Decays of Zb -> Upsilon + pi via triangle diagrams in heavy meson molecules
aa r X i v : . [ h e p - ph ] O c t KEK-TH-1677J-PARC-TH-30
Decays of Z b → Υ π via triangle diagrams in heavy meson molecules S. Ohkoda , S. Yasui , and A. Hosaka , Research Center for Nuclear Physics (RCNP),Osaka University, Ibaraki, Osaka, 567-0047, Japan KEK Theory Center, Institute of Particle and Nuclear Studies,High Energy Accelerator Research Organization,1-1, Oho, Ibaraki, 305-0801, Japan and J-PARC Branch, KEK Theory Center,Institute of Particle and Nuclear Studies,KEK, Tokai, Ibaraki, 319-1106, Japan
Abstract
Bottomonium-like resonances Z b (10610) and Z ′ b (10650) are good candidates of hadronicmolecules composed of B ¯ B ∗ (or B ∗ ¯ B ) and B ∗ ¯ B ∗ , respectively. Considering Z ( ′ ) b as heavy me-son molecules, we investigate the decays of Z ( ′ )+ b → Υ( nS ) π + in terms of the heavy meson effectivetheory. We find that the intermediate B ( ∗ ) and ¯ B ( ∗ ) meson loops and the form factors play asignificant role to reproduce the experimental values of the decay widths. We also predict thedecay widths of Z + c → J/ψπ + and ψ (2 S ) π + for a charmonium-like resonance Z c which has beenreported recently in experiments. PACS numbers: 12.39.Hg, 13.30.Eg, 13.20.Gd, 14.40.Rt Z b (10610) and Z ′ b (10650) were reported inthe processes Υ(5 S ) → Υ( nS ) π + π − ( n = 1 , ,
3) and Υ(5 S ) → h b ( mP ) π + π − ( m = 1 , I G ( J P ) = 1 + (1 + ), which indicates that the quarkcontent of Z ( ′ ) b must be four quarks as minimal constituents such as | b ¯ bu ¯ d i . The reportedmasses and decay widths of the two resonances are M ( Z b (10610)) = 10607 . ± . Z b (10610)) = 18 . ± . M ( Z b (10650)) = 10652 . ± . Z b (10650)) =11 . ± . B ¯ B ∗ (or B ∗ ¯ B ) and B ∗ ¯ B ∗ thresholds, respectively. In view of these facts, Z b and Z ′ b are likely molecular states of two B ( ∗ ) and ¯ B ( ∗ ) mesons [3–5].More recently, Belle reported the branching fractions of each channel in three-body decaysfrom Υ(5 S ) [6], the results of which are summarized in Table. I. They show a remarkablefeature of Z ( ′ ) b . One is that the dominant decay processes are channels to open flavor mesons,Br(Z +b → B + ¯B ∗ + B ∗ + ¯B ) = 0 .
860 and Br( Z ′ + b → B ∗ + ¯ B ∗ ) = 0 . h b ( mP ) π + decays are not suppressed in spite of their spin-flip processes ofheavy quarks from Υ(5 S ). In general, the spin-nonconserved decay in the strong interactionshould be suppressed due to a large mass of b quark. Nevertheless, the spin-conserveddecay Z ( ′ )+ b → Υ( nS ) π + and spin-nonconserved one Z ( ′ )+ b → h b ( mP ) π + occur in comparableratios. The previous studies suggest that molecular picture explains well this behavior [3, 5]:if the Z ( ′ ) b is a molecular state, the wave function is a mixture state of heavy quark spinsinglet and triplet. Then, Z ( ′ ) b is possible to decay into both channels. Secondly, the decayratios are not simply proportional to the magnitudes of the phase space. In particular, thebranching fraction of Z ( ′ )+ b → Υ( nS ) π + is only approximately ten percents of the one of Z ( ′ )+ b → Υ(2 S ) π + although the phase space of Υ(1 S ) π + is larger than the one of Υ(2 S ) π + .In fact, Γ( Z ( ′ )+ b → Υ(3 S ) π + ) is approximately half a size of Γ( Z ( ′ )+ b → Υ(2 S ) π + ), whichis still wider than the Γ( Z ( ′ )+ b → Υ(1 S ) π + ). The mechanism of this behavior is not stillelucidated completely and needs detailed considerations. In this paper, we focus on thestrong decays Z ( ′ )+ b → Υ( nS ) π + and analyze their decay widths as hadronic molecules.This study will also provide a perspective for the internal structure of Z ( ′ ) b . Our approachalso applies to the decays of Z c (3900), which is charged charmonium-like resonance reportedboth by the BESIII Collaboration [7] and by the Belle collaboration [8].2 ABLE I: Branching ratios (Br) of various decay channels from Z b (10610) and Z ′ b (10650).channel Br of Z b Br of Z ′ b ,Υ(1 S ) π + . ± .
