Hindered magnetic dipole transitions between P-wave bottomonia and coupled-channel effects

In the hindered magnetic dipole transitions of heavy quarkonia, the coupled-channel effects originating from the coupling of quarkonia to a pair of heavy and anti-heavy mesons can play a dominant role. Here, we study the hindered magnetic dipole transitions between two $P$-wave bottomonia, $\chi_b(n P)$ and $h_b(n^\prime P)$, with $n\neq n^\prime$. In these processes the coupled-channel effects are expected to lead to partial widths much larger than the quark model predictions. We estimate these partial widths which, however, are very sensitive to unknown coupling constants related to the vertices $\chi_{b0}(nP)B\bar B$. A measurement of the hindered M1 transitions can shed light on the coupled-channel dynamics in these transitions and hence on the size of the coupling constants. We also suggest to check the coupled-channel effects by comparing results from quenched and fully dynamical lattice QCD calculations.

counting. Thus, the triangle diagram in Fig. 1 scales as [28] where the factors 1/m b and E γ are due to the spin-flip of the heavy quark in M1 transitions and the P -wave coupling of the photon to the bottom mesons, respectively. One thus sees that the closer the bottomnia to the bottom-meson thresholds, the larger the coupled-channel effects. One remark is in order: v in the power counting is in fact the average of two velocities. This can be estimated  Fig. 1, M i(f ) is the mass for the initial (final) bottomonium, andm jk is the averaged value of m j and m k . However, unlike the case of charmonium hindered M1 transitions, the two-loop diagrams with a pion exchanged between two intermediate bottom mesons are not highly suppressed for the bottomonium transitions. From the power counting analysis in Ref. [28], the relative importance of the two-loop diagrams shown in Fig. 2 in comparison with the triangle diagram given in Fig. 1 can be described by a factor where M B is the bottom meson mass, Λ χ = 4πF π , with F π the pion decay constant, is the chiral symmetry breaking scale, and g 0.5 is the axial coupling constant for bottom mesons [37][38][39]. This ratio can be understood as follows (taking the left diagram in Fig. 2 as an example): the two more propagators and one more nonrelativistic loop integral measure, in comparison with the diagram in Fig. 1, together give the factor v = v 5 /(v 2 ) 2 in the above equation; g 2 /Λ 2 χ comes from the two pionic vertices and one more loop; M 2 B is introduced to make the ratio dimensionless. Taking the masses of the 1P , 2P and 3P bottomonia from Refs. [40,41], the velocity in the power counting may be estimated to be 0.31, 0.23 and 0.18 for the 2P → 1P , 3P → 1P and 3P → 2P radiative transitions, respectively. One then finds that the relative factor given in Eq. (2) is of order one, which means that the contribution of two-loop diagrams like the ones shown in Fig. 2 should be of similar size as the one-loop triangle diagram in Fig. 1. This is different from the charmonium case studied in Ref. [28] where M 2 B is replaced by the much smaller M 2 D and thus leads to a suppression. Nevertheless, we will only calculate the triangle diagram, and keep in mind that given the power counting of the two-loop diagrams such a calculation can only be regarded as an estimate, rather than a precise calculation, with a quantitative uncertainty analysis out of reach.
