Quantum chromodynamics (QCD)

Quantum chromodynamics, or QCD, is the non-Abelian $SU(3{)}_c$ gauge theory of quarks and gluons. Quarks transform in the fundamental color representation and the eight gluon fields in the adjoint representation. The non-Abelian field strength generates gluon self-interactions as well as quark–gluon interactions.

The strong coupling constant changes with scale: it becomes small at sufficiently high energies, permitting perturbative predictions, and grows toward the QCD scale. Low-energy color confinement and hadron structure are therefore not obtained by extrapolating a few weak-coupling diagrams without qualification. Parton distribution functions describe the long-distance input needed for many hard hadron-scattering predictions. Jets connect the calculated energetic partons with measured sprays of color-neutral particles after hadronization.

Expanding $\bar{q}i{\gamma }^{\mu }{D}_{\mu }q$ gives a vertex joining a quark, its conjugate line and a gluon. In the convention ${\mathcal{L}}_{\mathrm{int}}=-g_s\bar{q}{\gamma }^{\mu }{T}^{a}q{A}_{\mu }^a$, its rule is the displayed matrix. The gluon changes the color charge state through the generator $T^a$ but preserves the quark flavour. With distinct flavours in $q\bar{q}\to q^{\prime }{\bar{q}}^{\prime }$, this permits the tree-level annihilation graph but not a flavour-changing exchange along one quark line.

For distinct incoming and outgoing quark flavours, tree-level $q\bar{q}\to Q\bar{Q}$ has one gluon annihilation graph. Averaging the initial colours gives $2/9$, and averaging initial spins gives $1/4$. The outgoing fermion spin sum must use distinct particle and antiparticle projectors. For final mass $M$, $s+t+u=2M^2$ and the displayed result includes all outgoing colour and spin sums. Its amplitude squared remains finite at threshold even though the cross section vanishes through the final-state velocity.

For Hermitian fundamental representation generators normalized by $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, the Abelian-shaped one-loop Feynman diagram has colour factor ${\sum }_b{T}^b{T}^a{T}^b=(C_F-C_A/2)T^a=-T^a/(2N)$. Expand ${\sum }_b[T^b,[T^b,T^a]]=C_A{T}^a$ to prove the identity. With oriented structure tensor $f^{abc}$, the three-gluon graph has $f^{abc}{T}^b{T}^c=i{C}_A{T}^a/2$ by antisymmetrizing the generator product. The factor $i$ and its sign must be combined consistently with the chosen three-gluon Feynman rule. For $SU(3)$, the two real coefficients are $-1/6$ and $3/2$.

For massless distinguishable quarks exchanging one gluon, summing the squared generator contraction gives two with $\mathrm{Tr}(T^a{T}^b)={\delta }^{ab}/2$. Averaging nine initial colors and four initial spins leaves the coefficient $1/18$ multiplying the two Dirac traces. Their contraction is $8(s^2+u^2)$, yielding the displayed tree-level result. Identical quarks require an additional exchange amplitude and its interference.

Lattice QCD is a lattice gauge theory formulation of quantum chromodynamics. A Euclidean path integral on a discrete spacetime lattice provides a route to computing hadron observables numerically, particularly when weak-coupling expansions are inadequate. Physical predictions require control of lattice spacing, finite spatial volume and quark-mass effects.

The gauge-covariant kinetic term for two massless quarks separates into left- and right-handed parts. Independent constant unitary quark flavor rotations preserve these terms, giving classical $U(2{)}_L\times U(2{)}_R$ internal symmetry in addition to local color $SU(3)$. The singlet axial $U(1{)}_A$ has a chiral anomaly, while the nonsinglet chiral symmetry and vector baryon number remain exact quantum symmetries, up to their shared discrete identifications. Equal nonzero quark masses reduce the continuous flavor symmetry to $SU(2{)}_V\times U(1{)}_V$.

A collimated collection of final hadrons associated with an energetic parton and its radiation and hadronization. A precise perturbative prediction requires an infrared-safe observable that specifies how particles are grouped into jets.

