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Below the strange-quark threshold, the gauge-field and two-flavour quark terms arewhereThis is the low-energy quantum chromodynamics plus electromagnetism Lagrangian for the light flavours.

Every renormalizable gauge-invariant term contains equal numbers of quark and antiquark fields. The common phase symmetryis therefore automatic rather than imposed in constructing the Lagrangian. Its conserved charge is baryon number, making $U(1{)}_B$ an accidental symmetry of the renormalizable low-energy theory.

The strong-coupling scale $\Lambda$ is defined as the scale at which the one-loop inverse coupling vanishes. Setting $g^{-2}(\Lambda )=0$ givesEquivalently, in terms of the coupling measured at any scale $\mu$,This dimensional transmutation defines ${\Lambda }_{\mathrm{QCD}}$ for quantum chromodynamics.

For $q=(u,d{)}^T$, massless two-flavour quantum chromodynamics hasClassically its continuous global symmetry is $SU(2{)}_L\times SU(2{)}_R\times U(1{)}_V\times U(1{)}_A$, up to finite central quotients. The Adler-Bell-Jackiw anomaly destroys the continuous axial $U(1{)}_A$ in the quantum theory, leaving $SU(2{)}_L\times SU(2{)}_R\times U(1{)}_V$, together with an anomaly-preserved discrete axial subgroup.

All three interactions respect color charge; quark flavor is conserved by the strong interaction and the electromagnetic interaction, but can change in the weak interaction. These statements concern charges and net numbers, rather than an unchanging inventory of particles.

A quark transforms in the fundamental representation of the color charge gauge group $SU(3{)}_C$. The strong interaction can change an individual quark's colour by exchanging a gluon, which itself carries color charge. For example, the colour flow $q_r\to q_b+g_{r\bar{b}}$ is allowed: demanding that the outgoing quark remain red would be incorrect. At each interaction vertex the full colour flow is conserved. The colourless photon, W boson, and Z boson instead couple to a quark with an identity factor ${\delta }_{ab}$ in colour space. A weak charged current can therefore change $d_r$ to $u_r$, but cannot turn $d_r$ into $u_b$ without another coloured participant. External isolated hadrons are colour singlets; confinement prevents direct observation of a free coloured state.

For each quark flavor $f$, define the net quark flavor number $F_f=N_f-N_{\bar{f}}$. The strong interaction and electromagnetic interaction preserve every $F_f$: pair creation $g\to s\bar{s}$, for example, changes neither net strangeness nor any other $F_f$. Their quark couplings are diagonal in quark flavor. The weak charged current contains ${\bar{u}}_i{\gamma }^{\mu }(1-{\gamma }^5)V_{ij}{d}_j$, where $V$ is the Cabibbo-Kobayashi-Maskawa matrix, and changes these net numbers. Weak neutral currents are diagonal in tree-level Feynman diagrams; flavor-changing neutral currents can occur through higher-order Feynman diagrams.

The observed hierarchy of mean lifetimes and decay channels supplies evidence: associated strange-particle production through the strong interaction preserves net strangeness, whereas a kaon's weak interaction decay to pions changes it. The electromagnetic interaction process ${\pi }^0\to \gamma \gamma$ has zero net quark flavor before and after annihilation. Neutrino oscillations show that individual lepton family numbers are not exact conserved quantities. Within the usual effectively massless-neutrino approximation, charged current decays such as ${\mu }^{-}\to e^{-}{\bar{\nu }}_e{\nu }_{\mu }$ preserve family numbers only when the emitted neutrinos are counted.

Evidence for three colours includes the hadronic R ratio $R=3{\sum }_f{Q}_f^2$, the colour multiplicities in W boson decay widths, and three-jet events revealing gluon radiation. These test quantum chromodynamics and its colour structure together; they are not direct measurements of a named red or blue charge on a free quark.

The ideal two-shower event fraction is $B_e^2$, approximately $4\%$ in simple channel counting or $3.18\%$ using the measured tau lepton electronic branching fraction. The selected particle decay chain isThe electron and Positron each make a contained electromagnetic shower; muons penetrate, and generic hadrons do not deposit all their energy as a contained electromagnetic shower.

