Enhanced breaking of heavy quark spin symmetry

Heavy quark spin symmetry is useful to make predictions on ratios of decay or production rates of systems involving heavy quarks. The breaking of spin symmetry is generally of the order of $O({\Lambda_{\rm QCD}/m_Q})$, with $\Lambda_{\rm QCD}$ the scale of QCD and $m_Q$ the heavy quark mass. In this paper, we will show that a small $S$- and $D$-wave mixing in the wave function of the heavy quarkonium could induce a large breaking in the ratios of partial decay widths. As an example, we consider the decays of the $\Upsilon(10860)$ into the $\chi_{bJ}\omega\, (J=0,1,2)$, which were recently measured by the Belle Collaboration. These decays exhibit a huge breaking of the spin symmetry relation were the $\Upsilon(10860)$ a pure $5S$ bottomonium state. We propose that this could be a consequence of a mixing of the $S$-wave and $D$-wave components in the $\Upsilon(10860)$. Prediction on the ratio $\Gamma(\Upsilon(10860)\to\chi_{b0}\omega)/\Gamma(\Upsilon(10860)\to\chi_{b2}\omega)$ is presented assuming that the decay of the $D$-wave component is dominated by the coupled-channel effects.

A heavy quarkonium is a system consisting of a heavy quark and a heavy antiquark. The ground states and low-lying excited states below the open-flavor thresholds were well described in terms of potential quark models, e.g., the Godfrey-Isgur quark model [1], while the higher excited states are more complicated. The complexity comes from, e.g., the nearby strongly coupled thresholds, the existence of many new quarkonium-like states discovered in the last decade and so on. Because the heavy quark mass m Q is much larger than the scale of quantum chromodynamics (QCD), Λ QCD , the amplitude of changing the spin orientation of a heavy quark by interacting with soft gluons is small, suppressed by O (Λ QCD /m Q ) relative to the spin-conserving case [2]. The resulting heavy quark spin symmetry (HQSS) [3] can lead to important observable consequences. On one hand, heavy quarkonium states are organized into spin multiplets; on the other hand, the decay or production rate involving one heavy quarkonium can often be related to the one of its spin partner in the leading approximation. Breaking of HQSS is typically of the order of O (Λ QCD /m Q ) or even higher. In this paper, we will argue that the HQSS breaking could be much larger in certain processes. To be specific, we will show that a small mixing of S-and D-wave heavy quarkonia, which is at the order O (Λ QCD /m Q ) 2 , could result in a significant breaking of the spin symmetry relations when the decay amplitude of the D-wave component is enhanced. As an example, we will calculate the processes Υ(10860) → χ bJ ω (J = 0, 1, 2). Measurements for these transitions were done by the Belle Collaboration very recently, and the preliminary results for the branching fractions are [4] B (Υ(10860) → χ b0 ω) < 3.9 × 10 −3 , One sees that the branching fraction for the χ b1 ω mode is larger than that for the χ b2 ω, which is quite different from the HQSS prediction assuming the Υ(10860) to be the 5S bottomonium state, see Eq. (6) below. This indicates a very large spin symmetry breaking: As we will show later, a small mixture of a D-wavebb component in the Υ(10860) is able to cause the ratios of Γ(Υ(10860) → χ bJ ω) to be very different from the spin symmetry relations as observed.
Consequences of HQSS can be easily analyzed using heavy meson effective field theory (for a review, see Ref. [5]). Let us take the transitions from a vector heavy quarkonium into the χ J ω as an example, where χ J is a P -wave heavy quarkonium with quantum numbers J P C = J ++ . Here we will use the two-component notation in Ref. [6] which is convenient for nonrelativistic processes with negligible recoil effect. The fields for the S-wave, P -wave and D-wave heavy quarkonium states are denoted by J, χ i and J ij , respectively, which are J = ψ · σ, [5,[7][8][9], where σ are the Pauli matrices, and ψ, χ J and ψ D annihilate the S-, P -and D-wave heavy quarkonia, respectively. The states included in the above expressions have other spin partners which can be included as well, however, only the fields relevant for our discussion are shown.
