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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 CKM matrix appears in the charged current of b. For a tree semileptonic transition, a schematic extraction is $\Gamma =G_F^2|V_{ij}{|}^2\times \text{phase space}\times \text{hadronic factor}\times (1+\text{corrections})$. Measure an absolute rate or differential spectrum, calculate or independently determine the hadronic factor, and solve for the modulus. The process is not a direct count of quark flavours in a detector: form factors, decay constants, nuclear structure and radiation can limit the result.

Rate moduli alone do not determine the complex phase. CP-asymmetric decay distributions and interference in neutral-meson systems, or tree-level $B\to DK$ interference, constrain the phase and the CKM unitarity triangle. Unitarity supplies consistency checks and indirect determinations, but a fit that imposes it must be distinguished from a direct measurement of an element.

The parton model treats a fast proton as an ensemble of approximately collinear, nearly free quarks and antiquarks. A hard deep inelastic scattering collision resolves one parton on a time scale short compared with its binding motion. The probabilities are parton distribution functions; the simplest approximation neglects intrinsic transverse momentum, quark masses and the proton mass in the hard collision. Quantum chromodynamics introduces logarithmic violations of Bjorken scaling, so the parton distribution functions depend on a resolution scale as well as a parton momentum fraction.

Write the incident electron and proton four-momenta as $k,P$, the scattered electron four-momentum as $k^{\prime }$, and set $q=k-k^{\prime }$. With the $(+---)$ Minkowski metric, the Bjorken scaling variable and the inelasticity areAn incident parton with four-momentum $\xi P$ emerges on shell, giving $(\xi P+q{)}^2\simeq 2\xi P\cdot q-Q^2=0$. Thus $\xi =x_B$: the observable Bjorken scaling variable measures the struck parton momentum fraction in this approximation.

For clarity, measure the scattering angle $\vartheta$ from the incoming electron direction, and denote the outgoing electron energy by $E_e^{\prime }$. Direct four-momentum products giveThe expression uses the known beam energies and two measured quantities. An electromagnetic calorimeter measures $E_e^{\prime }$; a tracking detector, the collision interaction vertex and the position of the electromagnetic shower determine $\vartheta$. Coverage close to the incoming electron direction matters at small $Q^2$. A hadronic calorimeter supplies an independent consistency check and rejects backgrounds. If the angle is instead measured from the incoming proton, its sine and cosine half-angle factors interchange.

At the parton level, the dominant valence reactions are $e^{-}u\to {\nu }_{e}d$ through W boson exchange and $e^{+}d\to {\bar{\nu }}_{e}u$ through W boson exchange. The first also acts on $c$, while the second acts on $s$ with the appropriate Cabibbo-Kobayashi-Maskawa matrix factors; crossing supplies the antiquark reactions. The lines in the figure show fermion number flow; the Positron and antineutrino arrows point opposite to their momenta.

For unpolarized beams, the leading charged current cross sections, suppressing Cabibbo-Kobayashi-Maskawa matrix factors, have the formHere each parton distribution function is evaluated at $(x,Q^2)$. The proton has two valence up quarks and one valence down quark; moreover, its valence contribution in the Positron reaction carries the chiral angular factor $(1-y{)}^2$. Both effects favour $e^{-}p$. Increasing $Q^2$ at fixed collider energy enforces $x\ge Q^2/s$, progressively suppressing the sea relative to the valence quarks; large $y$ further suppresses the Positron valence term. There is no universal factor of two valid at every $(x,y)$.

The escaping neutrino prevents direct reconstruction from a scattered charged lepton. Instead use the Jacquet-Blondel reconstruction. With the $+z$ axis along the proton beam, measure the complete hadronic final state and formThese relations follow from four-momentum conservation, with the unobserved neutrino transverse momentum opposite to the measured hadronic sum. A nearly hermetic hadronic calorimeter, supplemented by tracking detectors, must measure jets, their energies and their directions. Forward losses, initial-state radiation and small $y_h$ cause significant reconstruction uncertainties.

Represent a quark-antiquark state by a matrix $F$ with $q_i{\bar{q}}_j\leftrightarrow E_{ij}$. Under flavor symmetry, $F\mapsto UF{U}^{†}$, so an infinitesimal generator acts by $[T,F]$. The minus sign in this commutator is the antiquark transformation law; acting on the quark alone would miss states.

