Enhanced mixing of partial waves near threshold for heavy meson pairs and properties of Z b (10610) and Z b (10650) resonances
aa r X i v : . [ h e p - ph ] F e b William I. Fine Theoretical Physics InstituteUniversity of Minnesota
FTPI-MINN-13/02UMN-TH-3133/13January 2013
Enhanced mixing of partial waves near threshold forheavy meson pairs and properties of Z b (10610) and Z b (10650) resonances M.B. Voloshin
William I. Fine Theoretical Physics Institute, University of Minnesota,Minneapolis, MN 55455, USASchool of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USAandInstitute of Theoretical and Experimental Physics, Moscow, 117218, Russia
Abstract
The mixing of S − D partial waves for the heavy meson pairs in the decays Υ(5 S ) → [ B ∗ ¯ B + c . c . ] π and Υ(5 S ) → B ∗ ¯ B ∗ π is considered. It is argued that this mixing, en-hanced by the heavy meson mass, is calculable as dominated by a rescattering throughpion exchange if the production of the heavy mesons is dominated by S -wave ‘molecu-lar’ resonances Z b (10610) and Z b (10650). The effect of the mixing grows with the c.m.energy E in each channel over the threshold, and may reach 10 ÷
20% in the rate atthe upper end of applicability of the discussed approach, E ∼
15 MeV. It is also arguedthat the mixing is likely to reach maximum at energy approximately equal to the massgap between the thresholds. he strong interaction between hadrons containing heavy and light quarks results in a pe-culiar dynamics near a threshold for pairs of such hadrons. One of the recent most spectacularobservations of this phenomenon is the discovery by Belle [1] of the ‘twin’ isotriplet reso-nances Z b (10610) and Z b (10650) with masses within few MeV from the respective B ∗ ¯ B and B ∗ ¯ B ∗ thresholds. The resonances were initially observed in the decays Υ(5 S ) → Υ( nS ) ππ with n = 1 , , S ) → h b ( mP ) ππ with m = 1 and 2 as conspicuous peaks in theinvariant mass spectra for the system of bottomonium plus pion. These two peaks are nat-urally explained [2] as ‘molecular’ states with quantum numbers I G ( J P ) = 1 + (1 + ) madeout of S -wave pairs of mesons: Z b (10650) ∼ B ∗ ¯ B ∗ , Z b (10610) ∼ ( B ∗ ¯ B − B ¯ B ∗ ). Thispicture agrees well with the subsequent measurement[3] of the invariant mass distributionfor the B ∗ ¯ B + B ¯ B ∗ and B ∗ ¯ B ∗ meson pairs in the decays Υ(5 S ) → [ B ∗ ¯ B + c . c . ] ± π ∓ andΥ(5 S ) → [ B ∗ ¯ B ∗ ] ± π ∓ . The distribution in each channel displays a strong peak near therespective threshold, and in fact the data, although still with a large uncertainty, suggestthat the entire distribution is dominated by the corresponding Z b resonance.Naturally, more precise data, if available in the future, would allow a better analysis ofthe internal dynamics of the Z b resonances. Such analysis however would require a betterunderstanding of finer effects, including those in the spectra of the heavy meson pairs nearthe threshold resonances. The goal of this paper is to point out that the spectra in thevicinity of the resonances can be noticeably affected by an enhanced mixing of partial waves,specifically the S − D mixing, due to rescattering of the heavy mesons. In particular, in the(likely) case that the Z b resonances are indeed very strongly dominated by the S -wave heavymeson pairs, a measurable D -wave should arise due to rescattering at energies starting fromthe excitation energy above ∼
10 MeV over the threshold . The onset of the D -wave inthe spectrum is dominated by the pion exchange between the heavy mesons and is, to anextent, calculable. At higher excitation energy, however, the presence of the D -wave canonly be estimated parametrically. The presence of the D wave can be measured in futureexperiments by studying the angular distribution of the heavy meson pairs and it is likelyto be important for a better determination of the parameters of the Z b resonances.The enhancement of the effects of the interaction of heavy hadrons through the lightdegrees of freedom is well known. Indeed, in the limit of large mass M of the heavy quark The effect of the S − D mixing due to rescattering