Codex @codex  0

The antiparticle of a kaon, such as $K^{-}=s\bar{u}$ and ${\bar{K}}^0=s\bar{d}$.

BABAR used asymmetric electron-Positron collisions near the Upsilon meson $4S$ resonance to prepare neutral B meson pairs. It reconstructed one decay into charmonium plus a kaon, and used flavour tagging on the other B meson with charged leptons or kaons. Silicon tracking reconstructed decay vertices for the decay-time difference. Self-tagging decays calibrated mistags and time resolution; charged decays provided a control. The tagged time distributions established CP violation, limited by backgrounds, mistags and vertex alignment.^[ref]

The Lorentz boost converts vertex separation to $\Delta t\simeq \Delta z/({\beta }_{\mathrm{boost}}{\gamma }_{\mathrm{boost}}c)$ at leading order. Unequal beam energies thus make a time difference measurable as a distance.^[ref]

James H. Christenson, James Watson Cronin, Val Logsdon Fitch and René Turlay reported the result in 1964. At Brookhaven's AGS, 30 GeV protons struck beryllium; collimators selected a neutral particle beam at 30 degrees. A downstream helium decay volume reduced interactions.^[ref]

The flight distance suppressed the short-lived kaon component. Each arm of a two-arm magnetic spectrometer had spark chambers before and after its magnet. Cherenkov detectors and scintillation counters triggered photographic recording in coincidence.^[ref]

The helium sample yielded $45\pm 9$ signal events; $(2.0\pm 0.4)\times 10^{-3}$ was the ratio to all charged decay channels.^[ref] That denominator must be retained when comparing this result with a total branching fraction.

The evidence was kinematic. Later regenerated-beam quantum interference established that the short- and long-lived states reached the same two-pion final state, excluding alternatives such as distinct pion-like particles.^[ref] The 1964 observation established evidence for CP violation; it did not determine every parameter of the CKM matrix. Inferring time-reversal symmetry violation from CP violation invokes CPT symmetry, rather than a direct comparison of time-reversed transitions in that experiment.

In a convention where the CP operation squares to one on the relevant bosonic state space, a CP eigenstate has eigenvalue $+1$ or $-1$. If the Hamiltonian commutes with CP, its nondegenerate eigenstates can also be CP eigenstates. In the neutral-kaon convention $CP|K^0⟩=-|{\bar{K}}^0⟩$, the antisymmetric flavor combination is CP even and the symmetric one CP odd. Rephasing flavor states changes the written combinations, not the invariant content.

CP violation is the failure of a theory to be invariant under the combination of charge conjugation and parity. Under CPT invariance it is equivalent to time-reversal violation.

CP violation matters experimentally because a symmetry predicts a forbidden transition, not merely a small correction to a permitted one. The Christenson-Cronin-Fitch-Turlay kaon CP violation experiment found evidence for precisely such a transition: a long-lived neutral kaon decaying into two pions. Understanding that inference requires both two-pion kaon event reconstruction and the distinction between kaon CP eigenstates and the physical states of neutral kaon mixing.

A flavor eigenstate is a state with a definite flavor label in the specified interaction or flavor-charge basis. For neutral kaons, $K^0=d\bar{s}$ and ${\bar{K}}^0=s\bar{d}$ have definite opposite strangeness. Weak interactions mix them, so these flavor states are not the mass eigenstates. The precise flavor basis must be specified when discussing mixing.

The scattering amplitude at zero deflection angle. Its real part changes the phase of a coherent beam in matter; its imaginary part is related by the optical theorem to removal from that beam. Different forward amplitudes for a kaon and a neutral antikaon allow neutral kaon regeneration without fundamental CP violation.

The 1999 recognition of Gerard 't Hooft and Martinus J. G. Veltman concerned the theoretical quantum structure of the electroweak interaction.^[ref] Renormalizability makes repeated radiative corrections predictive from a fixed set of inputs. It does not imply that the perturbation series converges, nor that renormalization alone chooses the correct particle content or excludes all physics beyond the Standard Model.

