Experiment: CP Violation

Tests Equation (105.46), Equation (105.47) and Phenomenon 103.31. Assuming Theorems 105.34, 105.36 and 105.38.

After parity fell in 1957 (Experiment: Parity Violation) the combined operation \(CP\) was expected to be exact, and the neutral kaon system was built around that expectation: the long-lived state was supposed to be \(CP\)-odd and therefore forbidden to decay to two pions. This chapter records the two-pion decay that Christenson, Cronin, Fitch and Turlay found at the Brookhaven alternating-gradient synchrotron in a branching ratio of about \(2\times 10^{-3}\) [Christenson:1964], and everything measured since: direct \(CP\) violation in kaon decay [Fanti:1999] [AlaviHarati:1999], the large asymmetries of the \(B\) mesons [Aubert:2001] [Abe:2001], and the small one in charm [Aaij:2019]. It sits after the weak-interaction and flavour chapters (Weak Interactions and Flavour Physics and Neutrinos) because it is the measurement that forced a third quark generation [Kobayashi:1973], and beside Discrete Symmetries and CPT, whose \(CPT\) theorem is what makes \(CP\) violation and \(T\) violation the same statement.

The division of labour with those two chapters is worth stating at the outset, because it decides what is and is not repeated here. The two-state formalism of the neutral kaon, the Wigner–Weisskopf elimination of the decay continuum, the definition of \(\epsilon\) and \(\epsilon'\), the double ratio, the Jarlskog invariant and the Sakharov conditions are all derived in Sections 105.3 and 105.6; the flavour structure that ties them to the quark mixing matrix is Section 103.5 and Section 103.4.3. What follows is the experimental side: what the beams and the detectors physically were, how each number was extracted, what background had to be excluded, and what the numbers are with their uncertainties in SI units. Where a result of the theory chapters is needed it is cited by its equation number rather than re-derived, and where the experimental argument requires a derivation of its own — the purity of the beam, the regeneration background, the geometry of the spectrometer, the extraction of \(\sin2\beta\) from a vertex separation — it is carried out here.

Its closing sections take the result out of particle physics and into cosmology: \(CP\) violation is one of Sakharov's three conditions for a baryon asymmetry [Sakharov:1967], and the honest arithmetic is that the Cabibbo–Kobayashi–Maskawa phase is many orders of magnitude too small to produce the asymmetry observed in the sky [Gavela:1994] — which is stated here as an open problem, not papered over.

Historical context and the prediction under test

The neutral kaon system

The experiment of 1964 was possible only because an unusual physical system had been identified nine years earlier, and the identification was made on paper before any of it was seen.

Gell-Mann and Pais asked what happens to a neutral particle that is not its own antiparticle when the interaction that produces it conserves a quantum number that the interaction destroying it does not [GellMann:1955]. The strong interaction makes the neutral kaon in a state of definite strangeness — \(\ket{K^{0}}\) with \(S=+1\), \(\ket{\bar{K}^{0}}\) with \(S=-1\) — and conserves that label, so the two are distinct particles. The weak interaction does not conserve it, and therefore connects them. The states of definite mass and lifetime are then not the strangeness eigenstates at all but the combinations that diagonalize the weak effective Hamiltonian, which under an exact \(CP\) symmetry are the \(CP\) eigenstates \(\ket{K_{1}}\) and \(\ket{K_{2}}\) of Equation (105.46).

The consequence Gell-Mann and Pais drew was a prediction of a new particle. Two pions have \(CP=+1\) and three neutral pions \(CP=-1\) (Equation (105.47)), so if \(CP\) is exact the \(CP\)-even combination may decay to \(\pi\pi\) and the \(CP\)-odd one may not. Since three pions barely fit inside the kaon mass, the second must be long-lived. Numerically, with the masses of Table 111.5, the energy released into \(3\pi^{0}\) is

against \(Q_{\pi\pi}=218.47\,\mathrm{MeV}\) for \(\pi^{+}\pi^{-}\). The non-relativistic phase-space volume for \(n\) bodies sharing a kinetic energy release \(Q\) grows as \(Q^{(3n-5)/2}\): the momentum delta function leaves \(3(n-1)\) independent components, the energy constraint restricts them to a sphere of radius proportional to \(\sqrt{Q}\) in that space, and the surface measure of that sphere divided by the gradient of the energy supplies \(R^{3(n-1)-1}/R\propto Q^{(3n-5)/2}\). Two bodies therefore give \(Q^{1/2}\) and three bodies \(Q^{2}\), so the three-body channel is far more strongly throttled near its own threshold, and it has less than half the energy release to work with. That is what produces the observed factor of several hundred in lifetime. Lande, Booth, Impeduglia, Lederman and Chinowsky found the long-lived partner at the Brookhaven Cosmotron the following year [Lande:1956], with a lifetime some six hundred times that of the known short-lived \(V\) particle. The modern values,

give a ratio \(\tau_{L}/\tau_{S}=571\) [Navas:2024].

The second consequence is the one that makes the system an instrument rather than a curiosity. Because the two mass eigenstates are superpositions of the two strangeness eigenstates, a beam prepared as \(K^{0}\) evolves into a mixture whose strangeness content oscillates, and the oscillation frequency is the mass difference divided by \(\hbar\). This is Equation (105.61). The measured splitting is

corresponding to a mass difference of \(\Delta m_{K}=6.211\times 10^{-42}\,\mathrm{kg}\), which is \(\Delta m_{K}/m_{K}=7.0\times10^{-15}\) of the kaon's own mass (Equations (105.60) and (105.63)). No spectrometer resolves a fractional mass difference of \(10^{-14}\); an interferometer whose fringe is a phase accumulated over a flight path of metres does, and that is what the neutral kaon is. Everything in this chapter rests on that fact: the quantities measured are interference terms, not energies.

Regeneration

The mundane explanation that had to be excluded is regeneration, and because it is simultaneously the cleanest demonstration of quantum superposition in the subject and the principal background of the 1964 experiment, it is derived here rather than assumed.

Pais and Piccioni observed that a long-lived beam traversing matter emerges with a short-lived component [Pais:1955]. The reason is that the strong interaction sees strangeness and not \(CP\): a \(\bar{K}^{0}\) can convert a nucleon into a hyperon, a \(K^{0}\) cannot, so the two forward scattering amplitudes differ. A medium that attenuates or phase-shifts the two strangeness components differently is, in the \(CP\) basis, a medium that rotates \(\ket{K_{L}}\) into \(\ket{K_{S}}\).

Two features of Equation (111.4) decide the design of the experiment. The first is that \(\rho\) is proportional to \(N\), so a dilute medium regenerates weakly and a vacuum not at all. The second, less obvious, is that the regenerated amplitude is coherent with the incident beam: it is forward, it carries the beam's momentum direction, and its decay to \(\pi^{+}\pi^{-}\) is therefore collinear with the beam — exactly like the signal. Coherent regeneration cannot be removed by an angular cut. Only its incoherent relatives can: diffractive regeneration off a single nucleus, and inelastic production, both leave the reconstructed parent momentum at a measurable angle to the beam, and those are what the collinearity requirement of Section 111.3 eliminates. The coherent part has to be made small, and made small by construction; that is what the helium bag is for, and Proposition 111.5 puts a number on it.

