Heavy Quark Symmetry Predictions for Weakly Bound B-Meson Molecules
HHeavy Quark Symmetry Predictions for Weakly Bound B-MesonMolecules
Thomas Mehen ∗ and Joshua W. Powell † Department of Physics, Duke University, Durham, NC 27708 (Dated: October 1, 2018)
Abstract
Recently the Belle collaboration discovered two resonances, Z b (10610) and Z b (10650), that lievery close to the B ¯ B ∗ and B ∗ ¯ B ∗ thresholds, respectively. It is natural to suppose that these aremolecular states of bottom and anti-bottom mesons. Under this assumption, we introduce aneffective field theory for the Z b (10610) and Z b (10650), as well as similar unobserved states thatare expected on the basis of heavy quark spin symmetry. The molecules are assumed to arise fromshort-range interactions that respect heavy quark spin symmetry. We use the theory to calculateline shapes in the vicinity of B ( ∗ ) ¯ B ( ∗ ) thresholds as well as two-body decay rates of the new bottommeson bound states. We derive new heavy quark spin symmetry predictions for the parametersappearing in the line shapes as well as the total and partial widths of the states. ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - ph ] S e p ecently the BELLE collaboration observed two resonances, Z b (10610) and Z b (10650),in the decays Υ(5 S ) → Υ( nS ) π + π − ( n = 1 , , or 3) and Υ(5 S ) → h b ( mP ) π + π − ( m = 1 or2) [1]. The Z b (10610) and Z b (10650) (which we will refer to as Z b and Z (cid:48) b below) have widthsof about 15 MeV, and their masses lie a few MeV above the B ¯ B ∗ and B ∗ ¯ B ∗ thresholds,respectively. However, an analysis in Ref. [2] concludes that an interpretation of the statesas bound states lying below the B ( ∗ ) ¯ B ∗ threshold is still consistent with the available dataon the decays Υ(5 S ) → Z ( (cid:48) ) b π → h b π + π − . This conclusion depends on using line shapesfor the Z ( (cid:48) ) b that account for the coupling of the Z ( (cid:48) ) b to the nearby B ( ∗ ) ¯ B ∗ thresholds ratherthan the Breit-Wigner form that was used in the experimental analysis. The experimentalanalysis favors the quantum numbers I G ( J P ) = 1 + (1 + ) for the Z b and Z (cid:48) b states. Argumentsbased on heavy quark spin symmetry [3, 4] indicate that there should be similar states called W b and W (cid:48) b with quantum numbers I G ( J P ) = 1 − (0 + ), as well as possibly W b and W b with quantum numbers 1 − (1 + ) and 1 − (2 + ), respectively.In this paper, we will assume that these states are weakly bound molecules of heavymesons. This is the approach adopted in Refs. [5–7]. For alternative interpretations of thesestates as tetraquarks, see Refs. [8–10]. Ref. [25] uses existence of the X(3872) and argumentsbased on heavy quark symmetry to argue that molecular states in the bottom sector mustexist. Assuming the states are weakly bound molecules means they can be studied using alow energy effective field theory (EFT) that consists of nonrelativistic kinetic terms for themesons and contact interactions whose coefficients are tuned to produce the bound stateswith energies close to threshold. A theory of this kind called XEFT has been developed for X (3872), which is thought to be a shallow S -wave bound state of D ¯ D ∗ + ¯ D D ∗ [11–14].The purpose of this paper is to construct the analogous theory for the isovector Z b , Z (cid:48) b , and W ( (cid:48) ) bJ states. This theory is similar in structure to the pionless effective theory used for verylow energy nuclear physics [15, 16]. It will be used to derive line shapes for the resonancesthat are valid near the relevant B ( ∗ ) ¯ B ( ∗ ) thresholds as well as calculate the two-body decaysof the resonances. The predicted line shapes and decay rates incorporate the constraintsimposed by heavy quark symmetry. New predictions for the parameters in the line shapesand the total and partial widths of these states are obtained. Experimental tests of thesepredictions should aid in interpreting the newly discovered Z b and Z (cid:48) b states and searchingfor their partners.The EFT of this paper can be applied when the relative momentum of the B ( ∗ ) ¯ B ( ∗ ) Q/ Λ NN where Q ∼ p ∼ m π and1 / Λ NN = g A M N / (8 πf π ) ≈ / (300 MeV), where M N is the nucleon mass, g A is the nucleonaxial coupling, and f π is the pion decay constant [17]. Perturbative treatment of pions failsrather badly in two-nucleon systems [18] when p ≥ m π . For pion exchanges between a B ∗ and B ( ∗ ) meson, the expansion parameter is Q/ Λ BB , where 1 / Λ BB = g M B / (8 πf π ). Here M B is the B meson mass and the axial coupling of heavy mesons, g , is between 0.5 and 0.7,which yields 160 MeV ≤ Λ BB ≤
320 MeV. Since Λ BB ≈ Λ NN we expect that perturbativetreatment of pions will fail in the B meson sector for p ∼ m π , but a pionless effective theoryshould work for p (cid:28) m π .The Lagrangian we will use to describe low energy B ( ∗ ) ¯ B ( ∗ ) scattering is L = Tr[ H † a (cid:32) i∂ + (cid:126) ∇ M (cid:33) ba H b ] + ∆4 Tr[ H † a σ i H a σ i ] (1)+ Tr[ ¯ H † a (cid:32) i∂ + (cid:126) ∇ M (cid:33) ab ¯ H b ] + ∆4 Tr[ ¯ H † a σ i ¯ H a σ i ] − C H † a H † a H b ¯ H b ] − C H † a σ i H † a H b σ i ¯ H b ] − C H † a τ Aaa (cid:48) H † a (cid:48) H b τ Abb (cid:48) ¯ H b (cid:48) ] − C H † a τ Aaa (cid:48) σ i H † a (cid:48) H b τ Abb (cid:48) σ i ¯ H b (cid:48) ] . Here a and b are SU (2) isospin indices, isospin matrices are normalized as τ Aab τ Bba = δ AB ,traces are over spin indices which are not explicit, and H a ( ¯ H a ) is the heavy meson (heavyanti-meson) superfield. In terms of components, H a = P a + V i σ i and ¯ H a = ¯ P a − ¯ V ia σ i ,where P a ( ¯ P a ) and V ia ( ¯ V ia ) are the pseudoscalar and vector ¯ B ( B ) mesons, respectively. Thetransformation properties of the fields under heavy quark spin and other symmetries aregiven in Ref. [12]. In Eq. (1), the mass M in the kinetic terms is the spin-averaged B meson mass, M = (3 M B ∗ + M B ) / B ( ∗ ) ¯ B ( ∗ ) scattering, as is conventional in nonrelativistic theory. The heavy mesons andanti-heavy mesons interact via the remaining terms in the Lagrangian which are contactinteractions that mediate S -wave heavy meson scattering. Contact interactions of this typefirst were written down in Ref. [23], where the operators considered are proportional to3r[ H † a H a ] Tr[ ¯ H † b ¯ H b ] and Tr[ H † a H a σ i ] Tr[ ¯ H † b σ i ¯ H b ]. It is easy to see that these operators canbe written in terms of the single trace operators given above using Fierz transformations.In the case of B ( ∗ ) ¯ B ( ∗ ) scattering, one can classify states in terms of the total spin of theheavy quark and antiquark ( S Q ¯ Q ), and the total angular momentum ( S q ¯ q ) and isospin ( I )of the light degrees of freedom. Since the light degrees of freedom in the B and ¯ B mesonare isodoublets and have spin-1 /
2, the possible states of the light degrees of freedom are: i) I = S q ¯ q = 0, ii) I = 0 and S q ¯ q = 1, iii) I = 1 and S q ¯ q = 0, or iv) I = S q ¯ q = 1. The fouroperators in Eq. (1) mediate S -wave scattering in each of these channels and the notationfor the coefficients is that the operator with coefficient C IS mediates scattering in the isospin I and spin S = S q ¯ q channel. In the heavy quark limit scattering should be independent of S Q ¯ Q .Ref. [3] classified possible bound states of B and ¯ B mesons and concluded that thereshould be at least four and maybe six such isotriplet states. The wavefunctions of thesestates in terms of their S Q ¯ Q and S q ¯ q quantum numbers are derived in Ref. [4]. (See alsoEq. (25) below.) W b and W b are pure states with S Q ¯ Q = S q ¯ q = 1. The remaining states aremixtures of S q ¯ q = 0 or 1 and S Q ¯ Q = 0 or 1. If the mechanism that leads to shallow boundstates operates in the S q ¯ q = 1 channel or both S q ¯ q = 0 and S q ¯ q = 1 channels, one expectsto find all six shallow bound states. If the mechanism operates only in the S q ¯ q = 0 channel,then only four shallow bound states ( Z