The nature of Zb states from a combined analysis of Υ ( 5 S ) → hb ( mP ) π + π − and Υ ( 5 S ) → B ( ∗ ) B̄ ( ∗ ) π

With a combined analysis of data on Υ(5S) → hb(1P, 2P)π+π− and Υ(5S) → B(∗) B̄(∗)π in an effective field theory approach, we determine resonance parameters of Zb states in two scenarios. In one scenario we assume that Zb states are pure molecular states, while in the other one we assume that Zb states contain compact components. We find that the present data favor that there should be some compact components inside Z (′) b associated with the molecular components. By fitting the invariant mass spectra of Υ(5S) → hb(1P, 2P)π+π− and Υ(5S) → B(∗) B̄∗π , we determine that the probability of finding the compact components in Zb states may be as large as about 40 %.

The discovery of the Z b states has inspired many interesting theoretical discussions.For example, it is suggested that these states can be molecular states of the B B * + c.c. or B * B * meson pairs [5][6][7][8][9][10][11][12].They are also proposed to be candidates of tetraquark states [13].In Ref. [14,15] the threshold enhancements are considered to be caused by cusp effects.
a e-mail: chengy@pku.edu.cnAlthough the masses of these Z b states determined from the experimental fits are slightly above the thresholds, one should note that the masses are extracted by the Breit-Wigner parametrization.As emphasized in [9,12], if an Swave shallow bound state exists below the threshold, the amplitude should not be parameterized in the Breit-Wigner form.Using the line shape for a pure bound state, Ref. [9] shows that the data on ϒ(5S) → h b (m P)π + π − are consistent with the bound state nature of Z ( ) b .Furthermore, the observed enhancements in ϒ(5S) → [B B * + c.c.]π and ϒ(5S) → B * B * π by Belle are very close to the thresholds of the B ( * ) B( * ) systems.It is also found that the masses of the Z b states can be below the corresponding thresholds if these masses are extracted from data on ϒ(5S) → B ( * ) B( * ) π [4].
As an observational fact, Z b states and their analogs in the charmonium sector Z c (3900) [16][17][18], Z c (4020/4025) [19,20], and also the famous X (3872) appear to be strongly correlated to the thresholds of either B ( * ) or D ( * ) pairs.This feature makes it natural to interpret these states as molecules.However, as was pointed out in Ref. [21,22], it is difficult to understand the large production rates of these states in B-factories, e.g.X(3872), if these states are assumed to be loosely bound molecular states.In particular, the recent LHCb measurement of the ratio R ψγ = B(X (3872)→ψ(2S)γ ) B(X (3872)→J/ψγ ) = 2.46 ± 0.64 ± 0.29 [23] seems not to support a pure D * 0 D0 molecular interpretation of X (3872), since R ψγ is predicted to be rather small for a pure D * 0 D0 molecule [24].Meanwhile, a compact component inside such states can compromise both threshold phenomena and sizable production rates.It is shown in Ref. [25,26] that the radiative decays of X (3872) are not only sensitive to long-range parts but also to short-range parts of the wave function.The search for a hidden-beauty counterpart of X (3872), which is usually denoted as X b , is important for understanding the structure of X (3872).An effective field theory study shows that if X (3872) is a molecular bound state of D * 0 and D0 mesons, the heavy-quark symmetry requires the existence of a molecular bound state X b of B * 0 B0 with mass of 10604 MeV [27].However, there is no significant signal of X b near the threshold of B * 0 B0 in X b → π + π − ϒ(1S) [28] and in X b → ωϒ(1S) [29].Reference [30] suggests that X b may be close in mass to the bottomonium state χ b1 (3P) and mixes with it.Therefore, the experiments which reported observing χ b1 (3P) might have actually discovered X b .
Obviously, more experimental data and theoretical development are required to clarify the nature of these near threshold states.In Ref. [31] an effective field theory (EFT) approach is proposed for the study of near threshold states (see also an independent study in Ref. [32]).In this framework the compositeness theorem can be incorporated with a determination of parameter Z , which is the probability of finding an elementary component in the bound state, and the nature of near threshold states can be described by the presence of both molecular and compact components in their wavefunctions.
The main purpose of this work is to study structure of Z b states by doing a combined analysis of data on ϒ(5S) → h b (m P)π + π − and ϒ(5S) → B ( * ) B( * ) π within EFT approach proposed in [31].Our work is organized as follows: in Sect.2, we recall the EFT approach proposed in Ref. [31].In Sect.3, we present the analysis of the ϒ(5S) → h b (m P)π + π − transitions and in Sect.4, the ϒ(5S) → B ( * ) B( * ) π .Our numerical results are presented in Sect. 5. Finally, a brief summary is given in Sect.6.
Here we recall some of the main points; more details can be found in Ref. [31].Consider a bare state |B with bare mass −B 0 and coupling g 0 to the two-particle state, where the bare mass is defined relative to the two-particle threshold.The two particles have masses m 1 , m 2 , respectively.If |B is near the two-particle threshold, then the leading two-particle scattering amplitude can be obtained by summing the Feynman diagrams in Fig. 1.Near threshold, the momenta of these two particles are non-relativistic.Therefore, the loop integral in Fig. 1 can be done in the same way as that in Ref. [35,36].The loop integral can be written as Fig. 1 Feynman diagrams for the two-particle scattering.The double lines denote the bare state where μ is the reduced mass of the two particles, and E is the kinematic energy of the two-particle system.Obviously, the above integral does not diverge in D = 4.Using the minimal subtraction (MS) scheme which subtracts the 1/(D − 4) pole before taking the D → 4 limit, one obtains It is interesting to note that with the MS scheme no counter term is needed in the renormalization.We then have the two body elastic scattering amplitude for Fig. 1: If a bound state exists, we have the following relations: where B is the binding energy, and Z is the probability of finding an elementary state in the physical bound state.Note that for the bound state, we mean a below threshold pole in the physical sheet.With Eq. ( 4), Eq. ( 3) can be re-expressed as where We can also express Eq. ( 5) in the form where G(E) is the complete propagator for the S-wave near the threshold state We have added a constant width in the propagator, which can simulate the decay channels other than the bottom and anti-bottom mesons.From Eq. ( 7), one can find that the Feynman rule for the coupling between the near threshold state and its two-particle component is ig 0 .Treating the binding momentum γ = (2μB) 1/2 and the three-momentum of the two-particle state p as small scales, i.e., γ, p ∼ O( p), one can then find that the leading amplitude Eq. ( 5) is at the order of O( p −1 ).

