On the emergence of heavy quark spin symmetry breaking in heavy quarkonium decays

Heavy-quark spin symmetry (HQSS) implies that in the direct decay of a heavy quarkonium with spin $S$, only lower lying heavy quarkonia with the same spin $S$ can be produced. However, this selection rule, expected to work very well in the $b$-quark sector, can be overcome if multiquark intermediate states are involved in the decay chain, allowing for transitions to the final-state heavy quarkonia with a different spin $S^{\prime}$. In particular, the measured decays $\Upsilon(10860)\to \pi Z_b^{(\prime)} \to \pi\pi\Upsilon(nS)$ $(n=1,2,3)$ and $\Upsilon(10860)\to \pi Z_b^{(\prime)} \to \pi\pi h_b(mP)$ ($m=1,2$) appear to have nearly equal strengths which is conventionally explained by a simultaneous presence of both $S_{b\bar{b}}=0$ and $S_{b\bar{b}}=1$ components in the wave functions of the $Z_b$'s in equal shares. Meanwhile, the destructive interference between the contributions of the $Z_b$ and $Z_b'$ to the decay amplitude for a $\pi\pi h_b$ final state kills the signal to zero in the strict HQSS limit. In this paper, we discuss how the HQSS violation needs to be balanced by the narrowness of the $Z_b^{(\prime)}$ states in the physical case, to allow for equal transition strengths into final states with different total heavy quark spins, and how spin symmetry is restored as a result of a subtle interplay of the scales involved, when the mass of a heavy quark becomes infinite. Moreover, we demonstrate how similar branching fractions of the decays into $\pi\pi h_b$ and $\pi\pi\Upsilon$ can be obtained and how the mentioned HQSS breaking can be reconciled with the dispersive approach to the $\pi\pi/ K\bar K$ interaction in the final state and matched with the low-energy chiral dynamics in both final states.

Heavy-quark spin symmetry (HQSS) implies that in the direct decay of a heavy quarkonium with spin S, only lower lying heavy quarkonia with the same spin S can be produced. However, this selection rule, expected to work very well in the b-quark sector, can be overcome if multiquark intermediate states are involved in the decay chain, allowing for transitions to the final-state heavy quarkonia with a different spin S . In particular, the measured decays Υ(10860) → πZ ( ) b → ππΥ(nS) (n = 1, 2, 3) and Υ(10860) → πZ ( ) b → ππh b (mP ) (m = 1, 2) appear to have nearly equal strengths which is conventionally explained by a simultaneous presence of both S bb = 0 and S bb = 1 components in the wave functions of the Z b 's in equal shares. Meanwhile, the destructive interference between the contributions of the Z b and Z b to the decay amplitude for a ππh b final state kills the signal to zero in the strict HQSS limit. In this paper, we discuss how the HQSS violation needs to be balanced by the narrowness of the Z ( ) b states in the physical case, to allow for equal transition strengths into final states with different total heavy quark spins, and how spin symmetry is restored as a result of a subtle interplay of the scales involved, when the mass of a heavy quark becomes infinite. Moreover, we demonstrate how similar branching fractions of the decays into ππh b and ππΥ can be obtained and how the mentioned HQSS breaking can be reconciled with the dispersive approach to the ππ/KK interaction in the final state and matched with the low-energy chiral dynamics in both final states.

I. INTRODUCTION
The spectroscopy of hadronic states containing heavy quarks remains one of the fastest developing and most intriguing branches of studies of the strong interaction. Many new states have been discovered in the spectrum of charmonium and bottomonium, which do not fit into the quark model scheme and qualify as exotic states. For example, the states Z ± b (10610), Z ± b (10650) [1], Z ± c (3900) [2,3], Z ± c (4020) [4], Z ± (4430) [5][6][7][8] are charged and decay into final states containing a heavy quark Q and its antiquarkQ plus light hadrons. Therefore, their minimal quark content is four quarks. The interested reader can find a comprehensive overview of the current experimental and theoretical status of the exotic hadrons with heavy quarks in the dedicated review papers, for example, in Refs. [9][10][11][12][13][14][15].
The bottomoniumlike states Z ± b (10610) and Z ± b (10650) (for brevity in what follows often referred to as Z b and Z b , respectively) are an ideal laboratory to get a better understanding of exotic states, since they exist as two resonances with the same J P C = 1 +− [16], split by only about 45 MeV and are seen simultaneously in several modes. Specifically, the Belle Collaboration observed them as distinct peaks (i) in the invariant mass distributions of the π ± Υ(nS) (n = 1, 2, 3) and π ± h b (mP ) (m = 1, 2) subsystems in the dipion transitions from the vector bottomonium Υ(10860) [1] and (ii) in the elastic BB * 1 and B * B * channels in the decays Υ(10860) → πB ( * )B * with the dominant branching fractions [17,18].
