CP Violation in Kaons and B Mesons

The CKM phase of the previous lesson is a number in a matrix until an experiment makes it visible. The place it shows up most cleanly is the neutral kaon system, where the weak interaction mixes a particle with its antiparticle and sorts them into states of definite CP that decay to two or three pions. In 1964 a decay that CP forbids was seen anyway, establishing that CP is not a symmetry of nature. This lesson develops the neutral-kaon story, distinguishes indirect from direct violation, carries the idea to the $B$ mesons where the effect is large and clean, and closes on why even this is not enough to explain the matter of the universe.

Neutral kaon mixing

The neutral kaon $K^0$ has quark content $d\bar{s}$; its antiparticle ${\bar{K}}^0$ is $s\bar{d}$. They have the same mass, $m_K\approx 498$ MeV, and are distinguished only by strangeness, which the strong interaction conserves but the weak interaction does not. Through a second-order weak process — a box diagram with two $W$ bosons and up-type quarks $u,c,t$ running in the loop — a $K^0$ can turn into a ${\bar{K}}^0$ and back:

This is a flavour-changing neutral transition, allowed only at loop level (the GIM mechanism forbids it at tree level) and therefore small. But it means the strong eigenstates $K^0$ and ${\bar{K}}^0$ are not the states that propagate with definite mass and lifetime; those are particular linear combinations, fixed by the requirement that they be eigenstates of the weak Hamiltonian.¹

CP eigenstates and their decays

If CP were an exact symmetry of the weak interaction, the propagating states would be CP eigenstates. Neither $K^0$ nor ${\bar{K}}^0$ is one — CP maps them into each other — but two combinations are:

Kaons decay to pions, and the pion final states carry definite CP. A two-pion state from a spin-zero kaon has $CP=+1$; a three-pion state has $CP=-1$ (the extra pion's intrinsic parity flips the sign, and the low available energy forbids compensating orbital angular momentum). If CP is good, then $K_1$ may decay to two pions and $K_2$ only to three:

The kinematics differ starkly. The two-pion decay has $m_K-2m_{\pi }\approx 220$ MeV of phase space; the three-pion decay only $\approx 80$ MeV. More phase space means a faster decay, so $K_1$ should be short-lived and $K_2$ long-lived. This is observed: the neutral kaons come in two lifetimes,

differing by a factor of about $600$.² The short state $K_S$ decays to two pions, the long state $K_L$ to three — if CP holds.

The Cronin–Fitch discovery

James Cronin and Val Fitch put this to the test in 1964. They let a kaon beam travel far enough that all the short-lived $K_S$ had decayed, leaving a pure $K_L$ beam, and then looked at what the survivors decayed into. If CP is exact, $K_L=K_2$ decays only to three pions. Instead, among $22,700$ decays they found $45$ going to two pions — a CP-forbidden mode occurring at the level of two per thousand.³

CP is violated. The long-lived physical state is not the pure CP eigenstate $K_2$ but carries a small admixture of $K_1$:

The parameter $ϵ$ measures the CP impurity in the mass eigenstates, and it traces directly to the imaginary parts of the CKM elements in the box diagram: the amplitudes for $K^0\to {\bar{K}}^0$ and ${\bar{K}}^0\to K^0$ differ because the CKM factors are complex, $\mathcal{M}(K\to \bar{K})\ne \mathcal{M}(\bar{K}\to K)$, and that inequality is CP violation.

Strangeness oscillation and indirect vs direct violation

Because a $K^0$ is a superposition of $K_S$ and $K_L$ with slightly different masses, its strangeness content oscillates as it propagates: a beam produced as pure $K^0$ develops a ${\bar{K}}^0$ component that grows, ebbs, and beats at the frequency set by the mass difference $\Delta m=m_L-m_S$, all under an overall exponential decay. Passing the beam through matter, which absorbs $K^0$ and ${\bar{K}}^0$ differently, regenerates the short-lived component — a striking confirmation of the coherent two-state picture.

The violation seen by Cronin and Fitch is indirect: it lives in the mixing, in the $ϵ$ that makes $K_L$ an impure CP state. A distinct effect, direct CP violation, occurs when the decay amplitude itself violates CP — when $K_2$ decays to two pions not through its $K_1$ admixture but intrinsically. It is parametrized by ${ϵ}^{\prime }$, and its measurement, $\mathrm{Re}({ϵ}^{\prime }/ϵ)\approx 1.7\times 10^{-3}$, established that CP violation is not confined to mixing but also afflicts the decay, exactly as the CKM phase predicts.⁴

CP violation in B mesons and $\sin 2\beta$

The neutral $B$ meson $B^0=d\bar{b}$ mixes with ${\bar{B}}^0$ through the same kind of box diagram, but with the top quark dominating the loop and CKM elements that make the CP-violating phase large rather than a two-per-mille perturbation. The signal is a time-dependent asymmetry. Producing $B^0$–${\bar{B}}^0$ pairs and reconstructing decays to a CP eigenstate such as $J/\psi K_S$, one measures the difference in decay rate between mesons that were born as $B^0$ versus ${\bar{B}}^0$ as a function of proper time. The interference between direct decay and decay-after-mixing produces a clean sinusoid,

whose amplitude is $\sin 2\beta$ — a direct readout of the interior angle $\beta$ of the unitarity triangle. The $B$ factories BaBar and Belle measured

a large, unambiguous CP asymmetry, in full agreement with the value the CKM triangle predicts from entirely independent measurements.⁵ The kaon and $B$ systems, decades and orders of magnitude apart in the size of the effect, are described by the same single phase.

Why it is not enough

CP violation is one of the three Sakharov conditions a theory must satisfy to explain why the universe contains matter but almost no antimatter, alongside baryon-number violation and a departure from thermal equilibrium. The CKM phase supplies CP violation — but far too little. Quantified by the Jarlskog invariant $J\approx 3\times 10^{-5}$, the Standard-Model effect falls short of the observed baryon-to-photon ratio by roughly ten orders of magnitude. The matter–antimatter asymmetry of the universe therefore demands a source of CP violation beyond the CKM matrix, one of the sharpest pointers to physics past the Standard Model, taken up in a later module.⁶ What the kaons and $B$ mesons establish is narrower but certain: CP is not a symmetry of nature, and its breaking is governed, at accessible energies, by the one phase of the CKM matrix.

Summary

Neutral kaons mix through weak box diagrams into short- and long-lived states that, if CP were exact, would be pure CP eigenstates decaying to two and three pions respectively. The 1964 Cronin–Fitch observation of $K_L\to \pi \pi$ at the two-per-mille level proved CP is violated, encoded in the mixing parameter $ϵ$; direct violation ${ϵ}^{\prime }$ later showed the decay amplitudes violate CP as well. The $B$ mesons carry the same physics with a large phase, and the $B$-factory measurement of $\sin 2\beta =0.70$ confirmed that one CKM phase governs CP violation across systems. Yet the effect is ten orders of magnitude too small to account for the cosmic matter–antimatter asymmetry — a shortfall that closes the weak-interaction module with an open question.