ϒ(5S) decays to h b (1 P, 2 P)π + π −
In this section, we study the decay b π , Z b states can be produced through both direct and indirect processes.In direct production processes, Z b states are produced directly via its compact component, while in indirect production processes a bottom and antibottom meson pair is produced first in the ϒ(5S) decay and then rescatters to Z The three-momenta of heavy mesons in decay ϒ(5S) → Z ( ) b π → h b (m P)π π are small compared with their masses.Therefore, these heavy mesons can be treated as nonrelativistic, and one can set up a power counting in terms of the small three-momentum p [31,[37][38][39].From the power counting, one can find that if Z ( ) b contains a compact component, its production will be driven by this compact component [31] (see also Refs.[21,22]).In Fig. 2  As we are only interested in low energy physics, it is convenient to collect B mesons in a 2 × 2 matrix [40,41], where σ i are the Pauli matrices, and a is the light flavor index, P * a and P a annihilate the vector and pseudoscalar heavy mesons, respectively, and P( * ) a annihilates the corresponding anti-particle.The leading effective Lagrangian describing the coupling of Z b states to the bottom and antibottom mesons can be written as that in Ref. [9], where Z ab annihilates Z ab , Z † ab creates Z ab , and g 0 is defined in Eq. ( 4).The Lagrangian for the coupling of the P-wave quarkonia and the B mesons reads [37] The chiral Lagrangian for the B mesons and the S-wave quarkonia can be written as [12] L where , and A μ is the axial vector pion current, which is given by We set g π = 0.25 as in [9,42].Note that our convention is different from that in [9], because a factor of √ 2M has been absorbed into the field operator of the heavy meson in our convention [31], then our g π is half of the value which is used in Ref. [9].The leading effective Lagrangian describing the Z b h b π interactions reads which describes the direct decay of Z Similar to Ref. [9], we use the same coupling g ϒ , g z for Z b and Z b .With the above effective Lagrangians and Eq. ( 8) as the propagator of Z ( ) b , one can then write out the amplitudes for all the Feynman diagrams in Figs. 2 and 3. We treat the loop integrals as was done in Ref. [37].We present the relevant one loop three-point functions in Appendix A, and give all the amplitudes of Figs. 2 and 3 in Appendix B. In the following we address several points before ending this section.
• As in Ref. [9], we assume that Z b only couples to B B * while Z b only couples to B * B * .We then find that there is a relative minus sign between iM 3a,3b,3c,3d for In this way, we can reduce the number of free parameters in our fitting.• We show the Feynman diagrams for non-resonant contributions to ϒ(5S) → h b (m P)π π in Fig. 4. Reference [43] shows that the non-resonant diagrams do not Hence their contributions will not be enhanced by the kinematic singularity.We do not include their contributions in the present work, since they are suppressed by the heavy-quark spin symmetry.The experimental fits also find no significant non-resonant contributions [1,2].