In inclusive electron-positron annihilation into hadrons, hard gluon radiation produces a quark, an antiquark and a gluon, giving three resolved particle jets. Observation of such events at PETRA in 1979 provided direct evidence for gluons. The institutional account is www.desy.de/news/news_search/index_eng.html?openDirectAnchor=1643&printversion=1 .

For an infrared-safe observable, loop corrections and integrated unresolved real emissions have compensating infrared divergences and collinear divergences. Keep a common regulator until they are added. At the first QCD correction to photon-mediated quark production, virtual gluon exchange corrects the two-parton state and real emission supplies the three-parton state. Their inclusive sum is finite after ultraviolet renormalization; resolved three-jet rates additionally depend on an infrared-safe resolution prescription.

Hadronization is the conversion of colored quarks and gluons into color-neutral hadrons. It affects exclusive final-state distributions even when an inclusive observable can be approximated through a short-distance partonic calculation. A tree-level partonic decay width does not by itself model this conversion.

A characteristic proper timescale for strong bound-state formation, estimated by dimensional analysis from the QCD scale. A scale of a few tenths of a GeV corresponds to a few $10^{-24}\mathrm{s}$. It is an order-of-magnitude guide rather than a universal sharply defined formation time.

Quark-hadron duality is the correspondence, in suitable inclusive or averaged observables, between a quark-level description and a sum over hadronic states. Asymptotic freedom supports perturbative short-distance production, but sharp resonances and threshold regions can invalidate a pointwise partonic approximation. Inclusive predictions can have both radiative and nonperturbative corrections.

In the large-color limit of QCD with $g_s^2{N}_c$ fixed, baryon masses scale as $N_c$ while meson masses remain of order one. A Skyrme model action with coefficients of order $N_c$ has soliton mass and rotational moment of inertia of order $N_c$, and rotor splittings of order $1/N_c$. This motivates a semiclassical mesonic description of baryons; it does not uniquely determine every term in a finite-color effective theory.

The strong coupling constant parametrizes the strength of Quantum chromodynamics interactions. Its dependence on the renormalization scale is governed by the renormalization-group beta function. It is dimensionless in four dimensions.

Color charge is the gauge group charge of Quantum chromodynamics, associated with its $SU(3)$ symmetry. A quark transforms in the fundamental three-dimensional representation, an antiquark in its conjugate, and a lepton is a singlet. The word color is a label for this charge, unrelated to visible light.

The group-theoretic tensor multiplying the kinematic part of a Feynman diagram involving color charge. It is obtained by contracting internal indices among gauge generators, invariant tensors and structure constants of a Lie algebra. For instance, one gluon exchange on a single quark line contributes ${\sum }_a{T}^a{T}^a=C_{F}1$; averaging or summing the squared tensor over external colour labels gives the corresponding color average in a squared scattering amplitude.

A quark-gluon Feynman vertex whose initial and final quark have the same chosen color charge basis label. Such vertices exist: the diagonal Gell-Mann matrices act nontrivially on the red basis vector. Their gluons are traceless colour-octet combinations, not colour singlets. A ninth singlet generator proportional to the identity is absent from SU(3).

A uniform incoherent initial mixture of two independently prepared fundamental color charges requires an average over their $N^2$ initial possibilities, together with a sum over unobserved final color charges. This differs from preparing a coherent color singlet. For quark-antiquark annihilation through one vector in the adjoint representation, $c_s=T_{ji}^a{T}_{kl}^a$ gives $N^{-2}\sum |c_s{|}^2=(N^2-1)/(4N^2)$. For the exchange contraction $c_t=T_{ki}^a{T}_{jl}^a$, $N^{-2}\sum c_s{c}_t^{\ast }=-(N^2-1)/(4N^3)$ by the SU(N) fundamental completeness relation. The negative overlap reverses the sign of the corresponding Abelian interference term.