CP as the symmetry that was supposed to survive

The parity experiments of 1957 (Experiment: Parity Violation) showed that the weak interaction distinguishes left from right maximally: in the beta decay of polarized cobalt the electrons come out preferentially opposite to the nuclear spin [Wu:1957], and in the pion–muon–electron chain the muon is fully polarized [Garwin:1957]. Charge conjugation fares no better, and for the same reason — applied to the left-handed neutrino it produces a left-handed antineutrino, which does not exist.

Landau's response was to propose that the true symmetry is the product [Landau:1957]. It is an economical suggestion: \(CP\) maps the left-handed neutrino onto the right-handed antineutrino, which does exist, so maximal violation of \(C\) and of \(P\) separately is exactly compatible with an exact \(CP\). Together with the \(CPT\) theorem of local Lorentz-invariant quantum field theory [Luders:1957] — proved as Theorem 105.34 — the proposal has three testable consequences, and it is worth listing them because the experiment tested the first and the other two survived:

1. \(K_{L}\to\pi\pi\) is strictly forbidden, the long-lived state being the \(CP\)-odd combination and the two-pion state \(CP\)-even.

2. A particle and its antiparticle have exactly equal masses and exactly equal total lifetimes (Theorem 105.36, Theorem 105.38), whether or not \(CP\) holds; these follow from \(CPT\) alone.

3. Any \(CP\) violation is, given \(CPT\), a \(T\) violation of the same size, so a direct comparison of a process with its motion reverse must show it too — which is the programme of Section 111.7.4.

The observable and what a measurement must deliver

The prediction under test is item (i), and stating it as an experimental requirement fixes the whole design.

A neutral kaon beam is prepared far enough downstream of its production target that the \(K_{S}\) component has decayed to an utterly negligible level. Any \(\pi^{+}\pi^{-}\) pair observed to emerge from the surviving beam must then satisfy two independent conditions if it is a genuine two-body decay of a beam particle: the invariant mass of the pair, computed under the pion hypothesis, must equal the kaon mass; and the vector sum of the two pion momenta — the reconstructed parent momentum — must be collinear with the beam. A measurement must therefore deliver

• a beam whose \(K_{S}\) contamination is calculably negligible,

• a decay region containing so little material that coherent regeneration cannot fake the signal,

• a spectrometer whose mass resolution separates \(\pi^{+}\pi^{-}\) from the three-body decays \(\pi e\nu\), \(\pi\mu\nu\) and \(\pi^{+}\pi^{-}\pi^{0}\), which are \(10^{3}\) times commoner,

• an angular resolution fine enough that a genuine two-body decay piles up in one bin of \(\cos\theta\) while three-body decays spread over many, and

• a normalization sample — the charged decays of the same beam in the same apparatus — so that the answer is a ratio in which the flux and most of the acceptance cancel.

Each of the five is met by a specific piece of the apparatus, and the next two sections take them in that order.

Apparatus

The Brookhaven beam and the decay volume

The source was the alternating-gradient synchrotron at Brookhaven, delivering protons of \(30\,\mathrm{GeV}\) — a kinetic energy of \(4.8\times 10^{-9}\,\mathrm{J}\) per proton — onto an internal target. A neutral beam was taken off at \(30\,^\circ\) to the circulating protons, passed through a collimator, swept clear of charged particles by a magnet, and re-collimated; the detector viewed the beam \(17.4\,\mathrm{m}\) (57 feet in the units of the original report) from the target [Christenson:1964]. The momentum spectrum of the surviving neutral kaons peaked near \(1.1\,\mathrm{GeV}/c\), which is the value used for every kinematic estimate below.

That distance is the first design requirement, and it is met with an enormous margin.

The decay volume itself was a bag of helium at atmospheric pressure. The reason is Proposition 111.3: coherent regeneration scales as the number density of scatterers times the difference of their forward amplitudes, and both factors are minimized by a light, dilute gas. A true vacuum over a volume of several cubic metres was impractical in 1964; helium was the practical approximation to one, and the following estimate is what justifies the choice.

The two-arm spectrometer

The detector was two symmetric arms viewing the helium bag, each consisting of a bending magnet with a spark chamber before it and a spark chamber after it, followed by a water Cherenkov counter and a scintillation counter [Christenson:1964]. Each arm measured one charged particle: its direction from the two spark chambers, its momentum from the deflection between them, and its identity, loosely, from the Cherenkov signal. The two arms together therefore delivered the two three-vectors \(\vect{p}_{1}\) and \(\vect{p}_{2}\) from which every quantity in the analysis is built.

The arms sat at about \(22\,^\circ\) to the beam on either side, and that angle is not arbitrary.

The two remaining pieces of the apparatus are there to reject the common decays. The Cherenkov counters used water, of refractive index \(n=1.33\), whose threshold velocity is \(\beta_{\mathrm{th}}=1/n=0.752\); for a pion this is a momentum of

so essentially every pion in the accepted momentum band radiates while the counters remain useful in the trigger. The scintillators supplied the fast timing that defined the coincidence.

The resolutions matter more than any of this, because the entire argument of the experiment is a two-dimensional separation. The spectrometer reconstructed the invariant mass of the pair to about \(4\,\mathrm{MeV}/c^{2}\) and the direction of the reconstructed parent to a few milliradians; the signal bin used in the analysis, \(\cos\theta>0.9999\), corresponds to \(\theta<14.1\,\mathrm{mrad}\) (Table 111.1).

Procedure

Trigger and event recording

An event was accepted when the scintillation and Cherenkov counters of both arms fired in coincidence, indicating one charged particle in each arm within the resolving time of the counters. The coincidence pulsed the spark chambers to a few tens of kilovolts, so that the ionization trails left by the two particles broke down into visible sparks, and triggered a camera which photographed the chambers [Christenson:1964]. The recorded object was therefore a photograph, and the analysis a measurement of positions on film, followed by a reconstruction.

This is worth stating plainly because it bounds the sample size. The experiment could not record a large fraction of the beam; it recorded about twenty-two thousand charged decays in total, and the signal it found is forty-five events. Everything about the argument — the use of a two-dimensional discriminant rather than a single cut, the reliance on sidebands for the background, the insistence on a control sample — follows from having a sample of that size and no possibility of enlarging it quickly.

Reconstruction in the mass-angle plane

From the two measured momenta the analysis formed two numbers. The first is the invariant mass of the pair under the hypothesis that both particles are pions,

The second is the angle \(\theta\) between the reconstructed parent momentum \(\vect{p}_{1}+\vect{p}_{2}\) and the beam direction, reported as \(\cos\theta\).

The discriminating power lies in requiring both at once, and the reason is a kinematic identity rather than a statistical convention.

For the semileptonic modes there is a second, independent handicap. Because the electron or muon is assigned the pion mass in Equation (111.6), its energy is overestimated, and the reconstructed mass is pushed further from \(m_{K}\) rather than towards it — so the misassignment cannot manufacture a signal, only remove background from the peak.

Backgrounds and the control samples

Three backgrounds had to be dealt with, and each was measured rather than modelled.

