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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]

The charm antiquark $\bar{c}$ is the antiparticle of the charm quark. It has electric charge $-2e/3$, baryon number $-1/3$ and the opposite charm flavor number, while retaining spin $1/2$ and the same mass. Its color transformation is the conjugate of a quark's.^[ref]

A charm–anticharm pair has zero net charm and can form charmonium. Zero net charm does not imply absence of charm constituents: it states that quark and antiquark flavor numbers cancel. This distinction is useful when balancing conserved flavor numbers at a strong interaction vertex or classifying a meson's minimal valence content.

Charmonium is a family of bound states of a charm quark and its antiquark, with zero net charm. It differs from open-charm D mesons, which carry a charm quark together with a light antiquark.

The constituent spins and relative orbital angular momentum distinguish states such as the J/psi meson, eta c meson and chi c mesons. Their level separations and radiative transitions test the interaction between heavy quarks. Decay can proceed through annihilation of the charm–anticharm pair, so net charm conservation does not imply that a charmonium state is stable. A valence quark label organizes the spectrum while additional gluonic and quark-pair contributions remain part of the interacting state.

The charm quark $c$ is the second-generation up-type quark, with electric charge $+2e/3$, spin $1/2$ and baryon number $1/3$. It is heavier than the strange quark and is commonly treated as a heavy quark.^[ref]

A charm–charm antiquark pair supplies the minimal valence content of charmonium, while a charm quark bound with a light antiquark gives an open-charm meson. These systems probe different aspects of quantum chromodynamics: internal heavy-pair motion in the former and heavy–light dynamics in the latter. The approximate flavor classification that includes charm does not make charm and the light quarks equal in mass.

The spin-triplet P-wave charmonium multiplet $1{}^3{P}_J$, with $J=0,1,2$ and $J^{PC}=0^{++},1^{++},2^{++}$. Radiative cascades between S-wave vector mesons and these states generate bands in three-body Dalitz plots.

The J/psi meson is the ground spin-triplet S-wave vector state of charmonium, with leading valence content $c\bar{c}$ and quantum numbers $J^{PC}=1^{--}$. It has zero net charm and can decay into hadrons or charged-lepton pairs.^[ref]

Its comparatively narrow decay width makes it a conspicuous resonance in reconstructed mass distributions. An electron–Positron mass peak in a fixed-target experiment and an enhanced event rate during an energy scan in an electron-positron collider were complementary observations in the Richter-Ting independent J/psi discovery experiments of 1974.^[ref][ref] A peak's measured width also includes detector resolution or beam-energy spread, so it must not automatically be identified with the intrinsic lifetime width.

A simple isolated particle resonance model starts with a survival amplitude proportional toIts survival probability is $|a(t){|}^2=e^{-\Gamma t/\hslash }$, giving mean rest-frame lifetime $\tau =\hslash /\Gamma$. The Fourier transform of this one-sided amplitude is proportional to $(\Gamma /2-i(E-E_0){)}^{-1}$; its squared modulus gives a Lorentzian line shape. Write its normalized density asAt $E-E_0=\pm \Gamma /2$ the density is half its peak, so its intrinsic full width at half maximum is $\Gamma$. This is a local isolated-line approximation, extending the offset coordinate over the real line; physical production thresholds, interference and energy-dependent decay widths can require a richer quantum scattering model.

Now stipulate a centered normal distribution for the measurement response, with standard deviation $\sigma$. A measured coordinate $x$ has densityThis convolution describes the smearing of an underlying line, with constant acceptance and no background. For a pair spectrometer the response summarizes invariant-mass resolution; for an idealized collider scan it summarizes incoming center-of-mass energy spread. These are different experimental sources of uncertainty even when the same mathematical response is useful.

Choose an arbitrary scale $E_{\ast }$, intrinsic decay width $\Gamma =0.20E_{\ast }$ and response standard deviations $0.40E_{\ast }$ and $1.20E_{\ast }$. Numerical integration gives apparent full width at half maximum values of approximately $1.05E_{\ast }$ and $2.93E_{\ast }$, respectively. The state and its rest-frame lifetime are identical in both cases. The normal distribution's width alone is $2\sqrt{2\ln 2}\sigma$; adding that width to $\Gamma$ in quadrature is not an exact rule for this convolution.

For a purely illustrative $E_{\ast }=1\mathrm{MeV}$, taking $\hslash \simeq 6.5821\times 10^{-22}\mathrm{MeV}\mathrm{s}$ gives $\tau \simeq 3.2911\times 10^{-21}\mathrm{s}$. These are invented line parameters, not J/psi meson measurements. Substituting the larger apparent width into $\hslash /\Gamma$ would incorrectly shorten the inferred lifetime by nearly a factor of fifteen. An unresolved width can support an upper bound on intrinsic decay width, and consequently a lower bound on lifetime, only after the instrumental response and model assumptions have been assessed.

