Isovector and hidden-beauty partners of the X(3872) Physics Letters B 732 (2014) 97–100 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Isovector and hidden-beauty partners of the X (3872) Hallstein Høgaasen a, Emi Kou b, Jean-Marc Richard c,∗, Paul Sorba d a Department of Physics, University of Oslo, Box 1048, NO-0316 Oslo, Norway b Laboratoire de l’Accélérateur Linéaire, Université Paris-Sud, IN2P3-CNRS, Centre Scientifique d’Orsay, 91898 Orsay Cedex, France c Université de Lyon, Institut de Physique Nucléaire de Lyon, UCBL–IN2P3-CNRS, 4, rue Enrico Fermi, 69622 Villeurbanne Cedex, France d LAPTh, Laboratoire d’Annecy-le-Vieux de Physique Théorique, CNRS, Université de Savoie, BP 110, 74941 Annecy-le-Vieux Cedex, France a r t i c l e i n f o a b s t r a c t Article history: Received 12 December 2013 Received in revised form 12 March 2014 Accepted 14 March 2014 Available online 19 March 2014 Editor: G.F. Giudice Keywords: Multiquarks Exotic hadrons Chromomagnetism The isovector partners of the X (3872), recently found at BES III, Belle and CLEO-c were predicted in a simple model based on the chromomagnetic interaction among quarks. The extension to the hidden- beauty sector is discussed. © 2014 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3 . 1. Introduction Recently, a new hidden-charm meson was seen at BES III and Belle [1,2]. Its remarkable feature, as compared to most previous X , Y , Z states is that it carries an electric charge. It is currently named X (3900)+. Shortly after its announcement, its existence was confirmed by the Northwestern group working on CLEO-c data [3], who also have some indication for the neutral member of the isospin triplet. Note that three other charged states with hidden charm have been observed, Z (4050)+ , Z (4250)+ and Z (4450)+ , but only by the Belle Collaboration. The Z (4050)+ and Z (4250)+ have been seen by Belle in the B decay [4], but not confirmed in a search by Babar [5]. The Z (4450)± was seen by Belle in the π ±ψ ′ in- variant mass of the B → K π ±ψ ′ decay [6,7], and the quantum numbers 1+ are favoured [8]. To our knowledge, this state was not confirmed in other channels or other experiments. Two charged states have been seen in the hidden-beauty sector, the Zb(10610) ± and the Zb(10650)± , again by the Belle Collabora- tion [9]. The latest result deals with the Zb(10610) 0 discovered by Belle [10], the neutral partner of the Zb (10610) ± . The X (3872) has J P C = 1++ as early indicated in several exper- iments (see, e.g., [11]), and confirmed recently at the Large Hadron Collider of CERN (LHC) (see, e.g., the analysis by LHCb [12]). The * Corresponding author. E-mail addresses: hallstein.hogasen@fys.uio.no (H. Høgaasen), kou@lal.in2p3.fr (E. Kou), j-m.richard@ipnl.in2p3.fr (J.-M. Richard), paul.sorba@lapth.cnrs.fr (P. Sorba). http://dx.doi.org/10.1016/j.physletb.2014.03.027 0370-2693/© 2014 Published by Elsevier B.V. This is an open access article under the CC simplest scenario is that the new X (3900)+ has the same J P quantum numbers as the X (3872), namely J P = 1+. A major issue is whether the X , Y and Z states are mostly molecules, i.e., bound states or resonances made of a flavoured meson and an anti-flavoured meson, or mostly a tetraquark states in which the quark interact directly. An analysis of the produc- tion rate of X (3872) in [13,14] indicates that the measured cross section at Tevatron is too large for a molecule interpretation, even after taking into account the re-scattering effect suggested in [15]. The problem is to find a simple explanation for the approx- imate degeneracy of the isospin I = 0 and I = 1 states. In the molecular model, the X (3872) is mainly a D D̄∗ + c.c. state, and an important contribution to binding comes from the one-pion ex- change, which includes an isospin-dependent factor τ 1.