Nonlocal form factor of chromomagnetic penguin in $B o Kll^+ll^-$ 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.

💡 Research Summary

The paper addresses the long‑standing tension between the measured branching fraction of the rare decay B → K ℓ⁺ℓ⁻ and the Standard‑Model (SM) predictions. In the SM the decay amplitude receives a dominant contribution from local operators (O₇, O₉, O₁₀) whose matrix elements are expressed through well‑studied B → K form factors, and a sub‑dominant but phenomenologically crucial non‑local contribution. The non‑local part arises from the time‑ordered product of the electromagnetic current with the effective weak Hamiltonian and is encoded in a single invariant function H(q²), the non‑local form factor, which depends on the squared momentum transfer to the lepton pair.

Previous calculations of H(q²) have combined QCD factorisation (QCDf) for hard‑gluon exchanges with light‑cone sum rules (LCSRs) for soft‑gluon effects, and have been performed in the space‑like region (q² < 0). The result is then analytically continued to the physical time‑like region using a hadronic dispersion relation. However, this hybrid approach treats hard and soft contributions with different theoretical frameworks and only includes the leading hard‑gluon term, leaving higher‑order and mixed contributions unaccounted for.

In this work the authors undertake a fully consistent LCSR calculation of the non‑local form factor associated with the chromomagnetic operator O₈g, the first step toward a complete LCSR treatment of all non‑local effects. The chromomagnetic operator, \