Nonlocal form factor of chromomagnetic penguin in $B\to K\ell^+\ell^-$ from QCD light-cone sum rules

Abstract

The branching fraction of the $B \to K\ell^+\ell^-$ decay has been measured recently by the LHC experiments, showing a deviation from theory predictions based on the Standard Model (SM). A major challenge in achieving a complete SM prediction and interpreting this discrepancy lies in the treatment of nonlocal hadronic effects. In $B \to K\ell^+\ell^-$, these effects are cast in a single nonlocal form factor, a function of squared momentum transfer $q^2$ to the lepton pair. One of the previously used methods provides this form factor in the region of spacelike momentum transfer, $q^2<0$, matching the result to the hadronic dispersion relation, which is then continued to the physical region. The calculation done so far was a combination of QCD factorisation for hard-gluon contributions with light-cone sum rules (LCSRs) for soft-gluon ones. In this work, we calculate for the first time the complete nonlocal form factor at $q^2<0$ for one of the effective operators, the chromomagnetic operator $O_{8g}$, applying the method of LCSRs with $ B$-meson distribution amplitudes. We compute, both analytically and numerically, the operator-product expansion (OPE) diagrams with hard-gluon exchanges, analyse their structure and hierarchy, and obtain their spectral density entering the LCSR together with soft-gluon contributions. This study paves the way for our next task, a complete calculation of nonlocal $B \to K\ell^+\ell^-$ form factor at spacelike $q^2$, including the dominant contributions of current-current operators, known as charm-loops.

Figures (3)

• Figure 1: Diagrams for the correlation function (\ref{['eq:corr']}). The chromomagnetic operator $O_{8g}$ is denoted by a shaded square, the $B$-meson DAs are shown with a shaded blob. The circled crosses indicate possible emission points of a virtual photon.
• Figure 2: Individual contributions to the rescaled non-local form factor for the operator $O_{8g}$, defined according to (\ref{['eq:H']}), in the spacelike region $q^{2}<0$. The blue and dashed-blue lines represent the real and imaginary parts of the form-factor-like contribution $H_{A+B}$, respectively. The magenta and gray curves correspond to the hard-scattering and soft-gluon contributions, $H_{C}$ and $H_{D}$. The total real part, $\mathrm{Re}\,H_{\text{tot}}(q^{2})$, obtained from their coherent sum, is shown in red.
• Figure 3: The blue (solid and dashed) lines show the real and imaginary parts of the factorisable term, while the magenta and gray curves represent the nonfactorisable hard- and soft-gluon pieces. The total ratio $\mathrm{Re}\,R_{\text{tot}}(q^{2})$ is shown in red.