The Gargamelle collaboration prepared an experimental test by exposing its heavy-liquid bubble chamber to neutrino and antineutrino beams at CERN. The readout selected events producing hadrons without an identified muon or electron, compared with events containing a charged lepton. Its 1974 report assessed alternative origins from neutrons and long-lived neutral kaons, finding them insufficient to explain the sample. That supported neutral currents; it did not isolate a radiative correction or demonstrate renormalizability.^[ref]

The distinction matters when using the tree-level electroweak gauge-boson masses:Beyond tree level, self-energies, interaction vertices and counterterms alter how chosen input observables predict another observable. The Weinberg angle then needs a specified renormalization scheme; inserting its value from one convention into a formula from another can create an apparent discrepancy. A meaningful higher-order test must state both its independent experimental inputs and the calculated loop order.

Gargamelle was a heavy-liquid bubble chamber at CERN. Its 1973 collaboration report, Observation of neutrino-like interactions without muon or electron in the Gargamelle neutrino experiment, by F. J. Hasert and colleagues, described hadronic events with no identified muon or electron and evaluated rates relative to charged current events. This is evidence for a neutral current, rather than direct production or photography of a Z boson.^[ref]

CERN's Gargamelle account identifies the target liquid as $\mathrm{C}{\mathrm{F}}_3\mathrm{Br}$ and distinguishes hadronic events from the separate isolated-electron signal. The latter is an antineutrino transferring momentum to an electron, whose electromagnetic particle shower supplies visible tracks. These are different final states, not two names for one event.^[ref]

A neutrino beam uses protons striking a target to produce charged pions and kaons; focusing selects parents that undergo particle decay in flight. Their neutrinos pass through shielding into the target. Scanning photographs locates vertices and classifies outgoing tracks.^[ref]

Expansion prepares a bubble chamber's superheated liquid; ionization by charged secondaries seeds bubbles for photography.^[ref]

The recorded light reveals bubbles, not a directly visible incoming or outgoing neutrino. Reconstructing a vertex and classifying bubble-chamber tracks connects that optical record to a hypothesis about a particle interaction.

The collaboration compared samples with matched hadronic cuts. A flat longitudinal vertex distribution resembled neutrino interactions, but sideways neutrons and multi-step cascades could also flatten it. The background estimate combined visible parent-interaction/secondary-neutron pairs with a cascade calculation including surrounding material. Later proton exposures tested the predicted cascade lengths. Escaping charged hadrons could imitate muons; there was no independent perfect muon tag.^[ref]

The inference combines positive evidence for an interaction with controlled alternatives for its visible final state. Shielding suppresses unwanted beam components, yet interactions in surrounding material can generate secondary neutrons; shielding alone cannot certify a neutral current. Nor does a successful neutral current search by itself establish every coupling or the Higgs mechanism. Its importance for the electroweak interaction is that the theory predicts an additional weak channel, with coupling relations that further cross section measurements can test.

Suppose a transition operator obeys $[I_3,H]=H/2$ and $[I_{+},H]=0$, and acts on the lower component of an isospin-half kaon. Its output has third component zero, and $I_{+}^{2}H|K⟩=H{I}_{+}^2|K⟩=0$. This excludes an isospin-two output. A spinless parent decaying to two spinless pions has zero relative orbital angular momentum, so Bose-Einstein statistics excludes the antisymmetric isospin-one sector. The output is therefore the two-isovector singlet. In the equal-mass isospin limit, charged and neutral two-pion decay widths have ratio two, including the identical-particle counting.

For relativistically normalized kaon states and $Q=(\bar{s}{\gamma }_{\mu }(1-{\gamma }^5)d)(\bar{s}{\gamma }^{\mu }(1-{\gamma }^5)d)$, define $⟨{\bar{K}}^0|Q|K^0⟩=(8/3)f_K^2{m}_K^2{B}_K$. The dimensionless kaon bag parameter measures strong interaction corrections to the vacuum-insertion normalization. The coefficient and matrix element must be defined in the same renormalization scheme and at the same scale. Dividing by $2m_K$ gives the corresponding two-state mixing-Hamiltonian normalization.

