Radiative transitions from Upsilon(5S) to molecular bottomonium
aa r X i v : . [ h e p - ph ] M a y William I. Fine Theoretical Physics InstituteUniversity of Minnesota
FTPI-MINN-11/13UMN-TH-3003/11May 2011
Radiative transitions from
Υ(5 S ) to molecularbottomonium M.B. Voloshin
William I. Fine Theoretical Physics Institute, University of Minnesota,Minneapolis, MN 55455, USAandInstitute of Theoretical and Experimental Physics, Moscow, 117218, Russia
Abstract
The heavy quark spin symmetry implies that in addition to the recently observed Z (10610) and Z (10650) molecular resonances with I G = 1 + there should exist two orfour molecular bottomonium-like states with I G = 1 − . Properties of these G -odd statesare considered, including their production in the radiative transitions from Υ(5 S ), byapplying the same symmetry to the Υ(5 S ) resonance and the transition amplitudes.The considered radiative processes can provide a realistic option for observing the yethypothetical states. he masses and the decay properties of the recently discovered [1] isotriplet resonances Z b (10610) ( Z b ) and Z b (10650) ( Z ′ b ) at the B ∗ ¯ B and B ∗ ¯ B ∗ thresholds strongly suggest thatthese are ‘molecular’ type [2] threshold singularities in the S -wave channels for the cor-responding heavy meson-antimeson pair. The properties of such threshold resonances arerelated by the heavy quark spin symmetry (HQSS) similarly to the relations between theproperties of the B and B ∗ mesons. In particular it has been argued [2] that the existenceof the observed ‘twin’ resonances Z b and Z ′ b resonances and the heavy quark spin sym-metry imply an existence of a larger family of ‘molecular’ peaks at the thresholds of the B and B ∗ meson-antimeson pairs, which family should consist of at least four, but morelikely of six isovector resonances. The observed Z b and Z ′ b states have quantum numbers I G ( J P ) = 1 + (1 + ). The rest of the isovector states in the family have negative G parity andare denoted here as W bJ with two states having J = 0 ( W b and W ′ b ) and one each with J = 1and J = 2: W b and W b . The W bJ states cannot be produced in single-pion transitions fromΥ(5 S ), but can be produced in hadronic transitions with emission of ρ meson from higherΥ-like bottomonium states with mass above approximately 11.4 GeV. At present however nodata are available on such bottomonium states, and we have to consider other possibilitiesfor a study of the expected new molecular resonances. In this paper I discuss the productionof the isotopically (and electrically) neutral components of the expected isovector multipletsin radiative transitions from the Υ(5 S ) resonance: Υ(5 S ) → W bJ γ . Specifically, will beconsidered the relations, following from HQSS, between the rates of such transitions to allthe W bJ resonances. Although a prediction of the absolute rate is currently quite uncertainand is limited to the general expectation that Γ[Υ(5 S ) → W bJ γ ] / Γ[Υ(5 S ) → Z b π ] ∼ α , anunderstanding of the relative rates could serve as a guidance in the searches for the expectedmolecular resonances.The relations between the discussed resonances arise due to the known suppression bythe inverse of the b quark mass of the strong interaction depending on the spin of the b quarkor antiquark. In particular in the limit, where this interaction is completely turned off thespin variables of the heavy quark and antiquark are purely ‘classificational’ in the sense thatthey define the quantum numbers of the states containing the b ¯ b pair, but the dynamicsproceeds as if the heavy quark had no spin at all. In particular in this ‘spinless b ’ (SLB)limit the B mesons behave as heavy particles with spin 1 /
