2 the second Cartesian component of the isospin a g3 a ⃗† AðΛc → Σcπ Þ¼ t S · q;⃗ ðA70Þ matrix. Now we will determine how G operates on the fπ different fields we consider here. For instance, pions have well-defined G-parity: a g3 a ⃗ AðΣc → Λcπ Þ¼ t S · q:⃗ ðA71Þ fπ Gjπi¼−jπi: ðA84Þ For the transitions involving Ξ we have c We can write it in terms of the components of the pion field a g3 τ for completeness: A Ξ0 → Ξ πa ffiffiffi σ⃗ ⃗ ð c c Þ¼p · q; ðA72Þ 0 1 0 1 3fπ 2 jπþi jπþi a B C B C g3 τ † B 0 C B 0 C 0 a ⃗ G jπ i ¼ − jπ i : ðA85Þ AðΞc → Ξcπ Þ¼ S · q;⃗ ðA73Þ @ A @ A fπ 2 − − jπ i jπ i τa 0 a g3 ⃗ AðΞc → Ξcπ Þ¼ S · q:⃗ ðA74Þ fπ 2 If we consider baryons instead, G will transform a baryon into an antibaryon in the same isospin state. If we consider Ξ0 Ξ 1 For the ones with the c and c, the amplitudes read nucleons or other isospin- 2 baryons, the G-parity trans- formation works as follows: a 2g2 τ A Ξ0 → Ξ0 πa ffiffiffi σ⃗ ⃗ ! ! ð c c Þ¼3 p · q; ðA75Þ fπ 2 2 jpi jn¯i G : A86 a ¼ ð Þ 2g2 τ jni −jp¯ i 0 0 a ffiffiffi ⃗ AðΞc → Ξc π Þ¼ p Σ · q;⃗ ðA76Þ 3fπ 2 2
a Now we can identify antiparticle states with isospinors as g2 τ 0 0 a ffiffiffi ffiffiffi ⃗ ⃗ follows: AðΞc → Ξcπ Þ¼p p S · q; ðA77Þ 3fπ 2 2
1 1 a jn¯i¼ þ ; ðA87Þ g2 τ † 0 a ffiffiffi ffiffiffi ⃗ 2 2 AðΞc → Ξc π Þ¼p p S · q:⃗ ðA78Þ I 3fπ 2 2 1 1 ¯ − − Σ Σ jpi¼ 2 2 ; ðA88Þ Finally for the transitions with the c and c, we have the I following amplitudes: where we use the subscript I to indicate that we are indeed a 2g2 T a ffiffiffi referring to isospinors. From the nucleon/antinucleon AðΣc → Σcπ Þ¼ p σ⃗· q;⃗ ðA79Þ 3fπ 2 example we can appreciate that the idea of a G-parity transformation is to have a good mapping between the a 2g2 T a ffiffiffi ⃗ isospin and the particle/antiparticle basis. The crucial point AðΣc → Σcπ Þ¼ p Σ · q;⃗ ðA80Þ 3fπ 2 in the G-parity transformation for the baryons is the relative a minus sign between the isospin vectors of the antineutron g2 T a ffiffiffi ffiffiffi ⃗ AðΣc → Σcπ Þ¼p p S · q;⃗ ðA81Þ and antiproton, which in turn allows for the use of the same 3fπ 2 SU(2) Clebsch-Gordan coefficients in the baryon and a antibaryon cases. g2 T † AðΣ → Σ πaÞ¼pffiffiffi pffiffiffi S⃗ · q:⃗ ðA82Þ For the heavy baryons the idea is the same as for the c c 3 2 fπ nucleons. But there is a subtlety: we are considering different 1 3 C-parity conventions for the spin- 2 and spin- 2 fields, 5. G-parity and heavy antibaryons ¯ The amplitudes in Eqs. (A69)–(A82) are for the heavy CjBi¼jBi; ðA89Þ baryons. Here we deduce the amplitudes for the heavy − ¯ antibaryons by working in the isospin basis and applying CjB6i¼ jB6i; ðA90Þ a G-parity transformation, which is a combination of 1 a C-parity transformation and a rotation in isospin where B ¼ B3¯ ;B6. For the antitriplet and sextet spin- 2 heavy space [47]: cascades, the transformation works exactly as in nucleons,
074026-27 LU, GENG, and VALDERRAMA PHYS. REV. D 99, 074026 (2019) ! ! þð0Þ ¯ 0ð0Þ ¯ jΞc i jΞc i where B ¼ B3;B6. The signs simply reflect the sign for the G ¼ ; ðA91Þ 0ð0Þ −ð0Þ G-parity transformation of the pion, plus the extra sign Ξ − Ξ j c i j c i involved in the B3¯ =B6 → B6 and B6 → B3¯ =B6 transitions. For a detailed example we consider while for the sextet spin-3=2 heavy cascades we have A ¯ → ¯ πa ¯ πa L ¯ Ξþ ¯ 0 ðB6 B6 Þ¼ihB6 j jB6i j c i jΞc i G ¼ − : ðA92Þ − πa L Ξ0 − ¼ ihGðB6 Þj jGB6i j c i −jΞc i a † ¼ −ihB6π jG LGjB6i For the isotriplet heavy baryons Σþþ; Σþ; Σ0 ,wehave f c c cg − πa L instead ¼ ihB6 j jB6i a 0 1 0 1 ¼ −AðB6 → B6π Þ; ðA98Þ Σþþ ¯ 0 j c i jΣci B C B C 2 1 −1 Ξ0 1 B Σþ C B − C where we have used that G jB6i¼ ( for Q, þ for G@ j c i A ¼ @ −jΣ i A; ðA93Þ c Σ 2 π 1 †L L Σ0 −− Q), G j i¼þ , and G G ¼ . j ci jΣc i plus the transformation for the excited isotriplet heavy 6. The OPE potential baryons, which will carry an extra minus sign. With the amplitudes of Eqs. (A69)–(A82) we can derive With the G-parity transformation we can deduce the the potential by using Eq. (A2). For simplicity we will amplitudes for the antibaryons from the ones we already consider the heavy quark limit, in which the B6 and B6 ¯ ¯ ¯ know for the baryons: heavy baryons are degenerate. For the TS ¼ ΛcΣc; ΛcΣc and TS¯ ¼ Ξ Ξ¯ 0 ; Ξ Ξ¯ potentials, we write the potential in A ¯ → ¯ πa −A → πa c c c c ðB B Þ¼ ðB B Þ; ðA94Þ the bases, A ¯ → ¯ πa A → πa ¯ ¯ ¯ ¯ ðB B6 Þ¼þ ðB B6 Þ; ðA95Þ B ¯ Λ Σ ; Σ Λ ; Λ Σ ; Σ Λ ; A99 ΛcΣc ¼f c c c c c c c cg ð Þ
