A Study of the Static-Light $B_B$ Parameter

TL;DR

This paper presents a lattice gauge theory calculation of the $B_B$ parameter, crucial for understanding B-meson mixing, employing improved techniques to reduce errors and uncertainties in the simulation.

Contribution

It introduces an optimized variational method for better signal extraction and linearizes renormalization to minimize higher-order uncertainties.

Findings

Reduced statistical errors with improved sources

Minimized excited-state contamination

Estimated renormalization with lower order uncertainties

Abstract

We calculate the $B_B$ parameter, relevant for $\overline{B}^0$--$B^0$ mixing, from a lattice gauge theory simulation using the static approximation for the heavy quark and the Wilson action for the light quark and gauge fields. Improved sources, produced by an optimized variational technique, {\sc most}, reduce statistical errors and minimize excited-state contamination of the ground-state signal. Renormalization of four-fermion operator coefficients, using the Lepage-Mackenzie procedure for estimating typical momentum scales, is linearized to reduce order $\alpha_s^2$ uncertainties.

Tables

Table 1. Table 1: “Typical” operator scales; using β 𝛽 \beta =6.0 and r 𝑟 r =1.

				$B_B$
$q_i^{\star }a$	$2.01$	$2.15$	$2.18$	$0.82$

Table 2. Table 2: The absolute changes from our preferred values of the coefficients Z B L subscript 𝑍 subscript 𝐵 𝐿 Z_{B_{L}} , Z B R subscript 𝑍 subscript 𝐵 𝑅 Z_{B_{R}} , Z B N subscript 𝑍 subscript 𝐵 𝑁 Z_{B_{N}} , and Z B S subscript 𝑍 subscript 𝐵 𝑆 Z_{B_{S}} as the parameters q ∗ ​ a superscript 𝑞 𝑎 q^{*}a , a − 1 superscript 𝑎 1 a^{-1} , m b subscript 𝑚 𝑏 m_{b} , and Λ c ( 5 ) superscript subscript Λ 𝑐 5 \Lambda_{c}^{(5)} are varied by 10%, and 20%, first individually, and then jointly (“All”), from our preferred input values. The coefficients are quite insensitive to the particular choice of input parameter.

	$q^{\star }a$	$a^{-1}$	$m_b$	${\Lambda }_c^{\left(5\right)}$	All
	2.18	2.1 GeV	4.33 GeV	0.175 GeV
$Z_{B_L}=1.070$
10%	$+0.002$ $-0.002$	$+0.003$ $-0.004$	$+0.003$ $-0.003$	$+0.0008$ $-0.0005$	$+0.004$ $-0.005$
20%	$+0.005$ $-0.003$	$+0.006$ $-0.009$	$+0.006$ $-0.005$	$+0.0019$ $-0.0009$	$+0.008$ $-0.009$
$Z_{B_R}=-0.0225$
10%	$+0.0005$ $-0.0006$				$+0.0005$ $-0.0006$
20%	$+0.0009$ $-0.0015$				$+0.0009$ $-0.0015$
$Z_{B_N}=-0.202$
10%	$+0.005$ $-0.006$				$+0.005$ $-0.006$
20%	$+0.008$ $-0.012$				$+0.008$ $-0.012$
$Z_{B_S}=-0.137$
10%			$+0.003$ $-0.003$	$+0.002$ $-0.003$	$+0.003$ $-0.004$
20%			$+0.006$ $-0.005$	$+0.005$ $-0.007$	$+0.006$ $-0.008$

Table 3. Table 5: The authors’ numbers, quoted at the listed scale, have been scaled to μ 𝜇 \mu =2.0 GeV and to m b subscript 𝑚 𝑏 m_{b} =4.33 GeV. If the authors quote a number which we used or reproduced, it is bold-faced in the table.