09 0 . ± . S ) π + . ± .
21 2 . ± . S ) π + . ± .
56 1 . ± . h b (1 P ) π + . ± .
10 7 . ± . h b (2 P ) π + . ± .
56 14 . ± . B + ¯ B ∗ + B ∗ + ¯ B . ± . − B ∗ + ¯ B ∗ − . ± . B + ( q ) ¯ B ∗ ( P − q, ǫ ) B + ( q − P + k ) Z b ( P, ǫ Z ) Υ ( p = P − k, ǫ Υ ) π ( k ) (a) i M ( B ) B ¯ B ∗ B + ( q ) ¯ B ∗ ( P − q, ǫ ) Υ ( p = P − k, ǫ Υ ) π ( k ) B ∗ + ( q − P + k, ǫ ) Z b ( P, ǫ Z ) (b) i M ( B ∗ ) B ¯ B ∗ B ∗ + ( q, ǫ ) ¯ B ( P − q ) Z b ( P, ǫ Z ) Υ ( p = P − k, ǫ Υ ) π ( k ) B ∗ + ( q − P + k, ǫ ) (c) i M ( B ∗ ) B ∗ ¯ B FIG. 1: Feynman diagrams for Z + b → Υ( nS ) π + . To start the discussion, we assume that the main components of Z b and Z ′ b are molecularstates of √ ( B ¯ B ∗ − B ∗ ¯ B )( S ) and B ∗ ¯ B ∗ ( S ), namely, | Z b i = 1 √ | B ¯ B ∗ − B ∗ ¯ B i , (1) | Z ′ b i = | B ∗ ¯ B ∗ i . (2) Υ ( p = P − k, ǫ Υ ) π ( k ) B ∗ + ( q, ǫ ) ¯ B ∗ ( P − q, ǫ ) B + ( q − P + k ) Z ′ b ( P, ǫ z ) (a) i M ( B ) B ∗ ¯ B ∗ B ∗ + ( q − P + k, ǫ ) Z ′ b ( P, ǫ z ) B ∗ + ( q, ǫ ) ¯ B ∗ ( P − q, ǫ ) Υ ( p = P − k, ǫ Υ ) π ( k ) (b) i M ( B ∗ ) B ∗ ¯ B ∗ FIG. 2: Feynman diagrams for Z ′ + b → Υ( nS ) π + . B ¯ B ∗ (or B ∗ ¯ B ) and B ∗ ¯ B ∗ thresholds, respectively, and the ratio of D -wave mixingis not large. In fact, the explicit calculations based on the hadronic model in our previousstudy indicate that the probability of the √ ( B ¯ B ∗ − B ∗ ¯ B )( D ) component is approximately9 % and the B ∗ ¯ B ∗ ( D ) component is approximately 6 % in the total wave function of Z b [4].In the hadronic molecular picture, the diagrams contributing to the decay Z ( ′ )+ b → Υ( nS ) π + are described with the intermediate B ( ∗ ) and ¯ B ( ∗ ) meson loops at lowest order [9, 10] as shownin Figs. 1 and 2. Since B + and ¯ B are interchangeable, the total transition amplitudes aregiven by the twice of the sum of each channel as follows, M Z b = 2( M ( B ) B ¯ B ∗ + M ( B ∗ ) B ¯ B ∗ + M ( B ∗ ) B ∗ ¯ B ) , (3) M Z ′ b = 2( M ( B ) B ∗ ¯ B ∗ + M ( B ∗ ) B ∗ ¯ B ∗ ) . (4)To calculate the transition amplitudes, we need the couplings from the effective La-grangians. We adopt the phenomenological Lagrangians at vertices of Z ( ′ ) b and B ( ∗ ) mesons,which are L ZBB ∗ = g ZBB ∗ M z Z µ ( BB ∗† µ + B ∗ µ B † ) , (5) L Z ′ B ∗ B ∗ = ig Z ′ B ∗ B ∗ ǫ µναβ ∂ µ Z ′ ν B ∗ α B ∗† β , (6)where the coupling constants g ZBB ∗ and g Z ′ B ∗ B ∗ are determined from the experimentally ob-served decay widths for the process to open heavy flavor channels from Z ( ′ ) b . The experimen-tal results are Γ( Z + b → B + ¯ B ∗ + B ∗ + ¯ B ) = 15 .
