Meson-antimeson mixing

TL;DR

This work surveys meson-antimeson mixing across $K$, $D$, $B_d$, and $B_s$ systems, foregrounding $| abla F|=2$ box diagrams that couple flavor-changing neutral currents to CP violation. It develops a unified time-evolution formalism based on the $2\times 2$ matrix $M-i\Gamma/2$, relating mass and width differences to $M_{12}$ and $\Gamma_{12}$, and shows how CKM phases enter mixing and decay through quantities like $\Delta M$, $\Delta \Gamma$, $a_{\rm fs}$, and $\lambda_f$. The review then connects these observables to SM predictions via the Operator Product Expansion, Heavy Quark Expansion, and lattice QCD, detailing the SM picture of $\epsilon_K$, $\Delta M_{d,s}$, and CP asymmetries, and outlining how precision flavor physics constrains or hints at BSM scenarios. It emphasizes the role of unitarity triangles and gold-plated modes in CKM metrology, highlights the historical validation of the KM mechanism, and underscores future potential to reveal heavy new physics through improved theory and CP-violating observables in mixing. Overall, meson-antimeson mixing provides a powerful, high-scale probe of flavor structure and CP violation, with current theory and experiments converging toward stringent tests of the SM and windows to new physics.

Abstract

Meson-antimeson transitions are flavor-changing neutral current processes in which the strangeness, charm, or beauty quantum number changes by two units. In the Standard Model (SM) these transitions originate from box diagrams with two W bosons. They permit the preparation of time-dependent, oscillating quantum states which are superpositions of a meson and its antimeson. By studying their decays one gains information on both the meson-antimeson mixing amplitude and the decay amplitude involved and one can measure complex phases quantifying the violation of charge-parity (CP) violation. I present a comprehensive overview on the topic, starting with phenomenological presentations of $K$-$\bar K$, $B_d$-$\bar B_d$, $B_s$-$\bar B_s$, and $D$-$\bar D$ mixing. Highlights are the discovery of the violation of CP and other discrete symmetries, the predictions of the charm quark and its mass and a heavy top quark, and the confirmation of the Kobayashi-Maskawa mechanism of CP violation. Further sections cover the theoretical formalism needed to describe meson-antimeson mixing and to calculate observables in terms of the fundamental parameters of the SM. I discuss the unitarity triangle of the Cabibbo-Kobayashi-Maskawa matrix, which is used to visualize how various CP-violating and CP-conserving quantities combine to probe the SM. I describe the emergence of precision flavor physics and the role of reliable theory calculations to link $K$-$\bar K$ mixing to $B_d$-$\bar B_d$ mixing, which was essential to confirm the Kobayashi-Maskawa mechanism, and present the current status of theory predictions. Today, the focus of the field is on physics beyond the SM, because meson-antimeson mixing amplitudes are sensitive to virtual effects of heavy particles with masses which are several orders of magnitude above the reach of current particle colliders.

Figures (9)

• Figure 1: Box diagrams for $K\!-\!\bar{K}\,$, $D\!-\!\bar{D}\,$, $B_d\!-\!\bar{B}{}_d\,$ and $B_s\!-\!\bar{B}{}_s\,$ mixing with zigzag lines representing W bosons. The diagrams show the transition from antimeson $\bar{M}$ meson entering the diagram from the left into meson $M$ leaving the diagram to the right. For each process there is also a second box diagram, obtained by a 90$^\circ$ rotation.
• Figure 2: Penguin diagram contributing to a $\Delta S=0$ decay of a $b$-flavored hadron.
• Figure 3: The (standard) unitarity triangle. The non-trivial sides are $R_u=\sqrt{\bar{\rho}^2+\bar{\eta}^2}$ and $R_t=\sqrt{(1-\bar{\rho})^2+\bar{\eta}^2}$.
• Figure 4: Left: generic self energy $\Sigma$ of a charged meson. Right: $M^0\!-\bar{M}{}^0$ mixing amplitude $\Sigma_{12}$.
• Figure 5: The four-quark operator $Q$ for $B_q\!-\!\bar{B}{}_q\,$ mixing with $q=d$ or $s$.