			scale		one-loop
Method	ref	$\beta$	(GeV)	$B\left(\mathrm{scale}\right)$	$n_f$	$\Lambda$	$B\left(2.0\right)$	$B\left(4.33\right)$	${\hat{B}}_B$
Static-Clover	[8]	6.2	$m_b$=5.0	0.69(4)	5	130	-	-	1.02(6)
		6.2	$m_b$=5.0	0.69(4)	4	200	0.75(4)	0.70(4)	0.98(6)
Static-Wilson	this	6.0	$m_b$=4.33	0.98(4)	5	175	-	-	1.40(6)
	work	6.0	$m_b$=4.33	0.98(4)	4	226	1.05(4)	0.98(4)	1.36(6)
Extrap. Static	[9]	5.7-6.3	$\mu$=2.0	1.04(5)	4	200	1.04(5)	0.97(5)	1.36(7)
		5.7-6.3	$\mu$=2.0	1.04(5)	4	226	1.04(5)	0.97(5)	1.34(6)
Extrap. Static	[10]	6.4	$\mu$=3.7	0.90(5)	0	200	0.94(5)	0.89(5)	1.21(7)
		6.4	$\mu$=3.7	0.90(5)	4	200	0.95(5)	0.89(5)	1.25(7)
Wilson-Wilson	[9]	5.7-6.3	$\mu$=2.0	0.96(6)	4	200	0.96(6)	0.90(6)	1.25(8)
		5.7-6.3	$\mu$=2.0	0.96(6)	4	226	0.96(6)	0.89(6)	1.24(8)
Wilson-Wilson	[9, 11]	6.1	$\mu$=2.0	1.01(15)	4	200	1.01(15)	0.94(13)	1.32(20)
		6.1	$\mu$=2.0	1.01(15)	4	226	1.01(15)	0.94(14)	1.30(19)
Wilson-Wilson	[12]	6.1	$m_b$=5.0	0.895(47)	0	239	0.96(5)	0.90(5)	1.21(6)
		6.1	$m_b$=5.0	0.895(47)	4	239	0.98(5)	0.91(5)	1.25(7)
		6.1	$m_b$=5.0	0.895(47)	5	183	-	-	1.29(7)
Wilson-Wilson	[12]	6.3	$m_b$=5.0	0.840(60)	0	246	0.90(6)	0.85(6)	1.14(8)
		6.3	$m_b$=5.0	0.840(60)	4	246	0.92(6)	0.85(6)	1.17(8)
		6.3	$m_b$=5.0	0.840(60)	5	189	-	-	1.20(9)
Wilson-Wilson	[10]	6.4	$\mu$=3.7	0.86(5)	0	200	0.90(5)	0.85(5)	1.16(7)
		6.4	$\mu$=3.7	0.86(5)	4	200	0.91(5)	0.85(5)	1.19(7)
Sum Rule	[13]		$m_b$=4.6	1.00(15)	5	175	-	-	1.43(22)
			$m_b$=4.6	1.00(15)	4	227	1.08(16)	1.00(15)	1.39(21)

Equations

$$B_B=\frac{{⟨{\overline{B}}_0\left|{\mathcal{O}}_L^{\mathrm{full}}\right|B_0⟩}_{}}{{⟨{\overline{B}}_0\left|{\mathcal{O}}_L^{\mathrm{full}}\right|B_0⟩}_{vs}}$$

$${\mathcal{O}}_L$$

$${\mathcal{O}}_R$$

$${\mathcal{O}}_N$$

$${\mathcal{O}}_S$$

$$B\left(t_1,t_2\right)$$

$$C_3\left(t_1,t_2\right)$$

$$C_2\left(t_1\right)$$

$$\chi \left(t,\vec{x}\right)=\sum _{\vec{r}}f\left(\vec{r}\right)\overline{q}\left(t,\vec{x}+\vec{r}\right){\gamma }_{5}b\left(t,\vec{x}\right)$$

$$B_B^{\mathrm{full}}\left(m_b\right)$$

$$Z_{B_L}$$

$$Z_{B_R}$$

$$Z_{B_N}$$

$$Z_{B_S}$$

$$B_B\left(\mu \right)={\left(\frac{{\alpha }_s^c\left(\mu \right)}{{\alpha }_s^c\left(m_b\right)}\right)}^{\left(p_{0,L}^c-2p_{0,f}^c\right)}{B}_B\left(m_b\right)$$

$${\hat{B}}_B={\left({\alpha }_s^c\left(m_b\right)\right)}^{-\left(p_{0,L}^c-2p_{0,f}^c\right)}{B}_B\left(m_b\right)$$

Full text

1 $B_{B}$ Parameter

Since the lattice static effective theory has fewer symmetries than the full continuum theory, when calculating the static-light $B_{B}$ parameter

[TABLE]

operators besides ${\cal O}_{L}^{{\rm latt}}$ must be included. These correspond to the following full-theory fermion operators (see Flynn et al. [1]):

[TABLE]

${\cal O}_{S}$ is generated at order $\alpha_{s}$ in the continuum due to the mass of the heavy quark. ${\cal O}_{R}$ and ${\cal O}_{N}$ are generated at order $\alpha_{s}$ from the chiral symmetry breaking Wilson mass term. The lattice calculation of the static-light $B_{B}$ uses the ratio of two- and three-point hadronic correlation functions.

