CP violation in kaons

The neutral kaon is the most eloquent two-state system in all of physics, and it speaks in three languages at once. The strong interaction that makes it knows only flavour: it produces a $K^0$ ( $d\bar{s}$ ) or its antiparticle $\bar{K}^0$ ( $\bar{d}s$ ), states of definite strangeness. The weak interaction that destroys it knows only CP: to a good approximation it sorts the decays into a CP-even combination $K_1$ that goes to two pions and a CP-odd combination $K_2$ that goes to three. And free flight through empty space knows only mass and lifetime: the kaon propagates as the eigenstates $K_S$ and $K_L$ , whose slightly different masses make an initially pure $K^0$ beam slosh back and forth into $\bar{K}^0$ — strangeness oscillation, a phenomenon that has nothing to do with CP violation and would occur even if CP were sacred. For nine years after the $K_L$ was found, physicists believed the three languages were secretly two: that the mass eigenstates were exactly the CP eigenstates, $K_S = K_1$ and $K_L = K_2$ . If so, the long-lived $K_L$ , being purely CP-odd, could never decay to the CP-even two-pion state. In 1964 Christenson, Cronin, Fitch and Turlay sent a pure $K_L$ beam down 17 metres of vacuum pipe — far enough that every $K_S$ had died — and caught it decaying to $\pi^+\pi^-$ about twice in every thousand. CP was not sacred after all. The physical states carry a tiny CP-impurity $\varepsilon$ : $|K_S\rangle \propto |K_1\rangle + \varepsilon|K_2\rangle$ and $|K_L\rangle \propto |K_2\rangle + \varepsilon|K_1\rangle$ , with $|\varepsilon|\approx 2.2\times10^{-3}$ . That number, and the near- $45^\circ$ phase it carries, is the subject of this article.

Key takeaways

The physical states $K_S,K_L$ are the CP eigenstates $K_1,K_2$ contaminated by a small admixture: $|K_S\rangle\propto|K_1\rangle+\varepsilon|K_2\rangle$ , $|K_L\rangle\propto|K_2\rangle+\varepsilon|K_1\rangle$ , with $|\varepsilon|=(2.228\pm0.011)\times10^{-3}$ .
CP violation is visible because $K_L\to\pi\pi$ occurs at branching ratio $\sim2\times10^{-3}$ , which the pure CP-odd $K_2$ could never do — this is indirect CP violation, living in the $K^0\!-\!\bar K^0$ mixing.
The phase of $\varepsilon$ is not free: it is the superweak phase $\phi_{SW}=\arctan(2\Delta m/\Delta\Gamma)\approx43.5^\circ$ , fixed entirely by the CP-conserving mass and width differences.
Strangeness oscillation between $K^0$ and $\bar K^0$ comes only from $\Delta m=m_L-m_S=3.484\times10^{-15}$ GeV and happens whether or not CP is violated — do not confuse the two.
A second, smaller effect — direct CP violation in the decay amplitude — is measured by $\mathrm{Re}(\varepsilon'/\varepsilon)=(1.66\pm0.23)\times10^{-3}$ , proving CP is broken in the decay itself, not only in the mixing.
The semileptonic charge asymmetry $\delta_L\approx2\,\mathrm{Re}\,\varepsilon\approx3.3\times10^{-3}$ gives an absolute, convention-free definition of matter versus antimatter.
Key idea: CP violation in kaons means the long-lived, nominally CP-odd state $K_L$ carries a tiny CP-even admixture $\varepsilon$ , so it can decay into two pions — an outcome forbidden if C and P together were exact symmetries of the weak force.

Where this lives today

The kaon is no longer the only place CP violation is seen — the $B$ -meson factories BaBar and Belle measured it at order unity in $B^0\to J/\psi\,K_S$ in 2001, and LHCb has now found it in $B_s$ and even $D$ mesons — but the kaon remains the sharpest laboratory for the subject. Two dedicated experiments defined the modern picture: CPLEAR at CERN (1990–1996) tagged the production flavour of each kaon by the charge of an accompanying $K\pi$ pair and its decay flavour by the sign of the semileptonic lepton, forming asymmetries that cancel detector biases to measure $\Delta m$ , $|\varepsilon|$ , and — most strikingly — a direct, model-independent observation of time-reversal (T) violation via the rate difference between $K^0\to\bar K^0$ and $\bar K^0\to K^0$ , exactly as CPT demands given CP violation. Meanwhile NA48 (CERN) and KTeV (Fermilab) fought a decade-long battle to nail the double ratio, converging on $\mathrm{Re}(\varepsilon'/\varepsilon)=(1.66\pm0.23)\times10^{-3}$ and killing the superweak model. The frontier now is the ultra-rare "golden" modes $K^+\to\pi^+\nu\bar\nu$ and $K_L\to\pi^0\nu\bar\nu$ , whose branching ratios (near $10^{-10}$ and $10^{-11}$ ) are calculable in the Standard Model with unusually small hadronic uncertainty; NA62 at CERN and KOTO at J-PARC are measuring them now, where a deviation would be a clean signal of new sources of CP violation beyond the CKM phase.