τ 2 whose absolute value is weaker for I = 1 than I = 0.1 In short, the molecular model of X , Y , Z states favours isospin I = 0 states, as did earlier the nucleon–antinucleon model of the baryonium reso- nances [16]. On the other hand, the quark model with a flavour-independent interaction gives a natural explanation to “exchange-degeneracy”, with, e.g., ω and ρ exactly degenerate as long as the quark– antiquark internal annihilation and the coupling to decay channels 1 There is also a change of sign for τ 1.τ 2 , which is + for I = 1 and − for I = 0, but the pion-mediated interaction is off-diagonal in the {D D̄∗, D∗ D̄} basis, and thus the attractive or repulsive character depends on which of the D D̄∗ ± D∗ D̄ combi- nation is considered. BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3 . http://dx.doi.org/10.1016/j.physletb.2014.03.027 http://www.ScienceDirect.com/ http://www.elsevier.com/locate/physletb http://creativecommons.org/licenses/by/3.0/ mailto:hallstein.hogasen@fys.uio.no mailto:kou@lal.in2p3.fr mailto:j-m.richard@ipnl.in2p3.fr mailto:paul.sorba@lapth.cnrs.fr http://dx.doi.org/10.1016/j.physletb.2014.03.027 http://creativecommons.org/licenses/by/3.0/ http://crossmark.crossref.org/dialog/?doi=10.1016/j.physletb.2014.03.027&domain=pdf 98 H. Høgaasen et al. / Physics Letters B 732 (2014) 97–100 are neglected. Thus if the X (3872) and the X (3900)+ have the same J P , it is tempting to seek an explanation in terms of quark dynamics, rather than in a molecular picture.2 Indeed, some mod- els based on quark dynamics have predicted the isospin I = 1 state X (3900)+ near its I = 0 partner X (3872). This is the case for the chromomagnetic model discussed below and for the diquark model of Ref. [17]. This property of exchange degeneracy illustrates the similari- ties and differences between QED and QCD. After the work of De Rújula, Georgi and Glashow [18], it has become widely ac- cepted that the pattern of spin–spin splittings in quark models is similar in structure to that of the hyperfine splittings in atomic physics, namely is due to an interaction among chromomagnetic moments. In the case of the positronium atom, the interaction between the magnetic moments explains only about half of the en- ergy difference between the spin-triplet and the spin-singlet states. The hyperfine splitting in positronium receives a substantial contri- bution from the annihilation diagram where the electron–positron pair goes into a single virtual photon and back to an electron– positron pair. For the usual quark–antiquark mesons, there is no such effect, as the gluon transforms as an octet in colour. But the effect can show up for multiquarks, in which a quark–antiquark pair can be in a colour octet state. We shall discuss later the role of the Pirenne potential, when suitably adapted from QED to quark models. The aim of this article is to revisit how the isospin I = 1 part- ner of the X (3872) was predicted in a simple model [19], and to discuss to which extent the model can be extended toward the hidden-beauty sector. 2. The X (3872) and the X (3900)+ in a chromomagnetic model Some years ago, three of us proposed a simple model for the X (3872) state, described as a (cc̄qq̄) tetraquark [19], in which both the (cc̄) pair and the (qq̄) pair are mostly in a colour-octet state. This structure prevents the state from dissociating freely into a charmonium and a light meson. More precisely, the dynamics of the (cc̄qq̄) in [19] is governed by the chromomagnetic Hamiltonian H = M + H CM = ∑ i mi − ∑ i, j C i j λ̃i · λ̃ j σ i .