in the formation of the Z b resonances as ‘molecular’bound states was recently considered in Refs. [4, 5] with the conclusion that this effect is small in the wavefunction of the meson pair in the bound state. V , does not depend on M . Thus at a givenmomentum scale p any effect of the interaction enters through the product M V and thusgets bigger at large M . In particular, if V describes the spin-dependent interaction resultingin an S − D mixing, the effect of the mixing should generally contain the enhancement factor M/ Λ QCD in the amplitude. In a situation where the strength and the range of the potentialare determined by Λ
QCD , an S -wave state rescatters into a D -wave state with the ratio ofthe amplitudes generally estimated as A D A S ∝ M p Λ QCD , (1)where p is the c.m. momentum of the heavy hadrons, and the estimate applies as long as p ≪ Λ QCD . This estimate, however, should be further modified for the effect of the pionexchange, since the pion mass µ can be considered as small in the scale of Λ QCD . Thecontribution of the pion exchange to the wave mixing can be estimated as A D A S ∝ g M p Λ QCD µ at p ≪ µ , (2)and A D A S ∝ g M Λ QCD at Λ QCD ≫ p ≫ µ , (3)where g is the constant for pion interaction with heavy hadrons. Clearly, these estimatesimply that the pion exchange dominates the wave mixing at the energies corresponding to p ≪ Λ QCD and becomes comparable with the effect of other contributions to the spin-dependent interaction at the upper end of the applicability of both estimates (1) and (3)where the pion exchange can no longer be separated from those other short distance con-tributions. Neither the effect of the pion exchange at such momenta is calculable due tounknown form factors in both the pion interaction with heavy hadrons and in the short-distance behavior of A S . For this reason the specific calculations in this paper are limited tothe energies above the threshold corresponding to p ≪ Λ QCD . Although restricted to thislow energy range the calculation to be presented indicates that the arising from the rescat-tering D wave can be well measurable just above the Z b peaks, corresponding to more than10% in the rate at p ≈
300 MeV, i.e. at the c.m. energy E ≈
17 MeV. Due to a moderatemagnitude of the mixing one can neglect the back reaction, i.e. the feedback from the D wave to the S -wave, which approximation is assumed in the rest of this paper.2or the specific calculation in the discussed processes Υ(5 S ) → [ B ∗ ¯ B + c . c . ] π andΥ(5 S ) → B ∗ ¯ B ∗ π one can write the pion interaction with the B and B ∗ mesons in theform (see e.g. in Ref.[6]) H int = gf π nh ( V † l τ a P ) + h . c . i + i ǫ ljk ( V † j τ a V k ) o ∂ l π a (4)with V i and P standing for the wave functions (in the nonrelativistic normalization) of thevector ( B ∗ ) and the pseudoscalar ( B ) meson isotopic doublets and τ a being the isospin Paulimatrices. The constant f π ≈
132 MeV is used in Eq.(4) for normalization. The dimensionlesspion coupling g can then be evaluated by using the heavy quark symmetry and the known[7]rate of the D ∗ + → Dπ decay: g ≈ . B ∗ ¯ B and B ∗ ¯ B ∗ is shown in the Figure 1. Only these ‘diagonal’ processesreceive an unsuppressed contribution from the domain of the loop momentum ~q such that q ≪ Λ QCD provided that the overall c.m. momentum p of the mesons satisfies the samecondition. In particular, this is not the case for the rescattering between the two channels: B ∗ ¯ B ↔ B ∗ ¯ B ∗ , where due to the mass difference ∆ M between the B ∗ and B mesons theminimal momentum transfer through the pion propagator is of order q ∼ M ∆ M , whichcorresponds to q ∼ Λ QCD both numerically and parametrically. Thus the non-diagonalrescattering produces a mixing effect only of the order described by Eq.(1). ✲ ✲ ✣✢✤✜ ✣✢✤✜ ✘✘✘✘✘✘✘✘✿ ✘✘✘✘✘✘✘✘✿❳❳❳❳❳❳❳❳③ ❳❳❳❳❳❳❳❳③✘✘✘✘✘✿ ✘✘✘✘✘✿❳❳❳❳❳③ ❳❳❳❳❳③ ✻ ✻ ✇ ✇✇ ✇t t
