Standard Model Benchmarks for $D^0\to K^- K^+, π^-π^+, K^0_{\rm S} K^0_{\rm S}$ Decays

TL;DR

This work analyzes charm SCS decays $D^0\to K^-K^+$ and $D^0\to\pi^-\pi^+$ within the Standard Model, treating the leading colour-allowed tree amplitude in factorisation using lattice-QCD inputs for decay constants and form factors. It shows that non-factorisable contributions and $U$-spin breaking at the $\mathcal{O}(50\%)$ level can reconcile the observed BRs, and that the charged-mode data imply a significant, but still SM-consistent, enhancement of penguin amplitudes necessary to account for direct CP violation. The $D^0\to K_S^0K_S^0$ channel provides a clean lab for non-factorisable exchange and penguin-annihilation topologies, and, together with isospin arguments, yields a coherent SM picture with moderate $U$-spin breaking; the predicted direct CP asymmetry in this mode reaches at most a few per mille in benchmark scenarios. Overall, the paper provides a concrete SM benchmark framework linking BRs and CP asymmetries across $D^0\to K^-K^+$, $D^0\to\pi^-\pi^+$, and $D^0\to K_S^0K_S^0$, with clear experimental targets for future high-precision tests of charm flavour dynamics.

Abstract

The non-leptonic $D^0\to K^- K^+$ and $D^0\to π^-π^+$ decays are powerful probes of the Standard Model and are related to each other through the $U$-spin symmetry of the strong interaction. Using lattice QCD inputs we calculate the corresponding colour-allowed tree amplitudes in factorisation and demonstrate that non-factorisable contributions and $U$-spin-breaking effects at the level of 50% allow us to accommodate the measured branching ratios in the Standard Model. An exciting direct probe of such non-factorisable and $U$-spin breaking effects is provided by the $D^0\to K^0_{\rm S} K^0_{\rm S}$ channel. This decay is governed by non-factorisable exchange topologies and essentially vanishes in the $U$-spin limit, although it is experimentally well established with a prominent branching ratio. Extrapolating our $D^0\to K^- K^+$ results using the isospin symmetry, we find a consistent benchmark picture. Specifically, we can accommodate the measured $D^0\to K^0_{\rm S} K^0_{\rm S}$ branching ratio with $U$-spin-breaking effects at the 50% level and exchange amplitudes at the level of 50% of the colour-allowed $D^0\to K^- K^+$, $D^0\to π^-π^+$ tree contributions. Finally, we explore the resulting range for direct CP violation in $D^0\to K^0_{\rm S} K^0_{\rm S}$, obtaining upper bounds in our benchmark scenarios of a few per mille, offering an exciting target for future measurements.

Figures (13)

• Figure 1: Examples of colour-allowed tree (left), exchange (middle), and penguin (right) topologies contributing in the SM to the decay amplitude of $D^0 \to K^-K^+$. Corresponding diagrams for $D^0 \to \pi^- \pi^+$ are obtained by replacing$s \to d$ and $K \to \pi$.
• Figure 2: Allowed ranges for the parameters $r_f$ and $\delta_f$ defined in Eq. \ref{['eq:Ampl-decomposition']} for $f = \pi\pi$ (left) and $f = KK$ (right). The curves have been obtained by fitting the expression in Eq. \ref{['eq:Br-P1mP2p']}, where the factorisation predictions are given in Eq. \ref{['eq:Br_fac_FNAL']}, to the corresponding experimental results in Eqs. \ref{['eq:Br_pipi_exp-0']} and \ref{['eq:Br_KK_exp-0']}. In each plot, the dotted and solid lines indicate the contours obtained using the central and $1\sigma$ values for the factorisation results. Note that the effect of including also the experimental uncertainties falls within the shown curves.
• Figure 3: Comparison of the contours for $r_f$ as a function of $\delta_{\pi\pi}$ for relative strong phase shifts $|\Delta|$ = $30^\circ$ (top left), $45^\circ$ (top right), $60^\circ$ (bottom left) and $90^\circ$ (bottom right). In each plot, solid blue lines indicate the results for $r_{\pi\pi}$ while the dotted green and orange curves correspond to $r_{KK}$ for negative and positive values of $\Delta$, respectively. The gray bands indicate the regions of $\delta_{\pi\pi}$ excluded by the analysis of Fig. \ref{['fig:BrP1P2']}. The red highlighted regions indicate the parameter space satisfying the constraints in Eq. \ref{['eq:constraints_P1P2']}.
• Figure 4: Constraints on the size of the non-factorisable $U$-spin-breaking ratio $|\tilde{\cal A}_{KK}/\tilde{\cal A}_{\pi\pi}|$ as a function of the strong phase $\delta_{\pi\pi}$ for fixed relative strong-phase shifts $|\Delta|$ = $30^\circ$ (top left), $45^\circ$ (top right), $60^\circ$ (bottom left) and $90^\circ$ (bottom right). In each plot the solid green and oranges lines correspond to negative and positive shifts, respectively. The red highlighted regions indicate the parameter space satisfying the constraints in Eq. \ref{['eq:constraints_P1P2']}.
• Figure 5: Zoomed-in view of the allowed values of the non-factorisable $U$-spin-breaking ratio $|\tilde{\cal A}_{KK}/\tilde{\cal A}_{\pi\pi}|$ as a function of the strong phase $\delta_{\pi\pi}$ shown separately for negative (left) and positive (right) relative shifts $|\Delta|$ = $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$. The horizontal gray line marks the $U$-spin symmetric limit. The red highlighted regions indicate the parameter space satisfying the constraints in Eq. \ref{['eq:constraints_P1P2']}.