As a result of the approximate heavy quark spin symmetry, one can classify the heavy-light bottom mesons according to the total angular momentum of the light degrees of freedom s and collect them in doublets with total spin J = s ± 1 2 . For instance, the pseudoscalar (P a ) and vector (V a ) bottom mesons are collected in the spin multiplet with s P = 1 2 − . The two-component effective fields [42] that describe the ground state heavy mesons in the heavy quark limit are H a = V a · σ + P a for annihilating bottom mesons andH a = − V a · σ +P a for annihilating anti-bottom mesons, where σ are the Pauli matrices and a is the light flavor index. Moreover, the P -wave bottomonia can be collected in a spin multiplet as As mentioned above, the leading order coupling of the P -wave bottomonium to the bottom and anti-bottom mesons is in an S-wave, and thus is given by [36,43] where Tr denotes the trace in the spinor space. We also need the magnetic coupling of the photon to the S-wave heavy mesons [26,42,44] where  Table 1 for the corresponding transitions. The pertinent transition amplitudes are given in the appendix. From these amplitudes, one clearly sees two sources of spin symmetry breaking: the terms from the bottom quark magnetic moment are explicitly proportional to 1/m b , and the sum of β-terms in each amplitude vanishes if the vector and pseudoscalar bottom mesons are taken to be degenerate. 1 The loops involved here are convergent, which means that the coupled-channel effects for the processes of interest are dominated by long-distance physics described in our NREFT. We do not  Fig. 1. For simplicity, the charge conjugation modes and the light flavor labels are not shown here.
need to introduce a counterterm here. The situation is different for the case of E1 transitions. The loop integrals involved there are divergent, and thus the contact term considered in Ref. [26] also serves as a counterterm and is necessary for renormalization.
Using the masses of the mesons given by the Particle Data Group [40], it is easy to get numerical results for the partial decay widths. As for the masses of the 3P bottomonia, we choose the quark model values from Ref. [6], which were obtained based on the measured χ bJ (3P ) mass by the LHCb Collaboration [41] with the predicted multiplet mass splittings, i.e. M h b (3P ) = 10.519 GeV, M χ b0 (3P ) = 10.500 GeV, M χ b1 (3P ) = 10.518 GeV and M χ b2 (3P ) = 10.528 GeV. These masses are very close to the ones in Ref. [7], where the coupled-channel effects are taken into account in a nonrelativistic quark model. We also take β = 1/276 MeV −1 [42], and m b = 4.9 GeV. The decay amplitudes are proportional to the product squared of the coupling constants of the bottom and anti-bottom mesons to the 1P , 2P and 3P bottomonia, denoted as g 1 , g 1 and g 1 , respectively. As the mass of the χ bJ (1P, 2P, 3P ) and h b (1P, 2P, 3P ) are below the bottom and anti-bottom meson threshold, the coupling constants cannot be measured directly. Here, we show the decay width of the hindered M1 transitions between two P -wave bottomonia in units of the coupling constants in the Table 2. The unknown parameters will get cancelled if we calculate ratios of the decay widths which are proportional to the same product squared of coupling constants. Furthermore, we also expect that these ratios are less sensitive to the two-loop diagrams in Fig. 2 as the numerator and denominator in the ratio, being related to each other via spin symmetry, would get a similar correction. The ratios in 0.59 0.03 1.1 0.5 1.4 9. 2   Table 3: Comparison of the ratios of the decay widths for the 2P to 1P bottomonia with the ones from the RQM [6].
our calculation can be easily obtained from Table 2. In order to show that the coupled-channel effects lead to very different values for some of these ratios, we show a comparison of ratios for selected decay widths of the hindered M1 transitions between the 2P to 1P bottomonia with those obtained in the quenched quark model of Ref. [6] in Table 3. These predictions can be tested in the future from experiments or lattice QCD calculations. In fact, radiative transitions of S-wave bottomonia, including the hindered M1 ones, have been studied by using lattice QCD [32,33,35]. As suggested in Ref. [28], one can check the coupled-channel effects directly in lattice QCD by comparing results in full and quenched calculations -the former includes the coupled-channel effects intrinsically while the latter does not.
As mentioned in Ref. [6], the numerical results of these hindered transitions in the quark model are very sensitive to relativistic corrections (these transitions do not vanish only when relativistic corrections are accounted for in quenched quark model). Nevertheless, they are tiny because the M1 transitions break heavy quark spin symmetry as well, and are in the ballpark of sub-eV to eV in Ref. [6]. If the partial widths really take such small values, an experimental observation of the bottomonium hindered M1 transitions would be impossible in the foreseeable future. In turn, this means that once such transitions are observed, the mechanism would be different from that in the quenched quark model, and would be caused by coupled-channel effects. Then, the measured partial widths can be used to estimate the involved coupling constants.
