$K\toπγ^*γ^*$ transitions at leading order and beyond

TL;DR

This work develops a comprehensive treatment of $K\to\pi\gamma^{(*)}\gamma^{(*)}$ transitions within Chiral Perturbation Theory, starting with a complete leading-order (one-loop) calculation for the doubly off-shell case and then extending to beyond-LO dynamics via a generic, gauge-invariant ansatz for the $K\pi PP$ vertex. Unitarity corrections from $K\to3\pi$ are incorporated to capture dominant rescattering effects, yielding a framework that simultaneously treats charged and neutral channels. The analysis introduces a refined diagrammatic notation that isolates gauge-invariant subsets, derives explicit LO form factors for one- and two-photon channels, and presents a robust structure for including higher-order counterterms and unitarity corrections through a small set of low-energy constants. The resulting amplitudes form crucial inputs for rare-kaon decays such as $K\to\pi\ell^+\ell^-$ and $K\to\pi\ell_1^+\ell_1^-\,\ell_2^+\ell_2^-$ and bear on ongoing NA62 measurements, while providing a foundation for extending the approach to related processes like $\eta^{(\prime)}$ decays. The paper thus delivers both a precise LO baseline and a practical route to systematic beyond-LO refinements in radiative kaon decays.

Abstract

The transition amplitude of a kaon to a pion and two off-shell photons is studied. First, it is computed at leading order (one-loop level) of the Chiral Perturbation Theory expansion. Explicit analytical results for the leading-order amplitude are presented, constituting the first complete calculation for the doubly off-shell case. Subsequently, it is reevaluated by employing a refined diagrammatic notation and a generic ansatz incorporating effects beyond leading order. The dependence on the underlying $KπPP$ vertex parameters is analyzed. This offers valuable insights into amplitude properties and allows inclusion of unitarity corrections from $K\to3π$, yielding the complete $K\toπγ^*γ^*$ amplitude structure. Both the charged and neutral channels are treated in parallel. The presented results provide crucial input for phenomenological studies of related rare decays like $K\toπ\ell^+\ell^-[γ]$ or $K\toπ\ell_1^+\ell_1^-\ell_2^+\ell_2^-$ and support ongoing precision measurements at experiments like NA62 at CERN. These results may also find application in other related processes, including $η^{(\prime)}$ decays.

Figures (8)

• Figure 1: The representation of the $K\to\pi\gamma^*$ transition amplitude in terms of Feynman diagrams emerging from the ChPT Lagrangian \ref{['eq:L_ChPT']}. The minimal electromagnetic coupling is denoted by a small dot and the weak coupling ($K\pi\pi\pi$ or $K\pi KK$) by an empty square; the combination of both is naturally represented by a square with a dot. Consequently, there are both (charged) pions and kaons in the loops; the contributions with neutral-meson loops cancel out at the considered orders. The counterterm contributions are represented by a square with a cross. The two one-particle-reducible (1PR) diagrams (those with photons attached to external legs) are absent in the neutral channel and placed within the gray dotted box.
• Figure 2: The contributions to the $K\to\pi\gamma^*\gamma^*$ transition. The first two diagrams (within the gray box) are derived from the $K^+\to\pi^+\gamma^*$ transition, which is denoted by three-particle vertices marked as filled black squares, and are absent in the neutral channel (they have photons attached to external legs). These relate to topology (1) of \ref{['sec:2g']}, and at LO in ChPT, the contributing diagrams are resolved by inserting those shown in \ref{['fig:K->pig_LO_ChPT_a']}. The third diagram is the remainder, which, resolved in LO ChPT, is shown in \ref{['fig:K->pigg_LO_ChPT']} and represents topology (2a), the "genuine" two-photon transition. The last diagram represents the contribution of the QCD anomaly and relates to topology (2b).
• Figure 3: The "genuine" two-photon contribution to the $K\to\pi\gamma^*\gamma^*$ transition. In the context of the decomposition in \ref{['fig:K->pigg']}, it is the remainder when the LO ChPT or its considered extensions are taken as a particular representation of \ref{['fig:K->pigg']}: Plugging the diagrams defined in \ref{['fig:K->pig_LO_ChPT_a', 'fig:K->pigg_LO_ChPT']} into \ref{['fig:K->pigg']} gives all the diagrams at LO ChPT. The minimal electromagnetic coupling is denoted by a small dot and the weak coupling ($K\pi\pi\pi$ or $K\pi KK$) by an empty square; the combination of both is naturally represented by a square with a dot. Consequently, both (charged) pions and kaons are in the loops; the neutral-meson loops do not contribute at considered orders. The counterterm contributions are represented by a square with a cross. The "+ cross" note denotes that an associated diagram in which the photons are interchanged (crossed), i.e., Bose symmetrization, should be added. Notice the minus sign as the associated contribution needs to be subtracted due to double-counting in \ref{['fig:K->pigg']}. The neutral channel does not feature the three (plus one crossed) 1PR diagrams (with photons attached to external legs) placed within the gray dotted box. The LO counterterm contribution is also absent in the neutral channel.
• Figure 4: The representation of the $K\to\pi\gamma^*$ transition amplitude in terms of LO Feynman diagrams emerging from the ChPT Lagrangian \ref{['eq:L_ChPT']}, while utilizing the diagrammatic notation of \ref{['sec:notation']}. The caption of \ref{['fig:K->pig_LO_ChPT_a']} applies. Furthermore, the one-particle-irreducible (1PI) (radiative) tadpole contribution is separated in two pieces (see also \ref{['sec:notation']}) for the diagrams in respective lines to form gauge-invariant subsets. Strictly speaking, the labeling 1PI and 1PR is thus violated; here, the 'R' in the latter can be thought of as standing for "reducible and remainder."
• Figure 5: The relevant contributions to the $K\to\pi\gamma^*\gamma^*$ transition at LO ChPT and its considered extensions. The first two diagrams within the gray box (present only in the charged channel) are derived from the $K^+\to\pi^+\gamma^*$ transition, which is denoted by vertices marked as filled black squares: Explicitly, the contributing diagrams are recovered by plugging in the 1PI vertices from \ref{['fig:K->pig_LO_ChPT_b']}, and to form a gauge-invariant subset, they are accompanied by the related diagrams (shown in the second line) where a photon is radiated directly from the vertices of 1PI diagrams of \ref{['fig:K->pig_LO_ChPT_b']} (associated with external meson radiation in the sense of \ref{['sec:notation']}). The genuine two-photon transition (structure-dependent part) is the 1PI remainder, which, resolved in LO ChPT and considered extensions, is shown in \ref{['fig:K->pigg_LO_1PI']}. The remaining tadpole diagrams are in \ref{['fig:K->pigg_LO_1PR']}. The diagrams within the gray dotted box are absent in the neutral channel.