In distinct-flavour quark annihilation, the one-photon exchange color charge tensor is ${\delta }_{ji}{\delta }_{kl}$, whereas the one-gluon tensor is ${\sum }_A{T}_{ji}^A{T}_{kl}^A$. Their inner product after the external color charge sum is ${\sum }_A\mathrm{tr}(T^A)\mathrm{tr}(T^A)=0$, since the fundamental gauge generators of $SU(3)$ are traceless. Thus the averaged squared scattering amplitude has no leading electromagnetic–strong interference term for this process. Fixed external color charge components, or processes with different exchange pairings, need not share this cancellation.

A colour singlet transforms in the trivial representation of the SU(3) colour group. Global colour rotations leave it invariant, and its colour generators annihilate it. A three-quark colour singlet uses the completely antisymmetric colour tensor; a quark–antiquark singlet contracts their colour indices.

For a colorless parent decaying through a color-diagonal interaction into a quark and antiquark, the three orthogonal matching-color states add incoherently. The inclusive flavor decay width is $N_c=3$ times the one-color width. In a parton scattering rate, an initial color average instead cancels the matching final color sum.

A parton is a constituent resolved inside a rapidly moving hadron by a hard scattering process. In quantum chromodynamics, the relevant partons are quarks, antiquarks and gluons. A parton distribution function describes the momentum-fraction distribution of a specified constituent at a stated resolution scale.

In deep inelastic scattering, a high-energy probe transfers enough momentum to resolve structure much smaller than the hadron. The parton model approximates the short collision as scattering from one constituent, with the remaining structure entering through its distribution. Which partons are resolved depends on the scale; increasing the resolution permits additional radiation and pair production. A parton is therefore not a permanently labeled classical bead inside the hadron, and the distributions are not probabilities for finding isolated free quarks.

The parton model approximates a high-energy collision with a hadron as a short interaction with one constituent carrying a fraction of the hadron's longitudinal momentum. In electromagnetic deep inelastic scattering, the probe couples to charged quarks and antiquarks.

The short collision is treated using a parton-level scattering probability, averaged over parton distribution functions. In the simplest massless collinear limit, the Bjorken scaling variable identifies the struck constituent's momentum fraction, and dimensionless structure functions approximately depend on that variable alone. QCD scaling violation adds logarithmic resolution dependence through radiation and splitting. The model explains why a composite target can display a simple short-distance pattern without requiring its constituents to be permanently isolated classical particles.

In the incoherent massless collinear parton approximation, $f_f(\xi )d\xi$ is the number of quarks in a momentum-fraction interval, not a covariantly normalized external state weight. A single-particle response divided by its external-state normalization is proportional to $w/(2k^0)$; the hadron normalization is $2P^0$. Thus converting a number-weighted parton sum to the target tensor contributes $P^0/k^0=1/\xi$. This gives the displayed convolution, including antiquarks. The spin-averaged electromagnetic parton tensor and $\delta (2\xi P\cdot q-Q^2)=\delta (\xi -x)/(2P\cdot q)$ then give $F_1=\frac{1}{2}{\sum }_f{Q}_f^2(f_f+{\bar{f}}_f)$ and $F_2=x{\sum }_f{Q}_f^2(f_f+{\bar{f}}_f)$, proving the Callan-Gross relation at this order. Target masses, transverse momentum and radiative effects are beyond this leading approximation.

At leading order in the naive parton model, dimensionless deep-inelastic structure functions depend only on Bjorken x. Quantum chromodynamics evolution introduces scale dependence.

Quantum chromodynamics makes parton distribution functions depend on resolution scale through gluon radiation and parton splitting. Thus Bjorken scaling and the linear-energy neutrino cross section are approximate leading-order patterns rather than exact laws.

Dependence of a deep-inelastic structure function on $Q^2$ at fixed Bjorken scaling variable. In QCD, radiation and gluon splitting evolve the parton distribution functions with the resolution scale, producing logarithmic departures from the simplest parton model scaling limit.

For massless spin-one-half partons at leading order, the unpolarized electromagnetic deep-inelastic structure functions obey this relation. Radiative quantum chromodynamics corrections and target-mass effects can modify it.

The leading high-energy approximation neglects quark and target masses and transverse constituent momentum. A lightlike reference momentum is used for the hadron; this does not describe a massless particle at rest.