The physical picture

Follow a single neutral kaon from birth to death and you watch it change languages twice. It is born a flavour eigenstate — say a $K^0$ from $\pi^-p\to\Lambda K^0$ . That $K^0$ is an equal quantum superposition of the two mass eigenstates $K_S$ and $K_L$ , which age at wildly different rates: $K_S$ lives $0.9\times10^{-10}$ s and dies to two pions, $K_L$ lives $500$ times longer and dies to three. A short way downstream the $K_S$ part has evaporated and what remains is an almost pure $K_L$ beam. If CP were exact that surviving beam could never make two pions. The first figure is that beam — the experiment Cronin and Fitch built — with the real decay lengths and the rare, forbidden-looking channel that broke the symmetry.

Figure

A pure Kꜱ-plus-Kʟ beam decaying along its flight path: the Kꜱ component dies within a few centimetres (cτ = 2.68 cm) while the Kʟ survives to 15.3 m, so far downstream the beam is pure Kʟ — and yet about 2 in 1000 of those Kʟ decay to π⁺π⁻, the CP-violating signal Cronin and Fitch found in 1964

Why is $\varepsilon$ a complex number, and why does its phase come out near $45^\circ$ ? Because $\varepsilon$ is built from the off-diagonal entries of the little $2\times2$ matrix that governs mixing — the dispersive part $M_{12}$ (virtual transitions) and the absorptive part $\Gamma_{12}$ (real common decays). When the CP-odd piece of $M_{12}$ is divided by the complex denominator that carries the mass splitting $\Delta m$ and width splitting $\Delta\Gamma$ , the result acquires a phase locked to those two CP-conserving quantities. The kaon system sits at the near-coincidence $2\Delta m\approx\Delta\Gamma$ , so that phase is almost exactly $45^\circ$ and $\mathrm{Re}\,\varepsilon\approx\mathrm{Im}\,\varepsilon$ . The second figure is that punchline drawn in the complex plane, with the measured numbers.

Figure

The CP-violation parameter ε in the complex plane: measured magnitude 2.23×10⁻³ at the superweak phase 43.5°, so its real and imaginary parts are nearly equal, Re ε ≈ 1.62×10⁻³ and Im ε ≈ 1.53×10⁻³ — the phase is fixed by arctan(2Δm/ΔΓ), not by the size of CP violation

The core equations

A neutral kaon is a two-component amplitude $\psi=(a,b)$ for being $K^0$ or $\bar K^0$ . Because the state decays, its evolution is not governed by a Hermitian Hamiltonian but by an effective one with a mass matrix $\mathbf M$ and a decay matrix $\mathbf\Gamma$ , both Hermitian. The Weisskopf–Wigner approximation gives the Schrödinger-like equation below, and CPT invariance forces the two diagonal entries to be equal.

i\frac{\mathrm d}{\mathrm dt}\begin{pmatrix}a\\ b\end{pmatrix}=\left(\mathbf M-\frac{i}{2}\mathbf\Gamma\right)\begin{pmatrix}a\\ b\end{pmatrix},\qquad \mathbf M-\frac{i}{2}\mathbf\Gamma=\begin{pmatrix}H_0 & H_{12}\\ H_{21} & H_0\end{pmatrix} \qquad (1)

where: $a,b$ — amplitudes to be $K^0,\bar K^0$ ; $\mathbf M$ — Hermitian mass matrix (dispersive, from virtual intermediate states); $\mathbf\Gamma$ — Hermitian decay matrix (absorptive, from real common final states); $H_{12}=M_{12}-\tfrac{i}{2}\Gamma_{12}$ ; $H_0$ — the common diagonal entry, equal for the two rows by CPT. Reading: everything about mixing and CP violation is hidden in the off-diagonal $H_{12},H_{21}$ . Diagonalizing (1) gives the mass eigenstates. Written in the CP basis $|K_1\rangle=\tfrac{1}{\sqrt2}(|K^0\rangle-|\bar K^0\rangle)$ (CP $=+1$ ) and $|K_2\rangle=\tfrac{1}{\sqrt2}(|K^0\rangle+|\bar K^0\rangle)$ (CP $=-1$ ), the physical states are the CP eigenstates plus a small cross-contamination $\varepsilon$ .

|K_S\rangle=\frac{|K_1\rangle+\varepsilon\,|K_2\rangle}{\sqrt{1+|\varepsilon|^2}},\qquad |K_L\rangle=\frac{|K_2\rangle+\varepsilon\,|K_1\rangle}{\sqrt{1+|\varepsilon|^2}} \qquad (2)

where: $K_S,K_L$ — short- and long-lived mass eigenstates; $K_1,K_2$ — CP-even and CP-odd combinations; $\varepsilon$ — the single complex CP-violation parameter. Reading: set $\varepsilon=0$ and $K_L=K_2$ is purely CP-odd, forbidden to decay to two pions; the observed $K_L\to\pi\pi$ measures $\varepsilon\neq0$ . The parameter $\varepsilon$ is not arbitrary — diagonalization ties it directly to the off-diagonal entries and to the mass and width splittings. Neglecting the small absorptive CP-odd part $\mathrm{Im}\,\Gamma_{12}$ , it takes the transparent form below.