2, and an S -wave B ¯ B pair can bein a state with total angular momentum zero or one, and these states are denoted here as0 − SLB and 1 − SLB respectively, where the superscript indicates the parity. Clearly, in the SLB1imit the dynamical properties, most importantly those arising from the interaction of thelight components of the B mesons, are determined only by these quantum numbers, so thatthere are only two independent channels for the B ¯ B meson pairs at a fixed total isospin. Inparticular, a threshold singularity in one or both channels gives rise to threshold ‘molecular’resonances for the physical meson-antimeson states. It can be seen [2] that for the case wherethe singularity at the threshold (a bound or a virtual state as appropriate for an S -wavescattering) exists only in the isovector 0 − SLB channel, the family of the physical resonancesconsists of two pairs of ‘twin’ peaks: the Z b and Z ′ b resonances, and an additional pair of‘twin’ resonances with the quantum numbers I G ( J P ) = 1 − (0 + ): W b and W ′ b , at respectivelythe B ¯ B and the B ∗ ¯ B ∗ thresholds. If, however, a threshold singularity is in the 1 − SLB channel,or in both channels, then, besides those four resonances, also arise two additional states: W b with I G ( J P ) = 1 − (1 + ) near the B ∗ ¯ B threshold and W b with I G ( J P ) = 1 − (2 + ) near the B ∗ ¯ B ∗ threshold. The resonances W b and W b contain the heavy b ¯ b quark pair in a pure spin-1 (ortho-) state, while each of the pairs of ‘twin’ resonances contains a mixture of the ortho-and para- (spin-0) states of the heavy quark pair. The Z b and Z ′ b resonances are two maximal(45 ◦ ) mixtures, while in the W b and W b resonances the ortho - para mixing is with a 30 ◦ angle. Therefore the HQSS requires that the W b and W b have an unsuppressed couplingto channels with ortho-bottomonium, while the Z b ( Z ′ b ) and W b ( W ′ b ) resonances couple tochannels with ortho- as well as with para- bottomonium with the relative coefficients in thesecouplings determined by the coefficients in the ortho - para mixing within these resonances.It should be mentioned that if only one of the possible SLB channels possesses a near-threshold singularity then it should be the 0 − SLB rather than 1 − SLB , since it is known [3] fromthe general properties of the QCD that the energy of the ground state in a pseudoscalarflavor-non-singlet channel is not larger than that in the vector channel.The described picture of the family of the threshold resonances can be established byconsidering the composition of the SLB spin states 0 − SLB and 1 − SLB with the spin states ofthe b ¯ b pair with the total spin 0 − H and 1 − H in terms of the physical meson-antimeson pairscontaining a B or B ∗ meson and an antimeson. One can find the appropriate compositionswith fixed overall quantum numbers by explicitly reconstructing the eigenstates of the spin-dependent Hamiltonian H s that lifts the SLB degeneracy between those physical states. TheHamiltonian can be written in therms of the spin operators ~s b ( ~s ¯ b ) for the b (¯ b ) quark and ~s q ( ~s ¯ q ) describing the B ( ¯ B ) mesons in the SLB limit (‘the spin of the light (anti)quark’): H s = µ ( ~s b · ~s ¯ q ) + µ ( ~s ¯ b · ~s q ) = µ ~S H · ~S SLB ) − µ ~ ∆ H · ~ ∆ SLB ) , (1)2here ~S H = ~s b + ~s ¯ b , ~S SLB = ~s q + ~s ¯ q , ~ ∆ H = ~s b − ~s ¯ b and ~ ∆ SLB = ~s q − ~s ¯ q . The first expressionin Eq.(1) is the standard phenomenological Hamiltonian for describing the masses of the B and B ∗ mesons: M ( B ) = ¯ M − µ/ M ( B ∗ ) = ¯ M + µ/
4, with ¯ M being the (common)mass of the B and B ∗ mesons in the SLB limit, so that µ = M ( B ∗ ) − M ( B ) ≈
46 MeV.The latter form of the expression in Eq.(1) is convenient for considering the states of themeson-antimeson pairs in terms of the total spin in the