82 MeV and Γ( Z ′ + b → B ∗ + ¯ B ∗ ) = 8 .
44 MeV.We obtain g BB ∗ Z b = 1 .
04 and g B ∗ B ∗ Z ′ b = 1 .
30 to reproduce the observed values.For the other vertices, we employ the effective Lagrangians reflecting both heavy quarksymmetry and chiral symmetry [11], L BB ∗ π = − ig BB ∗ π ( B i ∂ µ π ij B †∗ µj − B ∗ µi ∂ µ π ij B † j ) , (7) L B ∗ B ∗ π = 12 g B ∗ B ∗ π ǫ µναβ B ∗ iµ ←→ ∂ α ¯ B ∗ jβ ∂ ν π ij , (8) L BB Υ = ig BB Υ Υ µ ( ∂ µ BB † − B∂ µ B † ) , (9) L BB ∗ Υ = − g BB ∗ Υ ǫ µναβ ∂ µ Υ ν ( ∂ α B ∗ β B † + B∂ α B ∗† β ) , (10) L B ∗ B ∗ Υ = − ig B ∗ B ∗ Υ (cid:8) Υ µ ( ∂ µ B ∗ ν B ∗† ν − B ∗ ν ∂ µ B ∗† ν ) + ( ∂ µ Υ ν B ∗ ν − Υ ν ∂ µ B ∗ ν ) B ∗† ν + B ∗ µ (Υ ν ∂ µ B ∗† ν − ∂ µ Υ ν B ∗† ν ) (cid:9) , (11)4here B ( ∗ ) = ( B ( ∗ )0 , B ( ∗ )+ ). The two coupling constants g BB ∗ π and g B ∗ B ∗ π are expressed bya single parameter g thanks to heavy quark symmetry as follows: g BB ∗ π = 2 gf π √ m B m B ∗ , g B ∗ B ∗ π = g BB ∗ π √ m B m B ∗ , (12)where f π = 132 MeV is a pion decay constant. Since the decay B ∗ → Bπ is kinematicallyforbidden, it is impossible to determine the coupling g from experiments. Therefore, usingthe experimental information in the charm sector and the heavy quark symmetry, we adoptapproximately g = 0 .
59 when the observed decay width Γ = 96 keV for D ∗ → Dπ is used.The coupling g BB Υ( nS ) of Υ( nS ) and B is estimated on the assumption of vector mesondominance (VMD) [12]. VMD gives the coupling constant g BB Υ( nS ) = M Υ( nS ) /f Υ( nS ) , where f Υ( nS ) is a leptonic decay constant defined by h | ¯ bγ µ b | Υ( nS )( p, ǫ ) i = f Υ( nS ) ǫ µ . Here f Υ( nS ) is determined from the leptonic decays Υ( nS ) → e + e − as f Υ(1 S ) = 715 MeV, f Υ(2 S ) = 497 . f Υ(3 S ) = 430 . g BB Υ(1 S ) = 13 . g BB Υ(2 S ) = 20 . g BB Υ(3 S ) =24 .