\varepsilon\simeq\frac{\mathrm{Im}\,M_{12}}{\tfrac12\Delta\Gamma-i\,\Delta m}=|\varepsilon|\,e^{i\phi_{SW}},\qquad \phi_{SW}=\arctan\!\left(\frac{2\Delta m}{\Delta\Gamma}\right) \qquad (3)

where: $\Delta m=m_L-m_S>0$ ; $\Delta\Gamma=\Gamma_S-\Gamma_L>0$ ; $\mathrm{Im}\,M_{12}$ — the CP-odd part of the dispersive mixing, the only genuinely CP-violating quantity here; $\phi_{SW}$ — the superweak phase. Reading: the size of $\varepsilon$ is set by CP-violating physics, but its phase is fixed by the CP-conserving $\Delta m$ and $\Delta\Gamma$ . What experiments actually measure are the ratios of CP-violating to CP-conserving decay amplitudes. There are two, one for each pion charge state, and their difference exposes CP violation in the decay on top of the mixing.

\eta_{+-}=\frac{\langle\pi^+\pi^-|K_L\rangle}{\langle\pi^+\pi^-|K_S\rangle}=\varepsilon+\varepsilon',\qquad \eta_{00}=\frac{\langle\pi^0\pi^0|K_L\rangle}{\langle\pi^0\pi^0|K_S\rangle}=\varepsilon-2\varepsilon' \qquad (4)

where: $\eta_{+-},\eta_{00}$ — measured amplitude ratios; $\varepsilon$ — indirect CP violation (in the mixing, common to both); $\varepsilon'$ — direct CP violation (in the decay amplitude, weighted differently for the two charge states). Reading: if $\varepsilon'=0$ then $\eta_{+-}=\eta_{00}$ exactly; any measured difference is direct CP violation. Two further observables close the system. The semileptonic charge asymmetry gives $\mathrm{Re}\,\varepsilon$ cleanly, and the double ratio isolates the sign and size of $\varepsilon'/\varepsilon$ .

\delta_L=\frac{\Gamma(K_L\to\pi^-\ell^+\nu)-\Gamma(K_L\to\pi^+\ell^-\bar\nu)}{\Gamma(K_L\to\pi^-\ell^+\nu)+\Gamma(K_L\to\pi^+\ell^-\bar\nu)}=\frac{2\,\mathrm{Re}\,\varepsilon}{1+|\varepsilon|^2},\qquad \left|\frac{\eta_{00}}{\eta_{+-}}\right|^2\simeq1-6\,\mathrm{Re}\frac{\varepsilon'}{\varepsilon} \qquad (5)

where: $\delta_L$ — the lepton-charge asymmetry of $K_L$ decays; $\ell=e,\mu$ . Reading: $\delta_L>0$ means $K_L$ decays slightly more often to $\ell^+$ than $\ell^-$ , an absolute matter/antimatter label; the double ratio being below one measures direct CP violation $\mathrm{Re}(\varepsilon'/\varepsilon)>0$ .

Derivation ADiagonalizing the decaying two-state system

The goal is to start from the effective Hamiltonian $\mathbf M-\tfrac{i}{2}\mathbf\Gamma$ of equation (1), impose the discrete-symmetry constraints, and show that its eigenvectors are precisely the CP-contaminated states of equation (2) with a definite expression for $\varepsilon$ in terms of the off-diagonal entries. This is where the abstract statement "the mass eigenstates are not the CP eigenstates" becomes a concrete piece of linear algebra, and where the equality of the diagonal entries — a consequence of CPT — turns out to be exactly what makes a single parameter $\varepsilon$ suffice.

Step 1. Because $\mathbf M$ and $\mathbf\Gamma$ are separately Hermitian, $H_{21}=M_{12}^*-\tfrac{i}{2}\Gamma_{12}^*$ is fixed once $H_{12}=M_{12}-\tfrac{i}{2}\Gamma_{12}$ is known, so the matrix has the form of equation (1) with two free complex off-diagonal numbers related by conjugation. Why: Hermiticity of the physical mass and width matrices is what keeps the problem down to essentially one complex mixing amplitude rather than two independent ones.

Step 2. Impose CPT invariance. CPT relates a state to its antistate with equal mass and width, forcing the two diagonal entries to be identical, $H_{11}=H_{22}=H_0$ . Why: an unequal diagonal would signal CPT violation ( $m_{K^0}\neq m_{\bar K^0}$ ); setting it equal is the assumption we are testing to high precision, and it guarantees the eigenvectors have equal-magnitude $K^0$ and $\bar K^0$ content up to the small $\varepsilon$ .

Step 3. Solve the $2\times2$ eigenproblem. The eigenvalues are $\lambda_\pm=H_0\pm\sqrt{H_{12}H_{21}}$ and the eigenvectors are $\propto\big(1,\ \pm\sqrt{H_{21}/H_{12}}\big)$ in the $(K^0,\bar K^0)$ basis. Why: an off-diagonal matrix with equal diagonal always has eigenvectors that are symmetric and antisymmetric combinations twisted by the ratio $\sqrt{H_{21}/H_{12}}$ , and that twist is the whole story.

