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A meson containing a bottom quark or its antiquark partner. The neutral pair $B^0=d\bar{b}$ and ${\bar{B}}^0=\bar{d}b$ undergoes neutral meson mixing.

Bottom-quark tagging identifies jets compatible with a bottom hadron, often through displaced tracks, a secondary decay vertex or a lepton within the jet. The finite hadron lifetime gives the displacement. Efficiencies and mistaken tags from charm or light jets must be calibrated; a top quark itself decays before hadronization.

The Cabibbo-Kobayashi-Maskawa matrix $V=U_{uL}^{†}{U}_{dL}$ compares the left-handed unitary matrices that diagonalize the up-type and down-type Yukawa matrices. Its rows label up quarks, charm quarks and top quarks; its columns label down quarks, strange quarks and bottom quarks. In the weak charged current, the coefficient of ${\bar{u}}_{Li}{\gamma }^{\mu }{d}_{Lj}{W}_{\mu }^{+}$ is $V_{ij}$. A single CKM matrix therefore connects quark masses, flavour-changing decay amplitudes, and CP violation.

Write $s_{ij}=\sin {\theta }_{ij}$ and $c_{ij}=\cos {\theta }_{ij}$. An exactly unitary matrix with three mixing angles and one CP-violating phase isThis convention agrees with Makoto Kobayashi's account of three-generation quark mixing.^[ref]

The Christenson-Cronin-Fitch-Turlay kaon CP violation experiment established the phenomenon that required an explanation. Makoto Kobayashi and Toshihide Maskawa showed that two-generation quark mixing cannot supply it, whereas adding a third quark generation permits an irreducible weak phase. Their 1973 paper also considered additional fields as alternative mechanisms; the three-generation conclusion concerns CP violation from the minimal unitary quark mixing mechanism.^[ref]

A common weak phase makes CP violation across mesons testable through the BABAR time-dependent CP violation experiment. It does not explain the cosmic baryon asymmetry.^[ref]

For $q=u,c$ and a neutral photon or Z boson, the loop runs over down-type quarks. CKM unitarity cancels a common mass-independent loop contribution. The large bottom mass relative to down and strange masses normally makes the surviving bottom contribution largest, and $|V_{cb}|>|V_{ub}|$ favours final charm. Adding a resolved photon to a $qZ$ final state introduces another electromagnetic coupling and three-body phase space without removing the flavour cancellation. Some diagrams use the quartic gauge vertex, so the number of vertices need not increase.

Forbidden by electric-charge conservation. The initial bottom quark has electric charge $-1/3$, while the displayed final electric charges sum to $2/3+1+0=5/3$. Therefore no Standard Model Feynman diagram, at tree level or any loop order, can realize this channel.

This is an intentionally disallowed process, not a transcription correction. The charge-conserving semileptonic bottom decay instead has an electron and an antineutrino of electron flavour, while a Positron accompanies the conjugate antiquark decay. Replacing the displayed particles silently would change the requested classification.

A leading short-distance Standard Model contribution to neutral D-meson mixing is a box Feynman diagram with two W bosons and two internal down-type quark lines:

The incoming $c\bar{u}$ pair becomes $u\bar{c}$; the internal labels run over the down quark, strange quark and bottom quark. Each corner is a weak charged current vertex. Four such vertices make the contribution of order $g^4$. Summing the internal flavours produces Cabibbo-Kobayashi-Maskawa matrix factors and the Glashow-Iliopoulos-Maiani mechanism: the flavour-independent term cancels by ${\sum }_i{V}_{ci}^{\ast }{V}_{ui}=0$. This box Feynman diagram is one contribution; long-distance intermediate hadron states can also contribute to neutral D-meson mixing.

Use an inclusive spectator model: one heavy constituent decays through the Fermi interaction, and sum the open charged-current channels. In weak-decay channel counting, each massless final lepton pair gives basic widthA quark pair contributes three colors multiplied by its squared Cabibbo-Kobayashi-Maskawa matrix element. Do not multiply by three just because the initial quark has three colors: its lifetime is the width of one initial state, averaged over initial color when appropriate. Define

For the tau lepton, two leptonic channels contain an electron or a muon; light hadronic channels have $d\bar{u}$ or $s\bar{u}$. ThereforeNo charmed hadron can be produced: the lightest charmed meson exceeds the tau mass. ^[ref] The supplied charm-quark parameter is not a free asymptotic final-particle mass. The leading count neglects electron and muon masses and light hadronic threshold effects.