¯ ¯ a a AðB6 → Bπ Þ¼þAðB6 → Bπ Þ; ðA96Þ ¯ 0 0 ¯ ¯ ¯ B 0 ¯ Ξ Ξ ; Ξ Ξ ; Ξ Ξ ; Ξ Ξ ; A100 ΞcΞc ¼f c c c c c c c cg ð Þ A ¯ → ¯ πa −A → πa ðB6 B6 Þ¼ ðB6 B6 Þ; ðA97Þ in which the potential reads as
0 1 0 1 σ⃗ ⃗σ⃗ ⃗ 0 1ffiffi ⃗ ⃗σ⃗ ⃗ B 3 1 · q 2 · q p S1 · q 2 · q C B 3 C B 1 1 ⃗ C 2 B σ⃗1 · q⃗σ⃗2 · q⃗ 0 pffiffi σ⃗1 · q⃗S2 · q⃗ 0 C ¯ g3 1 3 3 1 VST ðq⃗Þ¼ τ B C þ O ; ðA101Þ OPE 2 2 2 B 1 † † † C fπ q⃗ þ μπ B 0 pffiffi σ⃗ ⃗⃗ ⃗ 0 ⃗ ⃗⃗ ⃗ C mQ B 3 1 · qS2 · q S1 · qS2 · q C @ A 1 † † pffiffi ⃗ ⃗σ⃗ ⃗ 0 ⃗ ⃗⃗ ⃗ 0 3 S1 · q 2 · q S1 · qS2 · q where the isospin factor τ is ¯ τðΛcΣcÞ¼1; ðA102Þ
τ⃗1 · τ⃗2 τ Ξ Ξ¯ 0 ð c cÞ¼ 4 : ðA103Þ
2 2 2 2 2 2 μ − − μ − 0 − The effective pion mass in the heavy quark limit is given by π ¼ mπ ðmΣc mΛc Þ or π ¼ mπ ðmΞc mΞc Þ , respectively. For the SS¯ potential we use the bases
0 ¯ 0 0 ¯ ¯ 0 ¯ B 0 ¯ 0 Ξ Ξ ; Ξ Ξ ; Ξ Ξ ; Ξ Ξ ; A104 ΞcΞc ¼f c c c c c c c cg ð Þ ¯ ¯ ¯ ¯ B ¯ Σ Σ ; Σ Σ ; Σ Σ ; Σ Σ ; A105 ΣcΣc ¼f c c c c c c c cg ð Þ in which the potential reads
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0 1 ⃗ ⃗ 2 ⃗ ⃗ B þσ⃗1 · q⃗σ⃗2 · q⃗ −λσ⃗1 · q⃗S2 · q⃗ þλS1 · q⃗σ⃗2 · q⃗ −λ S1 · q⃗S2 · q⃗C B C 2 ⃗ ⃗ 2 ⃗ ⃗† ⃗ ⃗ ¯ 2g 1 B −λσ⃗1 · q⃗S2 · q⃗ þσ⃗1 · q⃗Σ2 · q⃗ −λ S1 · q⃗S2 · q⃗ þλS1 · q⃗Σ2 · q⃗C 1 SS ⃗ 2 τ B C O VOPEðqÞ¼ 2 2 2 B † † C þ ; ðA106Þ 9fπ q⃗ þ mπ B λ⃗ ⃗σ⃗ ⃗ −λ2 ⃗ ⃗⃗ ⃗ Σ⃗ ⃗σ⃗ ⃗ −λΣ⃗ ⃗⃗ ⃗C m @ þ S1 · q 2 · q S1 · qS2 · q þ 1 · q 2 · q 1 · qS2 · q A Q 2 ⃗† ⃗† ⃗† ⃗ ⃗ ⃗† ⃗ ⃗ −λ S1 · q⃗S2 · q⃗ þλS1 · q⃗Σ2 · q⃗ −λΣ1 · q⃗S2 · q⃗ þΣ1 · q⃗Σ2 · q⃗
pffiffi 3 where λ ¼ 2 and where the isospin factor is Here we will write the coordinate space potential in coupled channels, in which case we obtain τ⃗1 · τ⃗2 τ Ξ0 Ξ¯ 0 ð c cÞ¼ ; ðA107Þ 2 4 0 g ¯ ¯ ¯ ð Þ ⃗ τ 3 CSTδ3 ⃗ − τ CST SST ˆ VST¯ ðrÞ¼ 2 12 ðrÞ ½ 12 WCðrÞþ 12 ðrÞWTðrÞ ; ¯ ⃗ ⃗ 3fπ τðΣcΣcÞ¼T1 · T2: ðA108Þ ðA112Þ 7. Coordinate space 2 2 ¯ ¯ ð0Þ g3 SS¯ 3 The general form of the ST and SS potential in V ¯ ðr⃗Þ¼τ C δ ðr⃗Þ SS 27 2 12 momentum space can be written as fπ 2 SS¯ SS¯ 2 − τ ½C12 WCðrÞþS12 ðrˆÞWTðrÞ ; ðA113Þ ¯ g a⃗1 · q⃗a⃗2 · q⃗ 9 TS − ¯ 3 ⃗ ⃗ VOBE ¼ R1R2 2 I1 · I2 2 2 ; ðA109Þ 2fπ q þ μπ where WC and WT are defined in Eqs. (64) and (65), and 2 ¯ g a⃗1 · q⃗a⃗2 · q⃗ with the central and tensor matrices given by SS − ¯ 2 ⃗ ⃗ VOPE ¼ R1R2 2 I1 · I2 2 2 ; ðA110Þ 2fπ q þ μπ 0 1 1 1 0 σ⃗ σ⃗ 0 pffiffi ⃗ σ⃗ ¯ B 3 1 · 2 3 S1 · 2 C where R1 and R2 are numerical factors; μπ is the effective B C B 1 1 ⃗ C pion mass at the vertices (as explained in the previous B σ⃗1 · σ⃗2 0 pffiffi σ⃗1 · S2 0 C ¯ B 3 3 C section); I1, I2 the appropriate isospin matrices; and a⃗1, a⃗1 CST ¼ ; 12 B 1 † † † C B 0 pffiffi σ⃗ ⃗ 0 ⃗ ⃗ C are the spin matrices acting on vertex 1 and 2. The specific B 3 1 · S2 S1 · S2 C @ A factors can be worked out easily from Eqs. (A101) and 1 † † pffiffi ⃗ σ⃗ 0 ⃗ ⃗ 0 (A106) to obtain the results of Table IV. The coordinate 3 S1 · 2 S1 · S2 space potential is obtained by Fourier transforming the momentum space potential: ðA114Þ
Z 3 0 1 d q ⃗ ⃗ 2 ⃗ ⃗ Vð0Þ r⃗ Vð0Þ q⃗e−iq⃗·r⃗ þσ⃗1 · σ⃗2 −λσ⃗1 · S2 þλS1 · σ⃗2 −λ S1 · S2 ð Þ¼ 2π 3 ð Þ B C ð Þ B ⃗ ⃗ 2 ⃗ ⃗† ⃗ ⃗ C ¯ B −λσ⃗1 · S2 þσ⃗1 · Σ2 −λ S1 · S2 þλS1 · Σ2 C 2 −μπ r SS g e C12 ¼ B C; ¯ i ⃗ ⃗ ⃗ ∇ ⃗ ∇ B ⃗† 2 ⃗† ⃗ ⃗ ⃗ ⃗ C ¼ R1R2 2 I1 · I2ða1 · Þða2 · Þ ; ðA111Þ @ þλS1 · σ⃗2 −λ S1 · S2 þΣ1 · σ⃗2 −λΣ1 · S2 A 2fπ 4πr 2 ⃗† ⃗† ⃗† ⃗ ⃗ ⃗† ⃗ ⃗ −λ S1 · S2 þλS1 · Σ2 −λΣ1 · S2 þΣ1 · Σ2 where gi ¼ g2 or g3 depending on the case. From this we obtain the expressions we already wrote in Eqs. (61)–(65). ðA115Þ
0 1 0 1 σ⃗ σ⃗ ˆ 0 1ffiffi ⃗ σ⃗ ˆ B 3 S12ð 1; 2; rÞ p S12ðS1; 2; rÞ C B 3 C B 1 1 C σ⃗ σ⃗ ˆ 0 pffiffi σ⃗ ⃗ ˆ 0 ¯ B 3 S12ð 1; 2; rÞ 3 S12ð 1; S2; rÞ C SSTðrˆÞ¼B C; ðA116Þ 12 B 1 † † † C B 0 pffiffi σ⃗ ⃗ ˆ 0 ⃗ ⃗ ˆ C B 3 S12ð 1; S2; rÞ S12ðS1; S2; rÞ C @ A 1 † † pffiffi ⃗ σ⃗ ˆ 0 ⃗ ⃗ ˆ 0 3 S12ðS1; 2; rÞ S12ðS1; S2; rÞ
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0 1 ⃗ ⃗ 2 ⃗ ⃗ þS12ðσ⃗1; σ⃗2; rˆÞ −λS12ðσ⃗1; S2; rˆÞþλS12ðS1; σ⃗2; rˆÞ −λ S12ðS1; S2; rˆÞ B C B ⃗ ⃗ 2 ⃗ ⃗† ⃗ ⃗ C ¯ B −λS12ðσ⃗1; S2; rˆÞþS12ðσ⃗1; Σ2; rˆÞ −λ S12ðS1; S2; rˆÞþλS12ðS1; Σ2; rˆÞ C SSSðrˆÞ¼B C: ðA117Þ 12 B ⃗† 2 ⃗† ⃗ ⃗ ⃗ ⃗ C @ þλS12ðS1; σ⃗2; rˆÞ −λ S12ðS1; S2; rˆÞþS12ðΣ1; σ⃗2; rˆÞ −λS12ðΣ1; S2; rˆÞ A 2 ⃗† ⃗† ⃗† ⃗ ⃗ ⃗† ⃗ ⃗ −λ S12ðS1; S2; rˆÞþλS12ðS1; Σ2; rˆÞ −λS12ðΣ1; S2; rˆÞþS12ðΣ1; Σ2; rˆÞ