To estimate $B_e$, use lepton universality and neglect final masses and quantum chromodynamics corrections. The electronic and muonic decay widths each carry weight one. The accessible inclusive hadronic channels have weights $3|V_{ud}{|}^2$ and $3|V_{us}{|}^2$, where three is the color multiplicity in a decay width. The charm quark is inaccessible in physical tau lepton hadronic decays. With $|V_{ud}{|}^2+|V_{us}{|}^2\simeq 1$, the total weight is $1+1+3=5$, givingThe 2021 measured value $B_e=0.1782$ instead gives $B_e^2=0.0317552$.^[ref] The difference reflects effects omitted by the simple free-channel estimate, especially the hadronic contribution to the total decay width.

There are two tau leptons per produced pair and two per selected pair, so the fraction of all produced tau leptons belonging to these two-shower events is also $B_e^2$. The unrestricted fraction of individual tau leptons decaying electronically is $B_e$; these are different selections. A real high-energy shower cut gives an additional acceptance factor: $N_{\mathrm{selected}}/N_{\mathrm{pairs}}=B_e^2{\epsilon }_{ee}$, which cannot be computed without the cut and detector response. No factor of two is needed for the unique $e^{+}{e}^{-}$ assignment.

For a purely electromagnetic interaction continuum with five active quark flavors, the usual estimate is $N_{e^{+}{e}^{-}}/N_{\mathrm{had}}=3/11$. The unspecified particle resonance and the word “total” require qualifications before this can be treated as a unique physical prediction.

In natural units, neglecting final masses, single-photon annihilation givesHere $\alpha$ is the fine-structure constant, $Q_f$ the quark electric charge in elementary-charge units, and the factor three counts orthogonal colour final states. Consequently the inverse hadronic R ratio isWith $u,d,s,c,b$ accessible, the denominator is $3[2(2/3{)}^2+3(1/3{)}^2]=11/3$. The corresponding values for $u,d,s$ and $u,d,s,c$ are $1/2$ and $3/10$. Inclusive quantum chromodynamics corrections increase the hadronic denominator, approximately by $1+{\alpha }_s/\pi$ away from thresholds and particle resonances. These are annihilation ratios before selection and detection efficiency corrections, not ratios of uncorrected detector counts.

A future muon-collider Higgs resonance is a plausible reading of part (c), since its narrow decay width makes a detuning of a few $\mathrm{MeV}$ relevant. Even then, two further assumptions are necessary. First, being a few $\mathrm{MeV}$ below the Higgs boson peak does not eliminate it: for detuning $\Delta =\sqrt{s}-m_H$, the ideal narrow Breit-Wigner resonance cross section retains the relative factorFor ${\Gamma }_H=4\mathrm{MeV}$ and $\Delta =-4\mathrm{MeV}$ this is $1/5$, not zero. Resonant hadronic production can still change the requested denominator substantially.

Second, at $\sqrt{s}\simeq 125\mathrm{GeV}$ the Z boson continuum is not negligible. A more complete annihilation estimate including both tree-level Feynman diagrams usesHere $s_W^2$ is the squared sine of the weak mixing angle, $T_3^f$ is weak isospin, and $N_c^f$ is one for a lepton and three for a quark. Thus the continuum ratio is $F_e/[3(2F_u+3F_d)]$, rather than necessarily $3/11$. For example, $s_W^2=0.231$, $m_Z=91.1876\mathrm{GeV}$ and ${\Gamma }_Z=2.4952\mathrm{GeV}$ give approximately $0.102$ at $125\mathrm{GeV}$ before quantum chromodynamics corrections.

If the unspecified particle resonance instead means the Z boson, detuning by a few $\mathrm{MeV}$ is tiny compared with its $2.5\mathrm{GeV}$ decay width. The resonant ratio remains approximately $\Gamma (Z\to e^{+}{e}^{-})/\Gamma (Z\to \mathrm{hadrons})=1/20.767\simeq 0.0482$.^[ref] This is a concrete counterexample to treating any slight downward detuning as permission to discard the particle resonance. The source supplies neither its identity nor the detuning and selection needed for a unique all-process ratio. The boxed continuum formula is the charge-counting answer under its stated approximation; a literal total must also include remaining resonant and other accepted hadronic channels.

For a three-body leptonic decay through the Fermi interaction, neglecting daughter masses and radiative corrections, lepton universality givesin natural units. Therefore scaling the muon decay rate by $m_{\tau }^5/m_{\mu }^5$ predicts only one leptonic channel, not the total tau lepton decay rate.