For the charm quark, $c\to s{W}^{+\ast }$ and $c\to d{W}^{+\ast }$ dominate. The emitted current has two light leptonic channels and the same light hadronic channels as above; a tau cannot be produced:

For the bottom quark, retain $b\to c{W}^{-\ast }$ with the supplied $V_{cb}$ and neglect the much smaller $b\to u$ channel. The emitted current can produce three lepton flavors and two up-type quark families, with three colors for each family. In particular $b\to c\bar{c}s$ is open at the given masses, although suppressed by phase space. Counting each open channel as massless givesSetting Cabibbo sums to one recovers the familiar counts $5$, $5$ and $9|V_{cb}{|}^2$. Despite its larger mass, the bottom quark is slowed by its small interfamily mixing factor.

These are leading estimates rather than precise lifetimes of free quarks. confinement makes charm and bottom observable in hadrons, whose lifetimes involve spectator interactions, Quantum chromodynamics corrections and bound-state effects. mass corrections are significant for bottom: even the single-charm channel with massless leptons has the three-body weak-decay mass correctionTau and double-charm channels have different suppressions, so multiplying the entire width by this one factor is inconsistent. Accurate hadron lifetimes require those channel-dependent factors and hadronic dynamics; the exercise supports the explicitly stated leading counts.

The nuclear determination of $|V_{ud}|$. A $0^{+}\to 0^{+}$ beta decay transition has no nuclear parity change and can proceed with zero lepton orbital angular momentum. The allowed scalar operator gives a Fermi transition; the rank-one Gamow-Teller transition operator cannot connect total nuclear spins zero to zero. The strong isobaric-analog transitions used for precision measurements are superallowed beta decays. A generic $0^{+}\to 0^{+}$ assignment alone would not guarantee superallowed strength if the states were nonanalog.

At the quark level the weak charged current converts $d\leftrightarrow u$ and carries $V_{ud}$ in the amplitude. The nuclear Fermi operator is an isospin ladder operator, givingHere $f$ denotes the known phase space dependence, with the dimensional factors included in the proportionality constant. In the isospin-symmetric limit, $M_F$ between analog states is fixed by isospin algebra, for example $|M_F{|}^2=2$ for the usual $T=1$ analog transitions. A measured partial half-life and the computed phase space factor give the comparative half-life; after radiative corrections and isospin breaking corrections, its inverse determines $|V_{ud}{|}^2$. The decay rate measures a magnitude squared, not the complex phase of $V_{ud}$.

Ds decay diagrams and CKM comparisons. The original PDF has $K^{+}{\bar{K}}^0$, with ${\bar{K}}^0=s\bar{d}$. The local TeX drops that overbar; using $K^0=d\bar{s}$ instead would give a different flavour channel. Use the PDF's neutral antikaon throughout.

In the representative Ds nonleptonic tree amplitudes, the Ds meson spectator is $\bar{s}$. The elementary weak transitions are

In the first two channels, the produced $u$ joins the spectator $\bar{s}$ to form $K^{+}$; the other quark joins $\bar{d}$. For the first channel this uses the $-d\bar{d}/\sqrt{2}$ component of the neutral pion. In the third, $s\bar{s}$ forms the phi meson and $u\bar{d}$ forms ${\pi }^{+}$. The two possible final pairings have different colour contractions and hadronic matrix elements even when their weak CKM factors coincide.

The purely CKM rate suppression of $c\to d$ relative to $c\to s$ isThus $K^{+}{\pi }^0$ is singly Cabibbo suppressed, while $K^{+}{\bar{K}}^0$ and $\varphi {\pi }^{+}$ are Cabibbo-favored at this level. If reduced hadronic strengths and phase space effects were equal, including the neutral-pion component would giveThe supplied branching fractions instead giveThe first observed ratio is $0.814$ times the neutral-pion-corrected simple estimate, requiring only an order-one remaining hadronic and phase space factor. Relative to $\varphi {\pi }^{+}$, that remaining factor is $0.525$. The two Cabibbo-favored channels differ by a factor $0.644$ despite equal CKM magnitudes. The CKM hierarchy therefore explains why one channel is much rarer, but it does not calculate the three exclusive branching fractions; spin, phase space, color charge and strong interaction final-state effects also matter.