Step 4. Define $\dfrac{1-\varepsilon}{1+\varepsilon}=\sqrt{\dfrac{H_{21}}{H_{12}}}$ . Then the antisymmetric-like eigenvector becomes $(1+\varepsilon)|K^0\rangle-(1-\varepsilon)|\bar K^0\rangle\propto|K_1\rangle+\varepsilon|K_2\rangle$ , and the symmetric-like one becomes $|K_2\rangle+\varepsilon|K_1\rangle$ — exactly equation (2). Why: rewriting the twist $\sqrt{H_{21}/H_{12}}$ as a small deviation from one is what packages all the CP violation into the single parameter $\varepsilon$ that measures how far $H_{12}$ differs from $H_{21}$ .

Step 5. Expand for small violation: $\sqrt{H_{21}/H_{12}}\approx1-2\varepsilon$ gives $\varepsilon\approx\tfrac14(H_{12}-H_{21})/H_{12}$ , whose numerator is $\propto i\,\mathrm{Im}\,M_{12}+\tfrac12\mathrm{Im}\,\Gamma_{12}$ and whose denominator is $\propto\lambda_S-\lambda_L=-\Delta m-\tfrac{i}{2}\Delta\Gamma$ . Collecting factors reproduces equation (3). Why: this is the payoff — $\varepsilon\neq0$ requires $H_{12}\neq H_{21}$ , i.e. a relative phase between the mass and width mixing, which is exactly a CP-odd $\mathrm{Im}\,M_{12}$ . *Verified: $\varepsilon\to0$ when $H_{12}=H_{21}$ (CP conserved); $|K_S\rangle,|K_L\rangle$ reduce to $|K_1\rangle,|K_2\rangle$ ; expression (3) reproduced in verify_aq-pp-cp-violation-kaons.py.*

Go deeper · Derivation B · 1 minWhy the phase of ε is nailed to 45 degrees

Derivation BWhy the phase of ε is nailed to 45 degrees

The goal is to explain the single most surprising number in the whole subject — that the phase of $\varepsilon$ comes out at $43.5^\circ$ , tantalisingly close to $45^\circ$ — and to show that this phase is not itself a measure of CP violation but a kinematic accident of the kaon system. We take equation (3), insert the measured mass and width splittings, and show both that the phase is $\arctan(2\Delta m/\Delta\Gamma)$ and that the near-equality $2\Delta m\approx\Delta\Gamma$ is what pins it near the diagonal of the complex plane.

Step 1. From equation (3), with $\mathrm{Im}\,M_{12}$ real, the entire phase of $\varepsilon$ comes from the complex denominator $\tfrac12\Delta\Gamma-i\Delta m$ , so $\arg\varepsilon=\arctan\!\big(\Delta m/\tfrac12\Delta\Gamma\big)=\arctan(2\Delta m/\Delta\Gamma)\equiv\phi_{SW}$ . Why: the CP-violating physics is a single real number in the numerator; it can only scale $\varepsilon$ , never rotate it, so the phase belongs entirely to the CP-conserving denominator.

Step 2. Insert the widths. $\Gamma_L\approx\Gamma_S/571$ is negligible, so $\Delta\Gamma\approx\Gamma_S=\hbar/\tau_S$ . With $\tau_S=0.8954\times10^{-10}$ s this is $\Delta\Gamma\approx7.35\times10^{-15}$ GeV. Why: because $K_L$ is so long-lived, the width difference is essentially just the $K_S$ width, which makes $\phi_{SW}$ depend on only two measured numbers, $\Delta m$ and $\tau_S$ .

Step 3. Insert the mass splitting $\Delta m=3.484\times10^{-15}$ GeV. Then $2\Delta m/\Delta\Gamma=2(3.484)/7.35=0.949$ , and $\phi_{SW}=\arctan(0.949)=43.5^\circ$ . Why: the kaon happens to sit at $2\Delta m\approx\Delta\Gamma$ , a coincidence of nature with no known deeper reason, and that coincidence is what puts the phase within a degree and a half of $45^\circ$ .

Step 4. Consequence: $\mathrm{Re}\,\varepsilon=|\varepsilon|\cos43.5^\circ\approx1.62\times10^{-3}$ and $\mathrm{Im}\,\varepsilon=|\varepsilon|\sin43.5^\circ\approx1.53\times10^{-3}$ are nearly equal. A model in which CP violation lived only in $\Delta S=2$ mixing (the "superweak" model of Wolfenstein) predicts $\varepsilon$ at exactly this phase and $\varepsilon'=0$ . Why: measuring the phase is therefore a test of where CP violation lives; the Standard Model's $\varepsilon'\neq0$ shows the superweak picture is not the whole truth even though the phase agrees. *Verified: $\phi_{SW}=43.5^\circ$ from $\Delta m$ and $\tau_S$ ; $\mathrm{Re}\,\varepsilon\approx\mathrm{Im}\,\varepsilon\approx1.6\times10^{-3}$ ; numbers reproduced in verify_aq-pp-cp-violation-kaons.py.*

Go deeper · Derivation C · 2 minIndirect versus direct: the meaning of ε prime

Derivation CIndirect versus direct: the meaning of ε prime

The goal is to separate the two physically distinct ways CP can be violated in $K\to\pi\pi$ — in the mixing that turns $K_L$ into a bit of $K_1$ (indirect, $\varepsilon$ ) and in the decay amplitude itself (direct, $\varepsilon'$ ) — and to show why it took thirty-five years and a factor-of-a-thousand-smaller measurement to prove the second exists. The tool is the decomposition of the two-pion final state into definite isospin, which is where the two charge modes $\pi^+\pi^-$ and $\pi^0\pi^0$ carry different information.