b , Z (cid:48) b , W b , and W (cid:48) b ) are expected. We will primarilyfocus on the former case, then comment on the latter at the end of the paper.The interpolating fields for these states are given by (we will drop the subscript b in whatfollows): Z A i = 1 √ V ia τ Aab ¯ P b − P a τ Aab ¯ V ib ) (2) Z (cid:48) A i = i √ (cid:15) ijk V ja τ Aab ¯ V kb W A = P a τ Aab ¯ P b W (cid:48) A = 1 √ V ia τ Aab ¯ V ib W A i = 1 √ V ia τ Aab ¯ P b + P a τ Aab ¯ V ib ) W A λ = (cid:15) λij V ia τ Aab ¯ V jb , where (cid:15) λij is a basis for symmetric traceless polarization vectors normalized as (cid:15) λij (cid:15) λ (cid:48) ij = δ λλ (cid:48) ,and a and b label flavor antifundamental and fundamental indices, respectively. It is also4ossible to define isoscalar interpolating fields that are obtained from those in Eq. (2) bydropping the index A and replacing τ Aab → δ ab / √
2. We will focus on isovector states in whatfollows, the generalization to isoscalars is straightforward. It is enlightening to rewrite thecontact interactions in terms of these interpolating fields. For the isovector fields these are L contact = − C (cid:32) W A i † W A i + (cid:88) λ W A † λ W A λ (cid:33) (3) − (cid:16) W A (cid:48) † W A † (cid:17) C + C √ C − C ) √ C − C ) C + 3 C W A (cid:48) W A − (cid:16) Z (cid:48) A i † Z A i † (cid:17) C + C C − C C − C C + C Z (cid:48) A i Z A i = − C (cid:32) W A † W A + Z A i † + Z A i + + W A i † W A i + (cid:88) λ W A † λ W A λ (cid:33) (4) − C (cid:16) W A † − W A − + Z A i †− Z A i − (cid:17) , where in the last line the interactions are diagonalized by defining the fields W A = W (cid:48) A + √ W A , W A − = √ W (cid:48) A − W A and Z A ± = √ ( Z A ± Z A (cid:48) ). The Lagrangian for isoscalar termsis obtained by dropping the superscripts A and replacing C i → C i , i = 0 or 1. Though wehave diagonalized the interactions in Eq. (4), we will not work in this basis because the Z A and Z (cid:48) A are split by the hyperfine splitting, ∆ = 46 MeV, and the W A and W (cid:48) A are splitby 2∆ = 92 MeV.It is straightforward to calculate the T-matrix for B ( ∗ ) ¯ B ( ∗ ) scattering in these channels.For the W A channel we find T W = 1 − / (2 C ) − Σ B ∗ ¯ B ∗ ( E ) , (5)where Σ B ∗ ¯ B ∗ ( E ) is computed from a one loop diagram containing nonrelativistic B ∗ and ¯ B ∗ propagators of total energy E and is given by:Σ B ∗ ¯ B ∗ ( E ) = M π (cid:16) Λ − (cid:112) M (2∆ − E ) − i(cid:15) (cid:17) . (6)The energy E is measured with respect to the B ¯ B threshold. The linear divergence inEq. (6) is cancelled by the coupling constant C = C (Λ) = 2 πM − Λ + γ , (7)5o the T-matrix is given by T W = 4 πM − γ + (cid:112) M (2∆ − E ) − i(cid:15) . (8)The T -matrix has a bound state pole at E = 2∆ − γ /M for γ >
0. A similar calculationfor the W A channel yields T W = 4 πM − γ + (cid:112) M (∆ − E ) − i(cid:15) , (9)which has a bound state pole at E = ∆ − γ /M . If shallow bound states W A and W A exist,heavy quark symmetry predicts their binding energies to be the same. On the other hand,if γ <
0, then there are no shallow bound states. In this case, heavy quark symmetrypredicts the S -wave scattering length for B meson scattering in these channels to be thesame.For Z A and Z (cid:48) A states we must solve a coupled channel problem. The T -matrix is givenby T − Z = − C − Z − Σ Z ( E ) , (10)where C Z and Σ Z ( E ) are matrices given by C Z = C + C C − C C − C C + C , (11)Σ Z ( E ) = Σ B ∗ ¯ B ∗ ( E ) 00 Σ B ¯ B ∗ ( E ) (12)= M π Λ − (cid:112) M (2∆ − E ) − i(cid:15)