8. The partial wave projection transitions are more complicated because in the general − 0 Heavy baryon-antibaryon bound states have well- case js s j can be even or odd. The exceptions are the ¯ ¯ ¯ defined JP quantum numbers. Hence we can simplify B3¯ B3¯ , B6B6, and B6B6 systems: these systems usually have 0 the OPE potential by projecting it into partial waves with well-defined C- or G-parity, which implies that js − s j is well-defined parity and angular momentum. For this we even. Even when C-/G-parity is not well defined, like in the Ξ Σ¯ define the states c c system, the tensor operator involves identical spin- X matrices and is symmetrical under the exchange of particles ¯ ˆ 1 and 2: jSTðjmÞi ¼ Ylml ðrÞjsmsihlmlsmsjjmi; ðA118Þ lm sm l s ¯ ¯ B3¯ B3¯ B6B6 X S12 ¼ S12 ¼ 3σ⃗1 · rˆσ⃗2 · rˆ − σ⃗1 · σ⃗2; ðA123Þ ¯ ˆ jSSðjmÞi ¼ Ylml ðrÞjsmsihlmlsmsjjmi; ðA119Þ ¯ lmlsms B6B6 ⃗ ⃗ ⃗ ⃗ S12 ¼ 3Σ1 · rˆΣ2 · rˆ − Σ1 · Σ2: ðA124Þ where j, m is the total angular momentum and its third As a consequence, the matrix elements for the tensor component for the heavy baryon-antibaryon pair, while operator vanish for odd js − s0j. But if we consider the l; ml and s; ms refer to the angular momentum and spin of ¯ ¯ ¯ 0 ¯ B3¯ B6 and B6B6 systems (for example, the ΞcΞc and ΣcΣc the pair. The product hlmlsmsjjmi is the Clebsch-Gordan coefficient for that particular combination of total, orbital, systems), it is perfectly possible to have a mix of even − 0 and spin angular momentum (notice that, when applied to and odd spin. The odd js s j transitions have a the coupling of angular momentum, the Clebsch-Gordan particularity that it is worth mentioning. The matrix − 0 coefficients are independent of whether we have particles element for the tensor operator for odd js s j changes or antiparticles). The spin wave function can be further sign as follows: decomposed as BB¯ Sj BB¯ − B B¯ Sj B B¯ ; A125 X h 6j sðs 1Þll0 j 6i¼ h 6 j sðs 1Þll0 j 6 i ð Þ jsmsi¼ js1m1ijs2m2ihs1m1s2m2jsmsi; ðA120Þ ¯ j ¯ ¯ j ¯ m1m2 S − S hBB6j sðs 1Þll0 jB6Bi¼ hB6Bj sðs 1Þll0 jBB6i; ðA126Þ with s1 and s2 the spin of the heavy baryon 1 and 2 (either ¯ 1 3 with B ¼ B3 or B6. In fact, the previous two equations 2 or 2). indicate that it is actually a good idea to take explicitly In this basis we can compute the partial wave projection into account the particle coupled channel structure. Now of C12 and S12 as if we consider the bases
0 0 0 0 hðs l Þj m jO12jðslÞjmi B0 ¯ ¯ Z ¼fB3¯ B6;B6B3¯ g; ðA127Þ 2 ˆ 0 0 0 0 O ˆ ¼ d rhðs l Þj m j 12ðrÞjðslÞjmi ¯ ¯ B ¼fB6B6;B6B6g; ðA128Þ j ¼ δ 0 δ 0 O 0 0 ; ðA121Þ jj mm ss ll then for B0 we can write the central and tensor operators as where the total angular momentum and its third component are conserved. The central force C12 conserves in addition ⃗† ⃗ 0 S1 · S2 the orbital angular momentum and the spin: C0 ¼ ; ðA129Þ 12 ⃗ ⃗† S1 · S2 0 Cj δ δ Cj ss0ll0 ¼ ss0 ll0 ls: ðA122Þ ! ⃗† ⃗ S12 S ; S2; rˆ 0 ð 1 Þ The tensor force is more involved. Owing to conservation S12ðrˆÞ¼ ; ðA130Þ ⃗ ⃗† ˆ 0 of parity, jl − l0j must be an even number. The spin S12ðS1; S2; rÞ
074026-30 HEAVY BARYON-ANTIBARYON MOLECULES IN EFFECTIVE … PHYS. REV. D 99, 074026 (2019) while for B they read 0 1 11 − 4 000 ⃗ ⃗† ⃗ B C σ⃗1 · Σ2 S1 · S2 B 11 C ¯ 0 − 00 C12 ¼ ; ðA131Þ B6B6 − B 4 C ⃗ ⃗† ⃗ C12 ð1 Þ¼B C; ðA144Þ S1 · S2 Σ1 · σ⃗2 B 00 9 0 C @ þ 4 A ! ⃗ ⃗† ⃗ 000þ 9 S12ðσ⃗1; Σ2; rˆÞ S12ðS1; S2; rˆÞ 4 S12ðrˆÞ¼ : ðA132Þ ⃗ ⃗† ˆ Σ⃗ σ⃗ ˆ S12ðS1; S2; rÞ S12ð 1; · 2; rÞ 0 1 − 15 000 This increases the complexity of the partial wave pro- B 4 C B 0 − 3 00C jection. However, in most cases we can define states with B B¯ B 4 C C 6 6 ð2−Þ¼B C; ðA145Þ good C- or G-parity, which effectively amounts to 12 B 00− 3 0 C reducing the previous coupled channel systems to a @ 4 A 3 single channel problem. 000− 4 The calculation of the matrix elements is straightforward ¯ in most cases. We begin with the BB system (with B ¼ B3¯ 0 1 11 or B6), for which we have the partial waves − 00000 B 4 C B 0 − 11 0000C ¯ − 1 B 4 C jBBð0 Þi¼fS0g; ðA133Þ B 9 C ¯ B 00þ 000C B6B6 − B 4 C ¯ − 3 3 C12 ð3 Þ¼B C; jBBð1 Þi¼fS1; D1g; ðA134Þ B 000 9 00C B þ 4 C B 0000 9 0 C which lead to the matrix elements @ þ 4 A 9 0 0000þ 4 BB¯ − C12 ð0 Þ¼−3; ðA135Þ ðA146Þ ¯ 10 CBBð1−Þ¼ ; ðA136Þ 12 01 ¯ 0 −3 B6B6 − S12 ð0 Þ¼ ; ðA147Þ BB¯ − −3 −3 S12 ð0 Þ¼0; ðA137Þ ffiffiffi ! p 0 pffiffi 1 ¯ 022 17 3 7 SBB 1− ffiffiffi 0 pffiffi − 0 12 ð Þ¼ p : ðA138Þ B 5 2 5 C 2 2 −2 B qffiffi qffiffi C 17 17 3 2 9 6 B pffiffi − − C ¯ B 5 2 10 5 7 5 7 C ¯ B6B6 − B qffiffi C In terms of complexity, the next case is the B6B6 system, S12 ð1 Þ¼B pffiffi pffiffi C; B − 3 7 3 2 − 108 9 3 C for which the partial waves are B 5 5 7 35 35 C @ A qffiffi pffiffi 9 6 9 3 45 ¯ − 1 5 0 − − jB6B6ð0 Þi¼fS0; D0g; ðA139Þ 5 7 35 14 ðA148Þ ¯ − 3 3 7 7 jB6B6ð1 Þi¼fS1; D1; D1; G1g; ðA140Þ 0 qffiffi qffiffiffiffi 1 3 2 2 0 − pffiffi 3 −9 ¯ 2− 1 5 5 5 B 5 7 35 C jB6B6ð Þi¼fD2; S2; D2; G2g; ðA141Þ B qffiffiffiffi C B 3 7 C B − pffiffi 03 0 C ¯ B 5 10 C ¯ − 3 3 7 7 7 7 B6B6 − jB6B6ð3 Þi ¼ f D3; G3; S3; D3; G3; I3g; ðA142Þ S12 ð2 Þ¼B qffiffi qffiffiffiffi C; B 2 7 9 18 C B 3 3 pffiffi C B 7 10 14 7 5 C which translate into the following matrices: @ qffiffiffiffi A 2 18 15 −9 0 pffiffi − 15 35 7 5 7 ¯ − 0 CB6B6 0− 4 12 ð Þ¼ 3 ; ðA143Þ ðA149Þ 0 − 4