Step 1. Bose symmetry allows the $\pi\pi$ system from a spin-0 kaon only isospin $I=0$ or $I=2$ . Write the decay amplitudes $A_I=\langle(\pi\pi)_I|K^0\rangle=|A_I|e^{i(\delta_I+\phi_I)}$ , splitting the strong rescattering phase $\delta_I$ from the weak (CP-odd) phase $\phi_I$ . Why: CP conjugation flips the weak phase $\phi_I\to-\phi_I$ but leaves the strong phase $\delta_I$ alone, so a difference between $K^0$ and $\bar K^0$ decay rates requires two interfering amplitudes with different weak phases — exactly $A_0$ and $A_2$ .

Step 2. Direct CP violation is measured by $\varepsilon'\propto\dfrac{i}{\sqrt2}\,e^{i(\delta_2-\delta_0)}\,\mathrm{Im}\!\left(\dfrac{A_2}{A_0}\right)$ , nonzero only if the two isospin amplitudes carry different weak phases. Why: if $A_2$ and $A_0$ shared a single weak phase it could be rotated away, and only mixing-induced $\varepsilon$ would survive — this is precisely the superweak scenario.

Step 3. Because the $I=2$ amplitude is suppressed relative to $I=0$ by the $\Delta I=1/2$ rule (the ratio $\mathrm{Re}(A_2/A_0)\approx1/22$ ), $\varepsilon'$ is intrinsically far smaller than $\varepsilon$ , and the two charge modes probe it with opposite sign through $\eta_{+-}=\varepsilon+\varepsilon'$ and $\eta_{00}=\varepsilon-2\varepsilon'$ . Why: the factor $-2$ versus $+1$ is pure Clebsch–Gordan bookkeeping for how $I=0$ and $I=2$ combine into $\pi^+\pi^-$ versus $\pi^0\pi^0$ , and it is what makes the double ratio $|\eta_{00}/\eta_{+-}|^2\simeq1-6\,\mathrm{Re}(\varepsilon'/\varepsilon)$ the clean observable.

Step 4. Experiments (NA48 at CERN, KTeV at Fermilab) measured the double ratio to be below one, giving $\mathrm{Re}(\varepsilon'/\varepsilon)=(1.66\pm0.23)\times10^{-3}\neq0$ . Why: a nonzero result rules out the pure superweak model and confirms the Standard Model prediction that CP violation acts in the decay vertex through the CKM phase, not only in the mass mixing. *Verified: $\eta_{+-}=\varepsilon+\varepsilon'$ , $\eta_{00}=\varepsilon-2\varepsilon'$ ; double ratio $\simeq1-6\,\mathrm{Re}(\varepsilon'/\varepsilon)$ to first order in verify_aq-pp-cp-violation-kaons.py.*

Limiting cases

Case	Result	Meaning
CP exact, $\varepsilon=\varepsilon'=0$	$K_S=K_1$ , $K_L=K_2$	$K_L$ purely CP-odd; $K_L\to\pi\pi$ strictly forbidden
Indirect only, $\varepsilon\neq0,\ \varepsilon'=0$	$\eta_{+-}=\eta_{00}=\varepsilon$	CP violation lives entirely in the mixing (superweak)
Direct present, $\varepsilon'\neq0$	$\eta_{+-}\neq\eta_{00}$	CP also violated in the decay amplitude itself
$\Delta\Gamma\gg2\Delta m$	$\phi_{SW}\to0$ , $\varepsilon$ real	phase set by width difference; not the kaon's regime
$2\Delta m=\Delta\Gamma$ (kaon)	$\phi_{SW}=45^\circ$ , $\mathrm{Re}\,\varepsilon=\mathrm{Im}\,\varepsilon$	the near-realized coincidence of the neutral kaon
CPT imposed	$\delta_L=2\,\mathrm{Re}\,\varepsilon>0$	absolute, convention-free matter/antimatter distinction

Common trap“CP violation and strangeness oscillation are the same phenomenon.”

They are not, and conflating them is the single most common error in this subject. Strangeness oscillation — the periodic sloshing of an initially pure $K^0$ beam into $\bar K^0$ and back — is driven only by the mass difference $\Delta m$ between $K_S$ and $K_L$ . It would happen with full amplitude even if CP were an exact symmetry, because $K_S$ and $K_L$ would still be distinct mass eigenstates ( $=K_1$ and $K_2$ ) beating against each other with frequency $\Delta m/\hbar\approx5.3\times10^{9}\,\mathrm{s}^{-1}$ . CP violation is the separate, far smaller fact that the mass eigenstates are not quite the CP eigenstates: $\varepsilon\neq0$ . You can have oscillation without CP violation (the world of 1955–1964, as everyone then believed), but not CP violation without a two-state system to violate it in. The clean tell: oscillation is an $O(1)$ effect governed by $\Delta m$ ; CP violation is an $O(10^{-3})$ effect governed by $\varepsilon$ . A related subtlety worth stating: the near- $45^\circ$ phase of $\varepsilon$ is likewise not a measure of how much CP is violated — it is fixed by the CP-conserving $\Delta m$ and $\Delta\Gamma$ , and it would sit at $43.5^\circ$ for any nonzero $|\varepsilon|$ .

