Precision lattice QCD determination of the $K\to\pi\pi$ decay amplitudes, the $\Delta I=1/2$ rule, and $\varepsilon'/\varepsilon$

In 1985 Bernard, Draper, Soni, Politzer and Wise proposed to determine the physical $K\to\pi\pi$ decay amplitudes — and thereby the $\Delta I=1/2$ rule and the direct-CP-violation parameter $\varepsilon'/\varepsilon$ — from lattice QCD by combining Monte Carlo computations of the simpler unphysical amplitudes $K\to\pi$ and $K\to 0$ (kaon to vacuum) with lowest-order chiral perturbation theory relations, the $K\to 0$ piece subtracting unwanted operator-renormalization mixing. The objective has been substantially achieved by a different route: direct calculation of $K\to\pi\pi$ with physical on-shell kinematics, enabled by the Lellouch–Lüscher finite-volume formalism. As of 2020 the RBC and UKQCD collaborations reproduce the $\Delta I=1/2$ enhancement from unquenched, physical-point lattice QCD ($\mathrm{Re}(A_0)/\mathrm{Re}(A_2)=19.9(2.3)(4.4)$ versus the experimental $22.45(6)$) and predict $\mathrm{Re}(\varepsilon'/\varepsilon)=21.7(2.6)(6.2)(5.0)\times10^{-4}$ versus $16.6(2.3)\times10^{-4}$. The surviving open core is the precision completion: a fully controlled determination including the continuum limit (the published results use a single lattice spacing, identified as the dominant systematic), so that the result can be quantitatively confronted with experiment.

Background

Non-leptonic kaon decays are described by two isospin amplitudes: $A_0$ for $K\to(\pi\pi)_{I=0}$, produced by the strangeness-changing weak transition with isospin change $\Delta I=1/2$, and $A_2$ for $K\to(\pi\pi)_{I=2}$ with $\Delta I=3/2$. Experimentally $|A_0|/|A_2|\approx 22.5$: the $\Delta I=1/2$ amplitude is enhanced roughly twenty-fold over its isospin partner (the "$\Delta I=1/2$ rule"), an effect not explained by the known short-distance enhancement of the four-quark weak operators. The interference of the CP-violating parts of $A_0$ and $A_2$ determines $\mathrm{Re}(\varepsilon'/\varepsilon)\approx 16.6\times10^{-4}$, the measure of direct CP violation (CP violation intrinsic to the decay amplitude itself, as opposed to the indirect CP violation induced by kaon–antikaon mixing), whose Standard-Model prediction is a sensitive test of the CP-violation sector and of new physics (see Cirigliano et al., Rev. Mod. Phys. 84 (2012) 399, for the overall framework).

Chiral perturbation theory (ChPT) is the effective field theory of the pseudo-Goldstone octet ($\pi$, $K$, $\eta$) associated with the spontaneous breaking of the chiral $SU(3)_L\times SU(3)_R$ symmetry of QCD; weak transitions with strangeness change $\Delta S=1$ are organized in a chiral weak Lagrangian ordered in meson momenta and masses. Bernard, Draper, Soni, Politzer and Wise (Phys. Rev. D 32 (1985) 2343) showed that, to lowest nontrivial order (tree level, quadratic order in meson masses), the physical $K\to 2\pi$ amplitude can be written algebraically in terms of the unphysical amplitudes $K\to\pi$ and $K\to 0$, where the vacuum amplitude cancels unwanted renormalization contributions arising when the weak four-quark operators mix with two-quark operators through "penguin" diagrams (flavor-changing quark loops with gluon exchange). Since $K\to\pi$ and $K\to 0$ involve only one external meson, they are far easier to compute in lattice QCD — the discretized, nonperturbative formulation of QCD suited to Monte Carlo evaluation — raising the hope of a first-principles determination of $A_0$, $A_2$, the $\Delta I=1/2$ rule, and $\varepsilon'/\varepsilon$. Their exploratory test used the quenched approximation (omitting virtual sea-quark loops), a very small ensemble, and heavy pion masses, and left open: the size of higher-order chiral corrections; controlled unquenched simulations at lighter quark masses; a practical nonperturbative implementation of the operator renormalization and subtraction; and the control of lattice artifacts (finite lattice spacing $a$, finite volume, scale setting).

