Heavy Baryons and Exotics Spectrum
HHeavy Baryon Spectrum and New Heavy Exotics
Marek Karliner a , Harry J. Lipkin b , c and Nils A. Törnqvist da Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Israel b Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel c High Energy Physics Division, Argonne National Laboratory Argonne, IL 60439-4815, USA d Department of Physical Sciences, University of Helsinki, POB 64, FIN-0014 Finland
We discuss several highly accurate theoretical predictions for masses of baryons containingthe b quark which have been recently confirmed by experimental data. Several predictionsare given for additional properties of heavy baryons. We also discuss the two charged exoticresonances Z b with quantum numbers of a ( bbud ) tetraquark, very recently reported by Bellein the channel [ Υ ( nS ) π + , n =
1, 2, 3 ] . Among possible implications are deeply bound I = Z b -s and existence of a Σ + b Σ − b dibaryon, a beauteron . QCD describes hadrons as valence quarks in a sea of gluons and qq pairs. At distancesabove ∼ − quarks acquire an effective constituent mass due to chiral symmetrybreaking. A hadron can then be thought of as a bound state of constituent quarks. In thezeroth-order approximation the hadron mass M is then given by the sum of the masses ofits constituent quarks m i , M = ∑ i m i . The binding and kinetic energies are “swallowed"by the constituent quarks masses. The first and most important correction comes from thecolor hyper-fine (HF) chromo-magnetic interaction, M = ∑ i m i + ∑ i < j V HF ( QCD ) ij ; V HF ( QCD ) ij = v ( (cid:126) λ i · (cid:126) λ j ) (cid:126) σ i · (cid:126) σ j m i m j (cid:104) ψ | δ ( r i − r j ) | ψ (cid:105) (1)where v gives the overall strength of the HF interaction, (cid:126) λ i , j are the SU ( ) color matrices, σ i , j are the quark spin operators and | ψ (cid:105) is the hadron wave function. This is a contactspin-spin interaction, analogous to the EM hyperfine interaction, which is a product of themagnetic moments, V HF ( QED ) ij ∝ (cid:126) µ i · (cid:126) µ j = e (cid:126) σ i · (cid:126) σ j / ( m i m j ) . In QCD, the SU ( ) c generatorstake place of the electric charge. From eq. (1) many very accurate results have been obtainedfor the masses of the ground-state hadrons. Nevertheless, several caveats are in order.First, this is a low-energy phenomenological model, still awaiting a rigorous derivationfrom QCD. It is far from providing a complete description of the hadronic spectrum, butit provides excellent predictions for mass splittings and magnetic moments. The crucialassumptions of the model are: (a) HF interaction is considered as a perturbation which doesnot change the wave function; (b) effective masses of quarks are the same inside mesonsand baryons; (c) there are no 3-body effects. [email protected] a r X i v : . [ h e p - ph ] S e p IV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany
Constituent quark mass differences depend strongly on the flavor of the spectator or“neighbor" quark [1]. For example, m s − m d ≈
180 MeV when the spectator is a light quarkbut the same mass difference is only about 90 MeV when the spectator is a b quark. Sincethese are effective masses , we should not be surprised that their difference is affected by theenvironment, but the large size of the shift is quite surprising and its quantitative derivationfrom QCD is an outstanding challenge for theory. We can extract the ratio of the constituentquark masses from the ratio of the the hyperfine splittings in the corresponding mesons.The hyperfine splitting between K ∗ and K mesons is given by(2) M ( K ∗ ) − M ( K )= v (cid:126) λ u · (cid:126) λ s m u m s [( (cid:126) σ u · (cid:126) σ s ) K ∗ − ( (cid:126) σ u · (cid:126) σ s ) K ] (cid:104) ψ | δ ( r ) | ψ (cid:105) = v (cid:126) λ u · (cid:126) λ s m u m s (cid:104) ψ | δ ( r ) | ψ (cid:105) ,and similarly for hyperfine splitting between D ∗ and D with s → c everywhere. From (2)and its D analogue we then immediately obtain M ( K ∗ ) − M ( K ) M ( D ∗ ) − M ( D ) ≈ m c m s (3) As an example