Emergence of kaonium as a sharp resonance in photon-photon to meson-meson cross-sections

TL;DR

This work investigates the hypothetical kaonium ($K^+K^-$ bound state) by computing its binding energies from the $K^+K^-\to K^+K^-$ elastic amplitude and incorporating isospin breaking and Coulomb effects. It then embeds kaonium formation into chiral perturbation theory-based amplitudes for $\gamma\gamma\to\pi^0\pi^0$ and $\gamma\gamma\to\pi^0\eta$, using isospin-breaking corrections and analytic continuation to reveal a sharp resonance near $M\simeq 992$ MeV that accompanies the $f_0(980)$/$a_0(980)$. The kaonium resonance induces noticeable, channel-dependent modifications of the cross-sections, notably producing a pronounced peak in $\gamma\gamma\to\pi^0\eta$ and yielding a cross-section ratio around $9$ at the resonance energy. These results provide a consistency check with prior kaonium studies and offer a potential, indirect experimental signature via photon-photon initiated processes, while highlighting the crucial roles of isospin breaking and electromagnetic interactions in exotic-atom phenomenology.

Abstract

We calculate the binding energies of the hypothetical mesonic atom, $K^+ K^-$ (kaonium), using the $K^+ K^- \to K^+ K^-$ elastic scattering amplitude. Our findings are in line with previously reported results, which involve solving an eigenvalue equation of the Kudryavtsev-Popov type. Using chiral perturbation theory, we show that kaonium manifests itself as a sharp resonance around 992 MeV accompanying $f_0 (980)$ or $a_0 (980)$ in cross-sections for processes $γγ\to π^0 π^0$ or $γγ\to π^0 η$. The latter process is particularly striking: the peak at the kaonium resonance energy is highly pronounced, with the ratio of the cross-sections $σ(γγ\to π^0 η) / σ(γγ\to π^0 π^0) \approx 9$. Due to the short lifetime of kaonium ($\sim 10^{-18}$ s) and its small decay width ($\sim 0.4$ keV), direct detection of this exotic atom poses a significant challenge and requires high experimental resolution. However, we show that once the formation of kaonium is considered in the cross-section, a better fit to the available experimental data is obtained.

Figures (5)

• Figure 1: Resonances in the $K^+ K^- \to K^+ K^-$ elastic scattering amplitude squared, $|T_{K^+ K^-}|^2 \times 10^{-6}$, where $T_{K^+ K^-}$ is given in (\ref{['esT']}), corresponding to the formation and decay of the 1s and 2s bound states of kaonium.
• Figure 2: The lowest 1s kaonium resonance in the $K^+ K^- \to K^+ K^-$ elastic scattering amplitude squared, $|T_{K^+ K^-}|^2 \times 10^{-4}$, where $T_{K^+ K^-}$ is given in (\ref{['esT']}), is plotted on an enlarged scale relative to that used in Fig. \ref{['TB']} (blue curve). The associated Breit--Wigner (BW) curve obtained from (\ref{['bw0']}), using the pole position of the $T_{K^+ K^-}$ for its parameters, is slightly shifted to the right (orange curve).
• Figure 3: Illustration of complex vectors $s$, $s - 4 m_K^2$, and $s - 4 (m_K^2 + 2 m_K \Delta)$, where $-\pi < \phi < \pi$ and $0 < (\theta, \psi) < 2 \pi$. The zigzag lines represent the branch cuts.
• Figure 4: Cross-section for $\gamma \gamma \to \pi^0 \pi^0$ as a function of the center-of-mass energy $\sqrt{s} = P_0$. The black curve represents $\sigma ( \gamma \gamma \to \pi^0 \pi^0 )$ with conserved isospin, corresponding to (\ref{['crp0c']}). The blue curve represents $\sigma ( \gamma \gamma \to \pi^0 \pi^0 )$ with broken isospin, corresponding to (\ref{['crp0']}) with $T_{K^+ K^-; \pi^0 \pi^0}$ given in (\ref{['TKPi']}). The isospin breaking is only significant in the close vicinity of the kaonium resonance around 992 MeV. The cutoff parameter is $\Lambda = 1351$ MeV.
• Figure 5: Cross-section for $\gamma \gamma \to \pi^0 \eta$ as a function of the center-of-mass energy $\sqrt{s} = P_0$. The black curve represents $\sigma ( \gamma \gamma \to \pi^0 \eta )$ with conserved isospin, corresponding to (\ref{['crpec']}). The blue curve represents $\sigma ( \gamma \gamma \to \pi^0 \eta )$ with broken isospin, corresponding to (\ref{['crpe']}) with $T_{K^+ K^-; \pi^0 \eta}$ given in (\ref{['tfke']}). The kaonium resonance manifests itself as a sharp peak around 992 MeV, where its width is smaller by a factor of $10^{-5}$ than that of the strong-interaction resonance width. The experimental data of the JADE and Belle Collaborations JADE1990Belle2009 are shown in red and green, respectively. The cross-section curve, when the formation of kaonium is taken into account, gives a better fit to the experimental data. The cutoff parameter is $\Lambda = 1351$ MeV.