Three-body decays under the peak. The overwhelming majority of the recorded events are \(\pi e\nu\), \(\pi\mu\nu\) and \(3\pi\), which populate the whole \((m_{\pi\pi},\cos\theta)\) plane. Their density in the signal region was fixed empirically from mass sidebands: the \(\cos\theta\) distribution was formed in the mass intervals immediately below and above the kaon mass, where no two-body signal can exist, and interpolated across. The interpolation is safe because the background varies smoothly in \(m_{\pi\pi}\) on the scale of the mass resolution while the signal does not.

Incoherent regeneration. Diffractive and inelastic regeneration in the residual material produce a \(K_{S}\) whose momentum is not collinear with the beam, and are therefore removed by the same angular requirement that defines the signal, as Proposition 111.7 makes explicit.

Coherent regeneration. This one is collinear and cannot be cut away, so it was measured. Slabs of material were deliberately placed in the beam, producing a large and unmistakable coherent \(K_{S}\to\pi^{+}\pi^{-}\) signal at \(m_{\pi\pi}=m_{K}\) and \(\cos\theta=1\); that sample served two purposes at once. It calibrated the mass and angular resolution of the spectrometer on real two-body decays of the right mass and the right momentum spectrum — the ideal control, since it is the signal in every respect except its origin — and it fixed the constant of proportionality in Equation (111.4) for the materials involved, from which the residual regeneration in helium follows by scaling. The observed forward peak did not scale with the amount of material in the beam, which is the direct experimental statement that it is not regeneration.

Finally, the normalization. The denominator of the published ratio is the number of all charged decays of the same beam recorded in the same apparatus. Taking a ratio to that sample cancels the beam flux, the live time, the film-scanning efficiency and a large part of the geometric acceptance, which is why the result could be quoted to twenty per cent from forty-five events.

Observations and data

The forward peak at the kaon mass

In the mass interval containing the kaon mass, and only there, the \(\cos\theta\) distribution shows a sharp excess in the single bin \(\cos\theta>0.9999\). In the two adjacent mass intervals, ten \(\mathrm{MeV}\) below and above, the same bin shows nothing above the smooth background. The excess amounts to \(45\pm9\) events, and the sample of all charged decays recorded in the same apparatus numbers \(22\,700\) [Christenson:1964].

The 1964 numbers and their modern successors are collected in Table 111.2. One comparison in that table deserves comment, because it is the only place where the original measurement can be checked against a modern one and the two are not quoted in the same variable. The published ratio has “all charged decays” in its denominator, whereas the modern branching fraction has all decays. The charged modes of the \(K_{L}\) — \(\pi^{\pm}e^{\mp}\nu\), \(\pi^{\pm}\mu^{\mp}\nu\), \(\pi^{+}\pi^{-}\pi^{0}\) and \(\pi^{+}\pi^{-}\) itself — account for \(0.8033\pm0.0014\) of the total width [Navas:2024], the quoted uncertainty being the quadrature combination of the uncertainties on those four branching fractions. It is negligible beside the twenty per cent statistical error on the 1964 ratio, so

against the present \((1.967\pm0.010)\times10^{-3}\) [Navas:2024]: agreement at \(1.2\) standard deviations. The original measurement was correct, and it was correct to twenty per cent on forty-five events.

The branching ratio and the parameter epsilon

The branching fraction is not the natural variable. What the theory predicts is an amplitude ratio, because the two-pion final state is reached from the \(CP\)-even admixture in the long-lived state, and the natural comparison is with the same final state reached from the short-lived one. The definitions are Equation (105.65):

the second form following because a branching fraction is a partial width divided by a total width and the total widths are \(\hbar/\tau\). Inserting \(\mathcal{B}(K_{L}\to\pi^{+}\pi^{-})=1.967\times10^{-3}\), \(\mathcal{B}(K_{S}\to\pi^{+}\pi^{-})=0.6920\) and the lifetimes of Equation (111.2) gives

that is \(\abs{\eta_{+-}}=2.23\times10^{-3}\), which is the measured value in Table 111.3. The arithmetic is worth seeing once: the tiny branching fraction and the enormous lifetime ratio very nearly cancel, and what survives is a number of order \(10^{-3}\) that has nothing to do with either.

The phase of \(\eta_{+-}\) is measured independently, by observing the interference between the \(K_{S}\) and \(K_{L}\) amplitudes in a beam containing both, and it comes out at \(43.51\pm0.05^\circ\). That value is predicted, not fitted: Equation (105.59) gives \(\arg\epsilon=\arctan\left[2\Delta m_{K}c^{2}\tau_{S}/\hbar\right]\), and with Equation (111.3) and Equation (111.2),

where \(\hbar/\tau_{S}=7.3510\times 10^{-6}\,\mathrm{eV}\) is the \(K_{S}\) full width. Equivalently, dividing numerator and denominator by two, \(\arctan\left[\Delta m_{K}c^{2}/(\hbar/2\tau_{S})\right]\) with the half width \(3.6755\times 10^{-6}\,\mathrm{eV}\): the factor of two in Equation (105.59) is the one that turns the half width into the full width, and must not be applied twice.

The agreement of a measured phase with a number built out of a mass difference and a lifetime is one of the sharper consistency checks in the subject, and it is what licenses the identification of \(\eta_{+-}\) with the mixing parameter \(\epsilon\) of Equation (105.54) up to the small direct term of Section 111.6.

Charge asymmetry in semileptonic decay

The two-pion decay establishes that \(CP\) is violated; it does not by itself give a sign. The observable that does is the charge asymmetry of the semileptonic decays of the same long-lived state, and it deserves attention out of all proportion to its size.

The convention-free character of the statement is worth spelling out, since it is the one experimental result in this book that changes what can be said about the words used to describe the world. Before 1964 the labels “matter” and “antimatter” were conventional: \(C\) violation and \(P\) violation alone leave open the possibility that what we call matter is simply the mirror image of what a distant correspondent calls matter, because a mirror reflection can be compensated by an exchange of particles for antiparticles. Equation (111.11) closes that possibility. The instruction “prepare a long-lived neutral kaon beam, count the charge of the emitted lepton, and call the commoner sign positive” can be transmitted in words, needs no shared sample, and gives the same answer everywhere.

The measured CP-violating observables

The remaining sections of the chapter analyse one measurement at a time. Table 111.4 collects them all first, with the year and the source of each, so that the numbers can be read against one another; Table 111.5 gives the masses used throughout, and Table 111.6 the three mixing frequencies in SI. Every value is quoted as its source quotes it, with the uncertainties separated into statistical and systematic where the source separates them.

Interpretation

Indirect versus direct violation

The 1964 result establishes that the weak interaction does not commute with \(CP\). It does not say where the violation sits, and there are two places available.

It may sit in the state. If the long-lived eigenvector of the effective Hamiltonian is not the \(CP\)-odd combination \(\ket{K_{2}}\) but carries a small admixture \(\epsilon\ket{K_{1}}\) of the \(CP\)-even one, then the two-pion decay proceeds entirely from that admixture, using a decay amplitude that itself respects \(CP\) perfectly. This is \(CP\) violation in the mixing, and it is a single complex number \(\epsilon\) for the whole system.

