Latest Results from the SGO Collaboration

TL;DR

This paper reports on the spectrum and decay constants of B mesons calculated using NRQCD on dynamical lattice configurations with improved light quark actions, providing updated results in heavy meson physics.

Contribution

It presents new lattice QCD results for B meson properties using dynamical configurations and improved actions, enhancing the precision of heavy meson decay constants.

Findings

Results for B meson spectrum and decay constants

Use of NRQCD with dynamical configurations

Application of improved Clover action for light quarks

Abstract

We present results for the spectrum and decay constants of B mesons from NRQCD using dynamical configurations at $\beta=5.6$ with two flavours of staggered fermions. The light quarks are generated using the Clover action with tadpole improvement.

Tables

Table 1. Table 1: The coefficients of various spectral quantities as determined from first order perturbation theory in 1 / M 1 𝑀 1/M , where ⟨ 1 ​ S | bra 1 𝑆 \left<1S\right| represents an S 𝑆 S -state in the infinite mass limit. The results are in physical units (in powers of GeV). Note that those obtained for 2 ​ S − 1 ​ S 2 𝑆 1 𝑆 2S{-}1S are estimates and not obtained from a fit. E 0 ∞ superscript subscript 𝐸 0 E_{0}^{\infty} is the energy of a heavy quark in the static theory, calculated perturbatively. κ l = κ c subscript 𝜅 𝑙 subscript 𝜅 𝑐 \kappa_{l}=\kappa_{c} .

Quantity	Coefficient	Results	Expectation
$\overline{E}$	$C_0=E^{\infty }$	1.00(2)	Static Result
	$\overline{\Lambda }=E^{\infty }-E_0^{\infty }$	0.3(3)	$+{\Lambda }_{QCD}$
		$-0.1$(1)	$+{\Lambda }_{QCD}^2$
$E\left(2S\right)-E\left(1S\right)$	$C_0=E^{\infty }\left(2S\right)-E^{\infty }\left(1S\right)$	0.5	$+{\Lambda }_{QCD}$
		0.3	$+$ve
$E{(}^3{S}_1)-E{(}^1{S}_0)$	$C_0=0$	0.000(2)	0
		0.20(4)	$+{\Lambda }_{QCD}^2$

Table 2. Table 2: The results are in GeV. Note that renormalisation factors are not included. κ l = κ c subscript 𝜅 𝑙 subscript 𝜅 𝑐 \kappa_{l}=\kappa_{c} .

Quantity	$C_1/C_0$	Results
${\left(f\sqrt{M}\right)}_{PS}^{tot}$	$G_{kin}+6G_{hyp}+G_{cor}/2$	$-2.0$(4)
$\left(\overline{f\sqrt{M}}\right)$	$G_{kin}$	$-1.8$(2)
$\frac{{\left(f\sqrt{M}\right)}_{PS}}{{\left(f\sqrt{M}\right)}_V}$	$8G_{hyp}$	0.72(6)
$\frac{{\left(f\sqrt{M}\right)}_{PS}^{tot}}{{\left(f\sqrt{M}\right)}_V^{tot}}$	$8G_{hyp}+2G_{cor}/3$	$-0.24$
Quantity	$C_0$	Results
$\frac{2M_0\delta {\left(f\sqrt{M}\right)}_{PS}}{{\left(f\sqrt{M}\right)}_{PS}}$	$-G_{cor}$	$-1.24$(2)

Equations

$${\mathcal{L}}_{NR}$$

$${\mathcal{L}}_{stat}$$

$${\mathcal{L}}_{kin}$$

$${\mathcal{L}}_{hyp}$$

$${\mathcal{O}}_1=\bar{q}\Gamma Q{\mathcal{O}}_2=\bar{q}\Gamma {\gamma }_i{D}_{i}Q/2M_0$$

$$G_{t+1}$$

$$\bar{E},\text{mbox}\Delta E=C_0+C_1/M+C_2/M^2+\dots$$

$$f\sqrt{M}$$

$$\delta (f\sqrt{M})$$

$$(f\sqrt{M}{)}^{tot}$$

$$c_P$$

$$Z_A^{-1}{f}_B=216\pm (6)\pm (10)\pm (20)\text{mbox}MeV$$

Full text

1 INTRODUCTION

We present a study of the heavy-light meson spectrum and decay constants, with a focus on investigating heavy quark symmetry as well as providing quantitative predictions. Our approach is to use NRQCD for the heavy quark and in this analysis the action includes terms to $O(1/M_{0})$:

[TABLE]

Matrix elements must be calculated consistently to this order. Here, we restrict the analysis to the tree-level operators for the axial and vector current to $O(1/M_{0})$ [1],

[TABLE]

where $\Gamma=\gamma_{5}\gamma_{0}$ and $\gamma_{i}$ respectively. Tadpole improvement is used throughout and then $c_{B}$ is given the tree-level value of $1.0$.

The heavy quark propagators were generated using the evolution equation

[TABLE]

for all timeslices. Equation (6) is fully consistent to $O(1/M_{0})$. This differs from the evolution equation previously used in [1] by a factor of $1{-}\frac{a\delta H}{2}$ on the source timeslice, and our results had a systematic error of $O(a\Lambda_{QCD}/M)$. However, we found this error to be small [1] and our previous findings are not affected.

The simulation was performed on 100 $16^{3}\times 32$ lattices at $\beta=5.6$ with two flavours of dynamical staggered fermions, generated by the HEMCGC collaboration. The light quark propagators were computed using the Clover action with tadpole improvement, at three masses of light quark, $\kappa_{l}=0.1385$, $0.1393$ and $0.1401$. The heavy quark propagators were computed over a wide range of bare quark mass, $aM_{0}=0.8{-}10.0$, in order to study heavy quark symmetry. Full details of our analysis can be found in [1].

