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

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.