The tau lepton has two allowed charged-lepton channels, $e^{-}{\bar{\nu }}_e{\nu }_{\tau }$ and ${\mu }^{-}{\bar{\nu }}_{\mu }{\nu }_{\tau }$, as well as hadronic channels from $W^{-\ast }\to d\bar{u},s\bar{u}$. Each allowed quark channel has three color charge choices and its Cabibbo-Kobayashi-Maskawa matrix factor. Charm production is kinematically inaccessible in a physical hadronic final state at this mass. Treating the inclusive hadronic channels by a simple quark-hadron duality estimate,Thus, using $m_{\mu }\simeq 105.7\mathrm{MeV}/c^2$ and $m_{\tau }\simeq 1777\mathrm{MeV}/c^2$,Finite daughter masses, hadronic decay thresholds and resonances, quantum chromodynamics radiative corrections, and electroweak radiative corrections refine this estimate. The threefold color multiplicity in a decay width and extra decay channels are the main correction to naive fifth-power scaling.

Measured half-integer spin and fermionic behavior rule out a simple quark-antiquark pair: two spin-$1/2$ constituents have integer total spin, and adding integer orbital angular momentum does not change that. Their measured baryon number $B=1$, inferred through production and conservation bookkeeping, corresponds to $n_q-n_{\bar{q}}=3$. Their pattern of electric charges, strangeness and isospin then agrees with three light valence quarks.

Baryon magnetic moments provide a more detailed check. The additive quark model gives, for example,Approximately equal up and down constituent masses give ${\mu }_u\simeq -2{\mu }_d$, hence ${\mu }_p/{\mu }_n\simeq -3/2$, close to the measured ratio of about $-1.46$.

These observations support three valence quarks; they do not prove the absence of extra quark-antiquark pairs. For instance, $qqqq\bar{q}$ also has $B=1$ and can have half-integer spin. Ordinary quantum chromodynamics baryons also contain sea pairs and gluons. The stronger interpretation that the measurements exclude every antiquark component is therefore false.

A Feynman diagram specifies a term in the perturbative scattering amplitude, not a measured microscopic trajectory. Assign external spinors or polarization vectors, an interaction factor to each Feynman vertex and a propagator to each internal line; impose four-momentum conservation and integrate over independent loop momenta. Sum amplitudes for indistinguishable alternatives, then square, sum final spins and average unpolarized initial spins. A decay rate uses $d\Gamma =|\mathcal{M}{|}^{2}d{\Phi }_n/(2M)$ in natural units; a cross section also divides by incident flux. interference means probabilities of individual diagrams must not simply be added.

The electromagnetic fermion vertex is $f\bar{f}\gamma$, with coupling proportional to $e{Q}_f{\gamma }^{\mu }$; it preserves fermion flavour. There is no elementary three-photon or four-photon vertex in quantum electrodynamics. The weak charged current has $W\ell \nu$ and $W{u}_i{d}_j$ vertices, with a left-chiral projector and a Cabibbo-Kobayashi-Maskawa matrix element at the quark vertex. The weak neutral current has $Zf\bar{f}$ vertices with vector and axial couplings; it includes neutrinos and is flavour diagonal at tree level. Electroweak gauge interactions also include $WW\gamma$, $WWZ$ and quartic $WWWW$, $WWZZ$, $WWZ\gamma$ and $WW\gamma \gamma$ vertices. Higgs-fermion and Higgs-vector interactions complete the mass-related scalar sector.^[ref]

Quantum chromodynamics has $q\bar{q}g$ vertices proportional to $g_s{\gamma }^{\mu }{T}^a$, plus three-gluon and four-gluon Feynman vertices because gluons carry color charge. A quark-gluon vertex preserves quark flavour. Internal particles need not be on their mass shells, but electric charge and four-momentum are conserved at each vertex. The figures below show representative leading flavour topologies; binding and hadronization are implicit, and additional strong-interaction contributions can modify hadronic amplitudes.

Evidence for quarks comes from several kinds of observation that test different aspects of their interpretation. The light meson and baryon spectra organize into approximate flavour multiplets with the charges and strangeness expected for $q\bar{q}$ and $qqq$. In particular, the baryon octet and baryon decuplet encode the three light quark flavours and their spin couplings. This is an economical account of hadron quantum numbers, but spectroscopy alone would not establish that the constituents behave as localized scattering centers.