7. The other couplings g BB ∗ Υ( nS ) and g B ∗ B ∗ Υ( nS ) are related with g BB Υ( nS ) as g BB Υ( nS ) M B = g BB ∗ Υ( nS ) √ M B M B ∗ = − g B ∗ B ∗ Υ( nS ) M B ∗ . (13)All the above arguments are valid in the heavy quark mass limit. We neglect 1 /m Q correc-tions assuming that the mass of the bottom quark is sufficiently heavy.In terms of the effective Lagrangians, we derive explicitly the transition amplitudes for Z ( ′ ) b → Υ( nS ) + π + as follows: i M ( B ) BB ∗ = ( i ) Z d q (2 π ) [ ig ZBB ∗ M Z ǫ Z · ǫ ][ g BB Υ( nS ) ( ǫ Υ · (2 q − p ))][ g B ∗ B ∗ π ( ǫ · k )] × q ) − m B P − q ) − m B ∗ q − p ) − m B F ( ~q , ~k ) , (14) i M ( B ∗ ) BB ∗ = ( i ) Z d q (2 π ) [ ig ZBB ∗ M Z ǫ Z · ǫ ][ g BB ∗ Υ( nS ) iǫ αβγδ v α ǫ β Υ ǫ γ (2 q − p ) δ ] × [ iǫ abcd g B ∗ B ∗ π M B ∗ v a ǫ b k c ǫ d ] × q ) − m B P − q ) − m B ∗ q − p ) − m B ∗ F ( ~q , ~k ) , (15) i M ( B ∗ ) B ∗ B = ( i ) Z d q (2 π ) [ ig ZBB ∗ M Z ǫ Z · ǫ ][ g BB ∗ π ] × (cid:2) g B ∗ B ∗ Υ( nS ) { ( ǫ Υ · ǫ ) ( ǫ · (2 q − p )) + ( ǫ Υ · ǫ ) ( ǫ · (2 q − p )) − ( ǫ · ǫ ) ( ǫ Υ · (2 q − p )) } (cid:3) × q ) − m B ∗ P − q ) − m B q − p ) − m B ∗ F ( ~q , ~k ) , (16)5 M ( B ) B ∗ B ∗ = ( i ) Z d q (2 π ) [ ig Z ′ B ∗ B ∗ ǫ µναβ P µ ǫ νz ǫ α ǫ β ] × [ ig B ∗ B ∗ Υ( nS ) ǫ δτθφ v δ ǫ τ Υ ǫ θ (2 q − p ) φ ][ g BB ∗ π ( ǫ · k )] × q ) − m B ∗ P − q ) − m B ∗ q − p ) − m B F ( ~q , ~k ) , (17) i M ( B ∗ ) B ∗ B ∗ = ( i ) Z d q (2 π ) [ ig Z ′ B ∗ B ∗ ǫ µναβ P µ ǫ νz ǫ α ǫ β ][ ig B ∗ B ∗ π ǫ τθφ M B ∗ ǫ τ k θ ǫ ] × (cid:2) g B ∗ B ∗ Υ( nS ) { ( ǫ Υ · ǫ ) ( ǫ · (2 q − p )) + ( ǫ Υ · ǫ ) ( ǫ · (2 q − p )) − ( ǫ · ǫ ) ( ǫ Υ · (2 q − p )) } (cid:3) × q ) − m B ∗ P − q ) − m B ∗ q − p ) − m B ∗ F ( ~q , ~k ) , (18)where P ( p , k ) is the momentum of Z ( ′ ) b (Υ( nS ), π meson), and q is the momentum in theloop integrals. We use the polarization vectors ǫ Z and ǫ Υ for Z ( ′ ) b and Υ as well as ǫ , , forthe propagating B ∗ and ¯ B ∗ mesons in the loops. To calculate the square of the absolutevalue of the transition amplitudes, we use the approximation for the polarization vector ofthe B ∗ meson as ǫ B ∗ ≃ λ as P λ ǫ µB ∗ ǫ νB ∗ = δ µν ( µ, ν = 1 , ,
3) and 0 for other µ and ν , because the absolute value of three-momentum ~q isassumed to be much smaller than the mass of B ( ∗ ) meson in heavy quark limit [14].In the above loop calculations, in order to reflect the finite range of the interaction, weuse the form factor F ( ~q , ~k ) as follows, F ( ~q , ~k ) = Λ Z ~q + Λ Z Λ ~k + Λ Λ ~k + Λ . (19)The introduction of the form factor is important. Since Z ( ′ ) b is the loosely bound state of B ( ∗ ) ¯ B ( ∗ ) , the internal B ( ∗ ) and ¯ B ( ∗ ) mesons move slowly almost as on-mass-shell particles.According to this, the loop momentum ~q should be limited within a certain physical scale bythe momentum cutoff parameter Λ Z . In a similar reason, the final state momentum ~k (= ~p )will be controlled by a certain scale given by the momentum cutoff Λ at vertices of Υ B ( ∗ ) B ( ∗ ) and πB ( ∗ ) B ( ∗ ) . Thus, form factors with momentum cutoff are naturally introduced for eachvertex from the view of the molecular picture. Since the scale factors Λ Z and Λ are relatedto the range of the hadron interaction, they should be taken around the typical energy scaleof hadron dynamics. Thus, our formulation can include the finite range effects in a conciseway and regularize the amplitudes by the typical hadron scale.We obtain the decay widths from the given amplitudes in Eqs. (3) and (4). As numericalinputs, all the masses are taken from the data of PDG [13]. The numerical procedure6 ABLE II: The partial decay widths of Z b (10610) + for various cutoff parameters Λ Z in units ofMeV. Λ = 600 MeV is fixed. The left column shows the results without the form factors.