Or it may sit in the decay. If the amplitudes for \(K^{0}\to\pi\pi\) and \(\bar{K}^{0}\to\pi\pi\) differ in more than the strong rescattering phase they are obliged to share, then even an exact \(CP\) eigenstate would decay asymmetrically. This is \(CP\) violation in the decay, or direct \(CP\) violation, and by Equation (105.72) it is measured by a second complex number \(\epsilon'\).

The distinction is not academic, and for thirty-five years it was the open question of the subject, because Wolfenstein had immediately supplied a hypothesis in which the second number is exactly zero [Wolfenstein:1964]. Suppose there exists a new interaction, enormously weaker than the weak interaction, which changes strangeness by two units and violates \(CP\) maximally. Acting once, it contributes to the off-diagonal element \(M_{12}\) of the mass matrix (Equation (105.50)) and so to \(\epsilon\); it never contributes to a decay, because the decays are \(\Delta S=1\) transitions. Such a “superweak” interaction reproduces every kaon measurement then available with one free parameter and predicts

and the last of these is the phase relation verified numerically in Equation (111.10). That agreement is worth pausing on: the superweak hypothesis was not a desperate one, it fitted the phase correctly, and the experiment that killed it had to measure a difference between \(\eta_{00}\) and \(\eta_{+-}\) of a few parts in a thousand of a quantity that is itself a few parts in a thousand. That is a measurement at the \(10^{-6}\) level in the ratio of rates, and it is why it took thirty-five years. It is Section 111.6.

The CKM matrix and the Jarlskog invariant

The alternative to a new interaction is that the violation is already present in the interaction we have, and this is what Kobayashi and Maskawa proposed [Kobayashi:1973]. Their argument is arithmetic and is proved as Theorem 105.25: a unitary \(N\times N\) quark mixing matrix, after all the rephasings of the quark fields have been used up, retains \(N(N-1)/2\) real angles and \((N-1)(N-2)/2\) phases. For \(N=2\) — the Cabibbo case [Cabibbo:1963], one angle and no phase — the matrix can be made real, and a real Lagrangian conserves \(CP\). For \(N=3\) there is exactly one phase, and it cannot be rotated away.

The status of this as a prediction should not be softened. In 1973 three quarks were known. The measurement being explained was a branching ratio of \(2\times10^{-3}\) in a kaon. The inference was that three more quarks exist. The charm quark appeared the following year (Section 103.2.3), the bottom quark in 1977 and the top quark in 1995, and the Nobel award of 2008 to Kobayashi and Maskawa recognized an argument, not a discovery of a particle.

Two further pieces of apparatus, in the theoretical sense, make the phase measurable. The first is a parametrization in which the observed hierarchy of the mixing angles is manifest: Wolfenstein's expansion in \(\lambda=\sin\theta_{C}\) [Wolfenstein:1983], whose four parameters, measured, are Equation (103.24). The second is a rephasing-invariant measure, since the phase \(\delta\) itself depends on the convention chosen for the quark fields and is therefore not an observable. Jarlskog's quantity [Jarlskog:1985], defined at Definition 105.26, is the imaginary part of a quartet of matrix elements and is invariant; its measured value is Equation (105.78),

Geometrically \(J/2\) is the area of every unitarity triangle (Proposition 103.30), so the single statement “\(CP\) is violated” is the single statement “the triangle is not degenerate”. Numerically, \(J\) is small not because the phase is small — it is of order unity, \(\delta=65.70^\circ\) by Equation (103.25) — but because the mixing angles are. That distinction returns with consequences in Section 111.8.2.

Direct CP violation in the kaon system

NA31 and the first evidence

The observable that separates the two hypotheses of Section 111.5.1 is the double ratio of the four two-pion widths, and the reason it and not something simpler is that every quantity common to the four channels cancels in it: the kaon flux, the solid angle, most of the reconstruction efficiency. What does not cancel is the difference between reconstructing two charged tracks and reconstructing four photons, and that difference is the systematic error of every measurement in this section.

The first experiment to see a non-zero value was NA31 at the CERN Super Proton Synchrotron [Burkhardt:1988]. Protons of \(450\,\mathrm{GeV}\) struck a beryllium target; the resulting neutral beam was allowed to run down a long evacuated decay region for the \(K_{L}\) measurement, and for the \(K_{S}\) measurement a second target mounted on a movable train was brought close to the detector, so that the same apparatus saw \(K_{S}\) decays distributed over the same fiducial region. Alternating the two configurations, rather than running them at once, is what makes the acceptance cancel.

NA31 carried no magnetic spectrometer. Its detector was a liquid-argon and lead sampling calorimeter for the photons of \(\pi^{0}\pi^{0}\), an iron–scintillator hadron calorimeter, and wire chambers for the charged mode; the charged pion momenta were never measured individually, and the charged decays were identified from the calorimetric energy and the transverse position. The result,

is three standard deviations from zero and was the first evidence for direct \(CP\) violation.

It was not accepted, and the reason is that the contemporaneous Fermilab experiment E731, using a different technique — two parallel \(K_{L}\) beams with a regenerator placed alternately in one of them, so that \(K_{S}\) and \(K_{L}\) decays were recorded simultaneously — obtained a value consistent with zero, of order \(7\times10^{-4}\) with an uncertainty of similar size. The two results were about two standard deviations apart, which is not a crisis in itself; what made the decade that followed uncomfortable is that they disagreed about a qualitative question, the existence or not of \(\epsilon'\), and therefore about whether the superweak hypothesis had been excluded. Neither collaboration could settle it with the data in hand, and both built successors designed around the other's technique. That is the correct response to an irreconcilable pair of measurements, and it is worth recording as such.

NA48 and KTeV

Both successors reported in 1999, and both found a non-zero value.

NA48, at CERN, produced \(K_{L}\) and \(K_{S}\) simultaneously rather than alternately [Fanti:1999]. The \(K_{L}\) beam came from a target \(126\,\mathrm{m}\) upstream of the decay region. A small fraction of the protons that missed that target were deflected by a bent silicon crystal onto a second target \(6\,\mathrm{m}\) upstream, producing a \(K_{S}\) beam nearly collinear with the first. The two decay populations were then recorded in the same detector at the same time, in the same beam conditions, with the acceptance difference reduced to the small difference in the decay distributions. Distinguishing a \(K_{S}\) decay from a \(K_{L}\) decay was done by timing: an array of scintillator counters — the tagger — in the proton beam heading for the \(K_{S}\) target recorded the passage of each proton to about \(200\,\mathrm{ps}\), and a decay coincident with a tagged proton is a \(K_{S}\) decay.

The detector was a magnetic spectrometer of four drift chambers around a dipole magnet, for \(\pi^{+}\pi^{-}\), together with a liquid-krypton electromagnetic calorimeter of about \(10\,\mathrm{m}^{3}\) and some thirteen thousand cells, twenty-seven radiation lengths deep, for the four photons of \(\pi^{0}\pi^{0}\). The krypton calorimeter is the critical instrument: the neutral mode has no tracks, the kaon's decay point must be reconstructed from the photon energies and positions alone by imposing the \(\pi^{0}\) mass, and the energy resolution therefore propagates directly into the fiducial volume definition. The 1999 result was

the second uncertainty systematic and larger than the first.