Since the heavy quarkonia can be treated nonrelativistically, an expansion over low momenta can be done. To leading order of such an expansion, the Lagrangian for the decays of an S-wave or a D-wave heavy quarkonium into χ J ω reads where denotes the trace over the spinor space. With this Lagrangian, one is ready to obtain the ratios of decay widths of an excited S-wave heavy quarkonium into the χ J ω when the difference in phase space is neglected The ratios are completely different if the initial state is a D-wave heavy quarkonium. In this case, one obtains Therefore, the ratios of the decay widths of an excited heavy quarkonium into the χ J ω can be used to probe the spin structure of the initial state.
Replacing the ω by a photon, the above analysis still applies if we change the widths on the left side of Eqs. (3) and (4) by Γ/E 3 γ with E γ the photon energy in the rest frame of the initial state. The factor of the photon energy is required by gauge symmetry. As was shown long time ago in Ref. [10], the spin symmetry relations for the radiative transitions are generally in a quite good agreement with the experimental data, and the breaking of the spin symmetry relations is at the However, HQSS breaking for near-threshold vector quarkonium states could be enhanced due to the coupling to heavy meson pairs in a P -wave [11]. In the following, we will explore a different mechanism, and show that a small S-D mixing could result in a significant spin symmetry breaking because of an enhancement of the decays of the D-wave component due to coupled-channel effect.
Let us take the decays of the Υ(10860) into the χ bJ ω as an specific example. The Υ (10860) is often considered as the 5S vector bottomonium. It was argued that the HQSS breaking in the Υ(10860) decays into open-bottom mesons could be as large as 10% to 20% [12] (see also discussions in Ref. [13]). It is thus reasonable to assume that the wave function of the Υ(10860) contains a small mixture of a D-wave component, Υ D . The decay amplitude can be written as where θ is the mixing angle, and A S and A D are the decay amplitudes from the the S-wave and D-wave components, respectively. One sees from Eqs. (3) and (4) that the ratios of the partial widths of the S-wave and D-wave components are distinct. When the phase space is taken into account, the corresponding ratios for the Υ(10860) decays in question are . All of the measured decay modes into a lower bottomonium plus a pair of pions or kaons are of the order of 10 −3 [14]. The two-body decays Υ(10860) → χ bJ ω can happen in an S-wave. We expect that these decays should be of 10 −3 as well since no suppression of these decays in comparison with the above mentioned three-body decays can be anticipated, which is in agreement with the Belle measurement [4] given in Eq. (1). How large would be the decay widths for Υ(5S) → χ bJ ω (the notation 5S is used to refer to the pure S-wave nature)? They may be estimated as where q ω and E γ refer to the ω momentum and the photon energy in the corresponding decays, respectively. The above formula can be derived using the vector meson dominance model for the radiative decays. The partial widths of the χ b1,b2 (2P ) into both the Υ(1S)ω and Υ(1S)γ were measured [14], however, the radiative decay widths of the Υ(5S) → χ bJ γ are not known.
Nevertheless, we get where the first and second numbers in the parentheses are obtained using the data for the χ b1 (2P ) and χ b2 (2P ), respectively. For a rough estimate of Γ(5S) → χ bJ γ, we may check all the mea-  Here the charge conjugated ones are not listed but considered in the calculation.
For the diagram shown in Fig. 1, all three vertices are S-wave, and thus the loop amplitude is of where v 5 and (v 2 ) 3 account for the measure of the loop integral and three nonrelativistic propagators, respectively. Therefore, the negative power of the small velocity provides an enhancement to the coupled-channel amplitudes. For more discussion of the power counting, we refer to Refs. [8,[16][17][18][19]. Next, we will perform an explicit calculation of the coupled-channel effect based on the mechanism shown in Fig. 1.