00 Λ − (cid:112) M (∆ − E ) − i(cid:15) , Σ B ¯ B ∗ ( E ) = M π (cid:16) Λ − (cid:112) M (∆ − E ) − i(cid:15) (cid:17) . (13)The cutoff dependence in the T -matrix can be completely cancelled if the coupling C hasthe same form as C in Eq. (7), i.e., if C = C (Λ) = 2 π/ ( M ( − Λ + γ )). For the T -matrixwe find T Z = T Z (cid:48) Z (cid:48) T Z (cid:48) Z T ZZ (cid:48) T ZZ , (14)6ith the components given by T Z (cid:48) Z (cid:48) = 4 πM − γ + + (cid:112) M (∆ − E ) − i(cid:15) ( γ + − (cid:112) M (∆ − E ) − i(cid:15) )( γ + − (cid:112) M (2∆ − E ) − i(cid:15) ) − γ − (15) T Z (cid:48) Z = T ZZ (cid:48) = 4 πM γ − ( γ + − (cid:112) M (∆ − E ) − i(cid:15) )( γ + − (cid:112) M (2∆ − E ) − i(cid:15) ) − γ − (16) T ZZ = 4 πM − γ + + (cid:112) M (2∆ − E ) − i(cid:15) ( γ + − (cid:112) M (∆ − E ) − i(cid:15) )( γ + − (cid:112) M (2∆ − E ) − i(cid:15) ) − γ − , (17)where γ ± = ( γ ± γ ) / W (cid:48) and W channels, we obtain T W (cid:48) W (cid:48) = 4 πM − γ W + + √− M E − i(cid:15) ( γ W + − √− M E − i(cid:15) )( γ W (cid:48) + − (cid:112) M (2∆ − E ) − i(cid:15) ) − ( γ W − ) (18) T W (cid:48) W = T W W (cid:48) = 4 πM γ W − ( γ W + − √− M E − i(cid:15) )( γ W (cid:48) + − (cid:112) M (2∆ − E ) − i(cid:15) ) − ( γ W − ) (19) T W W = 4 πM − γ W (cid:48) + + (cid:112) M (2∆ − E ) − i(cid:15) ( γ W + − √− M E − i(cid:15) )( γ W (cid:48) + − (cid:112) M (2∆ − E ) − i(cid:15) ) − ( γ W − ) , (20)where γ W + = ( γ + 3 γ ) / γ W (cid:48) + = (3 γ + γ ) /
4, and γ W − = √ γ − γ ) / √ γ − / γ /M , then for γ (cid:28) m π we can expand the amplitudesin powers of γ/ √ M ∆, where √ M ∆ = 494 MeV. For example, the Z (cid:48) b state in the vicinityof the B ∗ ¯ B ∗ threshold will have energy E = 2∆ − γ /M . Expanding the denominator inthe expression for T Z (cid:48) Z (cid:48) to leading order in γ , we find T Z (cid:48) Z (cid:48) = 4 πM − γ + + γ + O ( γ / √ ∆ M ) , (21)The pole in the amplitude is at γ (cid:39) γ + or E (cid:39) − γ /M , corresponding to a Z (cid:48) mass m Z (cid:48) = 2 M B ∗ − γ /M . For the other states the binding energies are γ A /M , whereA= Z, Z (cid:48) , W , W (cid:48) , W or W , and are given by γ Z = γ Z (cid:48) = γ + (22) γ W = γ W = γ γ W = γ + 3 γ γ Z + γ W γ W (cid:48) = 3 γ + γ γ Z − γ W . These relations between the binding momenta are consequences of heavy quark symmetry.7e can incorporate the effects of decays of these resonances on the line shapes by explicitlyviolating unitarity. If the decays to other states also respect heavy quark spin symmetry,then we expect that incorporating these decays will just give imaginary components in thecouplings of Eq. (1). So we promote C , C , C and C to complex values, which alsomeans γ , γ , γ and γ are complex. For each of them, we can write γ IS = − /a IS + i Γ IS /
2, where a IS is the scattering length and Γ IS is the total width of the bound state inthe IS channel. The relations in Eq. (22) will still hold since they are a consequence ofheavy quark spin symmetry and now give relationships among their imaginary components,i.e., the total widths. One predictionΓ = Γ[ W ] = Γ[ W ] = 32 Γ[ W ] −
12 Γ[ W (cid:48) ] , (23)was first derived in Ref. [4]. In addition we also find thatΓ + = 12 (Γ + Γ ) = Γ[ Z ] = Γ[ Z (cid:48) ] = 12 (Γ[ W ] + Γ[ W (cid:48) ]) . (24)The relation Γ[ Z ] = Γ[ Z (cid:48) ] was first derived in Ref. [3], while the last equality of Eq. (24) isnew. One could derive this result, as well as similar predictions for partial decay rates, fromthe following decomposition of the wavefunction of the molecular states in terms of theircomponents of definite S Q ¯ Q ⊗ S q ¯ q [4]: W : 1 Q ¯ Q ⊗ q ¯ q (cid:12)(cid:12)(cid:12) J =2 (25) W : 1 Q ¯ Q ⊗ q ¯ q (cid:12)(cid:12)(cid:12) J =1 W (cid:48) b : √
32 0 Q ¯ Q ⊗ q ¯ q + 12 1 Q ¯ Q ⊗ q ¯ q (cid:12)(cid:12)(cid:12) J =0 W : √
32 1 Q ¯ Q ⊗ q ¯ q (cid:12)(cid:12)(cid:12) J =0 −
12 0 Q ¯ Q ⊗ q ¯ q Z (cid:48) : 1 √ Q ¯ Q ⊗ q ¯ q − √ Q ¯ Q ⊗ q ¯ q Z : 1 √ Q ¯ Q ⊗ q ¯ q + 1 √ Q ¯ Q ⊗ q ¯ q . The molecular states inherit their widths from those of their constituent states with definite S Q ¯ Q ⊗ S q ¯ q . Since the same constituent state appears in multiple molecules, by restricting todecays which are sensitive only to one choice of S Q ¯ Q ⊗ S q ¯ q , one can arrive at relations amongmolecular decays. For S Q ¯ Q ⊗ S q ¯ q = 1 ⊗