of hyperfine splitting in baryons, let us now discuss the HF splitting in the Σ ( uds ) baryons. Σ ∗ has spin , so the u and d quarks must be in a state of relative spin1. The Σ has isospin 1, so the wave function of u and d is symmetric in flavor. It is alsosymmetric in space, since in the ground state the quarks are in a relative S -wave. On theother hand, the u - d wave function is antisymmetric in color, since the two quarks mustcouple to a ∗ of color to neutralize the color of the third quark. The u - d wave functionmust be antisymmetric in flavor × spin × space × color, so it follows it must be symmetricin spin, i.e. u and d are coupled to spin one. Since u and d are in spin 1 state in both Σ ∗ and Σ their HF interaction with each other cancels between the two and thus the u - d pair doesnot contribute to the Σ ∗ − Σ HF splitting,(4) M ( Σ ∗ ) − M ( Σ ) = v (cid:126) λ u · (cid:126) λ s m u m s (cid:104) ψ | δ ( r rs ) | ψ (cid:105) we can then use eqs. (2) and (4) to compare the quark mass ratio obtained from mesons andbaryons:(5) (cid:18) m c m s (cid:19) Bar = M Σ ∗ − M Σ M Σ ∗ c − M Σ c = (cid:18) m c m s (cid:19) Mes = M K ∗ − M K M D ∗ − M D = (cid:18) m c m u (cid:19) Bar = M ∆ − M p M Σ ∗ c − M Σ c = (cid:18) m c m u (cid:19) Mes = M ρ − M π M D ∗ − M D = IV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany
We find the same value from mesons and baryons ± Σ − Λ mass difference is believed to be due tothe difference between the u − d and u − s hyperfine interactions. Similarly, the Σ c − Λ c mass difference is believed to be due to the difference between the u − d and u − c hyperfineinteractions. We therefore obtain the relation(7) m u − m u m c m u − m u m s Bar / Mes = M Σ c − M Λ c M Σ − M Λ = ≈ ( M ρ − M π ) − ( M D ∗ − M D )( M ρ − M π ) − ( M K ∗ − M K ) = ± b quark instead of the s quark, obtaining the prediction for splittingbetween Σ b and Λ b :(8) M Σ b − M Λ b M Σ − M Λ = ( M ρ − M π ) − ( M B ∗ − M B )( M ρ − M π ) − ( M K ∗ − M K ) = M ( Σ b ) − M ( Λ b ) =
194 MeV [1, 2]. This splitting was measured by CDF [3], withisospin-averaged mass difference M ( Σ b ) − M ( Λ b ) =
192 MeV. There is also the predictionfor the spin splittings, good to 5%(9) M ( Σ ∗ b ) − M ( Σ b ) = M ( B ∗ ) − M ( B ) M ( K ∗ ) − M ( K ) · [ M ( Σ ∗ ) − M ( Σ )] =
22 MeVto be compared with 21 MeV from the isospin-average of CDF measurements [3]. Thechallenge is to understand how and under what assumptions one can derive from QCDthe very simple model of hadronic structure at low energies which leads to such accuratepredictions. In Λ , Λ c and Λ b baryons the light quarks are coupled to spin zero. Therefore the magneticmoments of these baryons are determined by the magnetic moments of the s , c and b quarks,respectively. The latter are proportional to the chromomagnetic moments which determinethe hyperfine splitting in baryon spectra. We can use this fact to predict the Λ c and Λ b baryon magnetic moments by relating them to the hyperfine splittings in the same way asgiven in the original prediction [5] of the Λ magnetic moment. We obtain(10) µ Λ c = − µ Λ · M Σ ∗ c − M Σ c M Σ ∗ − M Σ = µ Λ b = µ Λ · M Σ ∗ b − M Σ b M Σ ∗ − M Σ = − IV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany
On top of the already discussed Σ b with quark content bqq , q = u , d . there are two additionalground-state b -baryons, Ξ b and Ω b : Ξ b : the quark content is bsq . Ξ b can be obtained from an “ordinary" Ξ ( ssd or ssu )by replacing one of the s quarks by a b , with one important difference. In the ordinary Ξ , Fermi statistics dictates that two s quarks must couple to spin-1, while in the groundstate of Ξ b the ( sq ) diquarks have spin zero. Consequently, the Ξ b mass is given by theexpression: Ξ b = m b + m s + m u − v (cid:104) δ ( r us ) (cid:105) / m u m s . The Ξ b mass can thus be predictedusing the known Ξ c baryon mass as a starting point and adding the corrections due to massdifferences and HF interactions:(11) Ξ b = Ξ c + ( m b − m c ) − v ( (cid:104) δ ( r us ) (cid:105) Ξ b − (cid:104) δ ( r us ) (cid:105) Ξ c ) / ( m u m s ) Since the Ξ b and Ξ c baryons contain a strange quark, and the effective constituent quarkmasses depend on the spectator quark, the optimal way to estimate the mass difference ( m b − m c ) is from mesons which contain both s and b or c quarks:(12) m b − m c = ( B ∗ s + B s ) − ( D ∗ s + D s ) = ± M ( Ξ b ) = ± M ( Ξ b ) = ± ± M ( Ξ b ) = ± ±
