Predicting positive parity $B_{s}$ mesons from lattice QCD

We determine the spectrum of $B_s$ 1P states using lattice QCD. For the $B_{s1}(5830)$ and $B_{s2}^*(5840)$ mesons, the results are in good agreement with the experimental values. Two further mesons are expected in the quantum channels $J^P=0^+$ and $1^+$ near the $BK$ and $B^{*}K$ thresholds. A combination of quark-antiquark and $B^{(*)}$ meson-Kaon interpolating fields are used to determine the mass of two QCD bound states below the $B^{(*)}K$ threshold, with the assumption that mixing with $B_s^{(*)}\eta$ and isospin-violating decays to $B_s^{(*)}\pi$ are negligible. We predict a $J^P=0^+$ bound state $B_{s0}$ with mass $m_{B_{s0}}=5.711(13)(19)$ GeV. With further assumptions motivated theoretically by the heavy quark limit, a bound state with $m_{B_{s1}}= 5.750(17)(19)$ GeV is predicted in the $J^P=1^+$ channel. The results from our first principles calculation are compared to previous model-based estimates.

Over the years experiments have uncovered a number of mesons involving heavy quarks that do not seem to fit the simple quark-antiquark picture suggested by quark models. Examples of these include states in the charmonium and bottomonium spectrum [1] as well as the charm-strange D * s0 (2317) and D s1 (2460) [2]. The latter states are identified with the j = 1 2 heavy-quark multiplet, where j is the total angular momentum of the light quark [3]. These were predicted to be broad states above thresholds in potential models [4,5,6,7]. However, the observed D * s0 (2317) and D s1 (2460) are narrow states below the DK or D * K thresholds [2], and it has been suggested that the thresholds play an important role in lowering the mass of the physical states [8]. In a recent lattice QCD simulation [9,10,11] these states are seen as QCD bound states below threshold with a mass in good agreement with experiment.
In the B s meson spectrum only two positive parity states are known from experiment [12,13,14], the B s1 (5830) and B * s2 (5840). The LHCb experiment should be able to see the remaining two states (0 + and 1 + ), which are expected to decay into s-wave states by emitting either a photon or a π 0 [15]. On the theory side there are a number of phenomenological model and EFT mass determinations [16,17,18,19,20,21,15,5,22,23], a determination using Unitarized EFT based on low energy constants extracted from lattice QCD simulations [24], and some lattice QCD calculations in the static limit [25,26,27,28,29]. The HPQCD collaboration has published Email addresses: christian.lang@uni-graz.at (C. B. Lang), dmohler@fnal.gov (Daniel Mohler), sasa.prelovsek@ijs.si (Sasa Prelovsek), rwww@triumf.ca (R. M. Woloshyn) a prediction [30] taking into account explicitly only quarkantiquark operators and extracting only the ground states in the system. This strategy can lead to inaccurate results in the vicinity of thresholds where meson-meson scattering can have a significant effect. None of the previous lattice simulations clearly establish the states in question as either QCD bound states below threshold or resonances above threshold. It is this gap which we aim to fill with the current publication.
In this letter we present results for masses of the p-wave states of bottom-strange mesons with spin and parity quantum numbers J P = 0 + , 1 + , 2 + . For the heavy-quark doublet with j P = 3 2 masses determined using only quark-antiquark operators agree with those of the observed B s1 (5830) and B * s2 (5840). This, as well as calculated mass differences between heavylight mesons, verifies our computational setup. Then we simulate B ( * ) K scattering in the scalar (axial) channel and extract the scattering matrix. Bound state poles are found below threshold and their location determines the masses of the B s0 and B s1 .
The gauge configurations are from the PACS-CS collaboration [31]. They have 2 + 1 flavors of dynamical quarks (up/down, strange); the bottom quark is implemented as a valence quark. The light and strange quarks are nonperturbatively improved Wilson fermions. The lattice spacing is 0.0907(13) fm and the Pion mass is 156(7)(2) MeV. The lattice size is 32 3 × 64 and we use stochastic distillation [32] for the quark propagation as in our analysis of the D s mesons [9,10,11]. This allows to include contributions with annihilation diagrams. Further details including the u, d, and s quark parameters can be found in [10].