Λ Z - 1000 1050 1100 1150 Exp.Υ(1 S ) π + . ± . S ) π + . ± . S ) π + . ± . is as follows: we integrate the amplitudes with q analytically and pick up poles in thepropagators. Since the masses of Z ( ′ ) b are located above the B ¯ B ∗ (or B ∗ ¯ B ) and B ∗ ¯ B ∗ thresholds, respectively, the integrals have singular points. To treat them properly, we dividethe integrals into real and imaginary parts by using the principle value of the integral. In theend, it becomes possible to integrate with three-momentum ~q numerically. This method canbe naturally applied to the calculations of the amplitudes with the form factor. To confirmour calculations, we also adopt another method by a formalism of the Passarino-Veltmanone-loop integral [15, 16]. We obtain an agreement in the numerical results between the twomethods under the condition of the large limit of scale factors (Λ Z , Λ → ∞ ).Tables II and III present the numerical results for the partial decay widths of Z ( ′ ) b .When the form factors are ignored, the decay widths are proportional to | ~k | , namelyΓ( Z ( ′ ) b → Υ( nS ) π + ) ∝ | ~k | . This is much inconsistent with the experimental fact, be-cause the loop integrals without form factors include the high-momentum contributionswhich are not acceptable in the low energy hadron dynamics. In contrast, given the formfactor, our calculations are qualitatively consistent with the experimental results: (i) thedecay to Υ(1 S ) π + is strongly suppressed, (ii) the decay to Υ(2 S ) π + occurs with the highestprobability and (iii) the branching fraction of the decay to Υ(3 S ) π + is smaller than the oneof Υ(2 S ) π + but is still larger than the one of Υ(1 S ) π + . We determine the cutoff parametersΛ Z = 1000 MeV and Λ = 600 MeV to reproduce the experimental values. To see the cutoffdependence, we change Λ Z as Λ Z = 1000, 1050, 1100 and 1150 MeV and verified that theresults do not change much. The main reason for the suppression of the Υ(1 S ) π + decay isin the form factor depending on the final state momentum ~k ( ~p ). In contrast, this effect isminor for Υ(3 S ) π + decay due to the small final state momentum.Finally, we briefly discuss the decays of Z c (3900) in the similar formalism, which hasbeen recently observed in the J/ψπ + invariant mass spectrum of Y (4260) → J/ψπ + π − ABLE III: The partial decay widths of Z b (10650) + . Λ = 600 MeV is fixed. The unit is MeV.Λ Z - 1000 1050 1100 1150 Exp.Υ(1 S ) π + . ± . S ) π + . ± . S ) π + . ± . Z + c . Λ = 600 MeV is fixed. The unit is MeV.Λ Z - 1000 1050 1100 1150 Exp. J/ψπ + ψ (2 S ) π + decay by the BESIII Collaboration [7]. The reported mass and decay width are M ( Z c ) =3899 . ± . ± . Z c ) = 46 ± ±