Deep inelastic scattering supplies that second test. In energetic electron–proton scattering the outgoing lepton measures the energy and momentum transferred through a virtual photon, while the broken-up target is summed over. Approximate Bjorken scaling means the structure functions depend primarily on $x=Q^2/(2P\cdot q)$ rather than varying rapidly with the resolving momentum scale $Q^2$. It indicates nearly pointlike constituents. In the leading spin-half parton model,The charge-squared weights and the second relation test the constituents' fractional charges and spin-half nature. Scaling violations reflect quark and gluon radiation rather than a failure of the constituent picture.^[ref]

Particle jets give complementary final-state evidence. A high-energy quark produces a collimated spray of hadrons because confinement prevents it from emerging as an isolated particle. Back-to-back jets fit quark–antiquark production; three-jet events show hard gluon radiation. Their angular correlations test a spin-one radiated parton. This distinguishes constituents of hadrons from a literal observation of free fractional-charge particles.

Three independent arguments identify an additional threefold colour degree of freedom. First, the spin-aligned $uuu$ state of a ground-state ${\Delta }^{++}$ would have symmetric space, spin and flavour factors. The Pauli exclusion principle requires an antisymmetric additional factor. The three-quark colour singlet supplies it. The same construction accommodates the spin-aligned $sss$ state of the ${\Omega }^{-}$ without altering its measured quantum numbers.

Second, the inclusive hadronic R ratio, away from thresholds and narrow resonances, is at leading electromagnetic orderEach quark flavour has $N_c$ distinguishable colour final states. With $N_c=3$, the predicted plateaux for three, four and five accessible flavours are $2$, $10/3$ and $11/3$; without the factor three they would be a third as large. The measured inclusive normalization is therefore a colour-counting test, with radiative and threshold corrections included in precision comparisons.^[ref]

Third, the neutral pion's two-photon amplitude contains a quark-loop colour sum, proportional to $N_c(e_u^2-e_d^2)$; its rate and the other evidence are consistent with three colours. Quantum chromodynamics identifies this degree of freedom as a gauge charge, with eight gluons. Observed hadrons are colour singlets; the evidence for colour is its quantitative and symmetry consequences, not visible colours or isolated coloured quarks.

The Standard Model is a gauge theory with group $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, matter quarks and leptons, gauge bosons and a Higgs field. It combines Quantum chromodynamics with the electroweak interaction. Its successes do not make its field content a complete account of nature. Established missing phenomena should be distinguished from theoretical questions about its parameters.

neutrino oscillation demonstrates nonzero mass-squared differences and mixing of flavor states. The minimal renormalizable Standard Model has no right-handed neutrino and no neutrino mass term. The atmospheric experiment compared zenith-angle distributions of muon-like and electron-like events in a large water Cherenkov detector. A direction-dependent muon-neutrino deficit, with different path lengths through Earth and comparison samples, discriminated against a uniform flux-normalization error.^[ref] One extension adds right-handed neutrinos and small Dirac Yukawa couplings; another has a Majorana mass term and seesaw mechanism, with light eigenvalue of order $m_D^2/M$. Oscillations alone do not decide whether neutrinos are Majorana neutrinos.

dark matter is inferred from gravitational observations such as galactic motions and gravitational lensing; Standard Model particles do not supply the required dominant cold component. Active neutrinos contribute some matter but cannot constitute all the cold dark matter. A new stable weakly interacting particle, an axion, or other hidden-sector states are possible explanations, with different scattering, production and astrophysical signals. Dark-matter evidence does not identify a particular new particle. The institutional account distinguishes gravitational evidence from proposed particle candidates.^[ref]

The matter-antimatter imbalance poses a different problem. Dynamical baryogenesis needs Sakharov conditions, including baryon-number violation, CP violation and departure from thermal equilibrium. Although the Standard Model contains relevant ingredients in principle, they do not generate the observed asymmetry in its conventional thermal history with measured parameters. Additional CP phases, a stronger electroweak transition, or leptogenesis from heavy Majorana neutrinos are possible extensions. Each needs a quantitative mechanism.

gravity is absent as a dynamical quantum interaction, motivating quantum gravity and unification. The hierarchy problem asks why the Higgs mass scale is small compared with much higher scales when scalar masses receive sensitive quantum corrections. supersymmetry can cancel particular contributions between partner fields if its breaking pattern is suitable; compositeness gives another interpretation of the Higgs sector. Neither motivation proves that the corresponding new particles exist. The Strong CP problem asks why the allowed QCD CP-violating parameter is so small; an axion mechanism is one possible answer.