Start from ${\pi }^{+}=E_{ud}$. Three useful first steps areFurther actions give $[I_{-},E_{us}]=E_{ds}=K^0$, $[I_{-},E_{sd}]=-E_{su}=-K^{-}$ and $[I_{-},{\pi }^0]=\sqrt{2}{E}_{du}=\sqrt{2}{\pi }^{-}$. Overall phases are conventional. FinallyThus the second neutral octet state is obtained by removing the known ${\pi }^0$ component. Equivalently it is the normalized diagonal state orthogonal to both ${\pi }^0$ and the singlet. The full meson octet isTheir flavour matrices are traceless; the independent singlet is ${\eta }_1=(u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}$. The source uses ${\eta }^{\prime }$ for this ideal singlet. That notation is a basis convention here: the physical eta and eta prime mesons is not an exactly pure singlet because physical neutral pseudoscalar mesons mix.^[ref]

The Cabibbo-Kobayashi-Maskawa matrix is the unitary change of basis between down-type quark mass eigenstates and the combinations appearing in the weak charged current. In particular, $u\bar{d}$ and $u\bar{s}$ production carry $V_{ud}$ and $V_{us}$. Exclusive tau lepton decays into a charged pseudoscalar meson giveup to small radiative corrections. The pseudoscalar decay constants specify the vacuum-to-meson axial-current matrix elements, $⟨0|\bar{q}{\gamma }^{\mu }{\gamma }^{5}u|P(p)⟩=i{f}_P{p}^{\mu }$.

The observed branching fraction ratio is $0.70/10.83=0.06464$, and the kinematic factor is approximately $0.8622$. Hence the directly determined combination isWith an independent, representative lattice input $f_K/f_{\pi }\simeq 1.193$, this becomes $|V_{us}/V_{ud}|\simeq 0.2295$, close to the source's rounded expectation $0.224/0.975=0.2297$. Conversely that expectation and the hadronic input predict a branching fraction ratio near $0.0648$. Without the decay-constant input the table does not isolate a pure Cabibbo-Kobayashi-Maskawa matrix ratio. Setting $f_K=f_{\pi }$ would give approximately $0.0455$, demonstrating the importance of flavor symmetry breaking.^[ref]

Assuming that the two listed leptonic decay categories exhaust the leptonic modes, the inclusive hadronic branching fraction is $100-17.83-17.41=64.76\%$. ThusAt the simplest massless quark level, the W boson couples to $u\bar{d}$ and $u\bar{s}$ in three color charge states, so equal quark and lepton weak coupling constant values predictThis is a rough check of color multiplicity in a decay width and universal weak-current normalization. It is much less precise than the two-lepton ratio because hadronic Quantum chromodynamics and electroweak interaction corrections are sizeable: the data require a net correction factor near $3.632/3.0024\simeq 1.21$. They must be included before extracting a quark-to-lepton coupling constant ratio. It would be incorrect to interpret the excess over three directly as a failure of weak universality. The displayed exclusive hadronic modes alone would underestimate the inclusive sum.

Treat a nucleon as incoherent collinear spin-half quarks and antiquarks carrying fractions $x$ of its momentum. Neglect their masses and transverse motion, use a charged-current four-fermion interaction, and take $Q^2\ll m_W^2$. The subprocess energy is $\hat{s}\simeq xs$. Antineutrino scattering on a quark has the same selected-current helicity factor $(1-y{)}^2$ as in part a; crossing to an antiquark changes the allowed spin pairing and gives a flat distribution. Multiplying the elementary strength by the respective parton densities givesHere $q,\bar{q}$ are the effective weak flavour combinations with the isoscalar proton-neutron average and CKM weights absorbed into their definitions. With only light flavours and charge symmetry, the relevant up and down combinations coincide on an isoscalar target. Rescaling those density definitions requires rescaling the prefactor consistently.

The equation is a leading-order scaling result: it neglects logarithmic QCD evolution, heavy-quark thresholds, higher-twist and target-mass corrections, nuclear modifications, and weak-propagator suppression. The current produces ${\mu }^{+}$ for an incident muon antineutrino. At higher momentum transfer the weak propagator supplies a factor approximately $[m_W^2/(m_W^2+Q^2){]}^2$, and at low energy resonance and quasielastic scattering invalidate a pure DIS description.