σ j , (1) where the mi are effective quark masses including the chromo- electric effects, and λ̃i and σ i the colour and spin operator acting on the ith quark, with suitable changes for an antiquark. Should one start from an explicit potential model, then ∑ i mi would stand from the expectation value of the mass and kinetic-energy term, and the last term in (1) represents the expectation value of the spin–spin interaction. Thus C i j includes the intrinsic strength of the chromomagnetic potential divided by the quark masses, and multiplied by the short-range correlation of the quarks i and j. In principle, these terms should vary from a ground state hadron to another one. An empirical observation is that the quantities mi and C i j are nearly constants for i or j denoting u, d, s or c, suggesting the possibility of extrapolating from simple to more complicated configurations. A good surprise in our attempt [19] is that one of the eigenstates of (1) has some of the key properties of X (3872). Moreover, Ref. [19] contains a prediction for the isospin I = 1 partner of X (3872), at 3900 MeV. In the discussion following Eq. (10) of [19], it is stated that “the mostly I = 1 state lies 31 MeV 2 Of course, in case of identical quarks, the Pauli principle can induce some isospin dependence from the spin dependence. This is the reason why the Λ baryon is lighter than the Σ one. But here, this effect is not present, as isospin is carried by a quark and an antiquark. above the mostly I = 0 state”. This calculation includes a mixing effect, as the quark masses mu and md are taken to be different. In the neutral sector, the I = 0 and I = 1 states are left degenerate by the chromomagnetic Hamiltonian (1). Introducing the contribution of the annihilation diagram and different masses for the u and d quark give an additional contribution in the {(cc̄uū), (cc̄dd̄)} basis which reads δ H = ( 2mu − a −a −a 2md − a ) . (2) We now have to fix the value of the parameter a governing the annihilation term. In the positronium atom, the virtual pro- cess e+ + e− → γ → e+ + e− contributes to the hyperfine splitting, in addition to the Breit–Fermi interaction. The effect is given by the Pirenne potential [20]. Its strength is three times that of the Breit–Fermi contact interaction. The analogue for QCD has been discussed in the context of studies on baryonium and other ex- otic states [21–23]. In the perturbative limit, there is an additional factor 2 due to colour, besides the factor 3 in QED. However, as stressed by Gelmini, the annihilation is substantially suppressed by the confinement of the gluons. So, instead of a = 6Cqq̄ , a choice a ∼ Cqq̄ is reasonable. In [19], the values a = 15 MeV and md − mu = 3.5 MeV were adopted, leading to a difference of about 31 MeV between the two eigenvalues, leading the prediction of about 3904 MeV for the neu- tral I = 1 partner of the X (3872). For the charged states of the I = 1 multiplet, δ H is simply re- placed by mu + md , and this puts the charged states about 0.4 MeV below the neutral, mostly isovector, one. 3. Extension to the hidden-beauty sector The difficulty in our model (1) consists in identifying a sin- gle effective mass for a flavoured quark in open-flavour mesons, flavoured baryons and hidden-flavour mesons. The combinations 3(Q q̄)S=1 + (Q q̄)S=0 = 4m Q + 4mq, 2Σ ∗Q + ΣQ + ΛQ = 4m Q + 8mq, 3(Q Q̄ )S=1 + (Q Q̄ )S=0 = 8m Q , (3) should be compatible, and in particular, one should verify δM = 12(Q q̄)S=1 + 4(Q q̄)S=0 − 4Σ ∗Q − 2ΣQ − 2ΛQ − 3(Q Q̄ )S=1 − (Q Q̄ )S=0 = 0. (4) In the charm sector, one gets δM � −200 MeV, which is rather sat- isfactory, but for the beauty sector, the result is δM � 1000 MeV. It indicates that the bottomonium states give an average quark mass mb = 4721 MeV, much lighter than the combination mb = (12B∗ + 4B − 4Σ ∗b − 2Σb − 2Λb )/8 = 4852 MeV deduced from heavy-light systems. This is due to the strong chromoelectric at- traction between two heavy quarks in (bb̄). We thus generalize our model to include a chromoelectric term, and replace (1) by H = M + H CE + H CM = ∑ i mi − ∑ i, j Ai j λ̃i · λ̃ j − ∑ i, j C i j λ̃i · λ̃ j σ i .