KTeV, at Fermilab, was the successor of E731 and kept its technique [AlaviHarati:1999]. Protons of \(800\,\mathrm{GeV}\) on a beryllium-oxide target produced two parallel neutral beams; a regenerator — a block of scintillator, itself instrumented so that inelastic interactions in it could be vetoed — sat in one of the two and was moved from one beam to the other about once a minute, so that the regenerated \(K_{S}\) sample and the \(K_{L}\) sample were recorded simultaneously and each beam served in turn as each. The detector was a drift-chamber spectrometer around an analysing magnet and an electromagnetic calorimeter of \(3100\) pure caesium-iodide crystals, whose energy resolution below one per cent for the photons of interest is what made the neutral mode competitive. The result was

The two 1999 numbers differ by \(9.5\times10^{-4}\) against a combined uncertainty of \(8.4\times10^{-4}\), that is by \(1.1\) standard deviations, which is unremarkable. Their distances from zero are not alike: combining each experiment's two errors in quadrature, KTeV is \(6.8\) standard deviations from zero and NA48 is \(2.5\). It is KTeV that established the effect in 1999; NA48 confirmed it, at a significance that on its own would not have settled the question, and the two together left no room for zero. With the later data of both experiments the world average is Equation (105.74), \(\mathrm{Re}(\epsilon'/\epsilon)=(1.66\pm0.23)\times10^{-3}\).

The Standard Model prediction and what it is worth

The measurement excludes the superweak hypothesis by itself, and that conclusion needs no theory. Comparing the measured value with the Cabibbo–Kobayashi–Maskawa prediction is a different and much weaker statement, and the reason should be stated plainly rather than left to a citation.

The short-distance part of the calculation is under control: the \(\Delta S=1\) effective Hamiltonian is a sum of local four-quark operators with coefficients computed perturbatively, and the phase enters only through the CKM combination \(\mathrm{Im}\left(V_{td}V_{ts}^{*}\right)\). The long-distance part is not: what is needed is the matrix element of each operator between a kaon and a two-pion state at the physical kinematics, including the final-state rescattering that produces \(\delta_{0}\) and \(\delta_{2}\), and there is no expansion parameter in which that is small. Lattice calculations of these matrix elements now exist and give a value compatible with Equation (105.74) with an uncertainty several times larger, dominated by the \(I=0\) channel where the rescattering is strongest; those calculations are not cited here because this treatise's bibliography carries no entry for them, and the reader should not take the agreement on the authority of this sentence.

What is established without any of that machinery is the qualitative result: \(\epsilon'\ne0\), hence \(CP\) is violated in a \(\Delta S=1\) decay amplitude, hence no interaction acting only in the mass matrix can account for the phenomenon. That is the content of Phenomenon 111.11 and it is secure.

CP violation beyond the kaons

The B factories

If one phase in one matrix accounts for all of it, then the kaon's part in a thousand and an effect of order unity somewhere else are the same physics. The Standard Model says where to look: in \(B^{0}\to J/\psi\,K^{0}_{S}\), where the interference is between decay after mixing and decay without mixing, and no small ratio of amplitudes suppresses it.

The machines. Two asymmetric-energy electron–positron colliders were built for this measurement, PEP-II at SLAC and KEKB at KEK. Both run at a centre-of-mass energy of \(10.58\,\mathrm{GeV}\), the mass of the \(\Upsilon(4S)\), which decays to a \(B^{0}\bar{B}^{0}\) pair with almost no energy to spare — with \(m_{B^{0}}c^{2}=5279.66\,\mathrm{MeV}\) each \(B\) has a momentum \(p=\sqrt{(m_{\Upsilon}c^{2}/2)^{2}-(m_{B}c^{2})^{2}}/c =331\,\mathrm{MeV}/c\) in the \(\Upsilon\) frame. Asymmetric energies are the whole point. PEP-II collided \(9.0\,\mathrm{GeV}\) electrons with \(3.1\,\mathrm{GeV}\) positrons, KEKB \(8.0\,\mathrm{GeV}\) with \(3.5\,\mathrm{GeV}\), so that the centre of mass moves in the laboratory with

The one line behind this: for head-on beams whose energies far exceed the electron mass, the total four-momentum has energy \(E_{-}+E_{+}\) and momentum \((E_{-}-E_{+})/c\) along the electron direction, so \(\beta=(E_{-}-E_{+})/(E_{-}+E_{+})\), while \((E_{-}+E_{+})^{2}-(E_{-}-E_{+})^{2}=4E_{-}E_{+}\) is the square of the invariant centre-of-mass energy, so \(\gamma=(E_{-}+E_{+})/\sqrt{4E_{-}E_{+}}\). The product is Equation (111.18), and \(\sqrt{4E_{-}E_{+}}\) is \(10.56\,\mathrm{GeV}\) for PEP-II and \(10.58\,\mathrm{GeV}\) for KEKB, as it must be.

The detectors. BaBar and Belle are conventional collider detectors of their generation, built around the interaction point inside a \(1.5\,\mathrm{T}\) solenoid: a silicon strip vertex detector of five layers immediately around the beam pipe, a gas drift chamber for momentum measurement, a Cherenkov device for separating kaons from pions — internally reflecting quartz bars in BaBar, aerogel threshold counters and time-of-flight in Belle — a caesium-iodide crystal calorimeter, and an instrumented flux return for muons. What the measurement demands of them is unusual, however: not energy resolution but position resolution along the beam axis, to better than \(100\,\mu\mathrm{m}\) per decay vertex.

The two 2001 results were \(\sin2\beta=0.59\pm0.14\pm0.05\) from BaBar, on a sample of about thirty-two million \(B\bar{B}\) pairs [Aubert:2001], and \(0.99\pm0.14\pm0.06\) from Belle, on about thirty-one million [Abe:2001]. They differ by roughly two standard deviations, which is a useful reminder that first measurements scatter and that the agreement of two simultaneous results is not to be expected as a matter of course. Both are far from zero, which is the discovery; the average settled at \(0.708\pm0.011\) as the samples grew by a factor of ten [Navas:2024].

Neutral $B_{s}$ mesons at the LHC

The \(B_{s}^{0}\) system, in which the light quark is strange rather than down, oscillates thirty-five times faster than the \(B^{0}\) (Table 111.6) and was for that reason the last of the neutral-meson systems to be brought under control.

The apparatus. LHCb is not a collider detector in the usual sense: it is a single-arm forward spectrometer, covering pseudorapidities from about \(2\) to \(5\), built on the observation that at the Large Hadron Collider the \(b\) and \(\bar{b}\) of a produced pair both go forward at small angles. Its elements, in order from the interaction point, are a silicon strip vertex locator whose sensors approach to within about \(8\,\mathrm{mm}\) of the beams, a tracking system around a warm dipole magnet of about \(4\,\mathrm{T}\,\mathrm{m}\) bending power, two ring-imaging Cherenkov detectors with gas radiators, which give kaon–pion separation over the momentum range from a few \(\mathrm{GeV}/c\) to about \(100\,\mathrm{GeV}/c\), calorimeters, and muon chambers. The two numbers that matter for flavour physics are an impact-parameter resolution of about \(20\,\mu\mathrm{m}\) and a decay-time resolution of about \(45\,\mathrm{fs}\) — the latter being what makes a \(B_{s}^{0}\) oscillation of period \(2\pi\hbar/(\Delta m_{s}c^{2})=3.5\times 10^{-13}\,\mathrm{s}\) resolvable at all.