20 MeV. Belle collaboration also has reported Z c (3900) with mass M ( Z c ) = 3894 . ± . ± . Z c ) = 63 ± ± Z c has the decay properties and the mass spectrum both of which are similarto the Z b case, it is expected that Z c would be the heavy-flavor partner of Z b . Thus, ourmodel can apply to the analysis of the decays Z c → J/ψπ + and ψ (2 S ) π + . In the presentsituation in experiments, branching fractions of Z c have not still been observed. Besides,the decay Z c → ψ (2 S ), which is allowed kinematically, is unconfirmed. For these reasons,the numerical predictions are of benefit to the future experiments.We apply the triangle diagram to the decays of Z c (3900). We assume that Z c is asuperposition state of D ¯ D ∗ and D ∗ ¯ D , namely | Z c i = 1 √ | D ¯ D ∗ − D ∗ ¯ D i . (20)The main difference between Z + c → ψ ( nS ) π + and Z + b → Υ( nS ) π + is the coupling constantsfor each vertex and masses of the hadrons. As numerical inputs for Z c , we use the averagedmasses and decay widths reported by BESIII and Belle. Considering that the branchingfraction of Z + b → B + ¯ B ∗ + B ∗ + ¯ B is known to be 86.0 %, we assume the one of Z + c → D + ¯ D ∗ + D ∗ + ¯ D is also approximately 86 % from the view of the heavy-flavor symmetry.Then, we have the coupling g Z c DD ∗ = 2 .
23 for Z c DD ∗ vertex. The couplings g DDJ/ψ = 7 . g DDψ (2 S ) = 12 . Z c . The width of Z + c → ψ (2 S ) π + is narrower than the one of8 + c → J/ψπ + , owing to the small final state momentum. The predicted branching fractionsare f ( Z + c → J/ψπ + ) = 1 . − . f ( Z + c → ψ (2 S ) π + ) = 0 . − .
33 %, which will betestable for future experiments. Although f ( Z + c → ψ (2 S ) π + ) and f ( Z + b → Υ(1 S )) π + arealmost same probabilities in our calculations, the main factors are different: the former isthe narrow final phase space, the latter is the suppression due to the form factor.In summary, we have studied the Z ( ′ )+ b → Υ( nS ) π + decays in a picture of the heavy mesonmolecule. Assuming that Z ( ′ ) b is the B ∗ ¯ B ( ∗ ) molecular state, we have considered the transitionamplitudes given by the triangle diagrams with B ( ∗ ) and ¯ B ( ∗ ) meson loops at lowest orderbased on the heavy meson effective theory. The couplings of g ZBB ∗ and g Z ′ B ∗ B ∗ are fixedto reproduce correctly the observed decay widths from Z ( ′ ) b to the open flavor channels. Totreat the effect of the finite range of the hadron interactions and regularize the loop integralsin the transition amplitudes suitably, we introduce the phenomenological form factors withthe cutoff parameters Λ Z and Λ. The numerical result with Λ z = 1000 MeV and Λ = 600MeV is qualitatively consistent with the experimental data. Our results suggest that, if Z ( ′ ) b have molecular type structures, the form factor should play a crucial role in the transitionamplitudes. Our model also applies the decays, Z + c → J/ψπ + , ψ (2 S ) π + . We roughlyestimate the branching fractions as f ( Z + c → J/ψπ + ) ∼ . f ( Z + c → ψ (2 S ) π + ) ∼ . Z ( ′ )+ b → η b ρ + , Z ( ′ )0 b → η b γ and so on, which can be also studiedin future experiments.This work was supported in part by Grant-in-Aid for JSPS Fellows (No. 15-5858 (S. O.))and Scientific Research on Priority Areas “Elucidation of New Hadrons with a Variety ofFlavors (E01:21105006) (S. Y. and A. H.)” [1] Belle Collaboration, I. Adachi, (2011), arXiv:1105.4583.[2] Belle Collaboration, A. Bondar et al. , Phys.Rev.Lett. , 122001 (2012), arXiv:1110.2251.[3] A. Bondar, A. Garmash, A. Milstein, R. Mizuk, and M. Voloshin, Phys.Rev. D84 , 054010 D86 , 014004 (2012),arXiv:1111.2921.[5] S. Ohkoda, Y. Yamaguchi, S. Yasui, and A. Hosaka, Phys.Rev.
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