σ j . (5) Introducing a few non-vanishing chromo-electric coefficients Ai j implies a change of the effective masses. A minimal solution is found with mq = 450 MeV, mc = 1530 MeV, mb = 4860 MeV, and all Ai j = 0, except for Abb = 53 MeV by fitting the spin-averaged ground-state masses of (cc̄), (cq̄), (cqq) and the c → b analogues. H. Høgaasen et al. / Physics Letters B 732 (2014) 97–100 99 Table 1 Colourmagnetic Hamiltonian −H CM in the basis (6). ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 16C 13 − 163 C 24 0 0 0 8 √ 2 3 (C 23 + C 12) 0 0 − 163 C 13 + 16C 24 0 8 √ 2 3 (C 23 + C 12) 0 0 0 0 − 163 (C 13 + C 24) 0 0 8 √ 2 3 (C 23 − C 12) 0 8 √ 2 3 (C 23 + C 12) 0 23 C 24 − 2C 13 283 C 23 − 83 C 12 0 8 √ 2 3 (C 23 + C 12) 0 0 283 C 23 − 83 C 12 −2C 24 + 23 C 13 0 0 0 8 √ 2 3 (C 23 − C 12) 0 0 23 (4C 12 + 14C 23 + C 13 + C 24) ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ Table 2 Parameters of the model: masses mi , non-vanishing chromoelectric Ai j and chro- momagnetic C i j coefficients (in MeV). mq mc mb Abb Cqq Cqc Ccc Cqb Cbb 450 1530 4860 52 20 6 5.2 1.9 3.2 Table 3 Masses or mass differences of ground state hadrons in the model (in GeV). State J /ψ J /ψ − ηc D D∗ − D Λc Σc − Λc Σ ∗c − Σc Exp. 3.10 0.117 1.87 0.141 2.29 0.166 0.065 Model 3.09 0.111 1.88 0.128 2.27 0.149 0.096 State Υ Υ − ηb B B∗ − B Λb Σb − Λb Σ ∗b − Σb Exp. 9.46 0.069 5.28 0.046 5.62 0.194 0.020 Model 9.46 0.068 5.28 0.041 5.60 0.193 0.030 A slightly better agreement is found by allowing both Acc or Abb to be non-zero, but we shall keep the minimal solution. We use the basis defined in [19], namely α1 = (q1q3)10 ⊗ (q2q4)11, α2 = (q1q3)11 ⊗ (q2q4)10, α3 = (q1q3)11 ⊗ (q2q4)11, α4 = (q1q3)80 ⊗ (q2q4)81, α5 = (q1q3)81 ⊗ (q2q4)80, α6 = (q1q3)81 ⊗ (q2q4)81, (6) where the superscript denotes the colour 1 or 8, and the subscript 0 or 1 denotes the spin, with an overall recoupling to a colour- singlet J P = 1+ state. The matrix elements of the colour-magnetic part have been given in [19], and are reminded in Table 1 for completeness. One should now supplement it by the matrix elements of the chromo-electric term, which are H CE = ⎛ ⎜⎜⎜⎜⎜⎝ Xa 0 0 Xb 0 0 0 Xa 0 0 Xb 0 0 0 Xa 0 0 Xb Xb 0 0 Xc 0 0 0 Xb 0 0 Xc 0 0 0 Xb 0 0 Xc ⎞ ⎟⎟⎟⎟⎟⎠ , (7) with Xa = − 16 3 ( A13 + A24), Xb = 4 √ 2 3 ( A12 + A34 − A14 − A23), Xc = 2( A13 + A24) − 4( A12 + A34) − 14( A14 + A23) 3 . (8) The parameters are summarized in Table 2. The ground-state masses of heavy quarkonia and heavy light mesons obtained using these parameters are listed in Table 3 4. Results The Hamiltonian is now diagonalized, using the parameters of Table 2 fitting some ground-state ordinary hadrons containing the same quarks, q, c, b and the associated antiquarks. In the (cc̄qq̄) sector, one obtains results identical to the ones reported in [19], with in particular, a state of mass very close to 3872 MeV which is a pure α6 state. It was then argued that if Cq̄c is taken slightly larger than Cqc , then a small α3 component is admixed, that is responsible for the observed decay of X (3872) into J /ψ and a light vector meson. In our model, when the wave function is expressed in the (cq̄)(c̄q) basis, it has a large colour- singlet–colour-singlet component which corresponds to a decay into D D̄∗ or c.c., which is, however, strongly suppressed by the lack of phase-space for the X (3872). The X (3900)+ is less known experimentally. We refer to a very recent review by Olsen [24].3 The width is given as 46 ± 22 MeV. The decay proceeds mainly through D D̄∗ + c.c., and benefits for this channel from a much more favourable phase-space than for the X (3872) [25]. Our model predicted a dominance of this decay into a charm-carrying vector plus a charmed pseudoscalar config- uration when phase-space