That a hadron collider can do this at all rests on two things. The production cross section for \(b\bar{b}\) pairs is large, so statistics are never the limitation; and the \(b\) hadron flies a measurable distance — of order a centimetre at LHC momenta — before decaying, so a vertex detached from the primary interaction is by itself a powerful trigger against the enormous light-quark background.

The result. The first observation of \(CP\) violation in the decays of \(B_{s}^{0}\) mesons was of the direct kind, in the decay \(B_{s}^{0}\to K^{-}\pi^{+}\) [Aaij:2013]. The measured asymmetry,

is non-zero at \(6.5\) standard deviations, and the same analysis measured \(A_{CP}(B^{0}\to K^{+}\pi^{-})=-0.080\pm0.007\pm0.003\). Two features are worth noting. The asymmetry is of order a quarter — \(CP\) violation in a decay amplitude can be large when the interfering amplitudes are comparable, as tree and penguin are here. And the two asymmetries have opposite signs, which is a prediction of the Cabibbo–Kobayashi–Maskawa structure and not an accident: the ratio of the two, weighted by the branching fractions and lifetimes, is predicted to be \(-1\) by \(U\)-spin symmetry, the interchange of the \(d\) and \(s\) quarks. The reason in outline is that \(U\)-spin maps \(B^{0}\to K^{+}\pi^{-}\) onto \(B_{s}^{0}\to K^{-}\pi^{+}\) while leaving the strong phases and the moduli of the interfering tree and penguin amplitudes invariant and flipping the sign of the weak phase difference, so the two \(CP\) asymmetries have equal magnitude and opposite sign once each is normalized to its own width.

The null test. The observable usually associated with the \(B_{s}^{0}\) is not Equation (111.19) but the mixing phase \(\phi_{s}\), the analogue of \(2\beta\) for the strange-beauty system, measured from the time-dependent asymmetry in \(B_{s}^{0}\to J/\psi\,\phi\) and related channels. Its Standard Model value is small and sharply predicted, because in the Wolfenstein expansion the relevant combination \(V_{ts}V_{tb}^{*}\) is almost real; the world average, \(\phi_{s}\approx-0.05\,\mathrm{rad}\) with an uncertainty of about \(0.02\,\mathrm{rad}\) [Navas:2024], agrees with it. A small, precisely predicted quantity is the ideal place to look for a new short-distance contribution, since anything new entering the box would shift it; nothing has been seen, and the measurement is therefore a constraint rather than a discovery.

Charm

The up-type quarks were the last to yield, twenty years after the \(B\) mesons and fifty-five after the kaon.

The apparatus and the sample. The measurement is LHCb's, using proton–proton collisions at centre-of-mass energies of \(7\), \(8\) and \(13\,\mathrm{TeV}\) and an integrated luminosity of about \(9\times 10^{43}\,/\mathrm{m}^{2}\). Two independent tagging methods were combined: the charge of the soft pion in \(D^{*+}\to D^{0}\pi^{+}\), and the charge of the muon in a semileptonic \(b\)-hadron decay producing the \(D^{0}\). The samples run to tens of millions of decays in each final state, and at that size the measurement is systematics-limited in practice — which is why the whole design is arranged so that the systematics cancel in a difference.

Direct T violation

Given the \(CPT\) theorem, every result above is also a statement about time reversal. But that inference assumes the theorem, whose hypotheses — locality, Lorentz invariance, the spin-statistics connection — are the deepest in the subject; a test of \(T\) that does not assume it is independent information. Two experiments have done it, and the difficulty in both is the same: a genuine \(T\) test compares a transition with its reverse, exchanging initial and final states, and conjugates nothing.

CPLEAR. The neutral kaon offers a transition and its exact reverse without any need to reverse momenta or spins, because both states are the same particle at rest: \(K^{0}\to\bar{K}^{0}\) and \(\bar{K}^{0}\to K^{0}\) are related by exchanging initial and final state, and nothing else [Kabir:1970]. The Kabir asymmetry Equation (105.84) must vanish identically if \(T\) is a symmetry.

The apparatus was built around the tagging problem, which is the whole difficulty: one must know the strangeness at production and at decay, in the same event. CPLEAR used the Low Energy Antiproton Ring at CERN, stopping antiprotons of about \(200\,\mathrm{MeV}/c\) in a gaseous hydrogen target at high pressure at a rate of order \(10^{6}\) per second. About two annihilations in a thousand give \(\bar{p}p\to K^{\mp}\pi^{\pm}K^{0}/\bar{K}^{0}\), and strangeness conservation in the strong production ties the sign of the accompanying charged kaon to the strangeness of the neutral one: the initial state is tagged by a charge measured in the same event. The final strangeness is tagged by the charge of the lepton in the semileptonic decay, using \(\Delta S=\Delta Q\) exactly as in Equation (111.11). The detector was a cylindrical assembly inside a solenoid of about \(0.44\,\mathrm{T}\) — tracking chambers, a threshold Cherenkov and scintillator sandwich for identifying the charged kaon, and a lead and gas-sampling electromagnetic calorimeter. The measured asymmetry Equation (105.86),

is the first direct observation of a time-reversal asymmetry [Angelopoulos:1998], and it agrees with the prediction \(A_{T}=4\,\mathrm{Re}\,\epsilon=6.46\times10^{-3}\) of Proposition 105.29 evaluated with \(\abs{\epsilon}=2.228\times10^{-3}\) and \(\arg\epsilon=43.5^\circ\).

BaBar. The cleaner measurement uses entanglement as a state-preparation device [Lees:2012]. From the antisymmetric \(B^{0}\bar{B}^{0}\) state Equation (105.87), observing one meson decay to a flavour-specific state projects the other onto a definite flavour at that instant, and observing a decay to \(J/\psi K^{0}_{S}\) projects it onto a definite \(CP\) label. The comparison is then available in both time orders: events in which the flavour tag comes first measure \(B^{0}\to B_{-}\), events in the opposite order measure \(B_{-}\to B^{0}\), and these are a process and its reverse with nothing conjugated. This is the entanglement machinery of Entanglement and Bell Tests used not to test locality but as an instrument. On a sample of \(468\) million \(B\bar{B}\) pairs BaBar found the \(T\)-odd parameters non-zero at fourteen standard deviations,

while the \(CPT\)-odd parameters extracted from the same fit were consistent with zero [Lees:2012].

The \(CPT\) tests obtained as a by-product are worth recording, because they are among the sharpest tests of anything. The same formalism that yields Equation (111.21) bounds the \(CPT\)-violating parameter of the kaon mass matrix, and the resulting limit on the fractional mass difference between \(K^{0}\) and \(\bar{K}^{0}\) is of order \(10^{-18}\) — the most precise equality of a particle and antiparticle property in physics. The neutral-meson interferometer is what makes it possible, for the reason set out at Equation (105.63): the observable is an accumulated phase, not an energy. The details belong to Section 105.5.3.