A grand unified theory embeds the separate gauge factors in a larger group and can relate couplings or fermion representations, sometimes predicting proton decay. Its tests differ from dark-matter or neutrino-mass searches. A useful extension must explain a specified deficiency, preserve successful low-energy predictions and supply distinguishable experimental consequences. Attractive new symmetries alone are not evidence.

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.

Theory. Quantum chromodynamics is a local, non-Abelian gauge theory with gauge group SU(3) acting on color charge. Its matter fields are spin-half quarks of six flavours, each in three colors; its eight massless spin-one gluons are the gauge fields. Unlike the neutral photon, gluons themselves carry gauge charge. Consequently there are three-gluon and four-gluon vertices as well as the quark-gluon vertex. A compact Lagrangian density iswhere $D_{\mu }$ is the gauge covariant derivative and the field strength contains the nonlinear term $g_s{f}^{abc}{G}_{\mu }^b{G}_{\nu }^c$.

The running coupling is weak at high momentum scales. At one loop, for $n_f$ active flavours,The positive coefficient $11-2n_f/3$ produces asymptotic freedom: short-distance, large-$Q$ processes admit perturbation theory. The gluon self-interaction is crucial to its sign. At low scales the coupling grows and isolated colored particles are not observed; the physical spectrum consists of color-singlet hadrons, the phenomenon of confinement. A hard-produced quark or gluon radiates and undergoes hadronization, giving a collimated jet. The one-loop formula is not valid at $Q\sim {\Lambda }_{\mathrm{QCD}}$ and is not, by itself, a mathematical proof of confinement.

Experimental tests. Different measurements test different ingredients of this account:

The convincing evidence is the quantitative consistency across these probes, with a common color group and coupling evolution. A jet by itself is evidence for a hard parton; its rates and correlations supply the more specific tests of quantum chromodynamics.^[ref]

Colour and identical-quark spin. A spatial ground-state baryon has a symmetric $\ell =0$ spatial wavefunction. For $uuu$, $ddd$ or $sss$, the quark flavour state is also symmetric, and the total-spin $3/2$ state of three spin-one-half quarks is symmetric. Without another degree of freedom, the whole state would be symmetric under exchange of identical fermions, contrary to the Pauli exclusion principle.

A three-quark colour singlet supplies a completely antisymmetric factor,using the three color charge states. The remaining spatial-spin-flavour factor can then be symmetric. These baryons therefore provide evidence for an additional internal degree of freedom with at least three states, realized as colour in quantum chromodynamics.

For the $uu$ pair in $uus$, the spatial and quark flavour factors are symmetric under exchange of the two up quarks, while the colour factor is antisymmetric. The pair's spin state must consequently be symmetric. The identical-quark pair spin in a ground-state baryon isa triplet state. This holds whether the complete baryon has total spin $1/2$ or $3/2$.

Origin of the pair interaction. The terms in the baryon hyperfine mass formula are effective colour-magnetic spin-spin interactions, arising from gluon exchange between the quarks' colour-magnetic moments. Each magnetic moment brings an inverse constituent quark mass, explaining the factor $1/(m_i{m}_j)$. The constant $A^{\prime }$ absorbs the strong coupling constant, colour factors and the spatial overlap. A common $A^{\prime }$ is a constituent-model approximation.

Spin algebra and the two masses. Set $\hslash =c=1$, so each dimensionless spin angular momentum operator satisfies ${\mathbf{S}}_i^2=3/4$. Label the two up quarks 1 and 2 and the strange quark 3. Squaring ${\mathbf{S}}_{12}={\mathbf{S}}_1+{\mathbf{S}}_2$ givesNext square the total spin $\mathbf{J}={\mathbf{S}}_{12}+{\mathbf{S}}_3$:Substituting these pair sums into the baryon hyperfine mass formula yieldsTheir difference is $3A^{\prime }/(2m_u{m}_s)$. Repeating the argument for the identical $ss$ pair in $uss$ givesThus the model predicts equal Sigma and Xi hyperfine splittings:For a like-flavour comparison using exactly the supplied masses, the $uus$ splitting is $1383-1189=194\mathrm{MeV}$ and the $uss$ splitting is $1532-1315=217\mathrm{MeV}$, giving $194/217=0.894$. Averaging within the supplied isospin multiplets instead gives $192/215=0.893$. The predicted equality captures the common scale but misses the observed ratio by about $11\%$ in this table. Different spatial overlaps, and hence different effective $A^{\prime }$ values, can account for departures from the common-coefficient approximation.