The discovery peak and its later interpretation answer different questions. The J/psi meson is understood as charmonium, a bound charm quark–charm antiquark system. Richter's 1976 lecture distinguishes the initial discovery from subsequent tests of competing explanations, including the related level structure and later evidence for particles carrying uncancelled charm. A narrow peak alone did not establish all these properties in 1974.^[ref]

TODO: the numerical resonance-area reading is unavailable from the retained source. Page 6 retains the upper data panels but loses the lower part of the scanned data image, including the numerical horizontal energy calibration. A vertical cross section scale and an apparent peak cannot supply $\int \sigma dE$ without that calibration. The decoded image contains 2,834,898 of 4,437,216 required RGB bytes; the archived bundle has the same truncation. No horizontal scale or natural decay width is inferred from the display width.

The recoverable numerical conversion is nevertheless definite. With $M\simeq 3.1\mathrm{GeV}$, $B_{ee}=0.060$, $ϵ=0.50$, and $1{\mathrm{GeV}}^{-2}=389400\mathrm{nb}$, b givesThis coefficient assumes the plotted exclusive electronic sample has not already been corrected for acceptance. A restored area, with continuum subtraction and acceptance convention checked, would complete the estimate. A tabulated modern J/psi meson width would be external information rather than a reading of these data.

The narrow scattering resonance indicates suppressed strong interaction decay. In the quark model, $c\bar{c}$ is below the open-charm quark $D\bar{D}$ threshold, so its constituents cannot simply emerge in separate charmed mesons. The OZI rule suppresses annihilation into light hadrons; for a $1^{--}$ color-singlet state the leading purely gluonic channel needs three gluons. The observed leptonic channels remain competitive. Its mass and $I=0$ fit charmonium; the small width alone would not uniquely identify the new quark flavour without the wider spectrum and threshold evidence.

For a neutral $c\bar{c}$ bound state, meson parity and charge conjugation give $P=(-1{)}^{L+1}$ and $C=(-1{)}^{L+S}$. In spectroscopic notation $n{}^{2S+1}{L}_J$, the integer $n$ labels radial excitation, $L$ is orbital angular momentum, $S$ the total constituent spin, and $J$ their total angular momentum. The surviving level diagram contains eight states:

The spin-dependent splittings distinguish the singlet and triplet levels and the three triplet P levels. The radial excitation changes neither parity nor charge conjugation.

Assume the charmonium state annihilates through a virtual photon, with lepton universality and negligible higher-order corrections. For a vector meson of mass $M$, the partial decay width into a charged lepton pair contains both the relativistic two-body decay factor ${\beta }_{\ell }$ and a vector-current spin factor:^[ref] For example, contracting the vector polarization sum with the Dirac trace for the two final fermions gives a factor proportional to $M^2+2m_{\ell }^2$, explaining why retaining only ${\beta }_{\ell }$ is insufficient. Using $m_{\tau }=1.777\mathrm{GeV}$, $m_{\mu }=0.1057\mathrm{GeV}$ and $M=3.686\mathrm{GeV}$ givesThe supplied total decay width also gives ${\Gamma }_{\mu \mu }=2.352\mathrm{keV}$ and ${\Gamma }_{\tau \tau }\simeq 0.91\mathrm{keV}$. It cancels from the branching fraction ratio.

The decisive change at $3.770\mathrm{GeV}$ is crossing the open-charm D meson-pair threshold, approximately $3.73\mathrm{GeV}$. The strong interaction can now create a light quark-antiquark pair and rearrange $c\bar{c}$ into $D\bar{D}$, without annihilating the heavy pair. This increases the total decay width from hundreds of keV to tens of MeV. The leptonic decay partial decay width is additionally smaller because $\psi (3770)$ is predominantly a $1{}^3{D}_1$ state, whereas $\psi (3686)$ is predominantly $2{}^3{S}_1$; the short-distance annihilation overlap is suppressed for nonzero orbital angular momentum. A roughly hundredfold increase in total decay width together with an order-ten reduction in leptonic partial decay width explains the order-thousand reduction in branching fraction. Attributing all three orders of magnitude to the threshold alone would miss the second effect.^[ref]

At an electron-positron collider tuned to the charmonium resonance, select the mixed leptonic tau lepton decaysand their charge conjugates. Require exactly two well-reconstructed oppositely charged tracks, one identified electron and one identified muon, little extra visible activity, visible energy below the beam total, and significant missing momentum pointing into the instrumented region. The multiple neutrinos prevent full reconstruction of each tau lepton, but the known initial four-momentum, lepton spectra and missing energy support a statistical signal extraction. Off-resonance data constrain nonresonant electron-positron annihilation into a tau lepton pair; that continuum is present at both beam energies.^[ref]

A resonant muon pair is one possible detector background: one muon can be misidentified as an electron, while radiation escaping along the beam or a badly measured track produces apparent missing energy. Particle-identification requirements and rejection of almost back-to-back tracks near the beam momentum suppress it. This background needs detector mismeasurement; an ideal measurement would distinguish it.

Above the open-charm threshold, $\psi (3770)\to D\bar{D}$ supplies an additional resonant background. Semileptonic decays of the two D mesons can give an electron, a muon and neutrinos; unreconstructed hadrons let some events pass the selection. This on-shell D meson-pair background is kinematically forbidden at $\psi (3686)$.