ϒ(5S) decays to B ( * ) B( * ) π
In this section, we will study the decay For the previous study one may refer to Ref. [12], where the Z b states are assumed to be molecules.Instead of fitting data directly, Ref. [12] constrains some parameters using data on ϒ(5S) → B The leading order Feynman diagrams for these two different production mechanisms are presented in Fig. 5.The Feynman diagrams for the non-resonant contributions are shown in Fig. 6.We give all the amplitudes for Figs. 5 and 6 in Appendix C.  a.We assume that Z b states are pure molecular states, then we have to set Z = 0 in the fit.In this way, only the diagrams in Figs.3a, c, 5b, c, and 6 give nonvanishing amplitudes.b.We assume that Z b states contain substantial compact components, i.e., a tetraquark component.It is shown in Ref. [22] that the production rate of a molecular state is proportional to its wave function squared at the origin | (0)| 2 .Because the wave function of the molecular component in a loosely bound state spreads far out in space, | (0)| 2 is quite small, then the production rate of Z ( ) b through the molecular component will be suppressed.Therefore, we further assume that Z ( )  b is mainly produced through the compact component, and we set g 1 = g 2 = 0 in the fitting.It is worth mentioning that Refs.[21,31] demonstrate that the production of a near threshold state (by which we mean a mixture of the compact component and molecular component) is driven by the compact component.On the other hand, the hadronic decays of Z ( ) b into h b (m P)π will mainly go through the molecular component.This can be found from the power counting analysis.We treat the binding momentum γ , the three-momentum of the bottom meson p B and the four momentum of the pion p π as small scales, i.e., they are all at the order of O( p).Note that in the non-relativistic effective field theory, the propagator of the heavy meson is at the order of O( p −2 ), and the measure of the one loop integration is at the order of O( p 5 ).One can then find that Fig. 2a is at the order of O( p −1/2 ), while Fig. 2b is at the order of O( p 0 ).Thus, as a leading order study, we set g z = 0 and neglect the contribution from Fig. 2b.Up to now, we have shown that while the production of Z ( ) b is driven by the compact component, its hadronic decays mainly go through the molecular component.It is interesting to note that similar features are adopted for X (3872) in Ref. [21].By setting g 1 = g 2 = g z = 0, one can find that only the diagrams Figs.2a and 5a give nonvanishing amplitudes, and the number of the relevant free parameters in this scheme is the same as that in scheme (a) (see Table 1).

