Density distributions in the $B$ meson

We report on a two-flavor lattice QCD study of the axial, charge, and matter distributions of the $B$ meson and its first radial excitation. As our framework is the static limit of heavy quark effective theory (HQET), taking their Fourier transform gives access to several form factors at the kinematical point $q^2=0$ while our previous computations in that framework were performed at $q_{\max }^2$. Moreover, they provide some useful information on the nature of an excited state, i.e., a radial excitation of a quark-antiquark bound state or a multihadron state.

Figure 1

Three-point correlation function computed to extract the density distribution $f_{\mathrm{\Gamma }}(\overrightarrow{r})$.

Figure 2

Left: Effective mass plot extracted from a $3\times 3$ GEVP for the lattice ensemble E5. Right: Energy difference ${\mathrm{\Sigma }}_{12}=E_2-E_1$ between the first radial excitation and the ground state. The value of $t_0$ is chosen such that $t>t_0/2$ to reduce the contamination of higher excited states. We also plot the plateau in the chosen fit interval.

Figure 3

Spatial component of the radial distributions of the axial density $a{r}^2{f}_{{\gamma }_i{\gamma }_5}^{(11)}(r/a)$, $a{r}^2{f}_{{\gamma }_i{\gamma }_5}^{(12)}(r/a)$, and $a{r}^2{f}_{{\gamma }_i{\gamma }_5}^{(22)}(r/a)$ for the lattice ensemble F6.

Figure 4

Spatial component of the radial distribution $a{r}^2{f}_{{\gamma }_i{\gamma }_5}^{(11)}(r/a)$ obtained using the ratio method [Eq. (10)] for different numbers of Gaussian smearing iterations $R_n$, for the lattice ensemble N6. For each curve, $t$ is chosen such that the radial distribution has reached a plateau ($t/a=17$, 14, 10 for $R_n=0$, 135, 338, respectively).

Figure 5

Curing of volume effects on the axial spatial density distributions for the lattice ensemble E5. Raw data are in black, results of the fit are in blue, and results after image corrections are in red.

Figure 6

Left: Results of the fit for the lattice ensemble E5 ($L/a=32$) using the method described in Sec. 3d. Raw data are in black, results of the fit are in blue, and results after image corrections are in red. Right: Results obtained for the lattice ensemble D5 ($L/a=24$) using the fit parameters of the ensemble E5 that has the same value of $\beta$ and a similar pion mass. It means that the blue and red points of the right panel are not obtained by a fit of the black points.

Figure 7

Plateaus of the axial density summations over $r$ for the lattice ensemble N6 using the GEVP method [grey points, Eq. (13)] and the sGEVP method [black points, Eq. (14)]: $g_{11}$ (left), $g_{12}$ (center), and $g_{22}$ (right).

Figure 8

Extrapolations of ${\overline{g}}_{11}$, ${\overline{g}}_{22}$, and ${\overline{g}}_{12}$ to the chiral and continuum limits using the fit function (18).

Figure 9

Density distributions $a{r}^2{f}_{{\gamma }_i{\gamma }_5}^{(11)}(r/a)$, $a{r}^2{f}_{{\gamma }_i{\gamma }_5}^{(12)}(r/a)$, and $a{r}^2{f}_{{\gamma }_i{\gamma }_5}^{(22)}(r/a)$ for our quenched lattice ensemble Q2. The blue curves correspond to results after a fit and the red curves to the distributions after removing volume effects.

Figure 10

Plateaus for the summation over $r$ of the axial densities computed in Sec. 3e for our quenched lattice ensemble Q2: $g_{11}$, $g_{12}$, and $g_{22}$.

Figure 11

Comparison of the renormalized distributions $r^2{f}_{{\gamma }_i{\gamma }_5}^{(mn)}(r)$ in physical units. The quenched result (ensemble Q1) is plotted in red and the dynamical case (ensemble D5) is plotted in black.

