Prospects for a lattice computation of rare kaon decay amplitudes: $K\to \pi {\ell }^{+}{\ell }^{-}$ decays

The rare kaon decays $K\to \pi {\ell }^{+}{\ell }^{-}$ and $K\to \pi \nu \overline{\nu }$ are flavor changing neutral current (FCNC) processes and hence promising channels with which to probe the limits of the standard model and to look for signs of new physics. In this paper we demonstrate the feasibility of lattice calculations of $K\to \pi {\ell }^{+}{\ell }^{-}$ decay amplitudes for which long-distance contributions are very significant. We show that the dominant finite-volume corrections (those decreasing as powers of the volume) are negligibly small and that, in the four-flavor theory, no new ultraviolet divergences appear as the electromagnetic current $J$ and the effective weak Hamiltonian $H_W$ approach each other. In addition, we demonstrate that one can remove the unphysical terms which grow exponentially with the range of the integration over the time separation between $J$ and $H_W$. We will now proceed to exploratory numerical studies with the aim of motivating further experimental measurements of these decays. Our work extends the earlier study by Isidori et al. [1] which focused largely on the renormalization of ultraviolet divergences. In a companion paper [2] we discuss the evaluation of the long-distance contributions to $K\to \pi \nu \overline{\nu }$ decays; these contributions are expected to be at the level of a few percent for $K^{+}$ decays.

Figure 1

One-loop contribution in ChPT to the decays $K_S\to {\pi }^0{\ell }^{+}{\ell }^{-}$ and $K^{+}\to {\pi }^{+}{\ell }^{+}{\ell }^{-}$. The dashed, circled and wiggly lines represent the pions, kaon and photon, respectively. There is a branch cut when $q^2>4M_{\pi }^2$.

Figure 2

The three major contributions to $K_L\to {\pi }^0{\ell }^{+}{\ell }^{-}$ decays: (a1,a2) short-distance dominated penguin and box diagrams, (b) long-distance dominated indirect $CP$ violating contribution occurring through $K^0-{\overline{K}}^0$ mixing, (c) long-distance dominated $CP$ conserving component involving two-photon exchange.

Figure 3

Diagrams contributing to the 3-point function ${\mathrm{\Gamma }}_H^{(3)}(t_H,\mathbf{p})$ defined in Eq. (14). The two black circles represent the currents in the four-quark operators $Q_{1,2}^q$ defined in (8). $\ell$ denotes a light ($u$ or $d$) quark propagator. The different topologies contain the operators $Q_1^q$ or $Q_2^q$ depending on whether the initial state is a charged or neutral kaon. For example, when the initial state is $K^{+}$, the $W$ and $S$ topologies contain an insertion of $Q_2^q$ while the $C$ and $E$ topologies contain an insertion of $Q_1^q$.

Figure 4

Two-pion intermediate state contribution to the rare kaon decay amplitude. The dotted and dashed lines represent, respectively, the kaon and pion propagators.

Figure 5

Examples of contributions from a three-pion intermediate state to rare kaon decays.

Figure 6

A potentially quadratically divergent insertion into the $S$ and $E$ classes of diagram. $J_{\nu }$ represents the conserved electromagnetic current and $J_{\mu ,ij}^L$ is the local vector current ${\overline{u}}_j{\gamma }_{\mu }{u}_i$ or ${\overline{c}}_j{\gamma }_{\mu }{c}_i$ from $Q_{1,2}$.

Figure 7

“Wing” class of diagram contributing to the rare kaon decay correlator (18). The diagrammatic conventions are the same as those in Fig. 3.

Figure 8

“Connected” class of diagram contributing to the rare kaon decay correlator (18). The diagrammatic conventions are the same as those in Fig. 3.The $\ell$ quark is an up or down quark depending on the charge of the initial and final states.

Figure 9

“Saucer” class of diagram contributing to the rare kaon decay correlator (18). The diagrammatic conventions are the same as those in Fig. 3.

Figure 10

“Eye” class of diagram contributing to the rare kaon decay correlator (18). The diagrammatic conventions are the same as those in Fig. 3.

Figure 11

“${\pi }^0$” class of diagram contributing to the rare kaon decay correlator (18). These diagrams only contribute to the decay of the neutral meson $K^0$. The diagrammatic conventions are the same as those in Fig. 3.