1, the result is Eq. (23). To derive the prediction inEq. (24), consider the two S zQ ¯ Q = 0 spin configurations of the two heavy quarks: | S Q ¯ Q = 1 , S zQ ¯ Q = 0 (cid:105) = 1 √ | ↑↓(cid:105) + | ↓↑(cid:105) ) , | S Q ¯ Q = 0 , S zQ ¯ Q = 0 (cid:105) = 1 √ | ↑↓(cid:105) − | ↓↑(cid:105) ) . (26)8f ˆ S zQ is the operator that measures the magnetic quantum number of the heavy quark only,then it is clear that 2 ˆ S zQ | , (cid:105) = | , (cid:105) , S zQ | , (cid:105) = | , (cid:105) . (27)Now let M s,s (cid:48) denote the interpolating fields with S Q ¯ Q = s and S q ¯ q = s (cid:48) , with all otherindices labeling other quantum numbers suppressed for compactness. Then, it follows fromthe above that [2 ˆ S zQ , M † , ] = M † , , [2 ˆ S zQ , M † , ] = M † , . (28)In what follows, h stands for any bottomonium state with allowed quantum numbers and S Q ¯ Q = 0, and (cid:96) is any allowed configuration of light hadrons. We define | (cid:101) h i (cid:105) = 2 S iQ | h (cid:105) , whichmeans (cid:101) h is a bottomonium state with S Q ¯ Q = 1. Then for the matrix element mediating thetransition W (cid:48) → h(cid:96) , we find (cid:104) h(cid:96) | W (cid:48) † | (cid:105) = √ (cid:104) h(cid:96) | M † , | (cid:105) (29)= √ (cid:104) h(cid:96) | [2 ˆ S zQ , M † , ] | (cid:105) = √ (cid:104) h(cid:96) | S zQ M † , | (cid:105) = √ (cid:104) (cid:101) h ( S z =0) (cid:96) | M † ( S z =0)1 , | (cid:105) = (cid:114) (cid:104) (cid:101) h ( S z =0) (cid:96) | Z † ( S z =0) | (cid:105) = − (cid:114) (cid:104) (cid:101) h ( S z =0) (cid:96) | Z (cid:48) † ( S z =0) | (cid:105) Rotational symmetry can be used to extend this result to other values of S z and moreoverimplies that (cid:104) (cid:101) h ( S z = m ) (cid:96) | Z † ( S z = m (cid:48) ) | (cid:105) ∝ δ mm (cid:48) . (30)Applying the result for h = η b , which means (cid:101) h = Υ, it follows that |M ( W (cid:48) → η b (cid:96) ) | = 32 × (cid:88) spins |M ( Z → Υ (cid:96) ) | . (31)A similar analysis can be performed starting with a W instead. We findΓ[ W → η b (cid:96) ] : Γ[ W (cid:48) → η b (cid:96) ] : Γ[ Z → Υ (cid:96) ] : Γ[ Z (cid:48) → Υ (cid:96) ] = 12 : 32 : 1 : 1 . (32)9his result is valid in the extreme heavy quark limit, in which Υ, η b , Z ( (cid:48) ) and W ( (cid:48) ) J are alldegenerate. In reality decay rates will also depend on the available phase space, whichcan be sensitive to the hyperfine splittings in each of the multiplets. For example, in thedecays to single pions calculated below, the rates are multiplied by a prefactor of E π k π or k π and these kinematic prefactors introduce large corrections to Eq. (32) when the splittingsbetween multiplets are not significantly larger than the hyperfine splittings of the multiplets.For decays to P -wave bottomonia, a similar derivation shows thatΓ[ W → χ b (cid:96) ] : Γ[ W (cid:48) → χ b (cid:96) ] : Γ[ Z → h b (cid:96) ] : Γ[ Z (cid:48) → h b (cid:96) ] = 32 : 12 : 1 : 1 , (33)in the heavy quark limit. Eqs. (32) and (33) together imply the relationships among totaldecay rates in Eq. (24) assuming decays to quarkonia dominate the decays of the molecularstates.One can explicitly check the claimed relationships among decay rates by computing therates to final states with one pion using HH χ PT. To obtain these rates, we will add to theLagrangian of Eq. (1) the following interactions L HH χ PT = g Tr[ ¯ H † a σ i ¯ H b ] A iab − g Tr[ H † a H b σ i ] A iba (34)+ 12 g π Υ ,n tr[Υ † n H a ¯ H b ] A ab + 12 g Υ ,n tr[Υ † n H a σ j i ↔ ∂ j ¯ H a ] + h.c.+ 12 g πχ,n tr[ χ † n,i H a σ j ¯ H b ] (cid:15) ijk A kab + i g χ,n tr[ χ † n, i H a σ i ¯ H a ] + h.c. . The first line above gives the πB ( ∗ ) and π ¯ B ( ∗ ) interactions. The next two lines are theinteractions of bottomonium states with the B-mesons. Heavy quark spin symmetry groupsthe S - and P -wave bottomonium states into the multipletsΥ n = σ i Υ i ( nS ) + η b ( nS ) , (35) χ in = σ (cid:96) (cid:16) χ i(cid:96)b ( nP ) + 1 √ (cid:15) i(cid:96)m χ mb ( nP ) + 1 √ δ i(cid:96) χ b ( nP ) (cid:17) + h ib ( nP ) . The degeneracy of Z ( (cid:48) ) and W ( (cid:48) ) is not a consequence of heavy quark