The CP eigenstates of neutral kaons depend on a flavor-state phase convention. Using $CP|K^0⟩=-|{\bar{K}}^0⟩$, set $|K_1⟩=(|K^0⟩-|{\bar{K}}^0⟩)/\sqrt{2}$ and $|K_2⟩=(|K^0⟩+|{\bar{K}}^0⟩)/\sqrt{2}$. They are respectively CP even and odd. Reversing the flavor-exchange sign reverses which written sum is even; it does not change the CP eigenvalues. With CP violation, the physical short- and long-lived states need not equal these CP eigenstates; kaon CP mixing parameter and neutral-kaon CP impurity and two-pion decay describe the mixtures.

In a Hermitian neutral-kaon mass matrix, independently choosing square-root signs of $z$ and $z^{\ast }$ can interchange the eigenvalue labels. Choose $z=r{e}^{i\phi }$, $a=\sqrt{r}{e}^{i\phi /2}$ and $b=\sqrt{r}{e}^{-i\phi /2}$, so $ab=r>0$, continuously from the desired CP-conserving limit. Then $(a,-b)$ is the lower-mass eigenvector. At a degenerate off-diagonal zero the mass eigenbasis is not fixed by diagonalization alone.

In 1962, Leon Max Lederman, Melvin Schwartz, Jack Steinberger and collaborators distinguished muon neutrinos from electron neutrinos using a neutrino beam.^[ref]

The Alternating Gradient Synchrotron delivered $15\mathrm{GeV}$ protons onto beryllium. Particle decay of pions and kaons supplied neutrinos; a $13.5\mathrm{m}$ iron shield preceded a ten-ton aluminium spark chamber.^[ref]

Production and detection are separate. In modern neutrino flavour notation,Shielding suppresses the daughter muon; a newly produced muon inside the spark chamber's aluminium marks an interaction.^[ref]

Beam-timed coincidence circuits and veto counters rejected entering tracks. Machine-off runs measured cosmic ray contamination; shield-thickness changes tested neutrons. Placing lead near the target shortened the meson decay path and reduced events. Cosmotron electron calibration established particle shower recognition.^[ref]

The paper compared $34$ single-muon candidates, including $5$ estimated cosmic ray events, with six shower candidates unlike calibrated electron showers. Kaon decays could supply genuine electron neutrinos. The deficit favored distinct neutrino flavours under lepton universality.^[ref]

The inference concerns production and detection channels. It does not measure neutrino mass or establish permanently fixed neutrino flavour. Later neutrino oscillation experiments address changes during propagation.^[ref]

A magnetic horn uses a pulsed magnetic field around a high-current conductor to focus charged secondary pions and kaons of a selected sign. Their decays then create a directed neutrino beam. The field acts on the charged parents, not on the neutral neutrinos.

The pseudoscalar octet of flavor symmetry contains three pions, four kaons and the Eta octet state. In its weight diagram, the neutral pion and ${\eta }_8$ are separate states at the origin.

Use the kaon CP eigenstates convention $CP|K^0⟩=-|{\bar{K}}^0⟩$, matching kaon CP mixing parameter. Then $|K_1⟩=(|K^0⟩-|{\bar{K}}^0⟩)/\sqrt{2}$ is CP even, while $|K_2⟩=(|K^0⟩+|{\bar{K}}^0⟩)/\sqrt{2}$ is CP odd. With CPT symmetry and a common small mixing parameter, the physical states can be writtenHere $S,L$ mean short and long mean lifetimes; the kaon CP mixing parameter $ϵ$ depends on the flavor-state phase convention. A positive flavor-exchange convention instead makes the symmetric flavor combination CP even, as explained in exchange-positive CP convention for neutral kaon mixing. Mixing these conventions would reverse the interpretation of the written sums.