With the amplitudes given in
We then compare our fitting schemes with that used in Ref. [9].Although scheme (b) and Ref. [9] use the same decay mechanism for ϒ(5S) → Z b π → h b ππ as shown in Fig. 2a, there are some differences between them.The main difference is that Ref. [9] sets Z = 0, while in scheme (b) we let Z to be a free parameter which satisfies 0 < Z < 1.As shown explicitly in Appendix B, the amplitude for Fig. 2a is zero by setting Z = 0. Physically, by setting Z = 0, one assumes the Z b states as pure molecular states which do not contain compact components, hence they cannot be produced through the compact components.Therefore, if one uses Fig. 2a to describe the decay mechanism of ϒ(5S) → Z b π → h b ππ, one cannot set Z = 0 as in Ref. [9].The consistent treatment is to let Z to be a free parameter which satisfies 0 < Z < 1.On the other hand, if one assumes Z ( ) b to be a pure molecular state, i.e., Z = 0, one should note that it can only be produced through an indirect process.Therefore, for the pure molecular scenario, one should use Fig. 3a, c, i.e., scheme (a), instead of Fig. 2a to describe the decay mechanism of ϒ(5S) → Z b π → h b ππ.Now we come to discuss the applicability of EFT.In the decay Z ( )  b → h b (2P)π , the momentum of the pion is around 300-400 MeV in the energy region of our con-  b → h b (1P)π , the momentum of the pion is relatively large and around 600-700 MeV.Based on naive dimensional analysis, Ref. [10] warns that the EFT expansion may not be good enough for decay Z ( ) b → h b (1P)π due to the relatively large pion momentum.However, the results from the complete loop calculations can be more complex than the naive dimensional analysis.One may refer to Ref. [44] for an example.Generally, it is complex to study the convergence of the effective field theory, and reliable conclusions can only be reached once the complete higher loop contributions are available.Since such a kind of study is beyond the scope of this work, we take a more pragmatic approach with two options in the fit.We choose an individual normalization factor for each final state in the fit.In this way, we need not to fix values of g ϒ and g h .We present all the fitted parameters in Table .1, and we show the fitting results of fit (1a), fit (2a) and fit (1b) in Figs. 7, 8.Note that the width in Table 1 is not the total width, but the width defined in Eq. (8).
We give some brief discussions as regards our fitting results as follows: • It is found in the experimental fits that the relative phase between Z b and Z b in the h b (m P)π π channel is 180 • [1,2].In fitting scheme (a), the relative minus sign between iM 3a,3c for Z b and Z b can account for this relative phase.However, one cannot find such a relative phase in amplitudes which are used in scheme (b).
In our fitting, we find that scheme (b) gives a good fit only if such a relative phase is included.This may be attributed to g ϒ [defined in Eq. ( 15)] which has a relative minus sign between Z b and Z b .We note that a very recent paper, Ref. [45], proposed an explanation for this relative minus sign.• From the fitting results of fit (1b) and fit (2b), one can find that the fitted parameters in fit (1b) and fit (2b) are close to each other.This indicates that the fitting results in scheme (b) are not sensitive to data on ϒ(5S) → h b (1P)π + π − .Whether this means that the effective field theory can be successfully applied in ϒ(5S) → h b (1P)π + π − needs to be further investigated.Nevertheless, our numerical results show that such a possibility exists.It is also interesting to find that in scheme (b) the fitted binding energy and the width of Z b are close to those of Z b .This seems to be consistent with the heavy-quark spin symmetry.• With all data sets, scheme (1b) gives much better fitting quality than scheme (1a).Unfortunately, if data on ϒ(5S) → h b (1P)π + π − are dropped, the two schemes give almost equal fitting qualities.In this sense, it seems too early to claim conclusively that Z b states contain substantial compact components.However, a substantial compact component in Z ( ) b can explain its large production rates in experiments.In contrast, a pure molecular state with the tiny binding energy as determined in scheme (a) is not likely to have large production rates in ϒ(5S) decays.
• The binding energies of the Z b states from the fit are generally very small.If we fix B = 0.1 MeV, which is the case for X (3872), and Z = 0.4 in fit (1b), we get a fitting quality χ 2 = 90, which is still acceptable and better than fit (1a).The other fitting parameters are B = 0.23 (14) MeV, Z b = 6.5 (9) MeV and Z b = 5.6 (9) MeV.This result also seems to be consistent with the heavy-quark symmetry.• One can also analyze data on ϒ(5S) → ϒ(nS)π + π − in the EFT approach.However, different from h b (m P)π + π − and B ( * ) B( * ) π , the non-resonant contribution in ϒ(nS)π + π − is significant.It is impossible to consider the interference with the non-resonant contribution correctly in one-dimensional analysis.To analyze data on ϒ(5S) → ϒ(nS)π + π − , one needs to fit the two-dimensional Dalitz distribution, which is beyond the scope of the present manuscript.