The unitarity triangle

The individual measurements above are of very different quantities in very different apparatus. What ties them together is one closure condition, the orthogonality of the first and third columns of the mixing matrix,

which is Equation (105.80). Three complex numbers summing to zero close a triangle in the complex plane; dividing by the middle term puts its base on the real axis from \(0\) to \(1\) and its apex at \((\bar{\rho},\bar{\eta})\). Every measurement in this chapter is a constraint on that apex, and they come in two kinds:

• Sides, from decay rates that involve no \(CP\) violation at all: \(\abs{V_{ub}/V_{cb}}\) from semileptonic \(B\) decay fixes the distance from the origin, and the ratio of the \(B_{s}^{0}\) to the \(B^{0}\) oscillation frequency fixes the distance from \((1,0)\).

• Angles, from the \(CP\) asymmetries themselves: \(\sin2\beta\) from \(B^{0}\to J/\psi K^{0}_{S}\), \(\gamma\) from the interference of \(b\to c\) and \(b\to u\) amplitudes in \(B\to DK\), \(\alpha\) from \(B\to\pi\pi\) and \(\rho\rho\). The kaon's \(\epsilon\) is a constraint of its own, a hyperbola in the \((\bar{\rho},\bar{\eta})\) plane.

The measured angles are Equation (103.32), \(\alpha=85\pm5^\circ\), \(\beta=22.2\pm0.7^\circ\) and \(\gamma=66\pm3^\circ\); they sum to \(173^\circ\pm6^\circ\) against the identity \(\alpha+\beta+\gamma=\pi\), which is a consistency test the three measurements had no way of enforcing on one another. The global fits [Charles:2005] combine all of it, and the constraints cross in one region of the plane a few per cent across. The full account, with the fitted apex and every input, is Section 103.4.3 and Phenomenon 103.31; what belongs here is the experimental summary and the honest qualifications.

Those qualifications are three, and they are reported as they stand rather than smoothed over. The angle \(\alpha\) carries a \(5^\circ\) uncertainty dominated not by statistics but by the hadronic penguin amplitudes that must be subtracted from \(B\to\pi\pi\) — that is, by theory. The side \(\abs{V_{ub}/V_{cb}}\) inherits an unresolved tension: the inclusive determinations, which sum over all final states and use an operator product expansion, sit two to three standard deviations above the exclusive ones, which measure a single channel and need a lattice form factor, for both \(\abs{V_{ub}}\) and \(\abs{V_{cb}}\); choosing one or the other moves the apex by about one standard deviation. And the most precisely measured unitarity statement of all is not a triangle at all but the first-row relation \(\abs{V_{ud}}^{2}+\abs{V_{us}}^{2}+\abs{V_{ub}}^{2}=1\), which at present does not come out right at the \(10^{-3}\) level (Remark 103.32). The correct summary is that one phase accounts for all observed \(CP\) violation in the quark sector at the few-per-cent level, and that two specific tensions remain.

Cosmological relevance and open questions

The Sakharov conditions

The measurements of this chapter acquire a significance beyond particle physics because of an observation about the universe and an argument by Sakharov [Sakharov:1967].

Insufficiency of Standard Model CP violation

All three Sakharov conditions are formally present in the Standard Model. Two of them fail quantitatively, and the failures are independent of each other.

The phase is far too small, and it is the masses that make it so. \(CP\) violation in the quark sector is controlled not by \(J\) alone but by the rephasing-invariant combination Equation (105.79), which multiplies \(J\) by the six squared-mass differences and therefore carries the SI dimension \(\mathrm{J}^{12}\). An asymmetry generated at temperature \(T\) is a pure number, so it can depend on that invariant only through the dimensionless ratio Equation (105.106), obtained by dividing by \((k_{B}T)^{12}\). Evaluated with the running quark masses at \(k_{B}T\approx100\,\mathrm{GeV}\), where the sphaleron processes are active, this gives Equation (105.107),

against the measured \(\eta=6.12\times10^{-10}\). The ratio is

a shortfall of eleven orders of magnitude, and the detailed calculations [Gavela:1994] do not improve it. The reason is structural rather than numerical. \(CP\) violation in this theory requires all six quark masses to be distinct (Proposition 105.27); at \(100\,\mathrm{GeV}\) five of the six are negligible against the temperature, so the effect is suppressed by twelve powers of small mass ratios. The phase itself is of order unity — \(\delta=65.70^\circ\) — and the masses kill it. This is the same arithmetic as Equation (103.46), and it is quoted here in the same form so that the two chapters cannot drift apart.

The third condition fails outright. A departure from equilibrium at the electroweak scale requires the transition from the symmetric to the broken phase to be first order, proceeding by the nucleation of bubbles whose walls sweep the plasma — and strongly first order at that, since the sphalerons must switch off fast enough after the transition not to erase what was just made. Lattice computations locate the endpoint of the first-order line at a scalar mass of about \(72\,\mathrm{GeV}/c^{2}\) [Kajantie:1996]; beyond it the change is a smooth crossover with no bubbles, no walls and no departure from equilibrium. The measured Higgs mass, Equation (104.66), is \(125.20 \pm 0.11\,\mathrm{GeV}/c^{2}\) [Navas:2024], far above the endpoint (Remark 104.65). No amount of \(CP\) violation would help.

Electric dipole moments and the strong CP problem

Every measurement so far in this chapter is of a \(CP\) violation that exists. The last one is of a \(CP\) violation that does not, and the null result is the more constraining of the two kinds.

The number is worth converting once, because it is otherwise unimaginable. A dipole of \(2.9\times 10^{-47}\,\mathrm{C}\,\mathrm{m}\) corresponds to separating one elementary charge from its opposite by \(1.8\times 10^{-28}\,\mathrm{m}\), about \(10^{13}\) times smaller than the neutron itself, whose charge radius is of order \(0.8\,\mathrm{fm}\).

What the null result constrains. Two things, and they are of very different character. The first is new physics: the Cabibbo–Kobayashi–Maskawa phase generates a neutron dipole only at three loops with a severe chiral suppression, some six orders of magnitude below Equation (111.27).

The entire experimental range is one in which the Standard Model predicts nothing observable. A non-zero result would be unambiguous, and the null result excludes new \(CP\)-violating sources over a wide range with no background to subtract.

The second is a parameter of the Standard Model itself. Quantum chromodynamics admits a term proportional to \(\theta\,\tr\left(G_{\mu\nu}\tilde{G}^{\mu\nu}\right)\) which is a total derivative, invisible in perturbation theory, and physical nonperturbatively because the gauge field has topologically distinct sectors. Only the combination \(\bar{\theta}=\theta+\arg\det M_{q}\) is invariant under the chiral rotations that move phase between the gluon and Yukawa sectors (Equation (105.111)), and \(\bar{\theta}\) violates \(P\) and \(T\), hence generates a neutron dipole. The chiral estimates [Baluni:1979] [Crewther:1979] give \(d_{n}\approx\bar{\theta} \times1.6\times 10^{-37}\text{–}6.4\times 10^{-37}\,\mathrm{C}\,\mathrm{m}\) (Equation (105.112)), so Equation (111.27) implies

that is \(\abs{\bar{\theta}}\lesssim10^{-10}\), with a factor-of-a-few uncertainty inherited from the chiral coefficient (Equation (105.113)).