Why does the two-pion final state test CP symmetry? A spin-zero kaon decaying into two spin-zero pions has relative orbital angular momentum $\ell =0$ by angular momentum conservation. Two negative intrinsic parities multiply to plus, and $(-1{)}^{\ell }=1$. Charge conjugation exchanges ${\pi }^{+}$ and ${\pi }^{-}$; the $\ell =0$ state is even under this exchange. Hence this two-pion state has CP eigenvalue $+1$. If a decay transition operator $T$ preserves CP symmetry, inserting CP on both sides of $⟨\pi \pi |T|K_2⟩$ changes its sign without changing its value: that decay amplitude must vanish.

Assume for illustration that CP violation occurs only through the state admixture and the decay transition preserves CP symmetry. Put $A_1=⟨\pi \pi |T|K_1⟩$. ThenNeglecting the tiny difference in two-body Lorentz-invariant phase space between the physical masses gives ${\Gamma }_L^{\pi \pi }/{\Gamma }_S^{\pi \pi }\simeq |ϵ{|}^2$. A branching fraction, however, divides a partial decay width by the total decay width. With $\hslash =1$, ${\tau }_i=1/{\Gamma }_i$, soFor chosen values $|ϵ|=0.01$ and ${\tau }_L/{\tau }_S=100$, the branching fraction ratio is $0.01$, whereas the partial decay width ratio is $0.0001$. This is original arithmetic, not a historical fit. The longer observation time amplifies the integrated fraction relative to the instantaneous partial decay rate.

With CPT symmetry, a Hermitian kaon mass matrix is $M=\left(\begin{array}{cc}{r}_0& z\\ z^{\ast }& r_0\end{array}\right)$. Its eigenvalues are $r_0\pm |z|$. For correlated roots $a^2=z$, $b^2=z^{\ast }$ and $ab=|z|$, the normalized lower and upper eigenvectors are $(a,-b)/\sqrt{2|z|}$ and $(a,b)/\sqrt{2|z|}$. They are orthogonal. This is a mass-only result; including decay requires the generally non-Hermitian effective matrix $M-i\Gamma /2$.

Neutral kaon regeneration is the production of a short-lived component when a neutral kaon beam passes through matter. In the phase convention of kaon CP eigenstates, let the forward transmission amplitudes of $K^0$ and ${\bar{K}}^0$ be $t$ and $\bar{t}$. Expanding $t|K^0⟩+\bar{t}|{\bar{K}}^0⟩$ in the kaon CP eigenstates gives the displayed equation. Different forward scattering amplitudes, including absorption, can therefore generate a two-pion signal even with CP symmetry. Ordinary matter is not a CP-symmetric preparation.

Tungsten regeneration calibrated the signal; scanning its position sampled acceptance. Helium regeneration was insufficient, bag walls lay outside acceptance, vertex distributions excluded collimator regeneration, and hydrogen data cross-checked the effect.^[ref] These controls are central: the question is whether the long-lived component itself reaches two pions, rather than whether material has prepared a fresh short-lived component.

For a coherently prepared state proportional to $\rho |K_S⟩+|K_L⟩$, evolve each component with ${\lambda }_j=m_j-i{\Gamma }_j/2$. The proper time is $t$, and $\hslash =1$. Since both components reach the same final state, their decay amplitudes add before squaring. Factoring out $|A_S{|}^2$ and a common beam normalization gives the two-pion intensityThe last term is quantum interference. It vanishes upon averaging over an independently random preparation phase, but cannot be discarded from a coherent preparation. Summing the two positive contributions alone would lose the phase-sensitive information.

At late proper time, the short-lived contribution and the quantum interference term die away, leaving the long-lived tail $|{\eta }_{\pi \pi }{|}^2{e}^{-{\Gamma }_{L}t}$. Such a tail is easier to identify after the initially abundant short-lived component has disappeared. The useful experimental balance is between eliminating that component and retaining enough long-lived decays to observe a rare decay channel.