$B$, $B_s\to K$ form factors: An update of light-cone sum rule results

We present an improved QCD light-cone sum rule (LCSR) calculation of the $B\to K$ and $B_s\to K$ form factors by including SU(3)-symmetry breaking corrections. We use recently updated $K$ meson distribution amplitudes which incorporate the complete SU(3)-breaking structure. By applying the method of direct integration in the complex plane, which is presented in detail, the analytical extraction of the imaginary parts of LCSR hard-scattering amplitudes becomes unnecessary and therefore the complexity of the calculation is greatly reduced. The values obtained for the relevant $B_{(s)}\to K$ form factors are as follows: $f_{BK}^{+}(0)={0.36}_{-0.04}^{+0.05}$, $f_{B_{s}K}^{+}(0)={0.30}_{-0.03}^{+0.04}$, and $f_{BK}^T(0)=0.38\pm 0.05$, $f_{B_{s}K}^T(0)=0.30\pm 0.05$. By comparing with the $B\to \pi$ form factors extracted recently by the same method, we find the following SU(3) violation among the $B\to \mathrm{\text{light}}$ form factors: $f_{BK}^{+}(0)/f_{B\pi }^{+}(0)={1.38}_{-0.10}^{+0.11}$, $f_{B_{s}K}^{+}(0)/f_{B\pi }^{+}(0)={1.15}_{-0.09}^{+0.17}$, $f_{BK}^T(0)/f_{B\pi }^T(0)={1.49}_{-0.06}^{+0.18}$, and $f_{B_{s}K}^T(0)/f_{B\pi }^T(0)={1.17}_{-0.11}^{+0.15}$.

Figure 1

Diagrams corresponding to the leading-order terms in the hard-scattering amplitudes involving the two-particle (left) and three-particle (right) kaon DA’s shown by ovals. Solid, curly, and wavy lines represent quarks, gluons, and external currents, respectively. In the case of the $B_s\to K$ transition, $s$ and $d$ quarks are interchanged, and ${\overline{K}}^0$ is replaced by $K^0$.

Figure 2

Diagrams contributing to the hard-scattering amplitudes at $O({\alpha }_s)$.

Figure 3

Replacing the integration intervals by the contours in the complex planes of $a=u$, $X$, $X-\xi$, and $r2=s/m_b^2$ variables in the procedure of numerical integration of LCSR amplitudes.

Figure 4

Dependence of $f_B$ on (a) the Borel parameter ${\overline{M}}^2$ shown for $\mu =3\mathrm{GeV}$ (solid line) and $\mu =6\mathrm{GeV}$ (dashed line) and (b) the effective threshold parameter ${\overline{s}}_0^B$ using the central values of all other input parameters.

Figure 5

Dependence of $f_{B_s}$ on (a) the Borel parameter ${\overline{M}}_s^2$ shown for $\mu =3.4\mathrm{GeV}$ (solid line) and $\mu =6\mathrm{GeV}$ (dashed line) and (b) the effective threshold parameter ${\overline{s}}_0^{B_s}$ using the central values of all other input parameters.

Figure 6

Dependence of $f_{BK}^{+}$ on (a) the Borel parameter $M^2$ and (b) the effective threshold parameter $s_0^B$ using the central values of all other input parameters.

Figure 7

Dependence of $f_{B_{s}K}^{+}$ on (a) the Borel parameter $M_s^2$ and (b) the effective threshold parameter $s_0^{B_s}$ using the central values of all other input parameters.

Figure 8

The LCSR prediction for form factors $f_{BK}^{+}(q^2)$ (solid line), $f_{BK}^0(q^2)$ (dashed line), and $f_{BK}^T(q^2)$ (dash-dotted line) at $0<q^2<12{\mathrm{GeV}}^2$ and for $\mu =3\mathrm{GeV}$, $s_0^B=38{\mathrm{GeV}}^2$, $M^2=18{\mathrm{GeV}}^2$, and the central values of all other input parameters.

Figure 9

The LCSR prediction for form factors $f_{B_{s}K}^{+}(q^2)$ (solid line), $f_{B_{s}K}^0(q^2)$ (dashed line), and $f_{B_{s}K}^T(q^2)$ (dash-dotted line) at $0<q^2<12{\mathrm{GeV}}^2$ and for $\mu =3.4\mathrm{GeV}$, $s_0^{B_s}=39{\mathrm{GeV}}^2$, $M_s^2=19{\mathrm{GeV}}^2$, and the central values of all other input parameters.