spin symmetry, e.g., the bindingenergies predicted from Eq. (22) are not the same. However, these corrections to the masses of theseresonances are O ( γ /M ) and are expected to be a few MeV or less, and hence small compared to massdifferences inherited from the hyperfine splittings of their constituent mesons. This lack of degeneracy isbecause the Z ( (cid:48) ) and W ( (cid:48) ) are linear combinations of members of different heavy quark spin multiplets,cf. Eq. (25). The multiplets of heavy quark spin symmetry are the ( Z A + , W A + ) and ( Z A − , W A − ) defined inthe last line of Eq. (3) and in terms of these states heavy quark spin symmetry predicts Γ[ W − → η b (cid:96) ] =Γ[ Z − → Υ (cid:96) ] and Γ[ W + → χ b (cid:96) ] = Γ[ Z + → h b (cid:96) ]. S -wave bottomonia and a single pion are listed below. We will drop the quantum number n , labeling the radial excitation level, but it is important to keep in mind that the couplingconstants g π Υ , g πχ , g Υ and g χ will be different for distinct multiplets. The decay rates to S -wave bottomonia and a single pion areΓ[ W → πη b ] = m η k π E π πm W f π (cid:104) g π Υ − gg Υ k π E π ( E π + ∆) (cid:105) × O (36)Γ[ W (cid:48) → πη b ] = 3 m η k π E π πm W (cid:48) f π (cid:104) g π Υ − gg Υ k π E π (cid:16) E π − ∆ (cid:17)(cid:105) × O Γ[ Z → π Υ] = m Υ k π E π πm Z f π (cid:104)(cid:104) g π Υ − gg Υ k π E π (cid:16) − ∆3 E π − E π − ∆ (cid:17)(cid:105) + 29 (cid:104) gg Υ k π E π ∆ E π − ∆ (cid:105) (cid:105) × O Γ[ Z (cid:48) → π Υ] = m Υ k π E π πm Z (cid:48) f π (cid:104)(cid:104) g π Υ − gg Υ k π E π (cid:16) E π − ∆ (cid:17)(cid:105) + 29 (cid:104) gg Υ k π E π ∆ E π − ∆ (cid:105) (cid:105) × O . The decays to P -wave bottomonia and a single pion areΓ[ W → πχ b ] = m χ b k π πm W f π (cid:104) g πχ + 2 gg χ E π + ∆ (cid:105) × O (37)Γ[ W (cid:48) → πχ b ] = m χ b k π πm W (cid:48) f π (cid:104) g πχ + 2 gg χ E π − ∆ (cid:105) × O Γ[ Z → πh b ] = m h b k π πm Z f π (cid:104) g πχ + gg χ (cid:16) E π + 1 E π + ∆ (cid:17)(cid:105) × O Γ[ Z (cid:48) → πh b ] = m h b k π πm Z (cid:48) f π (cid:104) g πχ + gg χ (cid:16) E π + 1 E π − ∆ (cid:17)(cid:105) × O Γ[ W → πχ b ] = m χ b k π πm W f π (cid:104) g πχ + 2 gg χ (cid:16)
34 1 E π − ∆ + 14 1 E π + ∆ (cid:17)(cid:105) × O Γ[ W → πχ b ] = m χ b k π πm W f π (cid:104) g πχ + 2 gg χ E π (cid:105) × O Γ[ W → πχ b ] = 5 m χ b k π πm W f π (cid:104) g πχ + 2 gg χ E π + ∆ (cid:105) × O Γ[ W → πχ b ] = m χ b k π πm W f π (cid:104) g πχ + 2 gg χ E π − ∆ (cid:105) × O Γ[ W → πχ b ] = m χ b k π πm W f π (cid:104) g πχ + 2 gg χ E π (cid:105) × O . Decays not listed here can only proceed through higher-derivative interactions which aresuppressed compared to those listed. For compactness, we have defined the following non-11erturbative factors: O = 13 |(cid:104) | P a τ Aab ¯ P b | W A (cid:105)| (38) O = 13 |(cid:104) | √ V ia τ Aab ¯ V ib | W (cid:48) A (cid:105)| O = 19 |(cid:104) | √ ( V ia τ Aab ¯ P b − P a τ Aab ¯ V ib ) | Z A i (cid:105)| O = 19 |(cid:104) | √ i(cid:15) ijk V ja τ Aab ¯ V kb | Z (cid:48) A i (cid:105)| O = 19 |(cid:104) | √ ( V ia τ Aab ¯ P b + P a τ Aab ¯ V ib ) | W A i (cid:105)| O = 115 |(cid:104) | (cid:15) ijλ V ia τ Aab ¯ V jb | W λ (cid:105)| . Each of these has a prefactor of 1 / A , and an additional prefactor of 1 / (2 J + 1) for a molecular state with spin J toprevent overcounting when summing over the spin polarization. In the limit of exact heavyquark spin symmetry, using arguments like that given in Eq. (29) one can show that the O i ’s are all equal.The ratios of the decay rates in Eq. (36) and Eq. (37) match the predicted results fromEq. (32) and Eq. (33), respectively, when ∆ = 0 and E π is the same for all decays, which willbe the case when the heavy quark spin symmetry multiplets (Υ b , η b ), ( χ bJ , h b ), and the Z ( (cid:48) ) and W ( (cid:48) )0 are degenerate. Our explicit calculations of the decay rates allow us to incorporateimportant corrections to heavy quark spin symmetry predictions