The three structures are the radiative cascadesIn the quark model, the J/psi meson and $\psi (2S)$ are respectively $1{}^3{S}_1$ and $2{}^3{S}_1$ charmonium states. Thus $L=0$, $S=1$, $J=1$, and meson parity and charge conjugation gives $J^{PC}=1^{--}$. The intermediate chi c mesons are $1{}^3{P}_J$: $L=1$, $S=1$, and addition of angular momentum permits $J=0,1,2$. ConsequentlyThese are electric dipole transitions, changing $L$ by one and preserving $S$ to leading order. A singlet $1{}^1{P}_1$ state has $J^{PC}=1^{+-}$ and is not another member of these spin-triplet cascades.

The primary photon energy is $(M_{2S}^2-M_{{\chi }_J}^2)/(2M_{2S})$, about $261$, $171$ and $128\mathrm{MeV}$ for $J=0,1,2$. The secondary photon remains more energetic even after its boost into the parent rest frame. Hence $m^2(J/\psi {\gamma }_h)=M_{{\chi }_J}^2$: with this quantity on the vertical axis, each resonance gives a horizontal band. A finite decay width and detector measurement resolution broaden each band; photon ordering removes the mirrored duplication of an unordered Dalitz plot.

The faint band beside the negative-slope diagonal boundary is $\psi (2S)\to J/\psi {\pi }^0$, followed by ${\pi }^0\to \gamma \gamma$. The neutral pion is predominantly $(u\bar{u}-d\bar{d})/\sqrt{2}$, a ${}^1{S}_0$ pseudoscalar meson with $J^{PC}=0^{-+}$. This production transition violates isospin, accounting for its small rate. To locate the band, set $X=m^2(J/\psi {\gamma }_l)$ and $Y=m^2(J/\psi {\gamma }_h)$. Four-momentum conservation givesFor the neutral pion, $m_{\gamma \gamma }^2=m_{{\pi }^0}^2$ is small, putting the line just inside the $m_{\gamma \gamma }^2=0$ boundary. The allowed $J/\psi \eta$ channel, with the eta meson also $0^{-+}$, produces a parallel line further inside, at $m_{\gamma \gamma }^2=m_{\eta }^2$. The positive-slope edge $X=Y$ instead comes from ordering the two photon energies; proximity to that edge alone would not identify the two-photon state.

A quarkonium is a bound state of a heavy quark and its own antiquark. Examples are charmonium and the bottom-antibottom states called Upsilon mesons. Spectroscopic labels $n{}^{2S+1}{L}_J$ distinguish radial excitation, total quark spin, relative orbital angular momentum and total angular momentum.

With CPT symmetry, write the propagating states of neutral meson mixing as $p|B^0⟩\pm q|{\bar{B}}^0⟩$, defining the coefficient ratio $q/p$. Let $\Delta m$ be their mass splitting and use units $\hslash =1$.

For rates $f_{\pm }$ tagged as $B^0,{\bar{B}}^0$ and $\Delta t=t_f-t_{\mathrm{tag}}$, neglect the decay width difference and take $|q/p|=1$. DefineHere $A_f,{\bar{A}}_f$ are decay amplitudes. The tagged asymmetry is $(f_{+}-f_{-})/(f_{+}+f_{-})=S\sin (\Delta m\Delta t)-C\cos (\Delta m\Delta t)$ in this convention.^[ref]

For a single weak phase in a charmonium-kaon CP eigenstate with eigenvalue ${\eta }_f$, ${\lambda }_f\simeq {\eta }_f{e}^{-2i\beta }$ gives $S=-{\eta }_f\sin 2\beta$, $C\simeq 0$, withThis connects the CKM unitarity triangle to quantum interference between direct decay and decay after neutral meson mixing.^[ref]

For an independent teaching example choose $x=\Delta m\Delta t$, ${\lambda }_f=4i/3$ and an arbitrary common positive envelope $F(\Delta t)$. Then $S=24/25$, $C=-7/25$ andBecause $(24/25{)}^2+(7/25{)}^2=1$, both rates remain nonnegative. At $x=0$, the asymmetry is $7/25$; at $x=\pi /2$, it is $24/25$. These chosen values illustrate the sign convention and are not experimental estimates. More generally $S^2+C^2=1-[2\mathrm{Re}{\lambda }_f/(1+|{\lambda }_f{|}^2){]}^2\le 1$.

If a tag flips independently with probability $w$, exchanging $f_{+}$ and $f_{-}$ for that fraction of events gives $A_{\mathrm{observed}}=(1-2w)A_{\mathrm{tag}}$ by direct subtraction. The plotted toy choice $w=1/5$ reduces the amplitude to $3/5$ of its ideal value; at $w=1/2$ the tag contains no flavour information. Finite time resolution additionally averages over nearby oscillation phases. Extracting a weak phase consequently requires calibrated readout as well as a CKM matrix prediction.