opens up. In contrast to what happens for the X (3872), this superallowed decay becomes more important than the decays into (cc̄) + (qq̄). In this latter sector, the discovery channel of the X (3900)+ was J /ψ + π . In our model, as in the case of the X (3872), introducing Cq̄c = Cqc generates a small α3 component in the wave function of the X (3900)+ that induces a decay into J /ψ and a charged vector meson. The J /ψ + π decay involves an α2 component that is not provided in our simple model. Similarly, a decay involving ηc would require an α1 component, or a spin-flip in the decay, which is suppressed, as discussed, e.g., in [26]. In the hidden-beauty sector, one gets an analogue state of mass about 10.62 GeV, and a wave function ∑ i bi αi with {bi } ∝ {0, 0, 0, 0, 0, 1}. This means that this is a pure octet–octet state, so that the fall-apart decay into (bb̄) + (qq̄) is suppressed. This state is about 11 MeV above the B B̄∗ threshold, and thus slightly more un- stable with respect to this threshold, as compared to the X (3872) with respect to the D D̄∗ threshold. As for the X (3872), introducing some departure from Cbq = Cbq̄ would induce a small component consisting of a J /ψ and a light vector meson. As the breaking of exchange degeneracy and isospin symmetry occurs through light quarks, we except the same spacings between isospin I = 0 and I = 1 as in the hidden-charm sector, and same spacing among the neutral and charged states in the I = 1 triplet. Note that for the quartet of (bb̄qq̄) states predicted near 10.62 GeV, the chromoelectric term gives a repulsion of about 35 MeV. As, e.g., when deriving the short-range part of the nucleon–nucleon interaction [27], estimating the masses and prop- erties of multiquark states implies some speculation on the colour dependence of the effective interaction. The chromoelectric term in Eq. (5) corresponds to a colour-octet exchange, which is the most 3 This paper was posted after the first version of the present article. 100 H. Høgaasen et al. / Physics Letters B 732 (2014) 97–100 reasonable choice for a pairwise interaction, as a colour-singlet ex- change would confine everything together. But multi-body forces could be envisaged in more complicated models. 5. Summary and conclusions In this article, it was reminded that a simple quark model [19] predicted the existence of an I = 1 partner of the X (3872) at the right mass and thus anticipated the recent discovery by BES III, Belle and CLEO-c [1–3]. The model consisted of effective masses and a chromomagnetic interaction. It can be supplemented by a minimal chromoelectric term and then applied to the sector of states with hidden-beauty. The model predicts a nearly degenerate quartet (an I = 0 sin- glet and an I = 1 triplet, with some mixing of the neutrals) near 10.62 MeV. The charged states are possible candidates for either the Zb(10610) ± or Zb(10650)± states of Belle [4]. It is, however, very difficult in this approach to produce an isospin I = 1 state without a nearby I = 0 partner, and to arrange two nearly degen- erate isotriplets. It seems important to use the most advanced accelerators and detectors to investigate this sector of hadron physics. The Belle II facility [28] will of course provide us with crucial information. But the search is already active at the LHC, with in particular, a very recent search for the Xb by the CMS Collaboration [29], with no evidence in the Υ (1S)π +π − channel. It is hoped that the com- bined efforts at lepton and hadron colliders will definitely clarify the situation in the hidden-beauty sector. Acknowledgement We thank B. Grinstein for a useful correspondence. References [1] M. Ablikim, et al., BES III Collaboration, Phys. Rev. Lett. 110 (2013) 252001, arXiv:1303.5949. [2] Z. 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