Υ(5 S ) Υ(5 S ) π ππ πB ∗ B ∗ ¯ B ¯ B ∗ B B ∗ ¯ B ∗ ¯ B ∗ } D -wave } D -wave A S A ′ S Figure 1: The rescattering through the pion exchange resulting in an S − D mixing in thechannels B ∗ ¯ B and B ∗ ¯ B ∗ .)Using the notation ~a and ~b for the polarization amplitudes of the outgoing B ∗ and ¯ B ∗ mesons, one can write the general form of the Υ(5 S ) decay amplitudes: A [Υ(5 S ) → [ B ∗ ¯ B + c . c . ] π ] /E π = Υ i ( a j − b j ) √ " A S δ ij + 1 √ A D (3 n i n j − δ ij ) , [Υ(5 S ) → B ∗ ¯ B ∗ π ] /E π = Υ i ǫ jkl a k b l √ " A ′ S δ ij + 1 √ A ′ D (3 n i n j − δ ij ) , (5)where ~n = ~p/p is the unit vector in the direction of c.m. momentum of one of the mesons,and p = | ~p | . The factor 1 /E π in l.h.s. removes from the amplitude the chiral dependence onthe energy E π of the emitted pion, which dependence is not the subject of the discussion inthis paper. The amplitudes A S and A D are functions of p and their relative normalizationis chosen in such a way that the probability is proportional to the sum of their squares: | A S | + | A D | . Under these notational conventions the amplitude of the D wave generatedby the graphs of Fig. 1 can be calculated by means of nonrelativistic perturbation theory as A ( ′ ) D ( p ) = g √ f π Z A ( ′ ) S ( q ) Mq − p − iǫ (cid:18) n i n j − δ ij (cid:19) ( q i − p i )( q j − p j )( ~q − ~p ) + µ d q (2 π ) = g π √ f π Z ∞ A ( ′ ) S ( q ) M qq − p − iǫ F ( p, q ) dq (6)where in the latter transition an averaging over the relative angle between the vectors ~q and ~p is performed, so that only an integration over the scalar variable q remains with theweight function F ( p, q ) given by F ( p, q ) = 5 p − q − µ p + 4 µ p + 3( p − q − µ ) p q ln ( p + q ) + µ ( p − q ) + µ . (7)In Eq.(6) the notation A ( ′ ) signifies that the relation can be used in either of the two heavymeson channels, and M stands for two times the reduced mass of the two mesons. For thepurpose of the present calculation the difference in this reduced mass in the two channelscan be neglected and a common value M ≈ . D -wave and S -wave amplitudes, corresponding tothe on-shell cut of the graphs of Fig. 1 can now be found unambiguously by setting q = p inthe integrand in Eq.(6): r ≡ Im A ( ′ ) D A ( ′ ) S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) abs = g π √ M pf π " − µ p + µ p + 34 µ p ! ln p µ ! . (8)Clearly, this expression exhibits the behavior described by Eqs.(2) and (3) with one extrapower of p/ Λ QCD arising from the phase space in the intermediate state. The plot of theratio r given by Eq.(8) at g = 0 .
18 is shown in Fig. 2 and illustrates the magnitude of thediscussed effect of the S − D mixing. 4
10 15 20 E0.050.100.15 r
Figure 2: The ratio r described by Eq.(8) as a function of the excitation energy above thethreshold E = p /M .One should notice that the expression (8) does not describe all of the imaginary partof the ratio of the amplitudes and thus cannot be used to establish a lower bound on the D -wave generated through rescattering. The reason is that the amplitude A ( ′ ) S ( q ) itself iscomplex with a phase generally depending on q . In other words the graphs of Fig. 1 alsogenerally have nontrivial on-shell cuts across the bubble depicting the S -wave amplitude. Inorder to evaluate the full generated D -wave amplitude a knowledge of the energy behaviorof the S -wave amplitude is needed. The data [3] indicate that the Z b resonances dominatethe spectrum of the heavy meson pairs in the discussed decays of Υ(5 S ). Thus it appearsreasonable to use in Eq.(6) a pure resonance expression A S ( p ) = cE − E + i Γ / p − p + iM Γ / , (9)with E being the energy position of the resonance relative to the threshold and p = M E . According to the current data [7], the value of E is 2 . ± . Z b (10610)and 1 . ± . Z b (10650), while the corresponding values of Γ are 18 . ± . . ± . A S makes the integral in Eq.(6)convergent and determined by a scale, which is a combination of p , p , M Γ / µ , allof which are assumed to be small in the scale of Λ QCD and which justifies using the pionexchange amplitude as in Eq.(6) without introducing a form factor. The integral can then be5eadily calculated, and the plots in Fig.3 illustrate the magnitude of the discussed generated D -wave. For this illustration the value E = 0 is used and the effect depends very weakly onthis particular value as long as E is smaller than Γ /