What spin and parity establish. In either pseudoscalar-pseudoscalar final state, the total intrinsic spin is zero. Conservation of angular momentum for a spin-zero parent forces $L=0$, so the final parity isopposite to the parent $0^{-}$ Ds meson. ConsequentlyFor a vector meson plus a pion, the intrinsic final spin is one. Coupling to total $J=0$ requires $L=1$, giving $P_f=(-1)(-1)(-1{)}^1=-1$, equal to the parent parity. Thus spin and parity alone permit a parity-conserving $\varphi {\pi }^{+}$ transition. That is the limited sense in which it is not necessarily weak from this information alone. In reality it is also weak: the initial charm flavour disappears, which the strong interaction and electromagnetic interaction conserve.

W branching fractions. In the massless-daughter, tree-level estimate, each leptonic channel contributes one common unit of decay width. Each accessible quark pair contributes $3|V_{ij}{|}^2$ because of the three final colours. The charged-current W branching-fraction counting denominator isOnly $u$ and $c$ are accessible up-type quarks; the large $V_{tb}$ is irrelevant because a W boson cannot decay into the heavier top quark. HenceA final bottom quark arises in $W^{-}\to b\bar{u}$ or $b\bar{c}$, soThis counts the primary decay channels; finite daughter masses and QCD corrections refine the approximation.

The requested top-lifetime estimate and its limitation. Under the instructed equal-coefficient Sargent's rule scaling, $\Gamma \propto m^5$ and the mean lifetime is inversely proportional to that fifth power. Converting both masses to GeV givesThis is the requested order-of-magnitude estimate, assuming the same effective coefficient for the two total decay rates. It corresponds to $c\tau \simeq 54\mathrm{fm}$ and is longer than the hadronization time, roughly a few $10^{-24}\mathrm{s}$. Taken literally, it would allow top-flavored hadrons to form; it does not establish the physical statement that the top decays before hadronizing.

The extrapolation fails because Sargent's rule uses a low-energy Fermi interaction approximation, whereas $m_t>m_W+m_b$ permits $t\to bW$ with an on-shell W boson. The physical leading decay width instead scales asUsing $m_W\simeq 80.4\mathrm{GeV}$, the Fermi constant $G_F=1.16638\times 10^{-5}\mathrm{Ge}{\mathrm{V}}^{-2}$ and $|V_{tb}|=0.999$ gives about $1.55\mathrm{GeV}$ at the supplied top mass. Then ${\tau }_t=\hslash /{\Gamma }_t$, with $\hslash =6.58212\times 10^{-25}\mathrm{GeV}\mathrm{s}$, gives ${\tau }_t\simeq 4.2\times 10^{-25}\mathrm{s}$. This corrected regime explains top-quark decay before hadronization.^[ref]

The bottom quark has charge $-1/3$ and left-handed weak isospin $-1/2$, requiring a charge-$+2/3$ partner in a complete Standard Model weak doublet. Completing the third quark generation also maintains the quark/lepton cancellation of gauge anomalies. A three-generation CKM matrix permits an irreducible CP violation phase. Flavour constraints and loop corrections, including their strong top-mass dependence in electroweak precision observables, provided indirect evidence before direct production. These arguments identify the required partner and its interactions; they do not by themselves fix its mass.

The Tevatron collided protons and antiprotons at $\sqrt{s}=1.8\mathrm{TeV}$. Quantum chromodynamics produces $t\bar{t}$, mainly through $q\bar{q}$ annihilation there. Each top quark decays predominantly to $Wb$ before hadronization, giving $W^{+}{W}^{-}b\bar{b}$. The dilepton channel has two isolated oppositely charged leptons, missing transverse momentum from neutrinos, and bottom jets. The single-lepton channel has an isolated electron or muon, missing transverse momentum, and four leading jets, two consistent with bottom decay. The all-hadronic channel has more background. Secondary-vertex and soft-lepton tags identify bottom hadrons.

CDF and D0 reported independent excesses in 1995, with event kinematics and reconstructed masses consistent with a common very heavy quark. Fits assign jets to the two decay chains and compare with expected $W+$jets, multijet and instrumental backgrounds. Agreement of counts, bottom content and reconstruction across channels is stronger evidence than an excess in one event class alone. ^{[ref] [ref]}

The literal $t\to qZ\gamma$ final state has a three-vertex loop using the Standard Model quartic vertex $W^{+}{W}^{-}Z\gamma$. The two other vertices are $t\to d_i{W}^{+}$ and $d_i{W}^{+}\to q$, with $q=u,c$. The internal down-type quark joins those two vertices, while two charged-vector propagators join them through the quartic vertex emitting both neutral bosons. The bottom panel of the figure shows this topology, satisfying the three-vertex hint without dropping the photon. Its additional electromagnetic coupling is contained in the four-leg vertex; counting vertices alone does not determine coupling order.