SLB limit and of the total spin ofthe b ¯ b pair. The convenience of the latter form of presenting the Hamiltonian H s arises fromthe fact that the product ( ~S H · ~S SLB ) depends only on the overall total spin of the state | ~J | = | ~S H + ~S SLB | , while the operator ~ ∆ has only non-diagonal matrix elements between thespin-singlet and spin-triplet states. Considering non interacting meson-antimeson pairs, onecan use the Hamiltonian H s to find the eigenstates:1 − (2 + ) : (cid:16) − H ⊗ − SLB (cid:17)(cid:12)(cid:12)(cid:12) J =2 , µ ; (2)1 − (1 + ) : (cid:16) − H ⊗ − SLB (cid:17)(cid:12)(cid:12)(cid:12) J =1 , − µ ; (3)1 − (0 + ) : √ (cid:16) − H ⊗ − SLB (cid:17) + 12 (cid:16) − H ⊗ − SLB (cid:17)(cid:12)(cid:12)(cid:12) J =0 , µ ; (4)1 − (0 + ) : 12 (cid:16) − H ⊗ − SLB (cid:17) − √ (cid:16) − H ⊗ − SLB (cid:17)(cid:12)(cid:12)(cid:12) J =0 , − µ ; (5)1 + (1 − ) : 1 √ (cid:16) − H ⊗ − SLB (cid:17) − √ (cid:16) − H ⊗ − SLB (cid:17) , µ ; (6)1 + (1 − ) : 1 √ (cid:16) − H ⊗ − SLB (cid:17) + 1 √ (cid:16) − H ⊗ − SLB (cid:17) , − µ . (7)In these formulas the first expression indicates the quantum numbers I G ( J P ), the seconddescribes the composition in terms of the heavy and SLB spin states, and the third one givesthe energy of the state relative to the SLB B ¯ B threshold 2 ¯ M . Considering these energyshifts, one readily identifies the states (2), (4) and (6) as being at the B ∗ ¯ B ∗ threshold, thestates (3) and (7) at the B ∗ ¯ B threshold and, finally, the state (5) at the B ¯ B threshold.The major effect of the interaction depending on the spin of the heavy quark is thesplitting of the (otherwise degenerate) thresholds for the scattering channels described byEqs. (2) - (7). The actual interaction between mesons takes place in a finite volume, wherethe effect of the forces related to the spin of the heavy quark is small in comparison with theinteraction giving rise to near threshold singularities in either one or both the 0 − SLB and 1 − SLB channels. Thus one comes to the conclusion that the expected family of the near threshold3olecular resonances is as shown in Fig.1. The existence of the J P = 0 + states W b and W ′ b follows from the existence of the Z b ( Z ′ b ) resonances, while the existence of the W b and W b iscontingent on the presence of a near threshold singularity in the 1 − SLB channel. It can be alsonoted that the W b state is a pure isovector bottomonium-like analog of the charmonium-likeresonance X(3872), which is a pure (1 − H ⊗ − SLC ) state [4]. ❄ ❄ ❄❄❄ ❄ B ∗ ¯ B ∗ B ∗ ¯ BB ¯ B Z ′ b W ′ b W b W b W b Z b Υ π, h b π, η b ρ η b π, χ b π, Υ ρη b π, χ b π, Υ ρ χ b π, Υ ρ χ b π, Υ ρ Υ π, h b π, η b ρ I G ( J P ): 1 + (1 + ) 1 − (0 + ) 1 − (1 + ) 1 − (2 + ) ✴✌ ✎ ❲ Υ(5 S ) ✙ π γ Figure 1: The expected family of six isotriplet resonances at the B ¯ B , B ∗ ¯ B and B ∗ ¯ B ∗ thresh-olds and their likely decay modes to bottomonium and a light meson. The excited bottomo-nium states can be present in the decays instead of the shown lower states ( η b , Υ , h b , χ b ),where kinematically possible. The dashed arrowed lines show the discussed radiative transi-tions from Υ(5 S ). (The mass splitting to Υ(5 S ) is shown not to scale.)Clearly, the H ⊗ SLB spin structure described be Eqs. (2) - (7) also implies relationsbetween the total widths of the W bJ states:Γ( W b ) = Γ( W b ) = 32 Γ( W b ) −
12 Γ( W ′ b ) (8)4s well as relations for the rates of decays of the resonances to specific channels, e.g.Γ( W b → Υ ρ ) : Γ( W ′ b → Υ ρ ) : Γ( W b → Υ ρ ) : Γ( W b → Υ ρ ) = 34 : 14 : 1 : 1 , (9)with a possible slight modification due to the kinematical difference in the phase space. Suchexclusive decays can be used for an experimental identification of the resonances. This paperhowever concentrates on the radiative transitions in which the W bJ states can be produced.The details of the transitions from Υ(5 S ) crucially depend on its structure in terms ofits decomposition into heavy (H) and SLB spin states. The resonance Υ(5 S ) is produced in e + e − annihilation by the electromagnetic current, which creates a b ¯ b pair in an orhtho state1 − H . In the standard notation, the heavy quark pair is created in either a S state, or in a D one. In terms of the H ⊗ SLB decomposition the former is 1 − H ⊗ + SLB and the latteris 1 − H ⊗ + SLB , since the angular momentum of the b ¯ b pair is relegated to the ‘SLB’ system.However the D -wave contribution in the