The dynamic strange quark mass and the associated hop- ping parameter κ s used in [31] differs significantly from the physical value. We therefore use a partially quenched strange quark m val s m sea s . Different determinations agree very well and yield the value for κ s [10] which leads to the Kaon mass The bottom quark is treated as a valence quark and the Fermilab method [33,34] is used. See Ref. [35,10] for details of our implementation. In the simplified form that we use [36,37], only the bottom quark hopping parameter κ b is tuned non-perturbatively, while the clover coefficients c E and c B are set to the tadpole improved value c E = c B = c (h) sw = 1/u 3 0 , where u 0 denotes the average link. There are several ways of setting u 0 and we opt to use the Landau link on unsmeared gauge configurations. Within this simplified approach the static mass M 1 may have large discretization effects but mass differences are expected to be close to physical [38] and can be compared to experiment. Determining the bottom quark hopping parameter translates into determining the spin-averaged kinetic mass M 2 of 1S B s mesons from the lattice dispersion relation [37] where p = 2π L q for a given spatial extent L. After trying multiple forms a simplified form without a W 4 term is taken 1 and for the value κ b = 0.096 used in our simulation we obtain M 2,B s = 5086(135)(73) MeV. This value is significantly smaller than the physical value (m B s + 3m B * s )/4 = 5403.2 +1.8 MeV but the effects on the binding energies used in our analysis are small. This can be seen from the moderate difference between D s [10] and B s binding energies we obtain and will be accounted for in the systematic uncertainty. For the analysis of the phase shifts the dispersion relations for the Kaon (K) and the heavy meson (B or B * ) are needed. For the heavy B mesons we again take Eq. (1) with W 4 = 0 and the results are tabulated in Table 1. For the Kaon the relativistic dispersion relation E K (p) = m 2 K + p 2 is used. The discrete energy levels for our combined basis of quarkantiquark and B ( * ) K operators are extracted from time correlations using the variational method [39,40,41,42]. For a given quantum channel one measures the Euclidean cross-correlation between several operators living on the corresponding time slices. The generalized eigenvalue problem disentangles the eigenstates |n . From the exponential decay of the eigenvalues λ n (t) ∼ exp (−E n (t − t 0 )) one determines the energy values E n of the eigenstates by exponential fits to the asymptotic behavior. The overlap factors O i |n give the composition of the eigenstates in terms of the lattice operators. In order to obtain the lowest energy eigenstates and energy levels reliably one needs a sufficiently large set of operators with the chosen quantum numbers. All error values come from a jack-knife analysis, where the error analysis for the phase shift includes also the input from the dispersion relation (1).
To test our heavy quark approach we calculate a number of mass splittings involving heavy-light and/or heavy-heavy mesons, see Table 2. The quoted uncertainties are statistical and from scale-setting only and the values are not intended to be precision results. In particular our lighter than physical bottom quark mass strongly affects the spin-dependent splittings, but the effect tends to cancel with discretization errors. Estimates for both sources of uncertainty will be taken into account in our prediction of B s mesons.
Partial wave unitarity implies that the scattering amplitude T (s) for elastic B ( * ) K scattering can be written as where p(s) is the momentum and s = E 2 the energy squared in the center of momentum system. Assuming a localized interaction region smaller than the spatial lattice extent a relation between the energy spectrum of meson-meson correlators in finite volume and the infinite volume phase shift δ has been derived [40,43,44,45,46], which applies in the elastic region and in the rest frame. Z 00 denotes the generalized zeta function [44,45] This real function f (p) has no threshold singularity and the measured values can be found indeed above and below threshold. For s-wave scattering an effective range approximation (see Eq. (3)) may be used to interpolate between the closest points near threshold. The imaginary contribution to T −1 becomes real below threshold (responsible for a cusp in Re T ). When the two contributions cancel, T −1 (see Eq. (2)) develops a zero where That zero below threshold corresponds to a bound state pole of T in the upper Riemann sheet. For J P = 0 + we computed cross-correlations between four sb (in the form given in Table XIII of [10]) and three BK (irreducible representation A + 1 ) operators: where we assume that the closeness of the BKπ threshold can be ignored for our simulation. All operators are built according to the distillation method from quark sources that are eigenvectors of the spatial Laplacian, providing a smearing with a Gaussianlike envelope. The gauge links are four-dimensional normalized hypercubic (nHYP) smeared [47]. We omit B ( * ) s π interpolators since we work in the isospin limit where such decays cannot occur. We also neglect B ( * ) s η, partially motivated by the threshold lying O(140 MeV) above the B ( * ) K threshold. Inclusion would necessitate a coupled channel study which would need several volumes and considerably complicate the calculation.