The dominant semileptonic decay is $t\to b{W}^{+}\to b{\ell }^{+}{\nu }_{\ell }$. Replacing $b$ by $s$ or $d$ weights the amplitude by $V_{ts}$ or $V_{td}$ rather than $V_{tb}$. Neglecting the down-type masses,with the source's rounded matrix entries. For the $W$, three leptonic widths and two open up-type quark rows with three colours give nine common width units. Thus, ignoring QCD and final-mass corrections,Summing all three flavours gives approximately one third; summing only electron and muon modes gives approximately $22.2\%$. These are per-top branching ratios, not the separate branching ratio of a specified two-top event topology.

An isolated high-transverse-momentum charged lepton, missing transverse momentum from the neutrino, and a tagged bottom jet form a much cleaner signature than an entirely hadronic final state amid QCD jets. Electrons and muons are the easiest charged-lepton flavours to identify cleanly. Tau decays add neutrinos and often hadrons, making identification and reconstruction more difficult. The figure contains the quark/lepton-level charged-current diagram as well as the rare-loop examples used below.

The weak charged current couples an up-type quark to the rotated down field $d_{iL}^{\prime }={\sum }_j{V}_{ij}{d}_{jL}$. A neutral current is flavour universal within equal-charge quarks. Rotating it into the mass basis therefore givesThe off-diagonal terms vanish by CKM unitarity, so no tree-level flavour-changing neutral current is generated. In the two-generation description, the orthogonal strange-charm partner completes the rotation whose single down-strange combination alone would have produced an off-diagonal neutral coupling. The complete multiplet supplies the cancellation.

At loop level, a flavour-changing amplitude has the structureIf all internal masses were equal, $F$ would be common and ${\sum }_i{V}_{ai}{V}_{bi}^{\ast }=0$. Subtracting an arbitrary common value makes the suppression explicit:This is the Glashow-Iliopoulos-Maiani mechanism. Unequal masses leave small loop-level flavour-changing neutral currents, so the statement that they are never observed would be too strong: the mechanism suppresses them and forbids them at tree level. The cancellation requires complex CKM products; the rounded table of absolute values alone must not be summed as if every phase were positive.

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.

Assume only Standard Model decay channels, negligible final-state masses, three lepton families, three quark colours and a unitary CKM matrix. The top-quark channels are kinematically closed. Summing the allowed quark channels gives $3{\sum }_{i=u,c}{\sum }_{j=d,s,b}|V_{ij}{|}^2\simeq 6$ times one leptonic width, while the leptons contribute three. Neglecting QCD and electroweak radiative corrections,The source's rounded CKM entries give almost the same factor; their tiny nonunitarity is not physically significant. ConsequentlyThe propagated uncertainty uses $\delta g/g=\frac{1}{2}\delta {\Gamma }_W/{\Gamma }_W$, taking $m_W$ as fixed. It is the measurement uncertainty within the stated tree-level approximation; omitted radiative corrections are an additional modelling limitation. In particular QCD increases the hadronic width and lowers the inferred coupling when included.

The printed electromagnetic comparison $4\pi /137$ is $e^2$, not the probability amplitude coupling $e$. Compare like quantities:Equivalently ${\alpha }_W=g^2/(4\pi )\simeq 0.0347$ exceeds ${\alpha }_{\mathrm{EM}}\simeq 1/137$. The weak interaction is weak at low energy primarily because of the massive propagator, rather than an exceptionally small fundamental coupling.

The valence quark distributions describe net quark content: $\int u_v(x)dx=2$ and $\int d_v(x)dx=1$ for a proton. Their momentum densities $x{u}_v,x{d}_v$ are small at very small $x$, peak at moderate momentum fractions, and vanish toward $x=1$, where one parton carrying almost all proton momentum is unlikely. The larger $u$ distribution reflects two valence up quarks and its comparatively harder shape; its pointwise value need not be exactly twice the down distribution.