In addition, the Z b (10610) and Z b (10650) states could in principle be experimentally searched for (and hopefully seen) in the ρη b channel. More specifically, because of a large ρ-meson width, the relevant channel to look at would be Υ(10860) → πZ ( ) b → π(ππ) I=1 η b , which is however difficult experimentally because of a potentially large background related to the presence of the neutral pion in the final state. Meanwhile, no data for the given channel exist to date. On the other hand, the measured branching fractions Υ(10860) → (π + π − π 0 ) non−ω χ bJ (J = 1, 2) are by an order of magnitude suppressed [19] relative to the ππΥ(nS) and ππh b (mP ) hiddenflavour decay channels considered in this work. One may therefore expect that the analogous branching for the π(ππ) I=1 η b final state is also negligibly small. The fact that the ρ and χ bJ are in a relative P wave in contrast to the S-wave final state ρη b should not affect this estimate since both channels open far away from the relevant energy range near the B ( * )B * thresholds, so the centrifugal barrier suppression should not be operative any longer. Also, the measured elastic and inelastic branching fractions for Υ(10860) listed in the items (i) and (ii) above leave only very little room to other possible contributions.
The two most prominent explanations for the Z b 's claimed to be consistent with the data are provided by a tetraquark model and a hadronic molecule picture, see, e.g., Refs. [10][11][12]15] for review articles and references therein. A review of the sum rules approach to the exotic states with heavy quarks and relevant references on the subject can be found in Ref. [20]. It should be noted that a particularly close location of the Z b 's to the thresholds of the BB * and B * B * channels provides a strong hint in favour of their molecular interpretation.
Both the Z ± b (10610) and Z ± b (10650) contain a heavy bb pair, so it is commonly accepted that heavy-quark spin symmetry (HQSS) should be realised to high accuracy in these systems and indeed, as will be reviewed in Sec. II, HQSS is able to explain naturally the interference pattern in the channels Z [21]. In Sec. III we demonstrate how the experimental observation that Br[Υ(10860) → ππh b (mP )] is qualitatively similar or actually even larger than Br[Υ(10860) → ππΥ(nS)] [19] can be reconciled with HQSS and its violation. It appears as a result of a subtle interplay of the scales involved in the system -most relevant being the relationship between the widths of the Z b states and their mass difference. Also, we discuss the restoration of spin symmetry as the heavy quark mass is taken to be infinitely large.
Section IV is devoted to the inclusion of the ππ/KK interaction in the final state. In particular, in Refs. [22][23][24] it is explained how the ππ/KK final state interaction (FSI) can be included in the spin conserving heavy meson decays Υ → ππΥ by means of twice subtracted dispersion integrals. Here the structure of the subtraction terms can be fixed by the chiral structure of the transition amplitudes, giving rise to Υ → ππΥ contact interactions. While the chiral structure for the transitions of the kind Υ → ππh b is the same, such counter terms violate HQSS and thus are expected to be strongly suppressed quantitatively. How this pattern can be reconciled with the S-wave ππ/KK FSI also for this kind of transitions is discussed in Sec. IV. We summarise in Sec. V.

II. SPIN WAVE FUNCTIONS
The key that allows for a change of the heavy-quark spin in the presence of multiquark states lies in the fact that the heavy-quark spins get rearranged in the assumed compact building blocks. Indeed, let us stick to the strict HQSS limit of an infinite b-quark mass, m b → ∞ (quantities in this limit will be labelled by the superscript (0) ), and, following Ref. [21], assume for illustration a molec-ular substructure for the Z Then, taking the spin wave functions of the B and B * mesons in the form one arrives at the following spin structure of the I G (J P ) = 1 + (1 + ) molecular states . Finally, via a Fierz rearrangement, these structures can be rewritten as where the combinations (χ † b χ b ) and (χ † b σ i χ b ) correspond to the states with S bb = 0 and S bb = 1, respectively. Therefore, relations (4) entail two important messages 2 • in the strict HQSS limit both Z b states carry spin-0 and spin-1 components with equal weight and • the spin-0 and spin-1 components appear with different relative signs in the two states.