that come from phase spaceand kinematic factors. In HH χ PT there are two mechanisms that contribute to the decayof the bound states. There is a short-distance process, mediated by the contact interactions g π Υ and g πχ , in which the B ( ∗ ) ¯ B ( ∗ ) transition to the final state quarkonium and pion at apoint. If this process dominates, the predicted ratios of rates are the heavy quark symmetrypredictions in Eqs. (32) and Eqs. (33) weighted by factors of E π k π and k π , respectively.There is also a long-distance process B ( ∗ ) ¯ B ( ∗ ) → B ( ∗ ) ¯ B ( ∗ ) π followed by coalesence of the B and ¯ B meson into the final state quarkonium through the couplings g Υ and g χ . Theseprocesses lead to a more complicated dependence on the pion energy. When these processesdominate, the dependence is well approximated by k π /E π since in all cases E π (cid:29) ∆.For decays to S -wave bottomonium, the two processes appear at the same order in theexpansion. If contact interactions dominate, which is obtained when the dimensionless ratio12 (cid:45) FIG. 1: Ratio of Γ[ W → η b π ] (dashed), Γ[ W (cid:48) → η b π ] (dotted) and Γ[ Z → Υ π ] (solid) toΓ[ Z (cid:48) → Υ π ] as functions of g g Υ /g π Υ (cid:45) (cid:71) (cid:72) MeV (cid:76)
FIG. 2: Partial decay rates Γ[ W → η b (3 S ) π ] (dotted), Γ[ W (cid:48) → η b (3 S ) π ] (solid), Γ[ Z → Υ(3 S ) π ](dashed) and Γ[ Z (cid:48) → Υ(3 S ) π ] (dashed-dotted) as functions of g g Υ /g π Υ for one possible choice ofcurrently undetermined parameters ( O g π Υ = 10 − ). of couplings, λ Υ = gg Υ /g π Υ (cid:28)
1, then the ratios of partial decay rates are predicted to beΓ[ W → πη b (3 S )] : Γ[ W (cid:48) → πη b (3 S )] : Γ[ Z → π Υ(3 S )] : Γ[ Z (cid:48) → π Υ(3 S )]= 0 .
26 : 2 . .
62 : 1 ( λ Υ = 0) , (39)where all partial decay rates have been normalized to the rate for Γ[ Z (cid:48) → π Υ(3 S )]. In theopposite limit,Γ[ W → πη b (3 S )] : Γ[ W (cid:48) → πη b (3 S )] : Γ[ Z → π Υ(3 S )] : Γ[ Z (cid:48) → π Υ(3 S )]= 0 .
12 : 2 . .
41 : 1 ( | λ Υ | = ∞ ) . (40)In computing these ratios, we have assumed O = O = · · · = O holds without significantcorrections from symmetry violating terms. The values used for the masses of the bottomo-nium decay products, m Υ(3 S ) = 10355 MeV and m η b (3 S ) = 10328 MeV, were determined inRef. [24] using a relativistic quark model. 13he results do not differ greatly because ratios of the kinematic factors k π and k π /E π arenot that different. For intermediate values of λ Υ the results can depend rather dramaticallyon λ Υ when λ Υ ≈ .
6, as seen in Fig. 1. However, this wild variation occurs because of acancellation between the two processes mentioned above. For these values of λ Υ , all fourpartial rates are highly suppressed, as shown in Fig. 2. In this figure, we have chosen O i = (100 MeV) and g π Υ = 1 . − and calculated the rates as a function of λ Υ . Ourchoices of O i and g π Υ are based on naive dimensional analysis and are only intended to beestimates that should be accurate within a factor of 10. For this particular choice of O i and g π Υ , the branching fraction for Z → Υ π is less than 1%, for any value of λ Υ close enough to0.6 that the ratios deviate significantly from those given in Eqs. (39) and (40). Though thisbranching fraction has not been measured, the observation of this decay leads us to expectthat the parameter λ Υ does not take on values where such cancellations suppress the decayrates. Therefore, we expect that experimental measurement of the ratios will yield resultsclose to those in Eq. (39) or Eq. (40).For decays to P -wave bottomonium, the processes mediated by contact interactions aresuppressed in the power counting. The relative importance of leading order to contactinteraction mediated processes is controlled by the dimensionful parameter g πχ /g χ which weexpect to be ≈ − . In the limit g πχ /g χ = 0 we findΓ[ W → πχ b (2 P )] : Γ[ W (cid:48) → πχ b (2 P )] : Γ[ Z → πh b (2 P )] : Γ[ Z (cid:48) → πh b (2 P )]= 0 .
