Hidden-Beauty Charged Tetraquarks and Heavy Quark Spin Conservation
aa r X i v : . [ h e p - ph ] D ec Hidden-Beauty Charged Tetraquarks andHeavy Quark Spin Conservation
A. Ali ¶ , L. Maiani ∗ , A.D. Polosa ∗ , V. Riquer ∗ ¶ Deutsches Elektronen-Synchrotron DESY, D-22607 Hamburg, Germany ∗ Dipartimento di Fisica, Sapienza Universit`a di Roma, Piazzale Aldo Moro 2, I-00185 Roma, Italyand INFN Sezione di Roma, Piazzale Aldo Moro 2, I-00185 Roma, Italy
Abstract
Assuming the dominance of the spin-spin interaction in a diquark, we point out that themass difference in the beauty sector M ( Z ′ b ) ± − M ( Z b ) ± scales with quark masses as expectedin QCD, with respect to the corresponding mass difference M ( Z ′ c ) ± − M ( Z c ) ± . Notably, weshow that the decays Υ(10890) → Υ( nS ) π + π − and Υ(10890) → ( h b (1 P ) , h b (2 P )) π + π − arecompatible with heavy-quark spin conservation if the contributions of Z b , Z ′ b intermediate statesare taken into account, Υ(10890) being either a Υ(5 S ) or the beauty analog of Y c (4260). Belleresults on these decays support the quark spin wave-function of the Z states as tetraquarks. Wealso consider the role of light quark spin non-conservaton in Z b , Z ′ b decays into BB ∗ and B ∗ B ∗ .Indications of possible signatures of the still missing X b resonance are proposed.Preprint etraquark interpretation of the hidden charm and beauty exotic resonances has been ad-vanced and studied in considerable detail (see Refs. [1] [2], and [3]). In a recent contribution [4],a new scheme for the spin-spin quark interactions in the hidden charm resonances has been pro-posed, which reproduces well the mass and decay pattern of X (3872), of the recently discovered [5] Z ± , c (3900), Z ± , c (4020), and of the lowest lying J P C = 1 −− Y states.Tetraquark states in the large N c (color) limit of QCD have been considered in [6] and [7] (seealso the review [8] and references therein). Compact tetraquark mesons may have decay widthsas narrow as 1 /N c , contrary to previous beliefs, and therefore they are reasonable candidates fora secondary spectroscopic meson series, in addition to the standard q ¯ q one. Another route tomultiquark meson states is being explored in [9] within the Born-Oppenheimer approximation ofQCD , where examples of selection rules for hadronic transitions have been worked out.In this letter we consider the extension of the scheme presented in [4] for the hidden-charm tothe hidden-beauty resonances Z ± , b (10610) = Z b and Z ± , b (10650) = Z ′ b .These resonances are interpreted as S − wave J P G = 1 ++ states with diquark spin distribution(use the notation | s [ bq ] , s [¯ b ¯ q ] i for diquark spins) | Z b i = | bq , ¯ b ¯ q i − | bq , ¯ b ¯ q i√ | Z ′ b i = | bq , ¯ b ¯ q i J =1 (1)The J P = 1 + multiplet is completed by X b , which is given by the C = +1 combination | X b i = | bq , ¯ b ¯ q i + | bq , ¯ b ¯ q i√ X b and Z b to be degen-erate, with Z ′ b heavier according to M ( Z ′ b ) − M ( Z b ) = 2 κ b (3)where κ b is the strength of the spin-spin interaction inside the diquark. A similar analysis for thehidden-charm resonances has produced the value [4]2 κ c = M ( Z ′ c ) − M ( Z c ) ≃
120 MeV (4)The QCD expectation is κ b : κ c = M c : M b . The ratio can be estimated from the masses reportedin [10] M c M b ≃ . .
18 = 0 .