Importantly, as demonstrated in Ref. [21], the above nontrivial interference pattern is indeed consistent with the data, where the two Z b peaks appear to be equally strong in the decay chains with a constructive interference in the former and a destructive interference in the latter case. This underlines the fact that, although in general coupled-channel transitions mix the basis states (1), nevertheless, both wave functions of the physical Z b 's still contain the S bb = 0 and S bb = 1 components in comparable shares. The same pattern emerges in the tetraquark picture proposed in Ref. [25], where the driving subclusters are compact diquarks and anti-diquarks. In contrast to the two scenarios just sketched, it should be stressed that without sizable heavy-quark spin-0 and spin-1 clusters within the Z b states the observed pattern is unexplained and appears to be very unnatural.
In the limit of the heavy quark mass going to infinity, the heavy-quark spin needs to be conserved. It is thus interesting to investigate, how this limit is reached in the scenario outlined above. It turns out that what controls the importance of the h b final states is the ratio of the Z b 's mass splitting, which violates HQSS and thus vanishes in the heavy quark limit, and the Z b 's widths, which survive the heavy quark limit. This will be discussed in detail below.

III. SPIN-FLIP AND SPIN-CONSERVING TRANSITIONS
In this section we study the transition amplitudes for the reactions Υ → ππΥ and Υ → ππh b via two Z b states and discuss (i) the scales that make it possible for the two amplitudes to have signals of a similar size for the physical masses of the heavy states involved and (ii) how the heavy-quark limit can be restored.
Following Ref. [21], the transition amplitudes read and where Υ, Υ , and h b are the polarisation vectors of the J P = 1 ± states while C Υ , C h b are normalisation constants. The Green's functions G ± describe the contribution of the Z b 's in the intermediate state. The pion 4-momenta are taken in the form Importantly, the relations between the spin wave functions (4) imply that where G 1 and G 2 denote the individual propagators for the Z b and Z b in the intermediate state, respectively. The nonrelativistic Green's functions for the Z b 's read where n = 1, 2 and ξ = ±. Further, M 5 stands for the mass of the Υ(10860) and both Z b 's are treated as unstable particles with the masses m Z1 = m z and m Z2 = m z and the widths Γ Z1 = Γ z and Γ Z2 = Γ z , respectively. These widths should be understood as visible widths of the Z ( ) b peaks in the experimental line shapes. Moreover, following Refs. [21,26,27], we assume Γ z = Γ z . Here we emphasise that a quantitative treatment of the experimental line shapes requires a coupled-channel effective-field-theory (EFT) based framework manifestly consistent with unitarity, as given in Refs. [23,28], in which the widths are generated dynamically. However, for the purposes formulated above, that is, for studying a nontrivial interplay of various scales, the use of the simple analytic model of Ref. [21] is sufficient.
Thus it is easy to find for the Green's functions G ± (ω ξ ) that where we introduced two dimensionless parameters, and with the average massm z = (m z + m z )/2. To understand the behaviour of the amplitude in the heavy-quark limit, we need to discuss the scaling of its different ingredients as functions of the heavy-quark mass m b . First of all, it is easy to see that the width Γ z is (approximately) independent of the heavy-quark mass. Indeed, the contributions from the inelastic (hiddenbottom) and elastic (open-bottom) channels to Γ z behave as as the coupling of a compound state to its constituents [29], and the pion and B-meson momenta scale with the heavy mass as Then, according to the heavy-quark effective theory, the mass of an S-wave heavy-light B ( * ) meson scales as whereΛ Λ QCD is a universal parameter related to the light-quark dynamics while the m b -independent constants λ 1 and λ 2 parametrise the contribution of the spindependent interactions with The ellipsis in Eq. (15) denotes terms of a higher order in the 1/m b expansion. Using that the binding energies of Z b and Z b are the same in the HQSS limit [21,26,27], it is easy to see, therefore, that Finally, in the mass difference M 5 −m z the leading term 2m b is cancelled, so that this difference tends to a constant in the limit m b → ∞. Accordingly the parameter x ξ gets m b independent. Meanwhile, the parameter r introduced in Eq. (12) scales as r ∝ 1/m b and, therefore, can be used as a measure of the HQSS breaking effects. At the same time r is the parameter that controls the strength of the dipion transitions from Υ(10860). Thus changing the pion energy ω ξ (and accordingly changing x ξ ) allows one to scan through the Z b resonance structures to check their evolution with the variations of the parameter r.
