Charmed-Bottom mesons from Lattice QCD

Lattice QCD methods provide a unique opportunity to study hadronic physics, particularly the energy spectra of hadrons. Substantial progress has been made to extract the ground and the excited states of charmed hadrons, particularly for charmonia. However, the study of bottom hadrons with relativistic actions and controlled discretizations is still prohibitively computer intensive though recent progress in relativistic heavy quark actions is promising. Most studies involving bottom quarks are based on the non-relativistic QCD (NRQCD) formulation, heavy quark effective theory (HQET) and the static quark formulation.

Study of heavy mesons plays a very important role in understanding the nature of strong interactions. Though heavy quarkonia and heavy light mesons have been investigated in great detail, not much is known about the heavy mesons containing only charm and bottom quarks. It is expected that the physics of charmed-bottom mesons involves multiple scales : $1/m_{b}$ ($v_{b}=0.05$), $1/m_{c}$ ($v_{c}=0.4-0.5$), and $\Lambda_{QCD}$. It is interesting to investigate whether charmed-bottom mesons behave like heavy-light mesons or like quarkonia states. The information about hyperfine splittings and other spin splittings, in addition to the results on decay constants, can shed light on the structure of these states.

Experimentally only one state, $B_{c}(0^{-})$, is established with mass at 6275(1) MeV [1]. Recently ATLAS observed another $B_{c}$ meson with mass $6842\pm 4\pm 5$ MeV, which was interpreted as the excited state, $B_{c}^{0^{-}}(2S)$ [2]. However, this excitation has not been confirmed by other experiments yet. On the theoretical side, potential model predictions for these states vary widely. For example, the prediction for $1S$ hyperfine splitting varies in the range 40-90 MeV [3, 4, 5, 6, 7]. This is due to various ways of tuning the heavy quark potentials with spin dependent terms and less clarity of the wavefunction at the origin [3, 4, 5, 6, 7]. On the other hand, using lattice QCD methods, though heavy-light or heavy-heavy mesons were studied extensively not many calculations were carried out for charmed-bottom mesons and baryons. Only two collaborations, HPQCD [8, 9] and Wurtz et al. [10], have studied charmed-bottom mesons recently. Similarly, there are very few recent results for charmed-bottom baryons [11, 12].

In this report, we present our preliminary results on charmed-bottom mesons and baryons. For the bottom quark we use an NRQCD action with non-perturbatively tuned coefficients with terms up to $\mathcal{O}(v^{4})$, whereas the overlap action is utilized for the valence charm quark. Hyperfine splitting between $1S$ $B_{c}$ states as well as masses for other $B_{c}$ mesons are predicted. Moreover, ground state energy spectra of baryons with charm and bottom quarks are also predicted.

We use three sets of dynamical 2+1+1 flavours HISQ gauge field ensambles generated by the MILC collaboration : $24^{3}\times 64$, $32^{3}\times 96$ and $48^{3}\times 96$ lattices at gauge couplings $10/g^{2}=6.00,6.30$ and $6.72$, respectively. The details of these gauge configurations are summarized in Ref. [13]. We use the unphysical $\bar{s}s$ pseudoscalar mass equal to 685 MeV to tune the strange quark mass while the $\Omega_{sss}$ baryon mass is used for calculating the lattice spacings as mentioned in Refs. [14, 15]. The measured lattice spacings, 0.1192(14), 0.0877(10) and 0.0582(5) fm, are consistent with 0.1207(11) 0.0888(8) and 0.0582(5) fm, respectively, measured by MILC collaboration with this set of ensembles using the $r_{1}$ parameter.

We have adopted an NRQCD formulation for the bottom quark. The non-relativistic action that we use is discussed in Ref. [16]. We have considered all terms up to $1/M_{0}^{2}$ and the leading term of the order of $1/M_{0}^{3}$, where $M_{0}=am_{b}$ is the bare mass for bottom quark in lattice units. The NRQCD Hamiltonian is given by,

$$H=H_{0}+\delta H,$$

(1)

where $H_{0}$ is the kinetic term and is defined by,

$$H_{0}=-{{\Delta^{(2)}}\over{2M_{0}}},$$

(2)

whereas, $\Delta H$ contains interaction terms and is given by,

	$\displaystyle\Delta H$	$\displaystyle=$	$\displaystyle-c_{1}{{(\Delta^{(2)})^{2}}\over{8M_{0}^{3}}}+c_{2}{{i}\over{8M_{0}^{2}}}{\left(\tilde{\nabla}\cdot{\bf\tilde{E}}-{\bf\tilde{E}}\cdot\tilde{\nabla}\right)}-c_{3}{{1}\over{8M_{0}^{2}}}\bf{\sigma}\cdot{\left(\tilde{\nabla}\times{\tilde{\bf{E}}}-{\tilde{\bf{E}}}\times\tilde{\nabla}\right)}$		(3)
			$\displaystyle-c_{4}{{1}\over{2M_{0}}}{\bf{\sigma}}\cdot{\tilde{\bf{B}}}\,+c_{5}{{\Delta^{(4)}}\over{24M_{0}}}\,-c_{6}{{(\Delta^{(2)})^{2}}\over{8nM_{0}^{2}}}.$