30 (5)giving 2 κ b ≃
36 MeV, which fits nicely with the observed Z ′ b − Z b mass difference ( ≃
45 MeV).Next we consider another crucial prediction of QCD, namely conservation of the heavy quarkspin in hadronic decays. 1e recall that Z b , Z ′ b are observed in the decays of Υ(10890)Υ(10890) → Z b /Z ′ b + π → h b ( nP ) ππ (6)The Υ(10890) is usually reported as the Υ(5 S ) since its mass is close to the mass of the 5 S statepredicted by potential models. However, a different assignment was proposed in [11], namelyΥ(10890) = Y b , the latter state being a P − wave tetraquark analogous to the Y (4260). A reason forthis assignment is the analogy of Υ(10890) decay (6) with Y (4260) → Z c (3900) + π , with Y (4260)being the the first discovered Y state [12]. Current experimental situation about Υ(10890) is stillin a state of flux. In our opinion, the possibility that Υ(10890) is an unresolved peak involvingboth the Υ(5 S ) and Y b , reported by Belle some time ago [13], is plausible, providing a resolution ofthe observed branching ratios measured at the Υ(10890) [14]. However, this identification is not arequirement in the considerations presented below. In fact, following the assignment of Y (4260) asa P − wave tetraquark with s c ¯ c = 1 [4], one sees that in both cases the initial state in (6) correspondsto s b ¯ b = 1. As is well known h b ( nP ) has s b ¯ b = 0, pointing to a possible violation of the heavy-quarkspin conservation, as suggested in [14].We show now that the contradiction is only apparent. Expressing the states in the the basis ofdefinite b ¯ b and q ¯ q spin, one finds | Z b i = | q ¯ q , b ¯ b i − | q ¯ q , b ¯ b i√ | Z ′ b i = | q ¯ q , b ¯ b i + | q ¯ q , b ¯ b i√ b -systems, as the spin-spin dominant interaction is suppressed by the large b quark mass. In thiscase the composition of Z b , Z ′ b indicated in Eq. (7) would be more general: | Z b i = α | q ¯ q , b ¯ b i − β | q ¯ q , b ¯ b i√ | Z ′ b i = β | q ¯ q , b ¯ b i + α | q ¯ q , b ¯ b i√ M b → ∞ , e.g. | q ¯ q , b ¯ b i gives rise to η b , h b etc., but not Υ b , χ jb etc.. No similar constraintapplies to the light quark spin.We assume α and β to be both different from zero. If either one of the two vanishes, thedecay (6) would be altogether forbidden by heavy quark spin conservation, contrary to what isobserved for the distribution of M ( h b π ) in Ref. [15].Define g Z = g (Υ → Z b π ) g ( Z b → h b π ) ∝ − αβ h h b | q ¯ q , b ¯ b ih q ¯ q , b ¯ b | Υ i g Z ′ = g (Υ → Z ′ b π ) g ( Z ′ b → h b π ) ∝ αβ h h b | q ¯ q , b ¯ b ih q ¯ q , b ¯ b | Υ i (9)2inal State Υ(1 S ) π + π − Υ(2 S ) π + π − Υ(3 S ) π + π − h b (1 P ) π + π − h b (2 P ) π + π − Rel. Norm. 0 . ± . +0 . − . . ± . +0 . − . . ± . +0 . − . . ± . +0 . − . . +0 . . − . − . Rel. Phase 58 ± +4 − − ± +17 − − ± +11 − +44+3 − − +65+74 − − Table 1: Values of the relative normalizations and of the relative phases (in degrees), for s b ¯ b : 1 → → g are the effective strong couplings at the vertices Υ Z b π and Z b h b π . For both assignmentsof Υ(10890), Eq. (8) and heavy quark spin conservation require g Z = − g Z ′ (10)independently from the values of the mixing coefficients α, β .In Ref. [15] the amplitude for the decay (6) is fitted with two Breit-Wigners correspondingto the Z b , Z ′ b intermediate states. Table I therein, that we transcribe here in Table 1, shows therelative normalizations and phases obtained by the fit, for decays into h b (1 P ) and h b (2 P ). Withinlarge errors, consistency with Eq. (9), that is with the heavy-quark spin conservation, is apparent.It is interesting that the same conclusion was drawn using a picture in which Z b , Z b ′ have a“molecular” type structure [16] Z b = | B, ¯ B ∗ i − | ¯ B, B ∗ i√ Z ′ b = | B ∗ , ¯ B ∗ i J =1 (11)To determine α and β separately, one has to resort to s b ¯ b : 1 → → Z b /Z ′ b + π → Υ( nS ) ππ ( n = 1 , ,