The physical masses and widths are taken from the Review of Particle Physics (RPP) by the PDG [19] and read (in MeV) where the width comes as an average between Γ Z b and Γ Z b . Such parameters correspond to the physical value of the ratio which is used as a reference point in what follows.
As an illustration we show in Fig. 1 |f ± (x, r)| as functions of x (x = x + and x = x − result in identical curves) for different values of r. It can be clearly seen from this figure that the signals in the channels Υ → ππΥ and Υ → ππh b , which are controlled by the functions f ± (x, r), respectively, demonstrate quite different patterns in the limit r → 0. Indeed, while the constructive interference in the Υ channel (function f + (x, r)) provides the sum of two peaks at x = 0 for small values of r, the destructive interference in the h b channel (function f − (x, r)) results in a severe damping of the signal in this limit, namely, Thus the HQSS indeed gets restored in the heavy quark limit, and the rate of this restoration is controlled by the parameter r.
In the meantime, in the opposite regime of r 1, in which the effect of the HQSS violation is balanced by the narrow width [see Eq. (12)], both the peak positions and their strengths coincide for f + (x, r) and f − (x, r): It is therefore not surprising that already for r = r phys , the curves |f + (x, r)| and |f − (x, r)| demonstrate a noticeable similarity both in the shape and strength -see Fig. 2.
Finally, we study the behaviour of the differential rates for the dipion transitions from the Υ(10860). The doubly differential decay width for a three-body final state takes the form where, depending on the process considered, |M| 2 is the decay amplitude (6) or (7) squared and averaged over the polarisations of the vector (axial vector) particles. Then, expressing m 2 πh b /πΥ in terms of x = x + and performing the integration over the dipion invariant mass, it is straightforward to arrive at the differential rate in the form For the presentation purposes we fix the ratio of the unknown overall factors C Υ and C h b introduced in the amplitudes (6) and (7) such that the total decay widths, evaluated as the integrals from the differential width (23) over the region in the x covering the Z b peaks, obey the constraint in qualitative agreement with the data [19]. The resulting normalised line shapes (23) are depicted (in arbitrary units) in the left panel of Fig. 3 from which one can see a noticeable similarity of the curves in the shape.
We also define the ratio and plot it as a function of the parameter r in the right panel of Fig. 3. From this plot one can see that, in agreement with the considerations of this section, the probability of the decay Γ[Υ → ππh b ] vanishes in the strict HQSS limit (at r = 0) but then grows with r to reach Γ[Υ → ππΥ ] at the physical point (19), as prescribed by the normalisation condition (24). Notice also that, alternatively, the limit r → 0 can be reached by increasing Γ z -see the definition (12)so that the small Z b widths play an important role in explaining the observed large branching fractions for the HQSS violating decays Υ(10860) → ππh b .   (11) for r = r phys from Eq. (19). See the caption of Fig. 1 for further details.

A. General remarks
In the previous section it was demonstrated how the explicit HQSS breaking due to the Z b mass splitting could be balanced by their narrow widths to provide similar branching fractions for the reactions with and without the heavy-quark spin flip. Here we discuss consequences of yet another peculiar property of the HQSS violation in the reaction Υ(10860) → ππh b (mP ) as compared with the ππΥ(nS) final states. Namely, the tree level amplitude of the reaction Υ(10860) → ππΥ(nS) admits two momentum-dependent chiral contact terms (with two derivatives of the pion fields), which parameterize a short range interaction in the system, already in the strict heavy-quark limit [22][23][24]30]. These contact terms can be matched with two subtraction constants needed to include the ππ/KK FSI in this reaction. In contrast, the leading contact interactions in the decays Υ(10860) → ππh b (mP ), while having the same chiral structure, violate HQSS and therefore should be strongly suppressed relative to those for the transitions that conserve the heavy quark spin. In what follows, we discuss how this fact can be reconciled with the fact that in the individual reactions and Υ(10860) → πZ b → ππh b (mP ), the same number of subtractions is required to render the dispersive integrals finite as in the channel with the ππΥ(nS) final state. Specifically, we demonstrate that the destructive interference of the two amplitudes in the ππh b channel (as opposed to the constructive interference for the ππΥ(nS) final state) provides a better convergence of the resulting dispersive integral, so that a smaller number of subtractions is required.
To bring a quantitative twist to this discussion, we employ the coupled-channel EFT-based approach developed for the Z b states in Refs. [28,31] with the ππ/KK FSI for the transition Υ(10860) → ππh b (mP ) to demonstrate that its role in this channel is indeed suppressed, in contrast to the ππΥ(nS) final states [24].