Figure 12

Imaginary part of the time component $f_{{\gamma }_0{\gamma }_5}^{(12)}(r)$ of the axial density distribution for the lattice ensembles E5 and N6. Choosing $i=z$, we define $r_{\parallel }=r_z$ and $r_{\perp }=\sqrt{r_x^2+r_y^2}$.

Figure 13

The integrand in Eqs. (22) and (23) to obtain the form factor at $q^2=0$. The red curve is the product of the blue and dark curves and the normalization of the blue curve is arbitrary. For the time component (right plot), we plot the imaginary part and only points with $|r_{\perp }|\le 1$ are shown.

Figure 14

$q^2$ dependence of the spatial (left) and time (middle) contributions to the form factor $A_0^{12}(q^2)$ for the lattice ensemble E5. It should be stressed that the form factor $A_0$ is interesting per se only at $q^2=0$, where it is related to the $g_{B^{*\prime }B\pi }$ coupling.

Figure 15

Extrapolations of the form factor $A_0^{12}$ at $q^2=0$ to the chiral and continuum limits.

Figure 16

$q^2$ dependence of the spatial and time contributions to the form factor $A_0^{12}(q^2)$ for the quenched lattice ensemble Q2.

Figure 17

Left: Effective mass plot extracted from a $3\times 3$ GEVP for the lattice ensemble E5 using ${\gamma }_k$ interpolating operators. Right: Effective mass plot extracted from a $4\times 4$ GEVP for the lattice ensemble E5 using $\overline{q}{\gamma }_{k}b$ and $\overline{q}{\nabla }_{k}b$ interpolating operators.

Figure 18

Ratio $Z_{ni}(\mathrm{E}5)/Z_{ni}(\mathrm{D}5)$ for the first three levels. The two plain dark lines correspond to ratios of 1 and $4/3$.

Figure 19

Effective energies of the two-point correlation function (26) obtained by solving a $4\times 4$ GEVP for different values of $x$.

Figure 20

Unrenormalized vector charge obtained from $f_{{\gamma }_0}^{(nn)}(r)$ on the lattice ensemble E5, using $\overline{q}{\gamma }_{k}h$ and $\overline{q}{\nabla }_{k}h$ interpolating operators. The blue line corresponds to the expected plateau using the nonperturbative estimate $Z_V=0.750(5)$ extracted from Ref. [25].

Figure 21

Density distributions $a{r}^2{f}_{{\gamma }_0}^{(nn)}(r/a)$, $n=1$, 2, 3 (top) and $n=1$, 2, 4 (bottom) on the lattice ensemble E5, using only $\overline{q}{\gamma }_{k}h$ (top) and including $\overline{q}{\nabla }_{k}h$ interpolating fields (bottom) in the analysis.

Figure 22

Disconnected contribution to the three-point correlation function in the case $\mathrm{\Gamma }=\mathbb{1}$.

Figure 23

Plateaus of total vector charges $c_{11}$, $c_{12}$, and $c_{22}$ (top) and scalar charges $m_{11}$, $m_{12}$, and $m_{22}$ (bottom) for the lattice ensemble E5.

Figure 24

Top: Charge densities $a{r}^2{f}_{{\gamma }_0}^{(11)}(r/a)$, $a{r}^2{f}_{{\gamma }_0}^{(12)}(r/a)$, and $a{r}^2{f}_{{\gamma }_0}^{(22)}(r/a)$ for the CLS ensemble E5. Bottom: Matter densities $a{r}^2{f}_{\mathbb{1}}^{(11)}(r/a)$, $a{r}^2{f}_{\mathbb{1}}^{(12)}(r/a)$, and $a{r}^2{f}_{\mathbb{1}}^{(22)}(r/a)$ for the CLS ensemble E5.

Figure 25

Axial, matter, and charge densities $a{r}^2{f}_{\alpha }^{(11)}(r/a)$ obtained after the subtraction of volume effects for the CLS ensemble E5. The normalization is such that the area below the curve is one.