In the leading massless collinear parton approximation, $P^2=0$ and $q^2=-Q^2$ imply $(\xi P+q{)}^2=2\xi P\cdot q-Q^2$. With Bjorken x $x=Q^2/(2P\cdot q)$ and $P\cdot q>0$, the Dirac delta change-of-variable formula yields the displayed identity. The positive-energy branch of the final parton also gives $\delta (E_q+\xi E_P-E_k)/E_k=2\delta ((E_q+\xi E_P{)}^2-E_k^2)$ locally on physical support. Keeping target mass instead introduces ${\xi }^2{M}^2$ and changes the root and Jacobian.

A parton distribution function specifies a hadron's partonic content as a function of longitudinal momentum fraction and a stated parton factorization scale. In the leading collinear parton model, $f_i(x)dx$ has a number-density interpretation for partons of species $i$ with fractions in $dx$. It is not generally normalized to one.

For a proton, the integrals of the valence differences count two net up quarks and one net down quark. A momentum-weighted sum over all quark, antiquark and gluon species gives total momentum fraction one. These are different sum rules. Beyond the simplest approximation, the definition depends on factorization scheme and renormalized field correlations; a naive positive classical probability interpretation is not universal. The functions enter factorized cross sections and evolve with resolution, rather than being predictions of a free-quark model.^[ref]

A nuclear parton distribution describes parton momentum fractions in a bound nuclear target. Its normalization must specify whether it is per nucleus or per nucleon and how $x$ is defined. Nuclear binding, shadowing and other nuclear modifications mean it need not equal a simple sum of free-nucleon distributions.

In the collinear approximation, a parton carries a fraction $0<x<1$ of its parent hadron's longitudinal momentum. A parton distribution function $f_i(x,Q^2)$ gives its distribution at the specified factorization scale. For production of one on-shell particle with no transverse recoil, $x_{+}{x}_{-}s=M^2$ and $x_{\pm }=(M/\sqrt{s})e^{\pm y}$, where $y$ is its rapidity.

For charged current production $u\bar{d}\to W^{+}$, charge conjugation of the antiproton distributions gives $u(x_{+})d(x_{-})+\bar{d}(x_{+})\bar{u}(x_{-})$. With a common sea quark distribution $S$, the valence quark terms are $u_v{d}_v$, the mixed terms are $u_{v}S+S{d}_v$, and the two sea assignments give $2S(x_{+})S(x_{-})$. Omitting the second beam assignment undercounts the pure sea contribution by two.

The approximate equality of proton and neutron parton distribution functions after interchanging up quarks and down quarks, including their antiquarks. It follows from the corresponding isospin symmetry when electromagnetic and light-quark mass-difference effects are neglected. Isospin-singlet sea distributions are unchanged.

The parton distribution function is a number density. Integrating the quark distribution minus the antiquark distribution counts the net flavour charge of the hadron, since every sea quark-antiquark pair contributes zero. The corresponding flavour-vector conserved current counts particles minus antiparticles. For a proton, the net up/down numbers are two and one; for a neutron they are one and two. The rule fixes the difference of distributions, not their sum, and does not imply absence of a sea.

With only up/down valence quarks and no antiquarks, the quark number sum rule and leading parton model expression give the charge-squared number moment $\int F_2(x)dx/x={\sum }_f{Q}_f^2{N}_f$. The squared charges are $4/9$ for up and $1/9$ for down, so the proton has $8/9+1/9=1$ and the neutron has $4/9+2/9=2/3$. Restricting the flavour list to up/down does not itself remove antiquarks; the no-sea assumption is essential for these individual moments.

The scale at which collinear contributions are assigned to parton distribution functions rather than the hard scattering coefficient. The separate factors depend on this scale, while a consistently calculated physical differential scattering cross-section does not.

Scattering that resolves the constituents of a hadron at large spacelike momentum transfer, with an inclusive sum over final hadron states.