Radiative decay and recoil. In a radiative Sigma-to-Lambda transition, both baryons retain valence content $uds$. A photon couples to an electrically charged constituent while the final quarks are projected onto the lower baryon's spin-flavour wavefunction. The leading up- and down-quark diagrams are shown below; their contributions must be combined coherently. In the simplest model, the strange-quark spin operator alone cannot change the $ud$ pair from its Sigma triplet to the Lambda singlet. The transition is an $M1$ multipole radiation transition, consistent with $1/2^{+}\to 1/2^{+}$.

For the photon energy, let $M_{\Sigma }=1193\mathrm{MeV}$ and $M_{\Lambda }=1116\mathrm{MeV}$ in natural units. Four-momentum conservation gives equal opposite final momenta of magnitude $E_{\gamma }$, soThe recoil kinetic energy of the Lambda baryon is $77-74.5151=2.4849\mathrm{MeV}$; assigning the entire mass difference to the photon would miss it.

Which baryons decay strongly? The open strong-decay thresholds for baryons require both sufficient energy and conserved quark flavour, including strangeness. The dominant strong decayers in the supplied table areThe Delta baryon channels $\Delta \to N\pi$ have threshold near $940+140=1080\mathrm{MeV}$, below $1232\mathrm{MeV}$. The excited Sigma baryons can decay as ${\Sigma }^{\ast }\to \Lambda \pi$ or $\Sigma \pi$, and the excited Xi baryons as ${\Xi }^{\ast }\to \Xi \pi$, all with open thresholds and conserved strangeness. Their positive-parity baryon states can emit the negative-parity pion in an odd relative orbital angular momentum, with $L=1$ already allowing total $J=3/2$.

The last diagram illustrates ${\Delta }^{++}(uuu)\to p(uud)+{\pi }^{+}(u\bar{d})$: a gluon creates a $d\bar{d}$ pair and the quarks regroup into the two hadrons. No quark changes flavour at the gluon vertices. This is a representative strong pair-creation topology; quantitative hadronic decay rates require bound-state dynamics.

The Omega baryon is an important exception among the spin-$3/2$ states. Its $S=-3$ cannot become the $S=-2$ of $\Xi \pi$ through the strong interaction. Its flavour-conserving $\Xi K$ threshold is at least $1315+500=1815\mathrm{MeV}$, above its $1673\mathrm{MeV}$ mass, so it decays weakly. The ground-state strange octet also has no open strong channel: for example $\Sigma \to \Lambda \pi$ needs at least $1256\mathrm{MeV}$, and $\Lambda \to NK$ or $\Xi \to \Lambda K$ needs still more than the relevant initial mass. The neutral Sigma decays electromagnetically as above; the neutron and the other unstable octet members decay weakly, while the proton is stable within these processes.

Read the line styles and fermion-number arrows in the original PDF: a wavy line is a photon, and a coiled line is a gluon. The valid fundamental interaction vertices areBoth (a) and (f) have one electron-line arrow entering and one leaving a photon vertex; changing the drawing's orientation does not change the quantum electrodynamics interaction. The quantum chromodynamics vertices (b) and (g) are the cubic and quartic gluon self-interactions.

The remaining cases fail for different reasons. Diagram (c) couples an isolated electron line to gluons and is not an allowed fermion vertex. Diagram (d) changes an electron to a muon at a photon vertex, violating the flavour-diagonal electromagnetic coupling. Diagram (e) couples a colourless electron to a gluon. Diagram (h) changes an up quark into a down quark at a gluon vertex, whereas quantum chromodynamics preserves quark flavour; that assignment also fails electric charge conservation.

One measurement of the strong coupling constant uses the fraction of resolved three-jet events in $e^{+}{e}^{-}\to q\bar{q}g$: with a fixed jet-resolution definition, its leading ratio to two-jet production is proportional to ${\alpha }_s$, with calculable higher-order corrections.