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0 pffiffi pffiffi pffiffiffiffi 1 − 17 51 3 − 3 3 36 − 3 66 0 B 35 35 5 35 q35ffiffiffiffi qffiffiffiffi C B pffiffi pffiffi C B 51 3 − 17 0 − 3 3 9 2 −3 3 C B 35 14 35 7 11 11 C B pffiffi pffiffi C B − 3 3 009 3 00C ¯ B 5 5 C B6B6 − B ffiffi ffiffi ffiffiffiffi C S12 ð3 Þ¼B p p p C: ðA150Þ B 36 − 3 3 9 3 99 9 66 0 C B 35 q35ffiffiffiffi 5 70 35 qffiffi C B pffiffiffiffi pffiffiffiffi C B − 3 66 9 2 0 9 66 − 27 9 3 C B 35 7 11 35 77 11 2 C @ qffiffiffiffi qffiffi A 3 9 3 63 0 −3 11 0011 2 − 22
¯ L S We will continue with the B3¯ B6 system, where we have where C ¼ ηð−1Þ þ , we find the combinations the partial waves ¯ −þ 3 3 5 jB3¯ B6ð1 Þi¼fS1ð−Þ; D1ð−Þ; D1ðþÞg; ðA161Þ ¯ − 1 jB3¯ B6ð0 Þi¼fS0g; ðA151Þ ¯ −þ 3 5 5 5 B¯ B 2 D2 − ; S2 ; D2 ; G2 ; ¯ − 3 3 j 3 6ð Þi¼f ð Þ ðþÞ ðþÞ ðþÞg jB3¯ B6ð1 Þi¼fS1; D1g: ðA152Þ ðA162Þ
For these systems OPE is nondiagonal: the B3¯ B3¯ π vertex is ¯ ¯ ¯ −− 3 3 5 zero and OPE involves the B3¯ B6 → B6B3¯ transition. In fact jB3¯ B6ð1 Þi¼fS1ðþÞ; D1ðþÞ; D1ð−Þg; ðA163Þ 0 there is only one case, the ΞcΞc system, in which OPE does ¯ −− 3 5 5 5 not cancel. For this system either C- or G-parity is well jB3¯ B6ð2 Þi¼f D2ðþÞ; S2ð−Þ; D2ð−Þ; G2ð−Þg; ðA164Þ defined, for which we define the standard states where the number in parentheses is the value of η ¼ 1. ¯ 1 ¯ ¯ jB3¯ B6ðηÞi ¼ pffiffiffi ½jB3¯ B6iþηjB6B3¯ i ; ðA153Þ That is, there is the complication that each partial wave in a 2 particular channel can have a different η. Concrete calcu- with C ¼ ηð−1ÞLþS. With this definition in mind, the lations yield the following matrices: 0 1 projection of the central and tensor operators read 1 − 3 00 ¯ ¯ B C CB3B6 0− −3η; CE 1− @ 1 A 12 ð Þ¼ ðA154Þ 12ð Þ¼ 0 − 3 0 ; ðA165Þ 001 ¯ 10 B3¯ B6 − C12 ð1 Þ¼η ; ðA155Þ 01 0 1 1 − 3 000 B C ¯ ¯ B 0 100C B3B6 − E − S12 ð0 Þ¼0; ðA156Þ C ð2 Þ¼ B C; ðA166Þ 12 @ 0 010A pffiffiffi ¯ 022 0 001 B3¯ B6 − S12 ð1 Þ¼η pffiffiffi : ðA157Þ 2 2 −2 0 1 5 1 0 pffiffi pffiffi ¯ 3 2 2 The next case is the B3¯ B system. As happened with the B C 6 B 5 5 1 C ¯ ¯ SE 1− B pffiffi − C B3¯ B6 case, the B3¯ B6 system involves only nondiagonal 12ð Þ¼ @ 3 2 6 2 A; ðA167Þ ¯ ¯ OPE transitions, i.e., B3¯ B6 → B3¯ B6. We have the partial 1ffiffi 1 1 p 2 2 waves 2 0 qffiffiffiffi qffiffi qffiffiffiffi1 ¯ 1− 3 3 5 jB3¯ B6ð Þi¼fS1; D1; D1g; ðA158Þ 5 − 3 1 3 2 3 B 6 10 2 7 35 C B qffiffiffiffi qffiffiffiffi C ¯ 2− 3 5 5 5 B C jB3¯ B6ð Þi¼fD1; S2; D2; G2g: ðA159Þ B− 3 0 − 7 0 C B 10 10 C E − S12ð2 Þ¼ B qffiffi qffiffiffiffi C: ðA168Þ If we are considering states with well-defined C-parity B 1 3 7 3 6 C B − − − pffiffi C B 2 7 10 14 7 5 C 1 @ qffiffiffiffi A ¯ ¯ ¯ 3 6 5 jB3¯ B6ðηÞi ¼ pffiffiffi ½jB3¯ B6iþηjB6B3¯ i ; ðA160Þ 2 0 − pffiffi 2 35 7 5 7
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Here we have departed from the previous notation of 0 1 1 3 0 − pffiffi pffiffi indicating the channel with a superindex. Instead, we 2 2 E B C use the superindex to indicate that it is exchange or B 1 1 3 C SD 1− B − pffiffi C; A176 nondiagonal OPE. 12ð Þ¼@ 2 2 2 A ð Þ ¯ 3 3 3 6 pffiffi − Lastly, the most complex cases are the B B6 system. The 2 2 2 ¯ partial wave and C-parity structure is identical to the B3¯ B6 case, but now there is also a diagonal or direct piece of OPE 0 qffiffiffiffi qffiffi qffiffiffiffi 1 (besides the nondiagonal piece). The vertex factors R1 and − 1 −3 3 3 3 6 3 ¯ B 2 10 2 7 35 C R2 are different for the direct and exchange pieces, which B qffiffiffiffi qffiffiffiffi C B C has to be taken into account when writing down the B −3 3 037 0 C B 10 10 C potential: D − S12ð2 Þ¼B qffiffi qffiffiffiffi C: B 3 3 7 9 18 C B 3 pffiffi C D E B 2 7 10 14 7 5 C VOPE ¼ VOPE þ VOPE @ qffiffiffiffi A 3 18 15 D ¯ D⃗ ⃗ D D 6 0 pffiffi − ¼ R1 R2 I1 · I2½C12WC þ S12WT 35 7 5 7 E ¯ E⃗ ⃗ E E þ R1 R2 I1 · I2½C12WC þ S12WT : ðA169Þ ðA177Þ