Unfortunately, the values of the coupling constants g 1 , g 1 and g 1 cannot be estimated reliably. If one takes the model estimate made in Ref. [43], 2 g 1 = −2 m χ b0 /3/f χ b0 and uses the value f χ b0 ≈ 175 MeV from a QCD sum rule calculation [45], then one gets g 1 ∼ −20 GeV −1/2 . This value is so large that if the χ b0 is located only 1 MeV above the B 0B0 threshold it would have a huge width of 21 GeV. However, the quark model predictions for the open-bottom partial decay widths of the 4P bottomonia leads to |g 1 (4P )| ∼ 0.2 GeV −1/2 (the one for the 5P states is slightly smaller), which, although it is for the 4P states, is two orders of magnitude smaller than that from the former estimate. In Ref. [28], the product of the coupling constants (g 1 g 1 ) 2 is estimated to be of order O(10 GeV −2 ) in the charm sector, where the difference between the model estimate for g 1 [43] and the extracted value from quark model predictions of the 2P charmonium decay widths is much smaller. If we naively take the same estimate here, despite that there is no simple flavor symmetry between charmonia and bottomonia, then the partial decay widths of O 1 ∼ 10 2 keV could be large enough for a possible measurement in the future.
In principle, we can also calculate the decay widths for the isospin breaking transitions between the χ bJ (nP ) states with the emission of one pion. They would be proportional to the same combination of unknown coupling constants. The charmonium analogues from the coupled-channel effects have been analyzed in details in Ref. [24]. However, we refrain from such a calculation because the isospin breaking between the charged and neutral bottom mesons is one order of magnitude smaller than that in the charmed sector because of the destructive interference between the contributions from the up and down quark mass difference and the electromagnetic effect [46].
In summary, we studied the hindered M1 transitions between two P -wave bottomonia, χ b (nP ) and h b (n P ) (n = n ) assuming the mechanism is dominated by coupled-channel effects. Because of the suppression from heavy quark spin breaking and small relativistic corrections, such transitions have tiny partial widths from sub-eV to eV in quark model. In the mechanism underlying coupledchannel effects, the breaking of heavy quark spin symmetry can come from the different masses of bottom mesons within the same spin multiplet, and the problem of tiny matrix elements for transitions between bottomonia of different principal quantum numbers in the quark model does not exist as well. Therefore, it is natural to expect that the coupled-channel effects lead to much larger widths for such transitions than those predicted in the quark model. A future observation of such transitions at, e.g., Belle-II [29] may be regarded as a clear signal of the coupled-channel effects, and the measured widths could then be used to extract a rough value of the product of the so-far unknown coupling constants, e.g. At last, we want to emphasize again that the coupled-channel effects in heavy quarkonium transitions can be checked directly in lattice QCD by comparing results from quenched and fully dynamical simulations as we already suggested in Ref. [28]. A better understanding of coupled-channel effects would lead to new insights into the dynamics of heavy quarkonia. the initial bottomium and the photon, respectively, and m i (i = 1, 2, 3) are the mass of the intermediate mesons. In the deriving of Eq. (6), the nonrelativistic approximation has been adopted.
The pertinent amplitudes for the decays are listed here: where the initial bottomonium should be understood to be of higher excitation then the final one, ε i (γ), ε i (h b ) and ε i (χ b1 ) are the polarization vectors for the photon, h b and χ b1 , respectively, and ε ij (χ b2 ) is the symmetric polarization tensor for the χ b2 . One also needs to notice that a factor M i M f , with M i,f denoting the masses of the initial and final bottomonia, should be multiplied to each of the amplitudes to account for the nonrelativistic normalizations of the heavy quarkonium fields (similar factors for the intermediate heavy mesons have been obsorbed in the definition of the loop function).