In the two-component notation, the fields for the S-wave (s P = 1 2 − ) and P -wave (s = 3 2 + ) heavy mesons read H a = V a · σ + P a , and T i a = P ij 2a σ j + 2/3 P i 1a + i 1/6 ijk P j 1a σ k , where P a and V a annihilate the pseudoscalar and vector heavy mesons, respectively, with a = u, d labeling the light flavors, and P 1a and P 2a annihilate the axial and tensor heavy mesons, respectively. The fields annihilating their anti-particles areH a = − V a · σ +P a ,T i a = −P ij 2a σ j + 2/3P i 1a − i 1/6 ijkP j 1a σ k . The properties of these fields under symmetry transformations can be found in Refs. [7,15].
The Lagrangian, which is invariant under tranformations of parity, charge conjugation, HQSS and Galilean invariance, for the coupling of the P -wave and D-wave heavy quarkonia to the s = 1 2 − and s = 3 2 + heavy mesons to leading order of the nonrelativistic expansion can be written as [7,15,20] The S-wave coupling of the ω-meson to the S-wave and P -wave heavy mesons can be described by where isospin symmetry is assumed.
Denoting the diagram shown in Fig. 1 by [THH], the loops contributing to the processes Υ D → χ bJ ω are listed in Tab. I. Using the Lagrangians given in Eqs. (10) and (11), the decay amplitudes can be easily obtained. It is interesting to notice that if we take the same mass for the heavy mesons in the same spin multiplet, the spin symmetry relations are kept even if coupled channels are considered, that is, one would get the same ratios 20 One sees that the decays into the χ b0 ω and χ b1 ω are more enhanced than that into the χ b2 ω. The reason is that the Υ(10860) mass is closer to the B 1B threshold than to the B 1B * one, cf. Table I.
If we put the initial state at the mass of the Υ(11020), the heaviest known bottomonium, the ratios will be even larger, 27.5 : 18.4 : 1.
With the above preparation, we can now show quantitatively how a significant HQSS breaking effect can be obtained from a small S-D mixing. To be specific, let us take the mixing angle θ = 1 • for instance, which corresponds to sin θ = 0.017. That is the D-wave component is of 1 This provides a simple method to calculate the HQSS relations for partial decay widths of processes involving hadronic molecules of a pair of heavy mesons, and the results in, e.g., Ref. [21] calculated using 6-j and 9-j symbols can be checked in this way. Without taking into account the phase space factors which include E 3 γ for the radiative decays and setting mesons in the same spin multiplet to be degenerate, the ratios for the decay widths,  We want to emphasize that the mixing angle and A S /A D always appear together, and thus cannot be fixed from the measured branching fractions. However, when one of the ratios R 02 or R 12 is measured, the other can be predicted as shown in Fig. 3, and the uncertainty should be of where the first uncertainty is propagated from the measured uncertainty of R 12 , and the second one from v = 0.26 is inherent in our nonrelativistic framework. Both ranges are consistent with the Belle upper limit, and an examination of the HQSS breaking mechanism proposed here urges an improved measurement, especially for the R 02 . This can be done at the future super-B factory.
Similarly, we can make predictions for the decays Υ(11020) → χ bJ ω. The curves are similar with slightly larger values.
To summarize, we have discussed a new mechanism to produce a sizable breaking of HQSS. We showed that a small S-D mixing for the vector heavy quarkonium could result in a much larger spin symmetry breaking effect. The essential point is that the decays of the D-wave component are largely enhanced due to S-wave coupling to nearby thresholds of a P -wave and an S-wave heavy meson pair. In particular, when one of the ratios of branching fractions for the processes Υ(10860, 11020) → χ bJ ω is measured, the other can be predicted. Using the Belle measurement for R 12 , two possible ranges of R 02 was predicted. The prediction can be examined at the future super-B factory. Such measurements will be important to better understand the spin symmetry breaking as well as the nature of the Υ(10860) and Υ(11020).