3) (12)The effective couplings analogous to (9) are f Z = f (Υ → Z b π ) f ( Z b → Υ( nS ) π ) ∝ | β | h Υ( nS ) | q ¯ q , b ¯ b ih q ¯ q , b ¯ b | Υ i f Z ′ = f (Υ → Z ′ b π ) f ( Z ′ b → Υ( nS ) π ) ∝ | α | h Υ( nS ) | q ¯ q , b ¯ b ih q ¯ q , b ¯ b | Υ i The Dalitz plot of these decays indicate indeed that the transitions (12) proceed mainly through Z b and Z ′ b [14, 15], though the amplitude for the process Υ(10890) → Υ(1 S ) π + π − has a significantdirect component, which is expected in the tetraquark interpretation of the state Υ(10890) [3].This is also reflected in Table 1. A quantitative analysis of the Belle data including the direct andresonant components (i.e., via the intermediate resonant states Z b and Z ′ b ) is required to test the3nderlying dynamics. Leaving this for the future, we argue here that parametrizing the amplitudein terms of two Breit-Wigner as before, one determines the ratio α/β . Indeed, from the Belleresults [15] we find the following weighted average values : s b ¯ b : 1 → . Norm . = 0 . ± .
08 = | α | / | β | Rel . Phase = ( − ± ◦ (13)and s b ¯ b : 1 → . Norm . = 1 . ± . . Phase = (185 ± ◦ . (14)Within errors, the tetraquark assignment in Eqs. (1) and (7) with α = β = 1 is supported.As a side remark, we observe that a Fierz rearrangement similar to the one used in (7) putstogether b ¯ q and q ¯ b fields | Z b i = | b ¯ q , q ¯ b i J =1 | Z ′ b i = | b ¯ q , q ¯ b i + | b ¯ q , q ¯ b i√ b ¯ q and 1 b ¯ q could be viewed as indicating B and B ∗ mesons, respectively, leading to theprediction of the decay patterns Z b → B ∗ ¯ B ∗ and Z ′ b → B ¯ B ∗ [3]. This would not be in agreementwith the Belle data [14].This argument, however, rests on conservation of the light quark spin which, unlike the heavyquark spin, may change when the color octet pairs which appear in (15), evolve into pairs of colorsinglet mesons. Therefore predictions derived from (15) are not as reliable as those derived from (7).We also stress that the issue of the unaccounted direct production of the B ∗ ¯ B ∗ , B ¯ B ∗ and relatedstates in the Belle data, collected at and near the Υ(10890) resonance [14], once satisfactorilyresolved, may also reflect on the resonant contributions Z b → B ∗ ¯ B ∗ and Z ′ b → B ¯ B ∗ . We lookforward to an improved analysis of the Belle measurements.Finally we comment on the expected positive charge conjugation state, X b . On the basis ofthe assumed spin-spin interaction, one predicts M ( X b ) ≃ M ( Z b ) ≃ . < M < . . < M < . X b → Υ(1 S ) ππ (16)so far with negative results.In Ref. [2], it is noted that the near equality of the branching ratios for X (3872) → J/ψ π and X (3872) → J/ψ π can be understood if X (3872) is predominantly isosinglet. The isospin for simplicity we have computed the weighted averages with statistical errors only. J/ψ ω is phase space forbidden and the decay in the
J/ψ ρ mode, although isospinforbidden, is phase space favoured, leading to similar rates.In the X b decay, both ω and ρ channels are allowed by phase space, so that, if X b is isos-inglet, the dominant mode would be into Υ(1 S ) ω . The suggestion therefore is to look at thedecay X b (10600) → Υ(1 S ) 3 π with the 3 π in the ω mass band, in parallel with the search for the X b (10600) → Υ(1 S ) 2 π channel with the 2 π in the ρ band. A search for Y (10890) → γX b → γω Υ(1 S ) has been presented in Ref. [19] leading for the moment to an upper bound only. Acknowledgements
We thank C. Hanhart for interesting discussions.
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