--- title: CP Violation in Kaons and B Mesons module: The Weak Interaction moduleNumber: 7 lessonNumber: 4 order: 704 summary: > The neutral kaon is its own laboratory for CP. Weak box diagrams mix $K^0$ and its antiparticle into short- and long-lived states that should be pure CP eigenstates decaying to two and three pions. In 1964 Cronin and Fitch caught the long-lived kaon decaying to two pions — CP is violated, at the two-per-mille level of $\epsilon$. Direct violation ($\epsilon'$) followed, and the $B$ factories turned the CKM phase into a large, clean time-dependent asymmetry measuring $\sin 2\beta$. The effect is real but far too small to explain why the universe is made of matter. topics: [The Weak Interaction] draft: false sources: - book: Griffiths ref: "Ch. 4 §4.6 and Ch. 10 — the neutral kaon system" - book: Thomson ref: "Ch. 14 — CP violation" - book: Tong ref: "The Standard Model (Cambridge Part III), §6.4.4 Neutral Kaons" - book: PDG ref: "Kaon and B-meson CP parameters — pdg.lbl.gov" --- 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: $$ K^0 \;\rightleftharpoons\; \bar K^0 . $$ 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.[^tong-mixing] $$ % caption: The box diagram mixing a neutral kaon with its antiparticle. Two W bosons % and up-type quarks (up, charm, top) in the loop convert the d and anti-s of the K0 % into the s and anti-d of the K0-bar. The CKM factors at the vertices carry complex % phases, so the two directions have unequal amplitudes. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % left side incoming K0: d (top-left), anti-s (bottom-left) \node[black, font=\scriptsize, left] at (-2.7,1.1) {d}; \node[black, font=\scriptsize, left] at (-2.7,-1.1) {anti-s}; \draw[->, black, thick] (-2.6,1.1) -- (-1.0,1.1); \draw[->, black, thick] (-2.6,-1.1) -- (-1.0,-1.1); % box corners \coordinate (tl) at (-1.0,1.1); \coordinate (tr) at (1.0,1.1); \coordinate (bl) at (-1.0,-1.1); \coordinate (br) at (1.0,-1.1); % top W and bottom W (horizontal waves) \draw[acc, very thick] plot[domain=-1.0:1.0, samples=48] (\x, {1.1 + 0.08*sin((\x+1.0)/2.0*1800)}); \node[acc, font=\scriptsize, above] at (0,1.2) {W}; \draw[acc, very thick] plot[domain=-1.0:1.0, samples=48] (\x, {-1.1 + 0.08*sin((\x+1.0)/2.0*1800)}); \node[acc, font=\scriptsize, below] at (0,-1.2) {W}; % vertical quark lines in loop \draw[->, black, thick] (tl) -- (bl); \node[black, font=\scriptsize, left] at (-1.05,0) {u, c, t}; \draw[->, black, thick] (br) -- (tr); \node[black, font=\scriptsize, right] at (1.05,0) {u, c, t}; % outgoing K0-bar: s (top-right), anti-d (bottom-right) \draw[->, black, thick] (tr) -- (2.6,1.1); \draw[->, black, thick] (br) -- (2.6,-1.1); \node[black, font=\scriptsize, right] at (2.65,1.1) {s}; \node[black, font=\scriptsize, right] at (2.65,-1.1) {anti-d}; \end{tikzpicture} $$ ## 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: $$ |K_1\rangle = \tfrac{1}{\sqrt2}\big(|K^0\rangle - |\bar K^0\rangle\big), \quad CP = +1; \qquad |K_2\rangle = \tfrac{1}{\sqrt2}\big(|K^0\rangle + |\bar K^0\rangle\big), \quad CP = -1 . $$ 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: $$ K_1 \to \pi\pi \ (CP=+1), \qquad K_2 \to \pi\pi\pi \ (CP=-1) . $$ 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, $$ \tau_S \approx 0.90 \times 10^{-10}\ \text{s}, \qquad \tau_L \approx 5.1 \times 10^{-8}\ \text{s}, $$ differing by a factor of about $600$.[^tong-cp-kaons] 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.[^thomson-cronin] CP is violated. The long-lived physical state is not the pure CP eigenstate $K_2$ but carries a small admixture of $K_1$: $$ |K_L\rangle = \frac{1}{\sqrt{1+|\epsilon|^2}} \big(|K_2\rangle + \epsilon\,|K_1\rangle\big), \qquad |\epsilon| \approx 2.2 \times 10^{-3} . $$ The parameter $\epsilon$ 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, $\mathcal{M}(K\to\bar K)\neq\mathcal{M}(\bar K\to K)$, and that inequality is CP violation. $$ % caption: The Cronin-Fitch signature. The long-lived kaon should decay only to % three pions if CP holds. A small fraction decay to two pions instead, the % CP-forbidden mode, revealing that the long-lived state carries a CP-impurity % epsilon of about two parts per thousand. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \node[draw=black, thick, align=center, minimum width=22mm, minimum height=9mm] (kl) at (0,0) {long-lived kaon}; % dominant 3pi \node[draw=black, thick, align=center, minimum width=20mm] (t) at (4.4,0.9) {three pions}; \draw[->, black, very thick] (kl) -- (t); \node[black, font=\scriptsize, above] at (2.6,0.85) {allowed, common}; % rare 2pi \node[draw=acc, thick, align=center, minimum width=20mm] (h) at (4.4,-0.9) {two pions}; \draw[->, acc, thick, densely dashed] (kl) -- (h); \node[acc, font=\scriptsize, below] at (2.6,-0.85) {CP-forbidden, rare}; \end{tikzpicture} $$ ## 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 $\epsilon$ 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 $\epsilon'$, and its measurement, $\mathrm{Re}(\epsilon'/\epsilon)\approx 1.7\times10^{-3}$, established that CP violation is not confined to mixing but also afflicts the decay, exactly as the CKM phase predicts.[^pdg-epsilon] $$ % caption: Strangeness oscillation. A beam born as pure K0 develops an anti-K0 % component that oscillates at the frequency set by the mass difference of the long % and short states, all beneath an overall exponential decay. The interference of % two nearly degenerate mass eigenstates makes the strangeness beat. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.0,0) node[right, font=\scriptsize] {proper time}; \draw[->, black] (0,0) -- (0,2.6) node[above, font=\scriptsize] {anti-K0 fraction}; % decaying envelope (dotted) \draw[black, densely dotted] (0.2,2.2) .. controls (1.8,1.2) and (3.4,0.55) .. (5.6,0.25); % oscillating decaying curve (acc): a few damped humps \draw[black, very thick] (0.2,0.15) .. controls (0.7,1.9) and (1.2,1.9) .. (1.7,0.35) .. controls (2.1,1.15) and (2.6,1.15) .. (3.0,0.3) .. controls (3.4,0.75) and (3.9,0.75) .. (4.3,0.25) .. controls (4.7,0.5) and (5.2,0.5) .. (5.6,0.22); \node[black, font=\scriptsize, right] at (3.4,1.05) {decay envelope}; \end{tikzpicture} $$ ## 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, $$ A_{CP}(t) = \sin 2\beta \,\sin(\Delta m_B\, t) , $$ 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 $$ \sin 2\beta = 0.70 \pm 0.02 , $$ a large, unambiguous CP asymmetry, in full agreement with the value the CKM triangle predicts from entirely independent measurements.[^pdg-sin2beta] The kaon and $B$ systems, decades and orders of magnitude apart in the size of the effect, are described by the _same single phase_. $$ % caption: The time-dependent CP asymmetry in neutral B decays to a CP eigenstate. % The difference in decay rate between mesons born as B and anti-B oscillates as a % sine of the proper time, with amplitude sin two beta reading off the unitarity % triangle angle. Unlike the kaon case the asymmetry is order one. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.2,0) node[right, font=\scriptsize] {proper time}; \draw[black] (0,-1.6) -- (0,1.6); \node[black, font=\scriptsize, above left] at (0,1.6) {asymmetry}; % zero line already the axis; draw sine \draw[black, very thick] plot[domain=0:6.0, samples=60] (\x, {1.25*sin(\x r)}); % amplitude marker \draw[black, densely dotted] (0,1.25) -- (1.571,1.25); \node[black, font=\scriptsize, right] at (1.55,1.25) {amplitude sin two beta}; \end{tikzpicture} $$ ## 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 $J\approx 3\times10^{-5}$, 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.[^tong-wherefore] 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 $\epsilon$; direct violation $\epsilon'$ 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. [^tong-mixing]: Tong, _The Standard Model_ (Cambridge Part III), §6.4.4 — neutral-kaon box-diagram mixing with $u,c,t$ in the loop and the complex CKM factors at the vertices. [damtp.cam.ac.uk/user/tong/standardmodel.html](http://www.damtp.cam.ac.uk/user/tong/standardmodel.html) [^tong-cp-kaons]: Tong, _The Standard Model_ (Cambridge Part III), §6.4.4 — the CP eigenstates $K_{1,2}$, the two- and three-pion CP assignments, the phase-space argument for the two lifetimes, and the $K_S/K_L$ identification. Lifetimes from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov). [^thomson-cronin]: Thomson, _Modern Particle Physics_, Ch. 14 — the Cronin–Fitch experiment and the $\epsilon$ parametrization; original: J. H. Christenson, J. W. Cronin, V. L. Fitch, R. Turlay, _Phys. Rev. Lett._ **13**, 138 (1964). [^pdg-epsilon]: The values $|\epsilon|\approx 2.2\times10^{-3}$ and $\mathrm{Re}(\epsilon'/\epsilon)\approx1.7\times10^{-3}$ (direct CP violation) are tabulated by the Particle Data Group, kaon listings, [pdg.lbl.gov](https://pdg.lbl.gov). [^pdg-sin2beta]: The measurement $\sin 2\beta = 0.70\pm0.02$ from BaBar and Belle is reported in the Particle Data Group $B$-meson CP review, [pdg.lbl.gov](https://pdg.lbl.gov); see also Thomson, Ch. 14. [^tong-wherefore]: Tong, _The Standard Model_ (Cambridge Part III), §6.4.5 — the smallness of CKM CP violation relative to what baryogenesis requires; the Sakharov conditions and the baryon-to-photon ratio follow the Particle Data Group cosmology review, [pdg.lbl.gov](https://pdg.lbl.gov).