72 : 0 .
57 : 0 .
66 : 1 ( g πχ /g χ = 0 GeV − ) , (41)andΓ[ W → πχ bJ (2 P )] : Γ[ W → πχ bJ (2 P )] : 32 Γ[ W → πχ b (2 P )] −
12 Γ[ W (cid:48) → πχ b (2 P )]= 0 .
81 : 1 : 0 .
43 ( g πχ /g χ = 0 GeV − ) . (42)The masses used to compute these ratios are m χ b (2 P ) = 10233 MeV, m χ b (2 P ) = 10255 MeV, m χ b (2 P ) = 10269 MeV and m h b (2 P ) = 10261 MeV. The first three of these came from theParticle Data Group and the last is computed in Ref. [24]. For g πχ /g χ ≈ − theseratios change by only a few percent. For g πχ /g χ ≈ − (300 GeV) − , there is a cancellationbetween contact and leading order diagrams which suppresses the total rates and whichleads to modifications of the ratios similar to what was discussed above in the decays to S -wave bottomonium. Again the observation of Z (cid:48) → h b (2 P ) π disfavors such a suppression14f the decay rates. Such a cancellation between leading order and next-to-leading ordercontributions is also inconsistent with the power counting of the theory.Throughout this paper, it is assumed that all six isovector molecular states exist and areloosely bound. As stated earlier, it is also possible that binding occurs in the S q ¯ q = 0 channelonly so that only the W ( (cid:48) )0 and Z ( (cid:48) ) bound states exist. In this case, the treatment of thispaper is still applicable. One simply takes the binding momentum γ = 1 /a to be smallbut negative. In this case, resumming both the C and C interactions remains necessary.There is also the possibility that in these channels there are no shallow bound states or largescattering lengths. In this case, one can keep the summation in both channels as presentedin this paper but tune γ so there is no large scattering length. It should also be possibleto sum only the strong, binding interaction and give a perturbative treatment to the weakerinteraction. The T matrices computed using the latter approach should correspond to apower series expansion of those in this paper. We will save any investigation along theselines for future work.In this paper we introduced the Lagrangian describing heavy quark spin symmetric S -wave contact interactions among observed and hypothesized isovector B meson molecules.We derive the line shapes of these states in the vicinity of their respective B ( ∗ ) ¯ B ( ∗ ) thresholds,including coupled channel effects where mixture of states is possible. By doing so, wehave arrived at relationships among the binding energies and decay rates of the molecularstates. Some relationships among the widths of the W ( (cid:48) ) bJ and Z ( (cid:48) ) states derived in thispaper appear in Ref. [4], while the prediction in Eq. (24) is new. A confirmation of thesepredictions by explicit calculation of partial widths for strong two-body decays to S - and P -wave bottomonia using HH χ PT was performed. This allowed us to compute correctionsto the earlier predictions which arise from differences in the kinematics between the variousprocesses. Tests of these predictions will aid in interpreting the new states. Future workcould include an extension of these results to the isoscalar sector of B meson molecules, adetailed look at radiative decays, as well as a determination of currently unknown parameters λ Υ and g χ /g πχ using the angular distribution of decay products.15 cknowledgments This work was supported in part by the Director, Office of Science, Office of High EnergyPhysics, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231U.S(SF), Office of Nuclear Physics, of the U.S. Department of Energy under grant numbers DE-FG02-06ER41449 (S.F.), DE-FG02-05ER41368 (T.M.), and DE-FG02-05ER41376 (T.M.). [1] B. Collaboration, arXiv:1105.4583 [hep-ex].[2] M. Cleven, F. -K. Guo, C. Hanhart, U. -G. Meissner, [arXiv:1107.0254 [hep-ph]].[3] A. E. Bondar, A. Garmash, A. I. Milstein, R. Mizuk, M. B. Voloshin, [arXiv:1105.4473 [hep-ph]].[4] M. B. Voloshin, [arXiv:1105.5829 [hep-ph]].[5] Y. Yang, J. Ping, C. Deng and H. S. Zong, arXiv:1105.5935 [hep-ph].[6] D. Y. Chen, X. Liu and S. L. Zhu, arXiv:1105.5193 [hep-ph].[7] J. R. Zhang, M. Zhong and M. Q. Huang, arXiv:1105.5472 [hep-ph].[8] T. Guo, L. Cao, M. Z. Zhou and H. Chen, arXiv:1106.2284 [hep-ph].[9] A. Ali, arXiv:1108.2197 [hep-ph].[10] C. Y. Cui, Y. L. Liu and M. Q. Huang, arXiv:1107.1343 [hep-ph].[11] S. Fleming, M. Kusunoki, T. Mehen and U. van Kolck, Phys. Rev. D , 034006 (2007)[arXiv:hep-ph/0703168].[12] S. Fleming, T. Mehen, Phys. Rev. D78 , 094019 (2008). [arXiv:0807.2674 [hep-ph]].[13] E. Braaten, H. -W. Hammer, T. Mehen, Phys. Rev.
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