B. Kinematics and notations
The kinematics of the decays is conventionally introduced through the Mandelstam invariants, with where m π , m i , and m f are the masses of the pion, Υ ≡ Υ(10860), and Υ ≡ Υ(nS)/h b ≡ h b (mP ), respectively. As always, one has The chosen kinematics corresponds to considering the amplitude M(s, t, u) in a crossed channel, Υ(p i ) + Υ /h b (p f ) → π + (p + ) + π − (p − ). It proves convenient to pass over to the center of mass frame of the pions in the final state, so that their 4-momenta now read In the given kinematics, and with λ(m 2 1 , m 2 2 , m 2 3 ) for the standard Källen triangle function, Then, for z ≡ cos θ (here θ is the angle between the 3momenta p π and p f ) one finds C. Production amplitudes through Z b 's The amplitudes for the decays Υ(10860) → ππΥ(nS) and Υ(10860) → ππh b (mP ), which proceed through the Z b and Z b , can now be written as (36) and  (48), respectively. The figure shows that because of a very different interference patterns between the two Z b contributions (see especially the left panel), the resulting signal for the h b final state is much smaller than the one in the Υ channel, and will be further suppressed after "averaging" in the dispersive integral (43). To ease the comparison, in this figure, the masses of both quarkonia in the final states, Υ and h b , are taken equal to 9.899 GeV (the mass of h b (1P ) [19]).
respectively, wherep π = p π /p π , G ± , as before, include the contributions from both Z b states (see Eq. (9)), and the individual propagators of the Z b 's read with τ = t, u. Here we absorbed all slowly varying functions of momenta in the numerators into the normalization constants and neglected the widths of the Z b 's which do not play a role for the argument. Indeed, as one can see from the right plot in Fig. 3 (see also the discussion around Eq. (21)), the function R(r) is almost constant at r > r phys with the value of r phys quoted in Eq. (19). In other words, to pinpoint the main difference between the asymptotic (large-s) behaviour of the functions G + and G − , it is safe to take the limit of r → ∞ achieved for m z = m z and Γ z → 0 -see the definition of r in Eq. (12). We also note that the effect of a finite width will be included in the full calculation performed in Sec. IV D below. A nontrivial difference in the production of the Υ and h b states can be seen directly from the expressions for G ± (τ ) which read and found in a closed form, including anomalous pieces. Given that the left-hand cut amplitude M L 0 is known, the total amplitude including its part with the right-hand cuts due to the ππ and KK interaction in the final state can be restored dispersively aŝ withÎ whereΩ 0 (s) is the multichannel Omnès matrix 3 with the hats indicating multicomponent objects (vectors and matrices),σ(s) = diag{σ π , σ K } is a diagonal matrix with σ P (s) = 1 − s th P /s and s th P denoting the threshold in the corresponding channel (P = π, K). In particular, The S-wave meson-meson coupled-channel amplitudeT can be parametrised by the ππ scattering phase shift δ(s) [32][33][34][35] as well as the absolute value and phase of the ππ → KK transition [34,35]. For the reaction Υ → ππΥ the leading contribution comes from the Z b and Z b states, and thus one findŝ Consider now the S-wave-projected amplitude for the h b channel, where the factor z under the integral comes from the P -wave between the pions and the h b . According to Eq. (37), the Born amplitude reads FSI 4 . Furthermore, since the latter effect is expected to be small, we do not refit the data but employ all the parameters found in Ref. [28] from the fit to the data and only check how the line shapes change upon including the ππ FSI and varying the HQSS-breaking constant (52) within an interval consistent with naturalness. In particular, in Fig. 5 we demonstrate the modification of the line shapes in the channels π + π − h b (mP ) with m = 1, 2 when the coefficient c in Eq. (52) varies within a natural interval. For definiteness and keeping in mind that the values of c consistent with naturalness should be |c| ∼ 1, we choose a conservative interval |c| 10. Comparing the red band, which shows the spread with the parameter c of the line shapes with the ππ FSI included, and the black solid curve, which represents the fit built in Ref. [28], one can conclude that, in agreement with the arguments provided in this section, the modification of the line shapes due to the inclusion of the ππ interaction in the final state is marginal. To be more quantitative, in terms of the χ 2 /N dof this corresponds to a spread from 1.21 to 1.36 as compared with the value 1.29 obtained for the fit A from Ref. [28]. Moreover, if c is allowed to vary freely, the best fit to the data yields c(1P ) = 24 and c(2P ) = −24 with the χ 2 /dof =1.16, which is only marginally smaller than the lower bound within the natural interval (the red band in Fig. 5). However, such values of the parameter c returned by the best fit are formally somewhat larger than what we regard as natural.
