A study of beauty baryons with extended local hidden gauge approach A study of beauty baryons with extended local hidden gauge approach C. W. Xiaoa,∗, W. H. Lianga,b, E. Oseta aDepartamento de Fı́sica Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigación de Paterna, Apartado 22085, 46071 Valencia, Spain bDepartment of Physics, Guangxi Normal University, Guilin, 541004, P. R. China Abstract In present work we investigate the interaction of B̄N, B̄Δ, B̄∗N and B̄∗Δ states, together with their coupled channels. Taking into account the heavy quark spin symmetry for pion exchange and the results of the Weinberg Tomozawa term in the extended local hidden gauge approach, we search for states dynamically generated from the interaction, and find two states with small width, which we associate to the Λb(5912) and Λb(5920) states. In addition to these two Λb states, we find three more states with I = 0 and eight more states in I = 1, some of which are degenerate in different spin J. Keywords: coupled channels, box diagram, heavy quark spin symmetry 1. Introduction Recently, the discovery of the two Λb excited states by the LHCb collaboration [1], Λb(5912) and Λb(5920) with J P = 1/2−,3/2− respectively, drives more attention to the beauty sector, since the widths of these two states are very small, less than 1 MeV. The higher one is also confirmed by the CDF collaboration [2, 3]. In the present work, following the works [4, 5, 6, 7], we investigate the open beauty system of meson-baryon interaction, using the extended local hidden gauge for- malism and the coupled channel approach. The assump- tion that the heavy quarks act as spectators at the quark level automatically leads us to the results of the heavy quark spin symmetry for pion exchange. 2. Formalism We investigate the coupled channels πΣb, πΛb, ηΛb, ηΣb, B̄N with I = 0, 1. Similarly, we also study the B̄∗N and πΣ∗b, ηΣ ∗ b, B̄Δ, B̄ ∗Δ channels, belonging to a decuplet of 3/2+ states. Using the local hidden gauge ∗Presenter P P B B(a) V(q) V V B B(b) V(q) Figure 1: Diagrammatic representation of the pseudoscalar-baryon in- teraction (a) and vector-baryon interaction (b). approach, the meson- baryon interaction proceeds via the exchange of vector mesons as depicted in Fig. 1. Assuming that the heavy quarks act as spectators in the dominant terms of the interaction, with the extended local hidden gauge approach from the exchange of light vector mesons, the transition potential is given by Vi j = −Ci j 1 4 f 2 (2 √ s − MBi − MB j ) × √ MBi + Ei 2MBi √ MB j + E j 2MB j , (1) with f the pion decay constant (for light vector mesons exchange, taking f = fπ = 93 MeV), √ s the energy in the center mass frame, MBi, Ei (MB j, E j) the mass, energy of baryon of i ( j) channel. The Ci j coefficients Available online at www.sciencedirect.com Nuclear and Particle Physics Proceedings 273–275 (2016) 2726–2728 2405-6014/© 2016 Published by Elsevier B.V. www.elsevier.com/locate/nppp http://dx.doi.org/10.1016/j.nuclphysbps.2015.10.044 http://www.elsevier.com/locate/nppp http://dx.doi.org/10.1016/j.nuclphysbps.2015.10.044 http://dx.doi.org/10.1016/j.nuclphysbps.2015.10.044 http://www.sciencedirect.com are given in Refs. [8, 9] for the pseudoscalar meson- baryon interactions and in Refs. [10, 11] for the vector meson-baryon interactions. In coupled channels we use the Bethe-Salpeter equa- tion to evaluate the scattering amplitudes T = [1 − V G]−1 V, (2) where G is the diagonal matrix of the loop function for the propagating intermediate meson baryon channels, and the elements of the kernel V are given by Eq. (1). First, using the coupled channels approach, we inves- tigate the sector of B̄N and its coupled channels, and the sector of B̄∗N and its coupled channels, respectively. But, it is difficult to dynamically reproduce the two Λb excited states with a reasonable cutoff. More details can be seen in our recent paper [12]. Next, we break the degeneracy of the 1/2−, 3/2− states of the B̄∗N sector, by mixing states of B̄∗N and B̄N in both sectors. This means that it is sufficient to evaluate the contribution of the box diagrams as below: N N N B̄∗ B̄ B̄∗ π π Finally we obtain the following results for the B̄∗N → B̄N → B̄∗N