Worked examples

1 · The superweak phase and the real and imaginary parts of ε (fully worked)

Given the short-kaon lifetime $\tau_S=0.8954\times10^{-10}$ s, the mass splitting $\Delta m=3.484\times10^{-15}$ GeV, and the measured magnitude $|\varepsilon|=2.228\times10^{-3}$ , predict the phase of $\varepsilon$ and hence its real and imaginary parts and the semileptonic asymmetry $\delta_L$ .

Step 1. Turn the $K_S$ lifetime into a width: $\Gamma_S=\hbar/\tau_S=(6.582\times10^{-16}\ \mathrm{eV\,s})/(0.8954\times10^{-10}\ \mathrm{s})=7.35\times10^{-6}\ \mathrm{eV}=7.35\times10^{-15}$ GeV. Since $\Gamma_L\approx\Gamma_S/571$ is negligible, $\Delta\Gamma\approx\Gamma_S=\mathbf{7.35\times10^{-15}}$ GeV.

Step 2. Form the superweak phase from equation (3): $2\Delta m/\Delta\Gamma=2(3.484\times10^{-15})/(7.35\times10^{-15})=0.949$ , so $\phi_{SW}=\arctan(0.949)=\mathbf{43.5^\circ}$ . This matches the directly measured phase $\phi_{+-}=(43.5\pm0.5)^\circ$ with no CP-violation input at all — the phase is pure kinematics.

Step 3. Project onto the axes: $\mathrm{Re}\,\varepsilon=|\varepsilon|\cos43.5^\circ=2.228\times10^{-3}\times0.7254=\mathbf{1.62\times10^{-3}}$ and $\mathrm{Im}\,\varepsilon=|\varepsilon|\sin43.5^\circ=2.228\times10^{-3}\times0.6884=\mathbf{1.53\times10^{-3}}$ . They are equal to within $6\%$ — the hallmark of the $2\Delta m\approx\Delta\Gamma$ coincidence.

Step 4. Predict the semileptonic asymmetry from equation (5): $\delta_L\approx2\,\mathrm{Re}\,\varepsilon=2(1.62\times10^{-3})=\mathbf{3.23\times10^{-3}}$ . The measured value is $(3.32\pm0.06)\times10^{-3}$ ; the small excess over $2\,\mathrm{Re}\,\varepsilon$ is a real, tiny CPT-allowed correction, but the leading number is right. So from two lifetimes and a mass splitting we predicted $\phi_{SW}=43.5^\circ$ , $\mathrm{Re}\,\varepsilon\approx\mathrm{Im}\,\varepsilon\approx1.6\times10^{-3}$ , and $\delta_L\approx3.2\times10^{-3}$ .

2 · Extracting |ε| from the Cronin–Fitch rate

The 1964 experiment saw $K_L\to\pi^+\pi^-$ at a branching ratio $\mathrm{BR}(K_L\to\pi^+\pi^-)\approx2.0\times10^{-3}$ . Using $\mathrm{BR}(K_S\to\pi^+\pi^-)=0.692$ and the lifetime ratio $\tau_L/\tau_S=571$ , recover the amplitude ratio $|\eta_{+-}|$ and hence $|\varepsilon|$ .

Step 1. Relate the two rates through $|\eta_{+-}|^2=\Gamma(K_L\to\pi^+\pi^-)/\Gamma(K_S\to\pi^+\pi^-)$ . Convert to branching ratios by multiplying and dividing by the total widths: $\mathrm{BR}(K_L\to\pi^+\pi^-)=|\eta_{+-}|^2\,\mathrm{BR}(K_S\to\pi^+\pi^-)\,(\tau_L/\tau_S)$ , because $\Gamma_S/\Gamma_L=\tau_L/\tau_S$ .

Step 2. Solve for the amplitude ratio: $|\eta_{+-}|^2=\dfrac{\mathrm{BR}(K_L\to\pi^+\pi^-)}{\mathrm{BR}(K_S\to\pi^+\pi^-)\,(\tau_L/\tau_S)}=\dfrac{2.0\times10^{-3}}{0.692\times571}=\dfrac{2.0\times10^{-3}}{395}=5.06\times10^{-6}$ .

Step 3. Take the root: $|\eta_{+-}|=\sqrt{5.06\times10^{-6}}=\mathbf{2.25\times10^{-3}}$ . Since direct CP violation is a per-mille correction on top, $|\varepsilon|\approx|\eta_{+-}|\approx\mathbf{2.2\times10^{-3}}$ , in agreement with the world average $|\varepsilon|=2.228\times10^{-3}$ .