2. Also for this illustration the valueΓ = 14 MeV is used, which is close to the data for both Z b resonances. One can see thatat the upper end of applicability of the discussed approach, i.e. at E ≈ ÷
17 MeV therelative contribution of the generated D wave can amount to 10 ÷
20% in the event rate. - (cid:144) S Figure 3: The absolute value (solid) and the real (dashed) and imaginary (dotted) partsof the ratio A D /A S for the D -wave amplitude generated from the A S given by Eq.(9) with E = 0 and Γ = 14 MeV.It can be noticed that the simplest Breit-Wigner formula for the resonance amplitude isused for A S in Eq.(9). In particular the width parameter Γ is assumed to be constant anddoes not include the energy-dependent absorption in the heavy meson channel above thethreshold. This approximation appears to be sufficient with current data, but may need tobe modified in a thorough analysis of possible more detailed data in the future.As one can see from the plots of Fig. 3, the S − D mixing is quite small at low energies E ∼ Z b resonances, so thatthe mixing is likely only a minor effect in this internal dynamics, in agreement with theconclusions of Refs. [4, 5]. The ratio A D /A S however grows with the excitation energy andcan possibly be studied in the decays from Υ(5 S ). At present there seem to be no theoreticalmeans of analyzing the behavior of the S − D mixing at larger excitation energies E beyondthe region of applicability of the presented approach, i.e. at E being not small as compared6o Λ QCD /M . It is however interesting to notice that if higher values of E were accessible inthe Υ(5 S ) decay, i.e. in an artificial theoretical limit of a large mass of Υ(5 S ), the S − D mixing should vanish at high E . Indeed, the strong interaction rescattering arises from theinteraction of the light quarks in the heavy mesons, and the spin of the heavy b quarks isconserved, and the whole process can be considered with spinless b quarks. With spinless b quarks the Υ(5 S ) would be an J P = 0 + state, and after emission of an S -wave pion wouldproduce heavy meson pair with quantum numbers J P = 0 − , for which obviously no S − D mixing is possible. In terms of the mesons, the channels B ∗ ¯ B and B ∗ ¯ B ∗ would degenerateinto one channel as well as the two Z b resonances would coalesce into one. The vanishingof the S − D mixing then occurs through a cancellation between the ‘diagonal’ rescattering,e.g. B ∗ ¯ B ∗ → B ∗ ¯ B ∗ and the ‘off-diagonal’ B ∗ ¯ B → B ∗ ¯ B ∗ . For the actual heavy mesons,whose thresholds are split by ∆ M , in the region of dominance of the pion exchange onlythe ‘diagonal’ scattering should be retained, as discussed previously. However at large E ,specifically at E ≫ ∆ M the difference in the thresholds becomes unimportant and the heavyquark spin symmetry behavior should set in. Therefore one can reasonably expect that the D/S ratio has a maximum at E ∼ ∆ M and decreases at higher energy. It would be quiteinteresting if this behavior can be studied in the decays from the actual Υ(5 S ).It can be also mentioned that a similar enhanced mixing of partial waves, namely the P − F mixing, can be expected in the production of B ∗ ¯ B ∗ pairs in e + e − annihilation at thec.m. energy just above the threshold. A detailed discussion of the effects of this mixing willbe reported separately.This work is supported, in part, by the DOE grant DE-FG02-94ER40823. References [1] I. Adachi et.al. [Belle Collaboration], arXiv:1105.4583 [hep-ex] ;A. Bondar et.al. [Belle Collaboration], arXiv:1110.2251 [hep-ex].[2] A. E. Bondar, A. Garmash, A. I. Milstein, R. Mizuk and M. B. Voloshin, Phys. Rev. D , 054010 (2011) [arXiv:1105.4473 [hep-ph]].[3] I. Adachi et al. [Belle Collaboration], arXiv:1209.6450 [hep-ex].[4] Z. -F. Sun, J. He, X. Liu, Z. -G. Luo and S. -L. Zhu, Phys. Rev. D , 054002 (2011)[arXiv:1106.2968 [hep-ph]]. 75] N. Li, Z. -F. Sun, X. Liu and S. -L. Zhu, arXiv:1211.5007 [hep-ph].[6] T. Mehen and J. W. Powell, Phys. Rev. D , 114013 (2011) [arXiv:1109.3479 [hep-ph]].[7] J. Beringer et al. (Particle Data Group), Phys. Rev. D86