The quantities with a tilde assume an $\mathcal{O}(a)$ corrected form of discretization. $\nabla$ is the symmetric lattice derivative while $\Delta^{(2)}$ and $\Delta^{(4)}$ represent the lattice discretized versions of $\sum_{i}D^{2}$ and $\sum_{i}D^{4}$, respectively. More details of this Hamiltonian can be found in Ref. [17]. This Hamiltonian is improved by including spin-independent terms through $\mathcal{O}(v^{4})$. For the coarser two ensembles, we use the values of the improvement coefficients, $c_{1}$ to $c_{6}$, as estimated non-perturbatively by the HPQCD collaboration [18]. For the finer lattice, we use tree level coefficients. The NRQCD quark propagators are obtained by usual time evolution

$$G({\bf{x}},t+1)=\left(1-{\delta H\over 2}\right)\left(1-{\delta H_{0}\over 2n}\right)^{n}U^{\dagger}_{t}(x)\left(1-{\delta H_{0}\over 2n}\right)^{n}\left(1-{\delta H\over 2}\right)G({\bf{x}},t).$$

(4)

A wall source is utilized as smearing function for calculating these quark propagators.

The bottom quark mass is tuned by equating the lattice spin-average mass of $1S$ bottomonium to its experimental value. The lattice spin-average mass is obtained from the kinetic mass relation :

$$\bar{M}_{kin}(1S)={3\over 4}aM_{kin}(\Upsilon)+{1\over 4}aM_{kin}(\eta_{b}),$$

(5)

where the kinetic mass is calculated from the relativistic energy-momentum dispersion relation :

$$aM_{kin}={{a^{2}p^{2}-(a\Delta E)^{2}}\over{2a\Delta E}}.$$

(6)

Here $a\Delta E$s are extracted from the energy difference between the mesons with momenta $pa$ and zero. A momentum induced wall-source is utilized to obtain energy values from the correlators with finite momenta. This method was found to be very efficient compared to point or smeared sources [14] and helps to obtain kinetic masses precisely with significantly little statistics.

For valence charm quark propagators we use the overlap action as described in Refs. [14, 15]. The overlap action does not have $\mathcal{O}(ma)$ errors, and it is chirally symmetric at finite lattice spacings. The charm mass is tuned by equating the spin-averaged kinetic mass of the $1S$ charmonia states to its physical value. The tuned bare charm quark masses ($am_{c}$) are found to be 0.528, 0.425 and 0.29 on coarser to finer lattices respectively. Details of the charm tuning was described in Ref. [15].

We report the ground state energy spectra of charmed-bottom mesons. An NRQCD Hamiltonian with non-perturbatively improved coefficients is used for bottom quarks while a relativistic overlap action is adopted for charm quarks on a background of 2+1+1 flavours HISQ gauge configurations generated by the MILC collaboration. Results are obtained at three lattice spacings with physical volume of about 3 fm. We tune the bottom and the charm quark masses by equating the lattice spin-averaged kinetic masses of $1S$ states in bottomonia and charmonia with their experimental values. The hyperfine splittings between these $1S$ states are found to be 64(3) and $115(3)$ MeV for bottomonia and charmonia, respectively, which are quite consistent with their experimental values. Our result on hyperfine splitting between $1S$ states of $B_{c}$ mesons is $56^{+4}_{-3}$ MeV. This result is consistent with HPQCD [9] as well as Wurtz et al. [10] who found this splitting to be 54(3) and 57.5(3) MeV, respectively. It is interesting to note that all lattice results are inconsistent with many quark model results which vary in the range between 40-90 MeV [3, 4, 5, 6, 7]. Due to this rather small splitting, detection of $B_{c}^{*}(1^{-})$ from its $B_{c}\gamma$ decay may turn out to be challenging in experiments. Taking the PDG value of $B_{c}(0^{-})$ state as 6275(1) MeV, our prediction for the ground state mass of $B^{*}_{c}$ is $6331^{+4}_{-3}$ MeV. We also calculate the ground state energy of other $B_{c}$ mesons, namely, $B_{c}(0^{+})$ and $B_{c}(1^{+})$. Our preliminary results on the splittings between $(0^{+}-0^{-})$ and $(1^{+}-1^{-})$ states are found to be 411(16) MeV and 395(15) MeV, respectively. The ground state masses of baryons containing only charm and bottom quarks are also extracted. The hyperfine splittings between spin-3/2 and spin-1/2 $\Omega(ccb)$ and $\Omega(cbb)$ states are found to be 20(5) MeV and 21(5) MeV, respectively. Our prediction for the ground state spin-3/2 triply bottom baryon is $M_{\Omega_{bbb}}=14362^{+5}_{-4}$ MeV.

Computations were carried out on the Blue Gene/P of the Indian Lattice Gauge Theory Initiative, and on the Gaggle cluster of the Department of Theoretical Physics, TIFR. We thank A. Salve and K. Ghadiali for technical support. M. P. acknowledges support from the Austrian Science Fund FWF:I1313-N27 and the Deutsche and Forschungsgemeinschaft Grant No. SFB/TRR 55. R.L. is supported in part by NSERC of Canada. We are grateful to the MILC collaboration and in particular to S. Gottlieb for providing us with the HISQ lattices.