box diagram J = 1/2 : δV = F AC ( ∂ ∂m2π I′1 + 2 I ′ 2 + I ′ 3 ) , J = 3/2 : δV = F AC ( ∂ ∂m2π I′1 ) , (3) where I′1 = ∫ d3q (2π)3 4 3 �q 4 1 2ωB(�q ) MN EN (�q ) Num Den , I′2 = ∫ d3q (2π)3 2�q 2 1 2ωB(�q ) MN EN (�q ) Num Den , (4) I′3 = ∫ d3q (2π)3 3 2ωB(�q ) MN EN (�q ) × 1 P0in + K 0 in − EN (�q ) −ωB(�q ) + i� , with F AC = 9 2 g2 (mB∗ mK∗ )2(F + D 2 fπ )2 , Num = K0in − EN (�q ) − 2ωπ(�q ) −ωB(�q ) + P0in, Den = 2ωπ(�q )[P 0 in −ωπ(�q ) −ωB(�q ) + i�] ×[K0in − EN (�q ) −ωπ(�q ) + i�] ×[P0in + K0in − EN (�q ) −ωB(�q ) + i�], and P0in, K 0 in the energies of incoming B̄ ∗ and N, g = mV/2 fπ with mV ≈ 780 MeV, D = 0.75 and F = 0.51 [13]. For the other boxes, such as B̄N → B̄∗N → B̄N for I = 0, and for I = 1 B̄Δ → B̄∗Δ → B̄Δ, B̄∗Δ → B̄Δ → B̄∗Δ, B̄Δ→ B̄∗N → B̄Δ, B̄∗Δ→ B̄N → B̄∗Δ, we obtain similar results and more details can be referred to our paper [12]. 3. Results When we consider the mixing of B̄∗N and B̄N states by taking into account the contribution of the box dia- gram for the channels of B̄∗N and B̄N with their coupled channels, we obtain the energy splitting of 10 MeV be- tween the J = 1/2 and J = 3/2 in the sector of B̄∗N and its coupled channels, which is rather independent of the cutoff used. Thus, by tuning the cutoff qmax to get the right binding, we find qmax = 776 MeV leading natu- rally to two Λb states, where the energy of the J = 1/2 state is 5910 MeV, and the one of the J = 3/2 state 5920 MeV, associating them to the Λb(5912) and Λb(5920). Then, with this fixed cutoff, we search the generated states in other isospin I = 0 and I = 1 sectors. Finally, since we have many intermediate results, we summarize all the results that we obtain with qmax = 776 MeV. The results are shown in Table 1, where we also write for a quick intuition the main component of the states. Table 1: Energies and widths of the states obtained and the channels to which the states couple most strongly. channels J I (E, Γ) [MeV] Exp. B̄N 12 0 5820.9, 0 - πΣb 1 2 0 5969.5, 49.2 - B̄∗N 12 0 5910.7, 0 Λb(5912) B̄∗N 32 0 5920.7, 0 Λb(5920) ρΣb 1 2 0 6316.6, 2.8 - ρΣb 3 2 0 6315.7, 3.8 - B̄N, πΣb 1 2 1 6179.4, 122.8 - πΣb 1 2 1 6002.8, 132.4 - B̄Δ, πΣ∗b 3 2 1 5932.9, 0 - πΣ∗b 3 2 1 6063.8, 167.0 - B̄∗N 12, 3 2 1 6202.2, 0 - ρΣb 1 2, 3 2 1 6477.2, 10.0 - B̄∗Δ 12, 3 2, 5 2 1 6022.9, 0 - ρΣ∗b 1 2, 3 2, 5 2 1 6491.7, 1.6 - C.W. Xiao et al. / Nuclear and Particle Physics Proceedings 273–275 (2016) 2726–2728 2727 In the work of Ref. [5], the same interaction as here is used for the main diagonal channels, but the transi- tion between different coupled channels is not obtained through vector or pion exchange as done here, but in- voking a combined SU(6) and heavy quark spin sym- metry. Comparing the results with Ref. [5], we obtained some consistent results in I = 0. The two states asso- ciated to the Λb(5912) and Λb(5920) exhibit, as here, a substantial coupling to B̄∗N. There is also a 1/2− state in that work at 5797 MeV which we find at 5820 MeV, only 33 MeV higher, and another state at 6009 MeV that we find at 5969 MeV, 40 MeV below. The mostly ρΣb state found here at 6316 MeV, basically degenerate in J = 1/2, 3/2, was either not found or not searched for in that work because of its higher mass. The states of I = 1 are not investigated in that work. Thus, these states are our predictions with the same pa- rameters in I = 0, and we find quite a few, some of them narrow enough for a clear experimental observation. In summary, we predict 6 states with I = 0, two of them corresponding to the Λb(5912) and Λb(5920), and 8 states with I = 1. The energies of the states range from about 5800 MeV to 6500 MeV. 4. Conclusion The interaction of B̄N, B̄Δ, B̄∗N and B̄∗Δ states with its coupled channels are investigated in the present work by using dynamics mapped from the light quark sec- tor to the heavy one. We studied the interaction of the B̄N, B̄Δ, B̄∗N and B̄∗Δ with their coupled channels πΣb, πΛb, ηΣb (for the B̄N); πΣ∗b, ηΣ ∗ b (for the B̄Δ); ρΣb, ωΛb, φΛb, ρΣ∗b, ωΣ ∗ b, φΣ ∗ b (for the B̄ ∗N); and ρΣ∗b, ωΣ ∗ b, φΣ∗b (for the B̄ ∗Δ), and looked for poles of the scatter- ing matrix in different states of spin and isospin. 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