The subsequent history defines the present status. The next-to-leading-order (one-loop) weak chiral Lagrangian was computed by Kambor, Missimer and Wyler (Nucl. Phys. B 346 (1990) 17; Phys. Lett. B 261 (1991) 496), who found that the NLO corrections reduce the lowest-order octet coupling by about $30\%$ — the higher-order corrections to the lowest-order relations are large; Ecker, Kambor and Wyler (Nucl. Phys. B 394 (1993) 101) estimated the $O(p^4)$ weak counterterms from resonance exchange. Quenched implementations of the indirect $K\to\pi$/$K\to 0$ route with nonperturbative renormalization (Blum et al., Phys. Rev. D 68 (2003) 114506; Noaki et al., Phys. Rev. D 68 (2003) 014501) underpredicted the $\Delta I=1/2$ enhancement and obtained the wrong sign for $\varepsilon'/\varepsilon$, identifying quenching and lowest-order ChPT as the limiting approximations. The field therefore shifted to computing $K\to\pi\pi$ directly at on-shell kinematics, made possible by the Lellouch–Lüscher finite-volume relation (Commun. Math. Phys. 219 (2001) 31), which converts finite-volume transition matrix elements into infinite-volume decay amplitudes. Using unquenched $2+1$-flavor domain-wall fermions (a lattice discretization with good chiral symmetry, simulating two degenerate light-quark flavors plus the strange quark), nonperturbative operator renormalization (lattice operators matched to the continuum weak Hamiltonian, conventionally renormalized in the standard $\overline{\mathrm{MS}}$ modified-minimal-subtraction scheme), boundary-condition tuning (G-parity boundary conditions, where G-parity — charge conjugation combined with a $180^\circ$ isospin rotation — is imposed so that the two-pion state can carry the kaon's momentum) or variational excited-state resolution to reach the on-shell decay kinematics $E_{\pi\pi}=m_K$ (two-pion energy equal to the kaon mass), and all-to-all quark-propagator techniques (quark propagators computed between all lattice sites, needed for the disconnected contractions that dominate $A_0$), RBC/UKQCD computed $A_2$ at physical kinematics (Blum et al., Phys. Rev. Lett. 108 (2012) 141601), the $\Delta I=1/2$-rule mechanism (Boyle et al., Phys. Rev. Lett. 110 (2013) 152001), and $A_0$ and $\varepsilon'$ (Bai et al., Phys. Rev. Lett. 115 (2015) 212001; Abbott et al., Phys. Rev. D 102 (2020) 054509). The 2020 results — $\mathrm{Re}(A_0)=2.99(0.32)(0.59)\times10^{-7}\,\mathrm{GeV}$ (experiment $3.3201(18)\times10^{-7}\,\mathrm{GeV}$), $\mathrm{Re}(A_0)/\mathrm{Re}(A_2)=19.9(2.3)(4.4)$ (experiment $22.45(6)$), $\mathrm{Re}(\varepsilon'/\varepsilon)=21.7(2.6)(6.2)(5.0)\times10^{-4}$ (experiment $16.6(2.3)\times10^{-4}$) — demonstrate the Standard-Model origin of the $\Delta I=1/2$ rule, with central values differing from experiment by roughly $10$–$30\%$, within the total uncertainties. However, these published determinations were performed at a single lattice spacing ($a^{-1}=1.3784(68)\,\mathrm{GeV}$) and omit isospin breaking (electromagnetism and $m_u\neq m_d$), which is the third error quoted on $\varepsilon'/\varepsilon$; ongoing work identifies the continuum limit as the dominant lattice systematic and the isospin-breaking effects on $\varepsilon'$ as potentially as large as $25\%$.

Problem Statement

Produce a first-principles Standard-Model determination of the $K\to\pi\pi$ decay amplitudes — the isospin amplitudes $A_0$ ($\Delta I=1/2$) and $A_2$ ($\Delta I=3/2$), and thereby the ratio $\mathrm{Re}(A_0)/\mathrm{Re}(A_2)$ (the $\Delta I=1/2$ rule) and the direct-CP-violation parameter $\mathrm{Re}(\varepsilon'/\varepsilon)$ — based on unquenched lattice QCD at physical quark masses and on-shell kinematics ($E_{\pi\pi}=m_K$), with each of the systematic effects (1)–(4) below quantified and propagated into the error budget, so that the result can be quantitatively confronted with the experimental values $\mathrm{Re}(A_0)/\mathrm{Re}(A_2)=22.45(6)$ and $\mathrm{Re}(\varepsilon'/\varepsilon)=16.6(2.3)\times10^{-4}$. The determination must include and quantify: (1) the continuum limit $a\to 0$ with quantified scaling violations (the published 2015/2020 results use a single lattice spacing, identified by RBC/UKQCD as the dominant lattice systematic error); (2) the infinite-volume treatment of the two-pion final state (Lellouch–Lüscher or equivalent), including the $\pi\pi$ scattering phase shifts it requires; (3) nonperturbative renormalization of the $\Delta S=1$ four-quark operators with controlled matching to the $\overline{\mathrm{MS}}$ weak Hamiltonian and treatment of any mixing with lower-dimension two-quark operators; and (4) isospin-breaking effects (electromagnetism and the up–down quark-mass difference $m_u\neq m_d$). The determination should yield $\mathrm{Re}(A_0)$, $\mathrm{Im}(A_0)$, $\mathrm{Re}(A_2)$, $\mathrm{Im}(A_2)$, $\mathrm{Re}(A_0)/\mathrm{Re}(A_2)$ and $\mathrm{Re}(\varepsilon'/\varepsilon)$, each with statistical and systematic uncertainties. The determination should combine the cited multi-ensemble and isospin-breaking inputs into a continuum-limit result at on-shell kinematics, with the four systematics (1)–(4) quantified and propagated into the error budget.

References

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