15 MeV [8]. Ω b : for the spin-averaged Ω b mass we have(13) ( M ( Ω ∗ b ) + M ( Ω b )) = ( M ( Ω ∗ c ) + M ( Ω c )) + ( m b − m c ) B s − D s = ± M ( Ω ∗ b ) − M ( Ω b ) = ( M ( Ω ∗ c ) − M ( Ω c )) m c m b (cid:104) δ ( r bs ) (cid:105) Ω b (cid:104) δ ( r cs ) (cid:105) Ω c = ± M ( Ω b ) = ± M ( Ω ∗ b ) = ± Ω b mass [10], D0 collaboration publishedthe first measurement of Ω b mass [11]: M ( Ω b ) D = ± ( stat . ) ± ( syst . ) MeV . Thedeviation from the central value of our prediction was huge, 113 MeV. Understandably, wewere very eager to see the CDF result. CDF published their result about nine months later, inMay 2009 [12]: M ( Ω b ) CDF = ± ( stat . ) ± ( syst . ) MeV . Fig. 1 shows a comparisonof our predictions for the masses of Σ b , Ξ b and Ω b baryons with the CDF experimentaldata. We have made additional predictions [7, 10] for some excited states of b -baryons. Ourresults are summarized in Table 10 of Ref. [10].The sign in our prediction M ( Σ ∗ b ) − M ( Σ b ) < M ( Ω ∗ b ) − M ( Ω b ) , appears to be counterintu-itive, since the color hyperfine interaction is inversely proportional to the quark mass. Thisreversed inequality is not predicted by other recent approaches [13–15], but it is also seen inthe charm data, M ( Σ ∗ c ) − M ( Σ c ) = ± < M ( Ω ∗ c ) − M ( Ω c ) = ± SU ( ) symmetry breaking gives information about theform of the potential. It is of interest to follow this clue theoretically and experimentally.4 IV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany
Figure 1:
Masses of b -baryons – theoretical predictions [7, 10] vs. experiment. Ordinary hadrons contain either a qq pair or 3 quarks. The possible color representations ofquark combinations are then completely determined by confinement. In a meson the qq pair must couple to a color singlet and in a baryon any two quarks must couple to an anti-tripletof color, to neutralize the color charge of the third quark. The situation is very different inexotic hadrons which contain both qq and qq pairs, eg. a tetraquark with two heavy quarks Q and two light quarks q , QQqq . Such states have important color-space correlations thatare completely absent in ordinary mesons and baryons [16]. One also needs to keep inmind that the q - q interaction is much stronger than q - q interaction. The result is emergenceof color structures that are totally different from those in normal hadrons. In turn, thisleads to some very unusual experimental properties of such states. Until May 2011 theleading candidate has been the X ( ) , which is most likely either a ccqq or a thresholdbound state of D and D ∗ . Given that X ( ) exists, it is fascinating to explore possibleanalogues containing b quarks. General considerations suggest that such states should bemore strongly bound, since the attraction due to color forces is the roughly same, but therepulsion due to kinetic energy is smaller, as E k ∼ p / m Q . Using a simple model, we have5 IV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany suggested that bbqq might be below the BB threshold and bcqq might be below the BD threshold. A crucial difference vs. ordinary mesons is that ( Qq )( Qq ) can form a colorconfiguration which has much stronger binding than . Some of these states have exoticelectric charge, e.g. bdc u → J / ψπ − π − . Their decays have striking experimental signatures:monoenergetic photons and/or pions, e.g. bqc q with I = B c π threshold can decayinto B c π via isospin violation, or electromagnetically into B c γ , both very narrow.Hadrons containing two b quarks, such as double-bottom baryons bbq or bbqq and bbqq tetraquarks have a unique and a spectacular decay mode with two J / ψ -s in the final