The intermediate current is ${\gamma }^{\ast }$ or $Z^{\ast }$, and the extra gluon can be emitted from either the quark or the antiquark. Both Feynman diagrams contribute coherently. The observed jets arise from hadronization; a resolved-jet definition is needed because arbitrarily soft or collinear radiation is not a separate measurable third jet.

There are three vertex types, with all legs labeled in the following original drawings:

The gluon cubic interaction vertex follows from the non-Abelian gauge field strength of quantum chromodynamics. The electroweak Yang-Mills theory Lagrangian, after expressing the fields in the physical charged and neutral basis, produces the two charged-W boson vertices. In particular there are no fundamental $\gamma \gamma \gamma$, $ZZZ$, $ZZ\gamma$, or $Z\gamma \gamma$ vertices, and the separate colour and electroweak gauge factors do not generate a mixed $gg\gamma$ or $ggZ$ vertex. The count is by the boson labels requested, not by separate assignments of gluon colour indices.

The baryon hyperfine mass formula is a phenomenological constituent-quark model. The three constituent quark masses include effects of strong-interaction dressing and an approximate spin-independent confinement energy; they are not the small running quark masses in a short-distance quantum chromodynamics calculation.

The residual correction has the form of a colour-magnetic spin-spin interaction. A nonrelativistic reduction of one-gluon exchange produces a contact contribution proportional toAfter taking its expectation in the common spatial wavefunction and the colour singlet, the colour factor, coupling, and contact density are absorbed into $A^{\prime }$. Summing the three quark pairs gives the supplied formula. Positive $A^{\prime }$ raises aligned-spin configurations relative to lower-spin configurations. In natural units, the spin angular momentum operators are dimensionless, so $A^{\prime }$ has units of energy cubed, as in the given $\mathrm{Ge}{\mathrm{V}}^3$.

In a dense beam dump, long-lived pions and kaons usually interact and lose energy before decaying. Their residual stopped decays yield low-energy neutrinos, below the useful high-energy tau-interaction sample. Prompt charmed mesons have much shorter lifetimes and can decay before reinteraction. This suppresses the conventional muon neutrino background relative to the prompt component. The production sequence is quantum chromodynamics creation of $c\bar{c}$, principally $gg\to c\bar{c}$ with $q\bar{q}\to g^{\ast }\to c\bar{c}$ also allowed, followed by hadronization $c\bar{s}\to D_s^{+}$ and its charge conjugate. The $gg$ process includes both quark-exchange diagrams and the three-gluon diagram. Fragmentation also produces other hadrons; the proton does not turn directly into one isolated Ds meson.The charge-conjugate chain gives a prompt tau antineutrino and a secondary tau neutrino. Thus each leptonic Ds meson chain produces both tau-flavour signs. The corresponding Feynman diagrams are shown in a.

For the CKM matrix, diagonalizing the up/down quark mass matrices by $U_{uL},U_{dL}$ leaves $V=U_{uL}^{†}{U}_{dL}$ in the charged current:The three-generation unitary matrix has three physical mixing angles and one CP violation phase after field rephasings. At fixed kinematics and hadronic matrix element, rates are proportional to $|V_{ij}{|}^2$; phase space, parent lifetimes, form factors and helicity suppression must also be included when comparing branching fractions.

For a charged pseudoscalar meson $P^{+}=(c\bar{q})$, leptonic pseudoscalar decay width givesThe Ds meson channel is Cabibbo-favoured, $|V_{cs}|\simeq 0.974$, whereas the charged D meson channel uses $|V_{cd}|\simeq 0.224$. Taking $m_{D_s}\simeq 1.968$, $m_D\simeq 1.870$ and $m_{\tau }\simeq 1.777$ in GeV givesIncluding the approximate lifetime ratio ${\tau }_{D_s}/{\tau }_D\simeq 0.48$ and the given production ratio $5.2/11$ gives a tau yield ratio about $17(f_{D_s}/f_D{)}^2$, already much greater than one even with equal decay constants. This is an illustrative estimate using approximate additional mass/lifetime inputs, not a determination from the production cross sections alone. A neutral D meson cannot annihilate to the charged $\tau \nu$ pair by electric charge conservation; its semileptonic tau modes have insufficient phase space once the required strange or nonstrange recoil hadron is included. Light mesons are below threshold; bottom production is much smaller at this fixed-target energy. Consequently neither the larger $D^0$ production rate nor ordinary light-hadron production outweighs the Ds meson tau source. ^[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 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.