Summary
We have done a combined analysis of data on  [46,47].
Acknowledgments We would like to thank Qiang Zhao for a careful reading of the manuscript and valuable comments and Feng-Kun Guo for very useful discussions.We would also like to thank Guang-Yi Tang for the helpful discussions on the Belle result.This work is supported, in part, by National Natural Science Foundation of China (Grant Nos.11147022 and 11305137) and Doctoral Foundation of Xinjiang University (No. BS110104).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/),which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.Funded by SCOAP 3 .

Appendix A: One loop three-point functions
The three-point loop functions we will encounter are where μ i j = m i m j /(m i +m j ) are the reduced masses, I (1) (m 1 , m 2 , m 3 , q) is defined as .
For more details, one may refer to Ref. [37].

Appendix B: Amplitudes for ϒ(5S) → h b (1 P, 2 P)π + π −
The amplitudes for ϒ(5S) → Z + b π − → h b (m P)π + π − in Figs. 2 and 3 read E is the energy defined relative to the B B * threshold.B is the binding energy.g is defined in Eq. ( 4).E π is the energy of π − , p is the three-momentum of the π − , and q is the three-momentum of the π + .μ = m B m B * m B +m B * is the reduced mass.Note that the terms proportional to p k p m in M 3c and M 3d will disappear in the heavy-quark limit, i.e., m B = m B * .This indicates that in the heavy-quark limit, the D wave decay of ϒ(5S) → Z b π is forbidden.We neglect the terms proportional to p k p m in the fit, since they will be suppressed by the heavy-quark spin symmetry.
The amplitudes for ϒ 2 and 3 read E is the energy defined relative to the B * B * threshold.B is the binding energy.g is the renormalized coupling constant which is defined in Eq. ( 4).We use g here to distinguish from g, which is used in ϒ(5S) → Z + b π − → h b (m P)π + π − , since they may be different due to different binding energies.μ is the reduced mass of the B * B * system.Other notations are the same as that in ϒ(5S) → Z + b π − → h b (m P)π + π − .We also neglect the terms proportional to p k p m in the fit.
In the above, we have assumed that Z b only couples to B B * , while Z b only couples to B * B * .We also assume that the probability of finding an elementary state in Z b and Z b is the same.In other words, we use the same Z for Z b and Z b .