¯ If we ignore the fact that the direct and exchange pieces can Now there is a B6B system for which good G-parity μ 6 have a different effective pion mass π, we can merge states cannot be defined, which is together the direct and exchange central and tensor matri- ces. For this we notice the relation 00 0 ¯ ¯ B ¼fΞcΣ ; ΞcΣcg: ðA178Þ 3 E ¯ E − D ¯ D R1 R2 ¼ 4 R1 R2 : ðA170Þ This system is more involved than usual because we have to ¯ ¯ consider the two particle channels α ¼ ΞcΣc and β ¼ ΞcΣc separately. We can define the central and tensor matrices in This implies that we can reexpress the OPE potential as the particle basis as V VD VE OPE ¼ OPE þ OPE αα αβ αα αβ C12 C12 S12 S12 D ¯ D⃗ ⃗ C S C S ¼ R1 R2 I1 · I2½ 12WC þ 12WT ; ðA171Þ 12 ¼ βα ββ and 12 ¼ βα ββ : ðA179Þ C12 C12 S12 S12 where the central and tensor matrices C12 and S12 are a sum of the direct and exchange pieces In this particle basis, the diagonal central matrices are
3 αα − ββ − D − C PC CD PC − CE PC C12 ð1 Þ¼C12ð1 Þ¼C12ð1 Þ; ðA180Þ 12ðJ Þ¼ 12ðJ Þ 4 12ðJ Þ; ðA172Þ
3 Cαα 2− Cββ 2− CD 2− S PC SD PC − SE PC 12 ð Þ¼ 12ð Þ¼ 12ð Þ; ðA181Þ 12ðJ Þ¼ 12ðJ Þ 4 12ðJ Þ: ðA173Þ
D E while the nondiagonal are given by The C12 and S12 are identical to the ones we discussed in ¯ the B3¯ B6 case; see Eqs. (A165)–(A168). The direct 0 1 1 00 matrices are given by B 3 C αβ − βα − B 1 C 0 1 C12ð1 Þ¼C12ð1 Þ¼@ 0 0 A; ðA182Þ − 5 00 3 B 2 C 001 CD 1− B 5 C 12ð Þ¼@ 0 − 2 0 A; ðA174Þ 003 0 1 2 1 000 B 3 C 0 1 B C 5 B 0100C − 000 Cαβ 2− Cβα 2− B C B 2 C 12ð Þ¼ 12ð Þ¼B C: ðA183Þ B 3 C @ 0010A B 0 þ 2 00C CD ð2− Þ¼B C; ðA175Þ 0001 12 B 00 3 0 C @ þ 2 A 000 3 þ 2 For the tensor matrices we have
074026-33 LU, GENG, and VALDERRAMA PHYS. REV. D 99, 074026 (2019) 0 1 1 3 0 qffiffiffiffi qffiffi qffiffiffiffi1 0 − pffiffi pffiffi 2 2 1 3 3 3 3 B C − 2 3 10 − 2 7 −6 35 B 1 1 3 C B C Sαα 1− B − pffiffi C B qffiffiffiffi qffiffiffiffi C 12 ð Þ¼ 2 2 2 ; ðA184Þ B C @ A B 3 3 037 0 C 3ffiffi 3 3 ββ B 10 10 C p 2 − 2 − 2 S12ð2 Þ¼B qffiffi qffiffiffiffi C: ðA191Þ B 3 3 7 9 18 C B − 3 pffiffi C B 2 7 10 14 7 5 C 0 1 @ qffiffiffiffi A 5 1 3 18 15 0 − pffiffi pffiffi −6 0 pffiffi − B 3 2 2 C 35 7 5 7 αβ B 5 5 1 C S 1− B − pffiffi C 12ð Þ¼@ 3 2 6 2 A; ðA185Þ 1 1 1 − pffiffi − 2 2 2 APPENDIX B: THE ONE PSEUDOSCALAR 0 1 MESON EXCHANGE POTENTIAL IN HEAVY 5 1 0 − pffiffi − pffiffi HADRON CHIRAL PERTURBATION THEORY B 3 2 2 C βα B 5 5 1 C S 1− B − pffiffi − C 1. General form of the potential 12ð Þ¼@ 3 2 6 2 A; ðA186Þ 1 1 1 The potential generated from the exchange of the kaon pffiffi 2 2 2 and the eta can be derived from the same rules we have used to calculate the OPE potential. We will not present a 0 1 1 3 complete derivation here, but simply a quick overview. The 0 − pffiffi − pffiffi B 2 2 C one pseudoscalar meson exchange (OME) potentials are ββ − B 1ffiffi 1 3 C formally identical to the OPE potential. For a general S ð1 Þ¼B − p 2 − 2 C; ðA187Þ 12 @ 2 A pseudo Nambu-Goldstone boson P we can write 3 3 3 − pffiffi − − 2 2 2 2 ¯ g a⃗1 · q⃗a⃗2 · q⃗ VTS −RPR¯ P 3 FPFP ; OME ¼ 1 2 2 2 1 2 2 μ2 ðB1Þ 0 qffiffiffiffi qffiffi qffiffiffiffi 1 fP q þ P − 1 −3 3 3 3 6 3 B 2 10 2 7 35 C 2 B C ¯ g2 a⃗1 · q⃗a⃗2 · q⃗ B qffiffiffiffi qffiffiffiffi C VSS ¼ −RPR¯ P FPFP ; ðB2Þ B 3 7 C OME 1 2 2f2 1 2 q2 μ2 B −3 10 0310 0 C P þ P αα − B C S12 ð2 Þ¼B qffiffi qffiffiffiffi C; B 3 3 7 9 18ffiffi C f P RP R¯ P B 2 7 3 10 14 p C with P the weak decay constant for the meson ; 1 , 2 B 7 5 C P P @ qffiffiffiffi A numerical factors that depend on the meson P; F1 and F2 3 18 15 6 0 pffiffi − flavor factors (which for P π are the isospin factors 35 7 5 7 ¼ we have previously used for OPE); a⃗1 and a⃗1 spin ðA188Þ operators; and μP the effective mass of the meson P, μ2 2 − Δ2 which is P ¼ mP , where mP is the mass of the 0 qffiffiffiffi qffiffi qffiffiffiffi1 meson. Notice the analogy between Eqs. (B1) and (B2) and − 5 − 3 1 3 2 3 Eqs. (A109) and (A110) of Appendix A. B 6 10 2 7 35 C B qffiffiffiffi qffiffiffiffi C The numerical, flavor, and isospin factors of the one eta B C B 3 7 C 10 0 − 10 0 exchange potential are listed in Table XI for all vertices in αβ − B C S12ð2 Þ¼B qffiffi qffiffiffiffi C; ðA189Þ which eta exchange is allowed. The format is similar to B 1 3 7 3 6 C B − − − − pffiffi C Table IV, which was dedicated to OPE. There is a difference B 2 7 10 14 7 5 C @ qffiffiffiffi A worth mentioning: the G-parity of