The companion single-$Z$ penguin diagram illustrates $t\to qZ$: it instead attaches the neutral boson to the internal quark line and has three cubic vertices. These are different final states. A complete gauge-invariant amplitude requires all companion attachments; for $qZ\gamma$ these include photon radiation from charged lines as well as the quartic graph.

Both flavour sums have the formwhere the loop function also depends on the final-state kinematics. After cancellation of a mass-independent part by GIM mechanism, the bottom-quark mass gives the largest surviving contribution: $m_b^2\gg m_s^2,m_d^2$, while $|V_{tb}{V}_{cb}|\simeq 0.042$ is not unusually small compared with the strange combination. The larger $|V_{cb}|$ than $|V_{ub}|$ then favours final charm. Schematically,The same flavour preference applies to the separate two-body modes $t\to cZ$ and $t\to c\gamma$. This estimates a relative flavour preference, not an absolute branching ratio. Loop factors and mass-difference suppression remain, so these channels are fully compatible with GIM. A resolved extra photon also introduces another electromagnetic coupling and three-body phase space; an experimental $Z\gamma$ definition needs photon-energy and isolation cuts to distinguish hard radiation from the soft limit.

Away from narrow resonances and quark thresholds, the QED annihilation cross section into each accessible quark flavour is its squared fractional electric charge times the muon-pair cross section, multiplied by the number $N_c$ of distinguishable color charges. Thus the inclusive hadronic ratio isWith $N_c=3$, the continuum predictions are $R_{uds}=2$, $R_{udsc}=10/3$ and $R_{udscb}=11/3$ as successive charm quark and bottom quark channels become accessible. The measured normalization and steps support three color charges. At leading order these hadronic events form two back-to-back jets from the quark and antiquark; quantitative comparisons include QCD corrections and avoid the nonperturbative threshold region.

The extra resolved jet is explained by $e^{+}{e}^{-}\to {\gamma }^{\ast }\to q\bar{q}g$, with gluon radiation from either the quark or antiquark. One possible color charge routing is initial $q_r{\bar{q}}_{\bar{r}}$ followed by $q_r\to q_b+g_{r\bar{b}}$; the final $q_b$, ${\bar{q}}_{\bar{r}}$ and $g_{r\bar{b}}$ combine to a net colour singlet.

The three-jet cross section contains one additional strong interaction vertex in the amplitude, so its ratio to the two-jet rate is of order ${\alpha }_s=g_s^2/(4\pi )$ times a resolution-dependent kinematic coefficient. A resolved fraction of order $15\%$ is consistent with perturbative QCD and a strong coupling constant much larger than the fine-structure constant. It is not an equality ${\alpha }_s=0.15$: the jet definition and cuts are needed. Photon radiation has a much smaller electromagnetic rate. The large hadronic third-jet rate supports gluon radiation; identifying the radiated object specifically as a colour-octet gluon also uses the QCD vertex and colour-flow description, not merely the existence of three visible jets.

Here $x=m_W^2/m_t^2$, and the tree-level decay width neglects the bottom quark mass. The open two-body W channel gives a top quark lifetime of a few $10^{-25}\mathrm{s}$, shorter than the hadronization time. A fifth-power Sargent's rule extrapolation misses this physical regime.^[ref]

Here $x=m_W^2/m_t^2$, the bottom quark mass is neglected, and the W boson is on shell. The weak charged current gives the spin-averaged squared scattering amplitude $g_W^2|V_{tb}{|}^2(m_t^2-m_W^2)(m_t^2+2m_W^2)/(4m_W^2)$. The two-body Lorentz-invariant phase space has $p^{\ast }=(m_t^2-m_W^2)/(2m_t)$; multiplying by $p^{\ast }/(8\pi m_t^2)$ gives the displayed decay width. Its longitudinal-vector contribution explains the large-$m_t$ cubic growth, and its threshold factor makes the width vanish as $m_t\downarrow m_W$. The formula requires $m_t>m_W$ and receives finite-mass and radiative corrections.

A bottom quark-antiquark bound state. Its $4S$ resonance provides a source of B meson pairs.