Step 4. Sanity-check the beam geometry that made the measurement possible: the $K_S$ flight length is $c\tau_S=(3.0\times10^8)(0.8954\times10^{-10})=2.68$ cm, while $c\tau_L=15.3$ m. A detector even a metre downstream sees the $K_S$ population suppressed by $e^{-100}$ — an effectively pure $K_L$ beam — so every two-pion event is a CP-violating $K_L$ decay, not $K_S$ contamination. The rare $2\times10^{-3}$ channel yields $|\varepsilon|\approx2.2\times10^{-3}$ , the amount of CP-even $K_1$ hiding inside the long-lived kaon.

Physics of the portrait

CP violation was not predicted — it was stumbled upon by two experimenters who set out to test CP conservation and were startled to find it broken. The portrait is of the man who, with Val Fitch, turned a null-result experiment into one of the great surprises of twentieth-century physics.

Origins · 1 minJames Cronin and how this was discovered

James Cronin

1931–2016 · the 1964 discovery of CP violation in neutral kaons with Val Fitch

James Watson Cronin was a Chicago-trained experimentalist working at Princeton and Brookhaven when he and Val Fitch designed an experiment that almost everyone expected to see nothing. By 1964 the community had absorbed the shock of parity (P) violation (1957) and consoled itself with the belief that the combined operation CP was still an exact symmetry — so a pure long-lived $K_L$ beam, being CP-odd, must be forbidden from decaying to the CP-even two-pion state. Cronin, Fitch, and their students James Christenson and René Turlay built a two-arm spectrometer at the Brookhaven AGS, sent a neutral beam 17 metres downstream so that every $K_S$ had decayed away, and looked for the CP-forbidden $K_L\to\pi^+\pi^-$ . They found it: about 45 events in 22700 decays, a branching ratio near $2\times10^{-3}$ , unmistakably peaked at the kaon mass and in the beam direction. The result was so unexpected that they spent months hunting for a mundane explanation — regeneration in the helium bag, an overlooked three-body mode — before publishing the two-page Physical Review Letters paper that changed physics. Its consequences were enormous: CP violation is one of Sakharov's three conditions for a universe made of matter, and its need for a complex phase drove Kobayashi and Maskawa to predict a third generation of quarks nearly a decade before the bottom quark was found. Cronin and Fitch shared the 1980 Nobel Prize. Cronin later turned to the highest-energy cosmic rays, co-founding the Pierre Auger Observatory — but it is the small, stubborn $2\times10^{-3}$ that carries his name.

History

1955

— Gell-Mann and Pais predict two distinct neutral-kaon lifetimes from particle–antiparticle mixing

1956

— the long-lived neutral kaon is found at Brookhaven, as predicted

1964

— Christenson, Cronin, Fitch and Turlay observe Kʟ → π⁺π⁻, discovering CP violation

1973

— Kobayashi and Maskawa show that three quark generations can accommodate CP violation

1999

— NA48 and KTeV establish direct CP violation, Re(ε′/ε) ≠ 0

Going further

Texts. Thomson, Modern Particle Physics, §14.5 (neutral kaons, CPLEAR, and CP violation), Eqs. (14.35)–(14.51) · Griffiths, Introduction to Elementary Particles, 2nd ed., §4.4 and §12.3 · Perkins, Introduction to High Energy Physics, 4th ed., Ch. 9 · Halzen & Martin, Quarks and Leptons, §12.4 · Bigi & Sanda, CP Violation (the comprehensive monograph) · Branco, Lavoura & Silva, CP Violation.

Provenance. The two-state effective Hamiltonian and the mass eigenstates $K_S,K_L$ are recalled from aq-pp-neutral-kaon-system; the $O(1)$ beating that this article carefully separates from CP violation is owned by aq-pp-strangeness-oscillations; the discrete symmetries C, P, CP, T and their operator algebra come from aq-pp-symmetries-in-qm and aq-pp-parity-violation. A verifier should confirm: (1) diagonalizing $\mathbf M-\tfrac{i}{2}\mathbf\Gamma$ with equal diagonal gives equation (2) and $\varepsilon\to0$ when $H_{12}=H_{21}$ ; (2) $\phi_{SW}=\arctan(2\Delta m/\Delta\Gamma)=43.5^\circ$ from $\Delta m=3.484\times10^{-15}$ GeV and $\tau_S=0.8954\times10^{-10}$ s; (3) $\mathrm{Re}\,\varepsilon\approx\mathrm{Im}\,\varepsilon\approx1.6\times10^{-3}$ and $\delta_L\approx3.2\times10^{-3}$ from $|\varepsilon|=2.228\times10^{-3}$ ; (4) $|\eta_{+-}|=2.25\times10^{-3}$ from $\mathrm{BR}(K_L\to\pi^+\pi^-)=2.0\times10^{-3}$ , $\mathrm{BR}(K_S\to\pi^+\pi^-)=0.692$ , $\tau_L/\tau_S=571$ . All PDG values ( $|\varepsilon|$ , $\phi_{+-}$ , $\Delta m$ , $\tau_S$ , $\tau_L$ , $\mathrm{Re}(\varepsilon'/\varepsilon)$ ) are standard tabulated numbers.

Prerequisites. The Neutral Kaon System · Strangeness Oscillations · Symmetries in Quantum Mechanics · Parity Violation. Leads to. CP Violation in the Standard Model · The CKM Matrix · B-Meson Physics · The Matter–Antimatter Asymmetry.