the η meson is positive 3 6 5 −2 0 − pffiffi (in contrast to the negative G-parity of the pion), which 35 7 5 7 η η implies that the signs of the numerical factors R and R¯ are ¯ i i 0 qffiffiffiffi qffiffi qffiffiffiffi1 not the same as for the pion factors Ri and Ri of Table IV.In − 5 3 − 1 3 −2 3 particular we have that B 6 10 2 7 35 C B qffiffiffiffi qffiffiffiffi C B C η ¯ η 0 ⇒ ¯ 0 B 3 7 C Ri Ri > RiRi < ; ðB3Þ − 10 0 − 10 0 βα − B C S12ð2 Þ¼B qffiffi qffiffiffiffi C; ðA190Þ B 1 3 7 3 6 C and vice versa. Besides this, it is easy to see that the B − − − pffiffi C B 2 7 10 14 7 5 C η η @ qffiffiffiffi A strength of the flavor factors F1F2 is in general much 3 6 5 2 0 − pffiffi weaker than the corresponding isospin factors for OPE. For 35 7 5 7 ¯ η η example, in the ΣcΣc system we have that F1F2 ¼ 1=3 for
074026-34 HEAVY BARYON-ANTIBARYON MOLECULES IN EFFECTIVE … PHYS. REV. D 99, 074026 (2019)
TABLE XI. Numerical, isospin, and spin factor associated with TABLE XII. Numerical, isospin, and spin factor associated each vertex of the one eta exchange potential for the ST¯ and SS¯ with each vertex of the one kaon exchange potential for the ST¯ systems. The arrows are used to indicate the final baryon state in and SS¯ systems. The arrows are used to indicate the final baryon the vertex and P indicates the pseudo Goldstone meson in that state in the vertex and P indicates the pseudo Goldstone meson in ¯ vertex. The numerical factors RP and RP are for baryons and P ¯ P σ⃗ Σ⃗ ⃗ i i that vertex. The numerical and spin factors (Ri , Ri and i, i, Si) P σ⃗ P antibaryons respectively, while Fi are the flavor factors. i are are as in Table XI. The flavor factors F are in most cases a ⃗ ⃗ i the Pauli matrices, Σi are the spin S ¼ 3=2 matrices, and Si are number (implicitly multiplied by the identity in isospin space to 2 4 1 2 the × matrices that are used for the transitions from a spin- = guarantee isospin conservation). The exceptions are the Σc → Ξc to a spin-3=2 baryon. family of transitions in which the flavor factors are 2 × 3 matrices θa i , with a denoting which of the two isospin states of the kaon P P ¯ P ⃗ a⃗ Vertex Ri Ri Ii i have been exchanged. qffiffiffiffi q ffiffiffi Ξ → Ξ0 η 1 p σ⃗ c c 2 2 3 i P P ¯ P P ⃗ 3 3 2 Vertex Ri Ri Fi ai pffiffiffi pffiffiffi pffiffiffi qffiffiffiffi q Ξ → Ξ η 1 ⃗† Λ → Ξ0 σ⃗ c c 2 − 2 3 S c c K 2 2 1 i 2 i 3 3 ffiffiffi ffiffiffi Σ → Σ η 2 2 1ffiffi σ⃗ p p † c c 3 3 p i Λ → Ξ K 2 − 2 1 ⃗ 3 c c Si 1 1 1 qffiffiffiffi q Σ → Σ η pffiffi − pffiffi pffiffi ⃗ Σ → Ξ −θ σ⃗ c c 3 3 3 Si c c K 2 2 i i 3 3 1 1 1 † Σ → Σ η pffiffi − pffiffi pffiffi ⃗ pffiffiffi pffiffiffi c c 3 3 3 Si Σ → Ξ K 2 − 2 −θ ⃗ c c i Si Σ → Σ η 2 2 1ffiffi ⃗ qffiffiffiffi q c c 3 3 p Σ 3 i Ξc → Ωc K 2 2 1 σ⃗i 3 3 2 2 1 Ξ0 → Ξ0 η pffiffi σ⃗ pffiffiffi pffiffiffi c c 3 3 2 3 i Ξ → Ω ⃗† c c K 2 − 2 1 S 1 1 1 i Ξ → Ξ0 η pffiffi − pffiffi pffiffi ⃗ c c 3 3 2 3 Si 0 2 2 Σc → Ξc K θi σ⃗i 0 1ffiffi 1ffiffi 1 ffiffi † 3 3 Ξc → Ξc η p − p p ⃗ 1 1 3 3 2 3 Si Σ → Ξ0 K pffiffi − pffiffi θ ⃗ c c 3 3 i Si Ξ → Ξ η 2 2 1 ffiffi ⃗ c c p Σ 1 1 † 3 3 2 3 i Σ → Ξ K pffiffi − pffiffi θ ⃗ c c 3 3 i Si Ω → Ω η 2 2 2ffiffi σ⃗ 2 2 c c 3 3 p i Σ → Ξ K θ ⃗ 3 c c 3 3 i Σi 1 1 2 Ω → Ω η pffiffi − pffiffi pffiffi ⃗ 2 2 c c 3 3 3 S Ξ0 → Ω σ⃗ i c c K 3 3 1 i Ω → Ω η 1ffiffi − 1ffiffi 2ffiffi ⃗† 1 1 c c p p p S Ξ → Ω K pffiffi − pffiffi 1 ⃗ 3 3 3 i c c 3 3 Si Ω → Ω η 2 2 2ffiffi ⃗ 1 1 † c c 3 3 p Σ Ξ0 → Ω K pffiffi − pffiffi 1 ⃗ 3 i c c 3 3 Si Ξ → Ω K 2 2 1 ⃗ c c 3 3 Σi
⃗ ⃗ eta exchange in contrast to T1 · T2 ¼ −2, −1 and 1 for 1 100 0 pffiffi 0 0 þ1=2 −1=2 2 I ¼ , 1, 2 with the OPE potential. θ ¼ 1 ; θ ¼ : ðB5Þ i 0 pffiffi 0 i For the one kaon exchange potential we list the different 2 001 factors in Table XII for half the nonvanishing vertices. We θa write the vertices in order of decreasing strangeness for the The evaluation of the product of two i matrices can be heavy baryons, with a final kaon. The vertices with an done in terms of isospin matrices initial antikaon are indeed identical: X 1 θa†θa τ⃗ ⃗ 1 1 2 ¼ 2 ½ 1 · T2 þ ; ðB6Þ a AðB → B0KÞ¼AðKB¯ → B0Þ; ðB4Þ ⃗ where τ⃗1 and T2 refer to the isospin matrices as evaluated for the initial or final state. where B and B0 denote the initial and final heavy baryons. The flavor factors in Table XII are listed as numbers, or as θa 2. Strength of the eta and kaon exchange potentials the symbol i , which is a matrix in isospin space. When the flavor factor is a number, it is implicitly understood that The strength of eta and kaon exchange can be compared they are multiplied by the identity in isospin space (all to that of OPE using the techniques of Sec. IV. The central θa 1 scale ΛP for the OME potential can be defined as follows: vertices conserve isospin). The matrices i , where a ¼ 2 C refers to the isospin state of the kaon, mediate the 2 1 1 24πf transitions from isospin-1 to isospin- 2 baryons (e.g., ΛP P C ¼ P P P ¯ P 2 ; ðB7Þ 0 jσF1 F2 j μjR R jg Σc → ΞcK). Their explicit expressions are 1 2 i