The sea quark distribution is generated largely by gluon radiation and $g\to q\bar{q}$ splitting. It grows strongly toward small $x$ and falls at large $x$. The plotted sea curve is scaled by $0.05$, so it must be multiplied by twenty before numerical use. The total up-quark PDF isusing the valence-up distribution and the up-sea distribution. The summed sea curve alone does not fix $\bar{u}$ without a flavour assumption. Under the later stated four-flavour symmetry, $\bar{u}=\Sigma /8$.

Assume the usual semileptonic flavour selection rule $\Delta S=\Delta Q$, negligible wrong-sign decay probability amplitudes, and equal magnitudes for CP-conjugate allowed probability amplitudes. Then $f_{+}={\pi }^{-}{\ell }^{+}\nu$ is reached directly only from $K^0$, and $f_{-}={\pi }^{+}{\ell }^{-}\bar{\nu }$ only from ${\bar{K}}^0$. Put $|A_{f_{+}}|=|{\bar{A}}_{f_{-}}|=|A|$. The four rates areThere is only one path to each tagged semileptonic channel, so no direct/mixed decay interference survives. The semileptonic wrong-sign mixing asymmetry isIt isolates CP violation in mixing under the stated assumptions. Allowing direct semileptonic CP violation or wrong-sign probability amplitudes would spoil that isolation.^[ref]

The charged-current process iswith a $t$-channel $W^{-}$ emitted at the electron-to-neutrino vertex and absorbed at the antimuon-to-antineutrino vertex. It is not an $s$-channel Annihilation of the unlike-flavour charged leptons into one $W$.

At high momentum transfer the incident electron's left-chiral fraction tends to $4/5$. In the collision centre-of-mass frame, the incoming antimuon moves along $-z$ and its charged-current allowed right helicity has spin along $-z$. A rest-frame spin at Angle $\theta$ to $+z$ projects onto that spin direction with probability ${\sin }^2(\theta /2)$. Therefore, if ${\sigma }_0$ denotes the cross section for a pure left-helicity electron and fully selected antimuon spin,It vanishes at $\theta =0$ in the ultrarelativistic selection limit and is maximal at $\theta =\pi$. Finite-mass corrections can fill the suppressed orientation. The absolute normalization depends on beam energy and the exchanged-boson propagator; the spin-angle dependence is what is fixed here.

A proton contains up quarks on which the charged current can act:through $W^{-}$ exchange, weighted by $V_{ud}$. Spectator quarks complete the hadronic final state. At the exclusive nucleon level the charge assignment is $e^{-}p\to {\nu }_{e}n$; at the high momentum transfers specified, the generally dominant description is inclusive $e^{-}p\to {\nu }_{e}X$ with proton structure and hadronic breakup, not an unsuppressed pointlike proton-to-neutron transition. The quark-level diagram in the figure displays this allowed charge flow. The electron's left-chiral selection is the same as in part d.

Neglect the participating quark and lepton masses, so that the weak charged current selects a left-helicity $u$ and a right-helicity $\bar{d}$. Choose $+z$ along the proton beam in the partonic centre-of-momentum frame. Then the up quark moves along $+z$ with spin projection $-1/2$, while the down antiquark moves along $-z$ with helicity $+1/2$ and also has spin projection $-1/2$ along $+z$. ThusThe collinear two-body initial state has no orbital projection about its collision axis. In this approximation the produced W boson therefore has the indicated transverse polarization; its longitudinal spin state is absent. The unpolarized hadron beams contain other constituent spin states, but these do not participate in this massless charged-current subprocess.

In the final state the Positron has helicity $+1/2$ and the electron neutrino has helicity $-1/2$. Along the outgoing charged-lepton direction their combined angular-momentum projection is $+1$, because the oppositely moving neutrino has spin projection $+1/2$ on that axis. Finite masses, noncollinear production and sea contributions modify the ideal configuration.

Use the orthonormal basis $|u\bar{u}⟩,|d\bar{d}⟩,|s\bar{s}⟩$. The normalized flavour singlet isThe SU(3) invariant contraction pairs each fundamental flavour with its conjugate. Applying any supplied raising operator to the singlet gives zero because its quark and antiquark contributions cancel.

To fix the neutral isospin triplet phase consistently with the source's charged states, apply $I_{+}$ to $-|d\bar{u}⟩$:Thus one missing member of the meson octet isIndeed $I_{+}|T_0⟩=\sqrt{2}|u\bar{d}⟩$, the correct ladder operator strength for the $I=1$ triplet.