Ask the tutor: Ask me to walk through why the phase of $\varepsilon$ lands on $43.5^\circ$ from just two kaon lifetimes and a mass splitting, or how CPLEAR turned that phase into a direct sighting of time-reversal violation.

What is the difference between K⁰/K̄⁰, K₁/K₂, and Kꜱ/Kʟ?

They are three different bases for the same two-dimensional space, each natural to a different interaction. $K^0$ and $\bar K^0$ are flavour (strangeness) eigenstates — what the strong interaction produces and what a semileptonic decay tags. $K_1$ and $K_2$ are CP eigenstates — $K_1$ is CP-even and decays to two pions, $K_2$ is CP-odd and decays to three; these are what the weak decays would sort into if CP were exact. $K_S$ and $K_L$ are the mass/lifetime eigenstates — what actually propagates through space with a definite mass and exponential lifetime. If CP were conserved the last two bases would coincide ( $K_S=K_1$ , $K_L=K_2$ ); CP violation is precisely the statement that they differ by the small parameter $\varepsilon$ .

What is the difference between indirect and direct CP violation?

Indirect CP violation ( $\varepsilon$ ) lives in the mixing: the mass eigenstate $K_L$ is not a pure CP eigenstate, so it contains a little CP-even $K_1$ that decays to two pions. The CP violation happens in the $K^0\!\leftrightarrow\!\bar K^0$ transition, before the decay. Direct CP violation ( $\varepsilon'$ ) lives in the decay amplitude itself: the rate for a given weak vertex differs for particle and antiparticle because two interfering amplitudes (the $I=0$ and $I=2$ two-pion channels) carry different weak phases. Indirect CP violation was found in 1964; direct CP violation is a thousand times smaller and was only established in 1999. Their ratio, $\mathrm{Re}(\varepsilon'/\varepsilon)\approx1.7\times10^{-3}$ , is what distinguishes the real Standard Model from Wolfenstein's superweak model, in which $\varepsilon'$ would be exactly zero.

Why is the phase of ε so close to 45 degrees?

Because that phase is not set by CP-violating physics at all — it is $\phi_{SW}=\arctan(2\Delta m/\Delta\Gamma)$ , fixed entirely by the mass difference $\Delta m$ and the width difference $\Delta\Gamma$ , both of which are perfectly CP-conserving. It happens that in the neutral kaon $2\Delta m\approx\Delta\Gamma$ (a numerical coincidence with no known deeper cause), so the arctangent is close to $\arctan(1)=45^\circ$ . The precise value is $43.5^\circ$ . The upshot is that the phase of $\varepsilon$ carries essentially no information about how much CP is violated; only its magnitude does. Measuring the phase and finding it at $\phi_{SW}$ is instead a consistency check — and a test that CP violation really does reside dominantly in $\Delta S=2$ mixing.

Does kaon CP violation explain why the universe is made of matter?

It is the right kind of effect but nowhere near large enough. Sakharov showed in 1967 that generating a matter–antimatter asymmetry from a symmetric early universe requires three ingredients: baryon-number violation, C and CP violation, and a departure from thermal equilibrium. CP violation in kaons was the first laboratory demonstration that the second condition is met in nature, which is historically decisive. But quantitatively, the CP violation available from the CKM phase — the same phase that gives $\varepsilon$ — falls short of the observed baryon-to-photon ratio by roughly ten orders of magnitude. So kaon CP violation proves the mechanism exists but cannot be the whole story; a further, as-yet-undiscovered source of CP violation (perhaps in the neutrino or Higgs sectors) is needed. That is a major motivation for the rare-kaon and neutrino experiments running today.

If CP is violated, is CPT also broken?

No — and the two facts together are what make CP violation so interesting. The CPT theorem, a near-inviolable consequence of Lorentz invariance and locality in quantum field theory, guarantees that the combined operation CPT is an exact symmetry: particle and antiparticle must have identical mass and total lifetime. CPT invariance is exactly what let us set the diagonal entries of the effective Hamiltonian equal in the derivation. But CPT being exact means that if CP is violated, T (time reversal) must be violated by precisely the compensating amount. The CPLEAR experiment observed this T violation directly, showing that the rate for $\bar K^0\to K^0$ differs from $K^0\to\bar K^0$ — a genuine arrow of time at the level of fundamental interactions. So CP violation implies T violation, with CPT holding throughout.

What is kaon regeneration, and is it CP violation?

It is not CP violation — it is a beautiful but separate effect, and confusing it with CP violation is a classic trap that even Cronin and Fitch had to rule out. When a $K_L$ beam passes through matter, the $K^0$ and $\bar K^0$ components scatter differently off nuclei (the $\bar K^0$ has more open channels, like $\bar K^0 p\to\Lambda\pi$ ). This unequal attenuation re-mixes the beam and regenerates a $K_S$ component, which then decays to two pions with full strength. So two-pion decays downstream of a slab of material are usually just regenerated $K_S$ , not CP-violating $K_L$ decays. The 1964 experiment's central achievement was arranging a long vacuum (helium) flight path so that regeneration was negligible, proving the two-pion events were genuine $K_L$ decays. Regeneration is coherent, forward-peaked, and material-dependent; CP violation is none of those — which is how you tell them apart.