074026-35 LU, GENG, and VALDERRAMA PHYS. REV. D 99, 074026 (2019) which is analogous to Eq. (76), except for the changes ⃗0 ð0Þ ⃗ hp jVC jpi¼C; ðC1Þ → τ → P P fπ fP and F1 F2 ; ðB8Þ where p⃗ and p⃗0 are the initial and final center-of-mass momentum of the heavy baryon-antibaryon pair and C a ¯ P ¯ P while jR1R2j¼jR1 R2 j. The comparison is in fact direct if coupling. In principle the coupling C depends on a specific we write it as baryon-antibaryon system and its quantum numbers. But HQSS precludes C to depend on the heavy quark spin, τ f2 ΛP j j P Λ which will translate into a reduction of the number of C ¼ P P × 2 × C; ðB9Þ jF1 F2 j fπ couplings. We will explain in the following lines how to do that. which merely involves the evaluation of a few numerical Heavy baryons are jQqqi states with the structure factors. For the eta meson we have ¯ 1 2 B3 ¼jQðqqÞs ¼0ij¼ ; ðC2Þ τ fη L 2 j j 3 6 9 ∼ 1 5 η η ¼ ; ; and 2 . ; ðB10Þ jF F j fπ 1 2 B6 Q qq 1; C3 ¼j ð ÞsL¼1ij¼2 ð Þ where we have taken fη ≃ fK ∼ 160 MeV. By multiplying η Λ B6 ¼jQðqqÞ 1i 3; ðC4Þ these factors it is apparent that C is considerably harder sL¼ j¼2 than Λ . The conclusion is that one eta exchange is very C 0 suppressed with respect to OPE. For one kaon exchange the with sL ¼ , 1 the light quark pair spin and j the total spin. factors are The application of HQSS to the contact-range couplings C implies that they depend on the total light spin jτj f2 ⃗ ⃗ ⃗ ⃗ 1 2 K ∼ 1 5 SL ¼ s⃗L1 þ s⃗L2, but not on the total spin J ¼ j1 þ j2 or K K ¼ ; and 2 . ; ðB11Þ jF1 F2 j fπ ⃗ the total heavy spin SH ¼ s⃗H1 þ s⃗H2. To determine the which are not particularly large. The flavor factor suppres- exact structure we have to study the coupling of the light sion is larger for the lower isospin states, for which the OPE quark spin for different types of heavy baryon-antibaryon is stronger. The outcome is that the one kaon exchange molecules. In the following lines we will explain how to do ð Þ ¯ ð Þ ¯ ð Þ potential is perturbative. this for the B6 B6 , B3¯ B6 systems, ordered by decreasing ΛP Λ The comparison of the tensor scales T and T is similar degree of complexity. We did not list the system except for the existence of factors of the type emPRc and mπ R ¯ ⇒ 0 e c that are included to take into account that the OME B3¯ B3¯ SL ¼ ; ðC5Þ and OPE potentials cease to be valid below a certain distance; see Eq. (98). If we add these factors, we end up for which the light spin structure is trivial because 0 with sL1 ¼ sL2 ¼ . 2 ¯ jτj f emPRc 1. The SS contact potential ΛPðm Þ¼ × P × × Λ ðm Þ; T P P P 2 mπ Rc T p ¯ jF1 F2 j fπ e For a heavy SS baryon-antibaryon system (i.e., SL ¼ 1 ðB12Þ for both baryons) we can decompose the spin wave function into heavy and light components as follows: − where eðmP mπ ÞRc ∼ 3–5 and 2.5–4 for the eta and kaon, X ð Þ ¯ ð Þ − ⊗ respectively. The addition of this factor means that the jB6 B6 ðJ Þi ¼ DSH;SL ðJÞSH SLjJ; ðC6Þ ΛP SH;SL tensor scale T is in general considerably larger than the one for OPE. In particular it can be regarded as a hard scale. where D are the coefficients for this change of basis. They fulfill the condition APPENDIX C: THE CONTACT-RANGE POTENTIAL X 2 1 jDSH;SL ðJÞj ¼ : ðC7Þ The calculation of the contact-range potential will use a SH;SL different set of techniques than the one of the OPE potential. While we derived the OPE potential from the From this decomposition we can calculate the light spin lowest-dimensional Lagrangian compatible with HQSS and components of the contact-range potential: chiral symmetry, for the contact-range potential we will use 0 0 HQSS without any explicit reference to a Lagrangian. We hS ⊗ S jVjS ⊗ S i¼δ 0 δ 0 V ; ðC8Þ H L H L SHSH SLSL SL will first take into account that the lowest order contact- range potential is simply a constant in momentum space, where the light and heavy spin decouple.