state.To see this, recall that a b quark can decay via the hadronic mode b → ccs → J / ψ s . If both b quarks in a double-bottom hadron decay this way, for a bb baryon we get ( bbq ) → J / ψ J / ψ ( ssq ) → J / ψ J / ψ Ξ , and similarly for a tetraquark: ( bbqq ) → J / ψ J / ψ ( ssqq ) → J / ψ J / ψ K K , etc., with all final state hadrons coming from the same vertex. This uniquesignature is however hampered by a very low rate expected for such a process, especially ifone uses dimuons to identify the J / ψ -s. It is both challenge and a opportunity for LHCb [16]. Exotic double-bottom hadrons Z b : theoretical prediction and discovery by Belle In 2008 Belle reported [17] anomalously large (by two orders of magnitude) branching ratiosfor the decays Υ ( S ) → Υ ( mS ) π + π − , m =
1, 2. In [18] we suggested that the enhancementis due to an intermediate state of a tetraquark T bb = ( bbud ) and a pion, mediating thetwo-step process Υ ( S ) → T ± bb π ∓ → Υ ( mS ) π + π − We proposed looking for the ( bbud ) tetraquark in these decays as peaks in the invariantmass of Υ ( S ) π + or Υ ( S ) π + systems.Very recently Belle collaboration confirmed this prediction, announcing [19] the observa-tion of two charged bottomonium-like resonances Z b as narrow structures in π ± Υ ( nS )( n =
1, 2, 3 ) and π ± h b ( mP ) ( m =
1, 2 ) mass spectra that are produced in association with asingle charged pion in Υ ( S ) decays.The measured masses of the two structures averaged over the five final states are M = ± M = ± M and M are about 3 MeV above the respective B ∗ B and B ∗ B ∗ thresholds. This strongly suggests a parallel with X ( ) , whose mass is almostexactly at the D ∗ D threshold. It also raises the possibility that such states might have acomplementary description as deuteron-like “molecule" of two heavy mesons quasi-boundby pion exchange [20, 21].The attraction due to π exchange is 3 times weaker in the I = I = I = π contributes, whereas for I = π and π ± contribute. Consequently, in the charm system the I = D ∗ D thresholdand only the I = X ( ) is bound 2 MeV below the average of the isospin-related D + D ∗− IV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany and D D thresholds. The situation is likely to be different in the bottom system. Thisis because the attraction due to π exchange is essentially the same, but the B mesonsare much heavier than D mesons, so the kinetic energy is much smaller by a factor of ∼ m ( B ) / m ( D ) ≈ Z b states are almost bound (or quasi bound) B ∗ B and B ∗ B ∗ I = J P = + statesnear threshold; the neutral members of their isomultiplets have C = − G =+ I = I = π exchange we expect the corresponding I G = + ,J PC = ++ states tobe up to 40-50 MeV below the thresholds [22]. The I = Υ ( S ) . Their expected decay modes are Z b ( I = ) → Υ ( mS ) π + π − and Z b ( I = ) → Υ ( mS ) γ ,as well as Z b ( I = ) → BB γ via B ∗ → B γ , E γ =
46 MeV;which might well be within the reach of LHCb. A ( Σ + b Σ − b ) beauteron dibaryon? The discovery of the Z b states and their probable interpretation as B ∗ B and B ∗ B ∗ bound bypion exchange raises an interesting possibility that a strongly bound Σ + b Σ − b deuteron-likestate might exist, a beauteron . This is because Σ b is about 500 MeV heavier than B ∗ andhaving I =
1, it couples more strongly to pions than B and B ∗ which have I = . Theopposite electric charges of Σ + b and Σ − b provide an additional attraction. A possible decaymode of the beauteron is ( Σ + b Σ − b ) → Λ b Λ b π + π − which might be observable in LHCb. If the beauteron exists, it should also be seen in latticeQCD. Acknowledgements
The work on heavy baryons described here was done in collaboration with B. Keren-Zurand J. Rosner. It was supported in part by a grant from the Israel Science Foundation. Theresearch of H.J.L. was supported in part by the U.S. Department of Energy, Division of HighEnergy Physics, Contract DE-AC02-06CH11357.7
IV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany
References [1] M. Karliner and H.J. Lipkin,hep-ph/0307243, Phys. Lett.