As in earlier experience it turned out that the full set of operators gave noisier signals than suitable subsets so for the final analysis we use the operator set (1,2,4,5,7). The energy values resulting from correlated 2-exponential fits to the eigenvalues are given in Table 3.
In this channel B and K are in s-wave. If there is a bound state one expects an eigenstate with energy approaching the bound state energy from below in the infinite volume limit. The levels Table 4: Systematic uncertainties in the mass determination of the belowthreshold states with quantum numbers J P = 0 + , 1 + . The heavy-quark discretization effects are quantified by calculating the Fermilab-method mass mismatches and employing HQET power counting [34] with Λ = 700 MeV. The dominant contributions arise from mismatches in m B and m E and their size as a fraction of the reference scale Λ can be seen in Fig. 3 of [34]. The finite volume uncertainties are estimated conservatively by the difference of the lowest energy level and the pole position (see also Equations (9) and (28) of [48]). The last line gives the effect of using only the two points near threshold for the effective range fit. The third point might be affected more strongly by the B ( * ) s η threshold, so it is reassuring that the difference in results between two-point and three-point fits is minimal. The total uncertainty has been obtained by adding the single contributions in quadrature. above threshold then would be dominated by BK operators with back-to-back momenta. This is exactly what is seen from the overlap ratios: The lowest level is dominated by operators 1,2 and 4, level 2 by the B(0)K(0) operator 5 and level 3 by the B(1)K(−1) operator 7.
As shown in (3) we can use the values of p cot δ(p) from Lüscher's relation to determine the effective range parametrization near threshold. The energy eigenvalues give the points shown in Fig. 1 together with a linear fit. The value and slope at threshold can be related to the scattering length and effective range: Equation (4) gives the bound state position. From this the binding energy is estimated to be m B + m K − m B s0 = 64(13)(19) MeV; thus, using the physical threshold as input to minimize systematic effects, we predict a bound state B s0 with J P = 0 + at a mass of The first error is due to statistics and the effective range fit, and the second value is our estimate for the systematic error with the main contributions due to heavy quark discretization, unphysical Kaon mass, and finite volume effects. Details of this uncertainty estimate are provided in Table 4.
For J P = 1 + we computed cross-correlations between eight sb (in the form given in Table XIII of [10]) and three B * K (irrep Table 3: Energy levels for J P = 0 + (upper set), 1 + (middle set) and 2 + (lower set). A correlated 2-exponential fit is used andm = 1 4 (m Bs + 3m B * s ) with m = 1.62897 (43) in lattice units. t 0 denotes the reference point in the generalized eigenvalue problem. Energy 2 in the middle set corresponds to the B * s1 (5830). The lower set shows the naive energy level for the J P = 2 + and corresponds to the B * s2 (5840) using the same operator basis used in [10] Comparing various subsets of operators the most stable set was (3,4,6,9,11), where four energy levels could be determined ( Table 3). Based on the overlaps, levels 3 and 4 are dominated by operators 9 (B * (0)K(0)) and 11 (B * (1)K(−1)), respectively. The lowest energy level (dominated by operators 3 and 4) agrees with a bound state interpretation. A linear fit to the points corresponding to energy levels 1, 3 and 4 gives the scattering parameters This state has not (yet) been observed in experiments. Notice that our determination assumes that the effect of swave -d-wave mixing is negligible on the scale of our uncertainty. This is motivated theoretically by the heavy quark limit [3] (where such mixing is absent), which should be a good approximation for bottom-strange mesons. Level 2 (dominated by operator 6) lies just below threshold. This is interpreted, as in the case of the D s1 (2536) [10], to be the j = 3 2 state with J P = 1 + which does not couple to B * K in s-wave in the heavy quark limit [3]. The composition of the state with regard to the qq operators is fairly independent of whether the B * K operators are included or not. Assuming that the coupling to B * K in s-wave is indeed small, the "avoided level crossing" region is so narrow that this state may be treated as decoupled from the B * K scattering channel. Taking the mass difference with respect to the B s spin average and adding the physical value gives  Table 3; see analogous caption of Fig. 1.