Appendix C: Amplitudes for ϒ(5S) → B ( * ) B( * ) π
The amplitudes for ϒ(5S) → B + B * 0 π in Figs. 5 and 6 read × I (1) (m B * , m B * , m B , q π ) + I (1) (m B * , m B , m B * , q π ) q 2 π δ jm + I (1) (m B , m B , m B * , q π ) − I (1) (m B * , m B , m B * , q π ) q j π q m π × 1 (q i π p j B + q π • p B δ i j − q j π p i B ) .(35) E is the energy defined relative to the B B * threshold.E π is the pion energy, q π is the three-momentum of the pion.μ is the reduced mass of the B B * system.p B and p B are the three-momenta of B + and B * 0 , respectively.
is the hyperfine splitting of the B mesons.
The amplitudes for ϒ(5S) → B * + B * 0 π in Fig. 5 and Fig. 6 read × I (1) (m B * , m B * , m B * , q π )+ I (1) (m B , m B * , m B * , q π ) q 2 π δ km + I (1) (m B * , m B * , m B * , q π )− I (1) (m B , m B * , m B * , q π ) q k π q m π × 1 E is the energy defined relative to the B * B * threshold.E π is the pion energy, q π is the three-momentum of the pion.μ is the reduced mass of the B * B * system.p B and p B are the three-momenta of B * + and B * 0 , respectively.is the hyperfine splitting of the B mesons.
, the decay Z ( ) b → h b (m P)π can proceed through both direct and indirect processes.In direct decay, Z b states will decay to h b (m P)π directly.In indirect decay, Z ( ) b will first decay into a bottom and antibottom meson pair and then the meson pair rescatters into h b (m P)π .

Fig. 2
Fig. 2 Feynman diagrams for ϒ(5S) → Z ( ) b π → h b (m P)π π , where the Z b states are produced in direct production processes.Solid lines in the loop represent bottom and anti-bottom mesons

Fig. 3
Fig. 3 Feynman diagrams for ϒ(5S) → Z ( ) b π → h b (m P)π π , where the Z b states are produced in indirect production processes.Solid lines in the loops represent bottom and anti-bottom mesons ( )b → h b (m P)π .Finally, we come to the vertex describing decay of ϒ(5S) into Z ( ) b π .The corresponding Lagrangian to the leading order of the chiral expansion is given by[9] should not be surprising to find this relative minus sign, since if one assumes Z b (Z b ) couples to B * B * (B B * ) with the same strength as that of Z b (Z b ) couples to B B * (B * B * ), one would find that the meson loop amplitudes would be suppressed in a heavy-quark spin symmetry world as noticed in [37].• Assuming that Z b and Z b are spin partners of each other, we can use the same Z for Z b and Z b .

Fig. 4
Fig. 4 Feynman diagrams for non-resonant processes ϒ(5S) → h b (m P)π π.Solid lines in the loop represent bottom and anti-bottom mesons ( * ) B( * ) , and it then calculates the differential distribution for ϒ(5S) → B ( * ) B( * ) π as a function of the invariant mass of the B ( * ) B( * ) pair.In this work, we give the amplitudes for ϒ(5S) → B ( * ) B( * ) π in EFT and constrain parameters by fitting the data directly.Similar to ϒ(5S) → Z ( ) b π → h b (m P)π π , Z b states can be produced through both direct and indirect processes.

Fig. 7 M 2 Fig. 8
Fig.7 Comparison of the invariant mass spectra of h b (1P)π and h b (2P)π in fit (1a), fit (2a), fit (1b) and the experiment.The dotted line is the result of fit (1a).The dashed line is the result of fit (2a).The solid line is the result of fit (1b).Data are from[1]

Table 1
Parameters for four fits π and ϒ(5S) → B * B * π within EFT approach.With a combined analysis, we determine the resonance parameters of Z b states in two scenarios.In one scenario we assume that Z b states are pure molecular states, while in the other one we assume that Z b states contain compact components.It is found that by assuming that Z b states contain substantial compact components, one can have a better description of all data than by pure molecular assumption.By fitting the invariant mass spectra of ϒ(5S) → h b (1P, 2P)π + π − and ϒ(5S) → B ( * ) B( * ) π , we determine that the probability of finding a compact component in Z