2 RESULTS

A vital part of a lattice calculation of $B$ physics is a comprehensive calculation of the spectrum. This provides a firm foundation for the simulation of more complicated processes. Figure 1 presents the results for the low lying $B$ spectrum; $a^{-1}=2.0$ GeV from light spectroscopy has been used to convert to physical units and the $B$ meson mass to fix $aM_{0}^{b}$. For the mass splittings $B_{s}{-}B$ and $B^{**}{-}B$, the dominant systematic errors are estimated to be comparable to or less than the statistical errors. With the exception of $B^{*}{-}B$ we find good agreement with experiment, although, the experimental results for the $P{-}S$ splitting are preliminary. The discrepancy with the real world for the hyperfine splitting may be a residual quenching effect. However, it is also possible our value of $c_{B}$ may be too low by $10{-}20\%$; a nonperturbative determination of $c_{B}$ is needed.

Note that the gross shape of the spectrum supports the naive picture of a massive heavy quark with velocity $v{\sim}0.1$ surrounded by a light quark cloud. In this model the hyperfine spitting is expected to be $Mv^{2}{\sim}50$ MeV, while the $P{-}S$ and $2S{-}1S$ splittings are due to excitations of the light quark and so are of $O(\Lambda_{QCD}){\sim}200{-}500$ MeV. We can investigate this further by quantifying the change in these splittings with $M_{Q}$. Table 1 summarises the coefficients extracted from fitting the spin-averaged ground $S$-state energy ($\overline{E}$) and mass splittings to the function

[TABLE]

where $M$ is the $PS$ meson mass.

The results are compared with our expectations in Table 1. With the exception of the bare kinetic energy of the heavy quark (extracted from $\bar{E}$) the overall agreement is good. While $\left<1S\right|{\cal L}_{kin}\left|1S\right>_{bare}$ is small, it is opposite in sign to the physical result, which is positive. This is due to our use of tadpole-improved matrix elements [1]; using mean field theory the unimproved matrix element corresponds to a large positive result, $\left<1S\right|{\cal L}_{kin}\left|1S\right>_{bare}^{unimproved}{\sim}+0.7$. This latter result is in agreement with an explicit calculation by Crisafulli et al [5], and demonstrates that the matrix element is tadpole dominated. In fact, the renormalisation required to obtain the (positive) physical quantity from the tadpole-improved $\left<1S\right|{\cal L}_{kin}\left|1S\right>_{bare}$ is small and can be calculated perturbatively.

The slope of the $2S{-}1S$ splitting gives the difference between the physical kinetic energy of the heavy quark in the excited state and ground state mesons; it is a physical quantity since the renormalisation shift cancels in the difference. Note that the slope is positive, and its magnitude suggests $\left<2S\right|{\cal L}_{kin}\left|2S\right>$ is large compared to $\left<1S\right|{\cal L}_{kin}\left|1S\right>$.

Now consider a similar analysis of the vector and axial currents. We define the $V$ and $PS$ decay constants in the same way using

[TABLE]

where $\delta(f\sqrt{M})$ denotes the $O(1/M)$ correction to the current. The expansion of $f\sqrt{M}$ about the static limit is given in HQET to $O(1/M)$ [4] by

[TABLE]

where $d=3$ and ${-}1$ for $PS$ and $V$ mesons respectively. In equation (12) the slope is decomposed into the contributions from the kinetic energy of the heavy quark, $G_{kin}$, the hyperfine interaction, $G_{hyp}$, and the correction to the current, $G_{cor}$. Equation (12) suggests that these contributions can be isolated by finding the slope of various combinations of the $V$ and $PS$ decay constants.

Table 2 presents our results with the corresponding contributions to the slopes, obtained using a fit function of the form (7). The physical quantities $(f\sqrt{M})^{tot}_{PS}$, $(\overline{f\sqrt{M}})$ and $(f\sqrt{M})^{tot}_{PS}/(f\sqrt{M})^{tot}_{PS}$ are naively expected to have coefficients of the slope of $O(\Lambda_{QCD})$. However, we find a much larger slope than this for the total decay constant. The slope of the spin average of the decay constants, which only depends on $G_{kin}$, shows that it is the contribution from the kinetic energy of the heavy quark which is large and dominates $c_{P}$. Comparing with Neubert [4] we find good agreement for the slopes of the physical combinations of decay constants while for the unphysical slope of $(f\sqrt{M})_{PS}/(f\sqrt{M})_{V}$ and intercept of $2M_{0}\delta(f\sqrt{M})_{PS}/(f\sqrt{M})_{PS}$, where agreement is not expected, the results differ in magnitude and for the former also in sign [1].

Extracting the value of the $PS$ decay constant at $M_{B}$ we find

[TABLE]

The first error is statistical and the second is the systematic error arising from the clover light fermions estimated to be $O(a^{2})$. The third is due to the omission of $O(1/M_{0}^{2})$ terms in the NRQCD action, and is the dominant error in $f_{B}$. Using the static renormalisation factor of $Z_{A}^{stat}=0.8$ we find $f_{B}{\sim}180$ MeV.

3 CONCLUSIONS

We have performed a comprehensive study of the heavy-light meson spectrum and decay constants. The question that arises from this analysis is why there is such a large slope for the decay constant when the deviations from the static limit for spectral quantities are in agreement with naive expectation. A potential model approach [3] suggests that this is a consistent picture and that a large value for $c_{P}$ is coupled to a large value for $\left<2S\right|{\cal L}_{kin}\left|2S\right>$.

The authors acknowledge the support under grants from NATO and DOE. The computations were performed on the CM-2 at SCRI.