Reconciling hadronic and partonic analyticity in $b\to s\ell\ell$ transitions

Abstract

Rare $B$-meson decays mediated by $b\to s\ell\ell$ transitions constitute sensitive probes of physics beyond the Standard Model, and have triggered considerable interest due to hints for deviations from the Standard-Model prediction. To establish a discrepancy beyond a reasonable doubt, control over the nonlocal matrix elements involving charm loops is essential, which, for large spacelike virtualities, can be constrained by an operator product expansion with coefficients known at two-loop order. We observe that the analytic structure of this partonic calculation, whose understanding is important to put forward rigorous parameterizations, follows from simple triangle topologies and demonstrate explicitly how dispersion relations are fulfilled even in the case of anomalous thresholds. Crucially, these anomalous contributions match onto the ones expected when considering hadronic degrees of freedom, proving that the partonic calculation does not miss anomalous effects and justifying its use in regions of parameter space in which a perturbative description applies.

Figures (7)

• Figure 1: Basic partonic two-loop diagrams for $b\to s\gamma^*$ featuring charm loops, in the conventions of Ref. Asatrian:2019kbk: the straight lines indicate quarks, the black dots the insertion of the effective operator $\mathcal{O}_{1,2}$, and the curly lines gluons. In diagrams $(a)$ and $(b)$, the electromagnetic current couples in all possible ways to the $b$ or $s$ quark, in diagrams $(c)$, $(d)$, and $(e)$ to the $c$ quark (type-$(e)$ diagrams with current insertion at the $b$ or $s$ quark vanish).
• Figure 2: Discontinuities for diagrams $(a)$--$(d)$, comparing fits of our triangle-diagram-motivated parameterizations ("disc fit") to the exact results from Ref. Asatrian:2019kbk ("disc $2$-loop"). For diagrams $(a)$ and $(c)$ the discontinuities develop a real part due to the anomalous thresholds, whose position is indicated by the vertical dotted lines. The vertical dashed line refers to the normal threshold.
• Figure 3: Real and imaginary part of the FFs for diagrams $(a)$ and $(c)$, comparing our results from the dispersion relation ("DR") to the exact results from Ref. Asatrian:2019kbk ("$2$-loop"). The limits $s\pm i\epsilon$ disagree exactly where the respective discontinuity becomes nonzero; see Fig. \ref{['fig:discontinuities']}. The results for diagrams $(b)$ and $(d)$ are shown in Fig. \ref{['fig:dispersion_relation_bd']}. Vertical dashed and dotted lines indicate normal and anomalous thresholds, respectively.
• Figure 4: Trajectory of $s_+$ in the complex plane as a function of the parameter $\xi\in[0.8,10]$; see main text. The partonic threshold $s_\text{th}=4m_c^2$ and trajectory ($b\to s$, black) are given in units of $m_b$, the hadronic threshold $s_\text{th}=4M_D^2$ and trajectories ($B\to K$, red; $B\to K^*$, blue) in units of $M_B$. For $\xi=1$, $s_+$ moves onto the real axis. The crosses indicate the physical points, which occur at $\xi=1$ ($b\to s$, $m_1=m_c+m_s$), $\xi=2.04$ ($B\to K$, $m_1=M_{D^*_s}$), and $\xi=\{3.69,8.62\}$ ($B\to K^*$, $m_1=\{M_{D^*_s},M_{D_s}\}$), respectively. All masses are taken from Ref. ParticleDataGroup:2024cfk.
• Figure 5: Conventions for the triangle diagram.