074026-36 HEAVY BARYON-ANTIBARYON MOLECULES IN EFFECTIVE … PHYS. REV. D 99, 074026 (2019) rffiffiffi 1 2 2 The general way to carry on the heavy-light spin ¯ − −jB6B ð1 Þi ¼ þ 0 ⊗ 1 − 1 ⊗ 0 decomposition is to consider the spin wave function of 6 3 H L 3 3 H L rffiffiffi the heavy hadrons 1 1 5 ffiffiffi1 ⊗ 1 − 1 ⊗ 2 þ p H L 3 6 H L ; 2 J¼1 J¼1 jH1i¼js s j1i; jH2i¼js s j2i; ðC9Þ H1 L1 H2 L2 ðC14Þ rffiffiffi − 1 2 2 where sH1 , sH2 is the heavy spin; sL1 , sL2 the light spin; and ¯ 1 − 0 ⊗ 1 1 ⊗ 0 þjB6B6ð Þi ¼ 3 H L þ3 3 H L j1, j2 the angular momenta of the two hadrons. When we rffiffiffi couple the two hadrons, we have: 1 1 5 ffiffiffi1 ⊗ 1 1 ⊗ 2 þp H L þ3 6 H L ; 2 J¼1 J¼1 jH1H2i¼js s j1ijs s j2i XH1 L1 H2 L2 ðC15Þ ¼ D ðJÞjðs s ÞS ðs s ÞS ðj1j2ÞJi; SH;SL H1 H2 H L1 L2 L 1 1 SH;SL ¯ − −jB6B ð2 Þi¼þpffiffiffi 0 ⊗ 2 − pffiffiffi 1 ⊗ 1 6 3 H L 6 H L ðC10Þ J¼2
1 þ pffiffiffi 1 ⊗ 2 ; ðC16Þ 2 H L which is merely a detailed version of Eq. (C6), where J¼2 the notation indicates that the heavy spins coupled to SH, 1 1 ¯ − ffiffiffi ffiffiffi the light spins to SL, and the angular momenta to J. The þjB6B6ð2 Þi ¼ − p 0H ⊗ 2L þ p 1H ⊗ 1L 3 6 J¼2 coefficients DSH;SL ðJÞ can in fact be expressed in terms of 1 9-J symbols: ffiffiffi þ p 1H ⊗ 2L : ðC17Þ 2 J¼2 D J SH;SL ð Þ ¯ Finally, for the B6B6 case we have ¼hsH1 sL1 j1sH2 sL2 j2jjðsH1 sH2 ÞSHðsL1 sL2 ÞSLðj1j2ÞJi 8 9 rffiffiffi s 1 s j1 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi< H L1 = − ¯ 0− 0 ⊗ 0 − ffiffiffi 1 ⊗ 1 jB6B6ð Þi ¼ 3 H L p H L ; ðC18Þ 2j1 1 2j2 1 2S 1 2S 1 s 2 s j2 : 3 ¼ ð þ Þð þ Þð H þ Þð L þ Þ: H L2 ; J¼0 SH SL J pffiffiffi rffiffiffiffiffi 5 1 10 − ¯ 1− 0 ⊗ 1 1 ⊗ 0 ðC11Þ jB6B6ð Þi ¼ 3 H L þ 3 3 H L
rffiffiffi 1 2 Finally if we are considering antihadrons, we should − 1 ⊗ 2 3 3 H L ; ðC19Þ consider their behaviors under C-parity to define their spin J¼1 wave functions consistently: they might differ by a sign rffiffiffi 1 2 from the ansatz jsHsLji. − ¯ −jB B¯ ð2 Þi ¼ pffiffiffi 0 ⊗ 2 þ 1 ⊗ 1 ; ðC20Þ If we go back to the SS heavy baryon-antibaryon system, 6 6 3 H L 3 H L ¯ J¼2 for the B6B6 case we find the following: ¯ − −jB B ð3 Þi ¼ 1 ⊗ 2 j 3; ðC21Þ rffiffiffi 6 6 H L J¼ 1 2 ¯ − jB6B6ð0 Þi ¼ pffiffiffi 0 ⊗ 0 þ 1 ⊗ 1 ; ðC12Þ where we have included the minus sign to stress the 3 H L 3 H L J¼0 convention. ¯ Finally for the B6B6 we can also write the decomposition pffiffiffi in the basis with well-defined C-parity for those cases 2 1 ¯ − where it applies: jB6B6ð1 Þi ¼ 0 ⊗ 1 − pffiffiffi1 ⊗ 0 3 H L 3 3 H L rffiffiffi ¯ 1−þ 1 ⊗ 1 2 5 jB6B6ð Þi ¼ H LjJ¼1; ðC22Þ 1 ⊗ 2 þ 3 3 H L : ðC13Þ pffiffiffi J¼1 2 4 ¯ −− jB B6ð1 Þi ¼ 0 ⊗ 1 − pffiffiffi 1 ⊗ 0 6 3 H L 3 3 H L ¯ ¯ For the B6B and B B6 cases, we include a minus sign in rffiffiffi 6 6 1 5 front of the states containing a B¯ to highlight the C-parity − 1 ⊗ 2 6 3 3 H L ðC23Þ convention that we employ here: J¼1
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rffiffiffi ¯ 0− 1 ⊗ 1 2 1 jB3¯ B6ð Þi¼þ H q¯ q¯ jJ¼0; ðC29Þ ¯ −þ jB B6ð2 Þi ¼ 0 ⊗ 2 − pffiffiffi 1 ⊗ 1 : 6 3 H L 3 H L J¼2 ¯ 0− −1 ⊗ 1 jB6B3¯ ð Þi ¼ H qqjJ¼0; ðC30Þ ¯ 2−− 1 ⊗ 2 jB6B6ð Þi ¼ H LjJ¼2: ðC24Þ rffiffiffi 1 2 ¯ 1− − ffiffiffi0 ⊗ 1 1 ⊗ 1 jB3¯ B6ð Þi ¼ p H q¯ q¯ þ 3 H q¯ q¯ ; ðC31Þ From the previous decomposition and applying Eq. (C8) 3 J¼1 we obtain the contact-range potentials of Sec. III. rffiffiffi 1 2 ¯ ¯ ¯ 1− ffiffiffi0 ⊗ 1 1 ⊗ 1 2. The TS=ST contact potential jB6B3¯ ð Þi ¼ þp H qq þ 3 H qq ; ðC32Þ 3 J¼1 For a heavy TS=S¯ T¯ baryon-antibaryon system we have ¯ ¯ to pay attention to the fact that one baryon has SL ¼ 0 and while for the B3¯ B6 and B6B3¯ we include the minus sign in the other S ¼ 1. The expectation is that there will be a front of the states to make the C-parity convention manifest: L ¯ ¯ direct (exchange) contact term for the transition TS → TS rffiffiffi (TS¯ → ST¯ ). The heavy-light decomposition of the potential 2 1 − ¯ 1− 0 ⊗ 1 ffiffiffi1 ⊗ 1 is in this case jB3¯ B6ð Þi ¼ 3 H q¯ q¯ þ p H q¯ q¯ ; ðC33Þ 3 J¼1 rffiffiffi 0 0 0 ⊗ ⊗ 0 ⊗ ⊗ 2 1 hSH ðSL1 SL2 ÞS jVjSH ðSL1 SL2 ÞS i L L ¯ − ffiffiffi þjB6B3¯ ð1 Þi ¼ 0H ⊗ 1qq − p 1H ⊗ 1qq ; ðC34Þ 0 0 3 3 ¼ δ 0 δ 0 hS S jV jS S i; ðC25Þ J¼1 SHSH SLSL L1 L2 SL L1 L2 ¯ − −jB3¯ B ð2 Þi ¼ 1 ⊗ 1¯ ¯ j 2; ðC35Þ where now we have take into account that the light quark 6 H q q J¼ spin of particles 1 and 2 is different. If we have a particle- ¯ − þjB B3¯ ð2 Þi ¼ 1 ⊗ 1 j 2: ðC36Þ antiparticle system, C-parity implies 6 H qq J¼
In the decomposition above only the quark pair with SL ¼ 1 S0 S0 V S S S0 S0 V S S : C26 h L1 L2 j SL j L1 L2 i¼h L2 L1 j SL j L2 L1 i ð Þ is written. The other quark pair is implicitly understood, i.e., ¯ ¯ As a consequence, for the TS=ST case there are two contact 1qq ¼ 1qq ⊗ 0q¯ q¯ ; ðC37Þ couplings corresponding to 1q¯ q¯ ¼ 0qq ⊗ 1q¯ q¯ : ðC38Þ h01jV1j01i¼h10jV1j10i; ðC27Þ From the decomposition and the definitions
h01jV1j10i¼h10jV1j01i: ðC28Þ BD1 ¼h1qqjVj1qqi¼h1q¯ q¯ jVj1q¯ q¯ i; ðC39Þ
That is, a contact term that conserves the spin of particles 1 B 1 ¼h1 jVj1¯ ¯ i¼h1¯ ¯ jVj1 i; ðC40Þ ¯ E qq q q q q qq and 2 and a contact that exchanges it. For the B3¯ B6 and ¯ B6B3¯ the heavy-light spin decomposition reads we obtain the contact-range potentials of Sec. III.
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