production of the resonance is small inasmuch as theheavy quarks are nonrelativistic at the energy of the Υ(5 S ) resonance and can be neglected.It is thus reasonable to assume that the spin structure of Υ(5 S ) in terms of a H ⊗ SLB decomposition is dominated by 1 − H ⊗ + SLB .To a certain extent the assumed spin structure of the resonance can be tested againstthe available date on its decays. Namely, the relative yield of the meson-antimeson pairs B ∗ ¯ B ∗ , B ∗ ¯ B + B ¯ B ∗ , and B ¯ B significantly depends on this structure. In the limit, where theinteraction of the heavy quark spin is considered as small for a pure 1 − H ⊗ + SLB state the ratioof the yield in these channels is 7:3:1 [5] (see also in the review [6]). In the real world thereare finite effects due to the spin-dependent interaction both in the decay amplitudes and inthe kinematical P -wave factors p due to the mass splitting between the B ∗ and B mesonsdue to the same interaction. If only the phase space factors p are taken into accountthe ratio becomes 4.2:2.4:1. However, given the absence of a full calculation in the firstorder in the spin effects it may be more reasonable to compare the lowest order theoreticalresult with the experimental data. Thus the spread between the expected ratio with andwithout the kinematical factors illustrates the range of current theoretical uncertainty. Thefraction for the yield in each of the three meson-antimeson channels at the Υ(5 S ) resonanceas measured by Belle [7] corresponds to f ( B ∗ ¯ B ∗ ) = (37 . +2 . − . ± . , f ( B ∗ ¯ B + B ¯ B ∗ ) =(13 . ± . ± . , f ( B ¯ B ) = (5 . +1 . − . ± . − H ⊗ + SLB of the Υ(5 S ) resonance appears to not contradict the data.5nother test of the suggested spin structure of Υ(5 S ) is provided by its decays intothe channels B ∗ ¯ B ∗ π ( B ∗ ¯ B + B ¯ B ∗ ) π and B ¯ Bπ . The energy above the threshold for theheavy meson pair in these processes is small, so that only the lowest possible partial waveamplitude can be retained when considering these decays. This is in agreement with theobserved [7] suppression of the channel B ¯ Bπ : f ( B ¯ Bπ ) = (0 . ± . ± . S -wave unlike the two other. In the S wave the states of the heavymeson pair in terms of the H ⊗ SLB decomposition can be read off the formulas (6) for B ∗ ¯ B ∗ and (7) for ( B ∗ ¯ B + B ¯ B ∗ ). The heavy quark spin state is conserved, so that the decaysfrom Υ(5 S ) proceed only to the 1 − H ⊗ − SLB component of the states of these heavy mesonpairs. In other words the underlying process can be viewed as factorized into the transition(1 − H ) Υ(5 S ) → (1 − H ) final for the heavy spin and (0 + SLB ) Υ(5 S ) → (0 − SLB ) final + π for the rest degreesof freedom. Clearly, since the states (6) and (7) contain the spin state 1 − H ⊗ − SLB with thesame amplitude (up to the sign), the ratio of the decay amplitudes of the observed processescan be found as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A [Υ(5 S ) → B ∗ ¯ B ∗ π ( p )] A [Υ(5 S ) → ( B ∗ ¯ B + B ¯ B ∗ ) π ( p )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = E E , (10)where p and p are the momenta of the pion in these two decays, and E and E are thecorresponding energies. (The proportionality of the S -wave amplitude to the pion energy isdictated by the chiral algebra.) Also in Eq.(10) it is implied that state ( B ∗ ¯ B + B ¯ B ∗ ) is the G = +1 state of the heavy meson pairs normalized to one. In these processes the kinematicaleffect of the mass splitting between the B ∗ and B mesons is considerably enhanced by a verysmall released kinetic energy: about 75 MeV in the decay Υ(5 S ) → B ∗ ¯ B ∗ π and about120 MeV in Υ(5 S ) → ( B ∗ ¯ B + B ¯ B ∗ ) π . It thus appears reasonable to take this kinematicaleffect into account. The estimate for the ratio of the total decay rates further depends onthe distribution of the rate over the Dalitz plot. One can estimate the ratio of the relativeyield f ( B ∗ ¯ B ∗ π ) /f ( B ∗ ¯ Bπ + B ¯ B ∗ π ) as approximately 1/2 if the decay goes