The other missing state is orthogonal to both $|1⟩$ and $|T_0⟩$. For coefficients $(a,b,c)$, orthogonality requires $a=b$ and $a+b+c=0$. Choose the convenient overall phase $c>0$ and normalize:It can also be obtained by adding the supplied ladder operations $V_{+}[-|s\bar{u}⟩]=|s\bar{s}⟩-|u\bar{u}⟩$ and $U_{+}[-|s\bar{d}⟩]=|s\bar{s}⟩-|d\bar{d}⟩$. The three displayed states are mutually orthonormal. The meson octet here is the traceless adjoint component of $\bar{3}\otimes 3$; the source's word “antisymmetric” does not impose antisymmetry on all its quark-antiquark flavour wavefunctions.

The observed dominance of kaons supports a predominantly $s\bar{s}$ phi meson. In a connected strong decay, the original strange quark and antiquark survive into different kaons, while a gluon produces a light $q\bar{q}$ pair:This gives $K^{-}{K}^{+}$ or ${\bar{K}}^0{K}^0$ and is allowed by the OZI rule. A pionic final state contains no strange valence constituents, so the initial $s\bar{s}$ pair must annihilate. The strong annihilation of a colour-singlet $1^{--}$ state requires at least three gluons: one gluon cannot carry the singlet colour, and two have the wrong charge-conjugation quantum number. This disconnected flavour topology is suppressed by the OZI rule, although it need not vanish.

The leptonic mode proceeds through $s\bar{s}\to {\gamma }^{\ast }\to {\ell }^{+}{\ell }^{-}$ and is suppressed by electromagnetic couplings; its width is proportional to ${\alpha }^2$ times the vector-meson overlap. Unlike a strong hadronic decay, it cannot create its final particles through quark rearrangement. Hence a branching fraction of order $10^{-4}$ is qualitatively consistent with electromagnetic annihilation competing with strong modes.

Kaon dominance is especially suggestive because phase space is limited: using the supplied neutral-kaon and phi masses,The two spin-zero kaons have orbital angular momentum $L=1$ to match the vector parent, so their partial width has the additional near-threshold factor $p_K^3$. Pionic modes have more phase space yet remain subdominant because of their annihilation topology. The observations therefore support, but do not establish exact, ideal vector-meson mixing: a small nonstrange admixture, electromagnetic amplitudes and nonperturbative strong dynamics can all contribute to the pionic rate.

Use $B^0=d\bar{b}$ and ${\bar{B}}^0=\bar{d}b$. A box Feynman diagram with two W bosons and internal up-type quarks changes one neutral flavour into the other. The top quark contribution dominates the dispersive mixing amplitude and carries the squared CKM combination involving $V_{td},V_{tb}$. With a consistent meson phase convention,The illustration in b shows this box and the dominant decay topology. A suitable common final state is $f=J/\psi K_S$, reconstructed for example through $J/\psi \to {\mu }^{+}{\mu }^{-}$ and $K_S\to {\pi }^{+}{\pi }^{-}$. The leading $\bar{b}\to \bar{c}c\bar{s}$ tree transition gives $B^0\to J/\psi K^0$, while the conjugate gives ${\bar{B}}^0\to J/\psi {\bar{K}}^0$. Neutral kaon mixing projects both kaon flavours onto the observed $K_S$.

The final state is approximately a CP eigenstate with ${\eta }_f=-1$: $J/\psi$ and $K_S$ each have CP approximately $+1$, and their relative orbital angular momentum is $L=1$ to couple the vector charmonium to the spin-zero parent. Neglecting the small kaon CP impurity and subleading decay amplitudes,There are two indistinguishable paths, direct decay and mixing followed by decay, so their amplitudes interfere. For $\Delta {\Gamma }_d\simeq 0$ and negligible direct CP violation, the time-dependent neutral-B CP asymmetry isMeasure the production flavour and proper decay time, fit the oscillation coefficient, and extract $\sin 2\beta$. This determines $\beta$ up to the usual trigonometric ambiguities unless further information is used. The cancellation of the dominant decay phase is why this channel isolates the angle involving $V_{td}$ without needing $V_{ub}$.