This is the strong \(CP\) problem, and what makes it a problem is worth stating precisely, because it is unlike the other open questions in this chapter. \(\bar{\theta}\) is a free dimensionless parameter, periodic with period \(2\pi\), so there is no sense in which a small value sits near a boundary of its range; a value of order unity is the natural expectation for a number about which nothing else is known. It is measured to be smaller than \(10^{-10}\). No symmetry of the Standard Model requires it to vanish — the one candidate, a massless up quark, is excluded by the lattice determination \(m_{u}c^{2}=2.16 \pm 0.07\,\mathrm{MeV}\) [Navas:2024] — and no dynamical mechanism within the theory relaxes it. The problem is an observation, and it is recorded as such in Section 133.1.

The best-known response is to make \(\bar{\theta}\) dynamical: a global symmetry, spontaneously broken and itself anomalous, replaces it by a field whose instanton-generated potential is minimized at zero [Peccei:1977], at the cost of a light pseudo-Goldstone boson, the axion [Weinberg:1978] [Wilczek:1978]. No axion has been observed. The haloscope, helioscope and light-shining-through-walls searches report exclusions and nothing else, and the mechanism is a hypothesis with an attractive structure and no experimental support; it is listed as such in Section 133.7 and named here only to say so. What this chapter records as evidence is Equation (111.28): a free parameter that could have been of order unity is smaller than \(10^{-10}\), and nothing explains why.

Primary references

The literature of \(CP\) violation divides into papers that predicted a system, papers that measured something in it, and papers that supplied the framework in which the measurement means something. It is worth separating them, because the reader who goes to the sources will find that several of the most-cited ones do not contain what they are cited for.

The system. Gell-Mann and Pais [GellMann:1955] is the paper that invented the neutral kaon as a two-state problem and predicted the long-lived partner; it is a short paper and it is worth reading for the argument alone, which uses nothing beyond the superposition principle and a conservation law. Lande and collaborators [Lande:1956] reported the long-lived particle at the Brookhaven Cosmotron the following year. Pais and Piccioni [Pais:1955] predicted regeneration; the paper is the origin of the technique that produces \(K_{S}\) beams and of the background that Proposition 111.5 bounds.

The discovery. Christenson, Cronin, Fitch and Turlay [Christenson:1964] is two and a half pages of Physical Review Letters, and every quantitative statement about the 1964 apparatus in Sections 111.2 and 111.3 comes from it. The paper is unusual in modern eyes for how much of its argument is carried by one two-dimensional plot and how carefully it enumerates the alternatives it must exclude. Note that the published ratio has “all charged decay modes” in its denominator, not the total width; the conversion is Equation (111.8), and a reader comparing the 1964 number directly with a modern branching fraction will find a spurious twenty per cent discrepancy.

The frameworks. Landau [Landau:1957] proposed \(CP\) as the surviving symmetry after the parity results of Wu [Wu:1957] and Garwin [Garwin:1957]; Lüders [Luders:1957] proved the \(CPT\) theorem which is what makes \(CP\) violation and \(T\) violation the same statement. Wolfenstein [Wolfenstein:1964] supplied the superweak alternative in the same year as the discovery — a model that survived thirty-five years of measurement and was killed by \(\epsilon'\). Kobayashi and Maskawa [Kobayashi:1973] extended Cabibbo's angle [Cabibbo:1963] to three generations and showed a phase then becomes irremovable; the parametrization in which the hierarchy is manifest is Wolfenstein's second contribution [Wolfenstein:1983], and the convention-independent measure of the effect is Jarlskog's [Jarlskog:1985]. Kabir [Kabir:1970] is the source of the observable that CPLEAR measured.

The epsilon-prime programme. NA31 [Burkhardt:1988] reported the first evidence; the contemporaneous Fermilab result is described in Section 111.6.1 but not cited, since this treatise's bibliography carries no entry for it, and Remark 111.10 says so. The two 1999 papers that settled the question are Fanti and collaborators for NA48 [Fanti:1999] and Alavi-Harati and collaborators for KTeV [AlaviHarati:1999]. The world average is the Particle Data Group's [Navas:2024]; the lattice calculations to which it should be compared are not cited, for the same reason, and Section 111.6.3 says so.

Beyond the kaon. BaBar [Aubert:2001] and Belle [Abe:2001] reported the \(B^{0}\to J/\psi K^{0}_{S}\) asymmetry simultaneously in 2001; the two papers are worth reading side by side, since they disagree by two standard deviations while both establishing the effect. LHCb's first observation of \(CP\) violation in \(B_{s}^{0}\) decays [Aaij:2013] and in charm [Aaij:2019] complete the set of quark flavours. CPLEAR [Angelopoulos:1998] and BaBar [Lees:2012] are the two direct \(T\) tests. The global fit that combines all of it is CKMfitter [Charles:2005].

The cosmological side. Sakharov [Sakharov:1967] is a two-page note and states the three conditions in a form that has not needed amendment. 't Hooft [tHooft:1976a] showed that baryon number is anomalous in the electroweak theory; Kuzmin, Rubakov and Shaposhnikov [Kuzmin:1985] showed that the resulting rate is unsuppressed at high temperature. Gavela and collaborators [Gavela:1994] carried out the calculation that quantifies the shortfall, and Kajantie and collaborators [Kajantie:1996] showed by lattice computation that the electroweak transition is a crossover for any Higgs mass above about \(72\,\mathrm{GeV}/c^{2}\), which removes the third Sakharov condition independently. The measured baryon-to-photon ratio is Planck's [Aghanim:2020].

The null results. The current neutron electric dipole moment bound is from ultracold neutrons at the Paul Scherrer Institute [Abel:2020], improving the Institut Laue-Langevin result [Baker:2006]; the technique is Ramsey's [Ramsey:1950] and the argument for measuring the quantity at all was made by Purcell and Ramsey [Purcell:1950] at a time when parity was universally assumed. The chiral estimates relating the bound to \(\bar{\theta}\) are Baluni [Baluni:1979] and Crewther, Di Vecchia, Veneziano and Witten [Crewther:1979]. Peccei and Quinn [Peccei:1977] proposed the dynamical relaxation of \(\bar{\theta}\) and Weinberg [Weinberg:1978] and Wilczek [Wilczek:1978] pointed out the axion that follows; none of it has been observed.

The data. Every world-average number in this chapter is from the Particle Data Group's 2024 Review [Navas:2024], and the masses in Table 111.5 are from its machine-readable release, stored with this treatise as evidence. The SI conversions — the elementary charge, the speed of light, the Planck and reduced Planck constants — are CODATA 2022 [Mohr:2025], likewise stored; in the present SI the first three of those are exact, so the conversions in this chapter carry no uncertainty of their own.