dominantly intothe lowest invariant mass of the heavy meson pair (i.e. if it is in fact dominated by the Z b and Z ′ b resonances), and as approximately 1/4 if the spectrum of the invariant massesof the heavy pair is given by the phase space. Experimentally [7] the total fractional ratesare: f ( B ∗ ¯ B ∗ π ) = (1 . +1 . − . ± . , f ( B ∗ ¯ Bπ + B ¯ B ∗ π ) = (7 . +2 . − . ± . B ∗ ¯ B ∗ π one can expect, in the suggested here approach, that theDalitz distribution in these decays should be spread over the physical region, rather thanbeing dominated by the Z b and Z ′ b resonances.6he radiative decays Υ(5 S ) → W bJ γ can be considered in terms of the H ⊗ SLB decom-position in the same way as the decays into heavy meson pairs. Indeed, an emission of thephoton by the spin of the heavy quarks is negligible, so that in these decays, as before, the1 − H spin state ‘goes through’without a change in the orientation of the spin, while the photonemission occurs in the process (0 + ) Υ(5 S ) → (1 − SLB ) final + γ , whose polarization structure isdescribed by only one amplitude in terms of the polarization amplitudes of the 1 − SLB state( ~ψ ) and of the photon ( ~a ): A [(0 + ) Υ(5 S ) → (1 − SLB ) final + γ ] = C ω ( ~ψ · ~a ) , (11)where ω is the photon energy, and C is a constant, currently unknown, except its obviousdependence on fine structure constant: C ∝ α . Using this structure of the amplitude andthe amplitudes of the state 1 − H ⊗ − SLB in the W bJ resonances, described by the relations (2)- (5), one can readily find the ratio of the rates of the discussed radiative transitions: f ( W b γ ) : f ( W ′ b γ ) : f ( W b γ ) : f ( W b γ ) = 34 ω : 14 ω : 3 ω : 5 ω , (12)where ω , , are the photon energies in the corresponding transitions: ω ≈ M eV , ω ≈ M eV , and ω ≈ M eV . If the kinematical ω factors are taken into account in Eq.(12),then the ratio of the rates is estimated approximately as 8.5 : 1 : 21 : 20, rather than as3:1:12:20 in the case where one ignores these factors on the grounds that their difference isjust another effect of the heavy quark spin interaction. This in fact illustrates the range ofuncertainty in the predictions based on the heavy quark limit for the discussed transitions.However in spite of such an uncertainty, the presented estimates clearly indicate the relativefeasibility of observing the yet hypothetical W bJ resonances.It can be noted that the isoscalar counterparts of the W bJ resonances can in principle bealso sought for in the radiative transitions from Υ(5 S ). Such C-even molecular resonances X bJ were considered in Ref. [2]. One can expect however that the rates for the radiativetransitions are small as compared to those for the isovectors. Indeed, the b quark and theantiquark are slow in both the initial and the final state in the transition. Thus an emissionof the photon by the heavy quarks can be neglected, and the photon is radiated by thecurrent of the light u and d quarks. In the latter current the isoscalar part is only 1/3 of theisovector one in the amplitude, and one estimatesΓ[Υ(5 S ) → X bJ γ ]Γ[Υ(5 S ) → W bJ γ ] ≈ / . (13)7owever, unlike the W bJ states, the isoscalar resonances X bJ can contain an admixture of χ bJ bottomonium states, and thus can be possibly be produced with an observable rate inhigh-energy p ¯ p and pp collisions at the Tevatron and LHC [2], which provides a viable optionfor their discovery.In summary. The relations between the rates of radiative transitions from Υ(5 S ) tothe (yet hypothetical) isovector molecual bottomonium resonances W bJ with G = − S ) resonance is dominantly a 1 − H ⊗ + SLB stateand that the existing data do not contradict such an assignment. The relations betweenthe rates of the radiative transitions are then found by applying the HQSS to the radiativeprocesses.I am grateful to Alexander Bondar for many enlightening discussions. This work issupported, in part, by the DOE grant DE-FG02-94ER40823.
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