We note also, as a disclaimer, that the fit may use the freedom of varying c to partly compensate for other approximations made in the contact fit A of Ref. [28], for example, the effect of the neglected one-pion exchange. Since the resulting effect for the h b final states is small, we do not dwell on this any further in this work. Although the inclusion of the ππ/KK FSI in an EFT approach with non-perturbative pions (see, e.g., Ref. [28]) is technically more demanding because of the additional left-hand cut introduced in this way, this would be an important step in the future more quantitative studies.

V. SUMMARY
In this work we investigate the consequences of heavyquark spin symmetry and the pattern of its breaking for the measured decays Υ(10860) → πZ These decays appear to have nearly equal probabilities in spite of the fact that a direct transition from S bb = 1 Υ(10860) to S bb = 0 h b (mP ) changes heavy-quark spin and therefore violates HQSS. As argued in Ref. [21], this can be naturally understood in the molecular scenario for the Z b states, because in this case Z b (10610) and Z b (10650) appear as orthogonal combinations of the S bb = 0 and S bb = 1 components of equal strength. On the other hand, the interference between the contributions of the Z b (10610) and Z b (10650) to the decay amplitude for a ππh b final state is destructive which results in a vanishing signal in the strict HQSS limit. In contrast to this, the signal for a ππΥ final state is fully in line with HQSS due to a constructive interference between the two Z b states. Thus, in order to have signals of a similar strength in the ππΥ and ππh b final states, HQSS in the latter channel must be violated (m z − m z = 0), and this violation should be balanced by another quantity with dimension of a mass of a similar (small) size. We demonstrate that this balance is provided by the small widths of the Z b states Γ z . As a consequence, the transition amplitude to the ππh b final state strongly depends on the ratio of these two scales, r = (m z − m z )/Γ z , as given in Eqs. (11) and (12). Remarkably, while the ratio r vanishes in the strict HQSS limit, its value r phys ≈ 3 obtained for the physical masses of the Z b 's corresponds to the dynamical regime which is already quite close to the opposite limit of r → ∞ -in this limit the absolute values of the transition amplitudes to the ππΥ and ππh b final states are shown to have exactly the same peak positions and strengths.
Also, we pinpoint a crucial difference related to HQSS in the role played by the ππ/KK FSI in the dipion transitions Υ(10860) → πZ ( ) b → ππΥ(nS) and Υ(10860) → πZ ( ) b → ππh b (mP ). We demonstrate how the absence of the direct short-range HQSS-conserving operators in the bottomonium transitions Υ(10860) → ππh b (mP ) can be reconciled with a subtraction in the dispersive integral if the ππ/KK FSI is taken into account. In particular, it is shown that when both Z b states are included simultaneously, their leading contributions to the total amplitude cancel against each other. As a result, the dispersive integrals in the ππh b channels demonstrate a better convergence than those in the ππΥ channels and the remaining subtraction constants in the ππh b amplitudes are indeed suppressed by HQSS. We have verified that, in agreement with natural expectations, implementing this scheme to take into account the ππ/KK FSI does not result in any appreciable change in the fitted line shapes in the ππh b (1P ) and ππh b (2P ) channels (contrary to the crucial role played by the ππ/KK FSI in the ππΥ(nS) channels), so that neglecting the ππ/KK FSI in the previous analysis of Ref. [28,31] is indeed justified, at least at the present level of accuracy dictated by the quality of the existing data. Meanwhile, our findings establish a way to improve on such an analysis when/if such improvements are finally called for by the experiment.
As a final remark, we notice that while the numerical analysis performed in this work relies on a particular fit to the data built in the framework of the molecular approach for the Z b 's, the qualitative arguments presented in this paper are solely based on the interplay of different scales (the mass splitting m z −m z and the width Γ z in the first place) and therefore can equally apply to any approach to the Z b 's which is consistent with HQSS and its leading breaking effects, in particular, to the tetraquark model.
Our findings concerning the role played by the ππ/KK FSI should be also important for understanding analogous charmonium transitions to the final states ππψ(nS) and ππh c (mP ) via the Z c (3900) and Z c (4020) states, though the effects of the HQSS violation in the charmonium sector are expected to be larger.