[TABLE]

where the required correlation functions are

[TABLE]

The three-point function has a fermion operator inserted at the spacetime origin, between two external $B$-meson interpolating fields. The times are restricted to the range $t_{2}>0>t_{1}$. The gamma matrices, $\Gamma_{\!\!I\,}$ and $\Gamma_{\!\!J\,}$ define the type of four fermion operator (see equation 2). A spatially extended $B$-meson operator

[TABLE]

is used, where $f$ is a smearing function produced by most [2] for our static $f_{B}$ study.

2 Scale Formulation

Using the integrand of the one-loop perturbative contribution from the coefficients as a weighting function, as per Lepage and Mackenzie [3], a “typical” momentum scale can be found (Table 1).

Our value for the scale relevant for $\left<A_{\mu}\right>$ agrees with that found by Hernández and Hill [4]. This is the scale which we claim is relevant for this calculation as well. We notice that the scale found for $B_{B}$ is singularly different than the others and claim that each of the other matrix elements is describing physics at essentially the same scale. However, when a ratio is considered, the integrands should cancel, but the scale should not. Since the other values are similar, we choose 2.18, as it has been used for the $f_{B}$ study.

3 Calculation of the Coefficients

The coefficients of the operators are calculated [1, 5, 6] by renormalization group techniques.

[TABLE]

where $p_{0,i}=\frac{\gamma_{0,i}}{2b_{0}}$, and $p_{1,i}=p_{0,i}\left(\frac{\gamma_{1,i}}{\gamma_{0,i}}-\frac{b_{1}}{b_{0}}\right)$. For $\alpha_{s}^{l}$, we use $\alpha_{V}(q^{\star})$=0.18 [3].

The statistical uncertainties for the coefficients are listed in Table 4. There is a systematic error due to the linearization of the coefficients which is not listed. See reference [7] for complete details.

4 Results of Simulation

The raw lattice $B$ parameters for the operators which appear in the lattice-continuum matching are determined from a Monte Carlo calculation of equation 3 and listed in Table 4. Table 4 lists the linear combination $B_{{\cal O}_{L}^{\rm full}}=B_{B}$ as a function of $\kappa$ and extrapolated to $\kappa_{c}$ using fully-linearized tadpole-improved coefficients. For both tables, the first errors are statistical (bootstrap) and the second are systematic due to choice of fit range.

We find $B_{B}(m_{b})$={0.98}^{+{4}}_{-{4}}{({3})}$${}^{+1}_{-2} as our calculated value. The first two errors are as mentioned above. The final error is due to the statistical uncertainties in the coefficients. If we run to a scale of $\mu$=2 GeV, with $n_{\!f}$=4, using

[TABLE]

we find $B_{B}(2\,{\rm GeV})$=$1.05(4)$. When we convert $B_{B}(m_{b})$ to a RG invariant quantity using

[TABLE]

with 4 flavors, we find $\hat{B}_{B}$=$1.36(6)$. With 5 flavors, we find $\hat{B}_{B}$=$1.40(6)$.

5 Comparison to Others

The simulations using Wilson quarks calculate the $B_{B}$ parameter for quark masses around charm and extrapolate up to the physical mass, using a fit model of the form $B=B^{0}+\frac{B^{1}}{M}$. The value of $B^{0}$ (“extrapolated static” in Table 5) should be the same as the static theory. We scale the authors’ numbers to 2.0 GeV and 4.33 GeV. The JLQCD collaboration cite their $\Lambda$ as $n_{\!f}$=0 values. When Abada et al. quote a $\hat{B}_{B}$ for the propagating Wilson quarks, they use $n_{\!f}$=0. When we scale these, we list values for both $n_{\!f}$=0 and $n_{\!f}$=4. Soni quotes numbers at 2.0 GeV, but no $\Lambda$ is given. We use our value for $\Lambda^{(4)}$ as well as 200 MeV. We calculate $B_{B}\!\left(\!m_{b}\!\right)$, scale it to 2.0 GeV using $n_{\!f}$=4, and calculate a $\hat{B}_{B}$ with both 4 and 5 flavors. Since all of the “raw” values are close to 1.0, differences between the estimates of the static $\hat{B}_{B}$ are due not to the choice of action, but to choices in the coefficients. See [7] for the justification of our choice.