In electron scattering from a proton, write the reaction as $e^{-}p\to e^{-}X$: the outgoing electron is measured, while $X$ denotes all allowed final hadron states. Large momentum transfer supplies a short-distance probe; a large final invariant mass separates the deep continuum from elastic scattering and individual particle resonances. The useful surprise is that an extended proton can have a response characteristic of much smaller constituents. Deep inelastic scattering turns measurements outside the proton into constraints on its internal parton distribution functions.

The SLAC-MIT deep inelastic electron scattering experiment connects this definition to an actual measurement. Its evidence came from distributions of outgoing electron energies and scattering angles, rather than a photograph of a quark. Bjorken scaling concerns deep-inelastic structure functions after known scattering factors have been accounted for; it does not say that the raw differential scattering cross-section is constant.

In a head-on lepton-proton collision with the proton along $+z$, hadronic four-momentum conservation gives $y_h={\sum }_h(E_h-p_{zh})/(2E_e)$ and $Q_h^2=p_{T,h}^2/(1-y_h)$. With $s\simeq 4E_e{E}_p$, the reconstructed Bjorken scaling variable is $x_h=Q_h^2/(s{y}_h)$. This is useful for charged current events whose final neutrino escapes. Initial-state radiation and unmeasured forward hadrons spoil the ideal closure.

In the operator product expansion for a hard scattering process, twist is operator dimension minus spin. Contributions beyond leading twist describe additional multiparton correlations and are suppressed by powers of the large momentum-transfer scale. A leading-order quark-parton formula omits these power corrections.

For an isoscalar target in the leading massless quark-parton approximation, integrate the two helicity weights over inelasticity: ${\int }_0^{1}dy=1$ and ${\int }_0^1(1-y{)}^{2}dy=1/3$. Then $R={\sigma }_{\bar{\nu }}/{\sigma }_{\nu }=(Q/3+\bar{Q})/(Q+\bar{Q}/3)$, where $Q=\int xq(x)dx$ and $\bar{Q}=\int x\bar{q}(x)dx$. Inversion gives the displayed ratio. These are effective weak-flavour momentum fractions; the data do not determine unweighted parton number integrals.

For an isoscalar valence target and massless leading-order scattering, $R_{\nu }=G_L^2+G_R^2/3$, where $G_{L/R}^2=(g_{L/R}^u{)}^2+(g_{L/R}^d{)}^2$. The factor $1/3$ is ${\int }_0^1(1-y{)}^{2}dy$. With Standard Model quark couplings and $w={\sin }^2{\theta }_W$, this becomes $1/2-w+20w^2/27$. Sea quarks and threshold, nuclear and QCD corrections matter beyond this limit.

A nuclear target with equal proton and neutron contributions, or data corrected to that average. Its per-nucleon PDFs are $q_N=(q_p+q_n)/2$. Charge symmetry gives $u_N=d_N=(u_p+d_p)/2$; using proton down-quark density alone is not an isoscalar average.

Starting in 1967, a collaboration at the Stanford Linear Accelerator Center and the Massachusetts Institute of Technology directed an electron beam at liquid hydrogen, providing proton targets. Jerome Isaac Friedman, Henry Way Kendall and Richard Edward Taylor participated in this experimental program.^[ref]

The 1969 paper Observed Behavior of Highly Inelastic Electron-Proton Scattering reports inclusive electron spectra at $6^{\circ }$ and $10^{\circ }$ with incident energies from $7$ to $17\mathrm{GeV}$. Its scaling comparison estimated ${\nu }_{\mathrm{lab}}{W}_2$ under a transverse-virtual photon assumption, finding approximate agreement as a function of $\omega =1/x$ for $Q^2>1{\mathrm{GeV}}^2$ and $\omega >4$. These are the paper's conditions, not a claim of exact Bjorken scaling at every momentum transfer.^[ref]

A movable magnetic spectrometer selected scattered electrons by momentum and scattering angle. Magnetic fields dispersed momenta at the focal plane. With relativistic electrons, the inferred momentum determines outgoing energy $E^{\prime }$. Commissioning included steering the beam through the spectrometer at zero angle to check its optics and accepted solid angle. Counters sorted events into momentum and angle bins. The acceptance entered the conversion from counts to differential scattering cross-sections.^[ref]

Radiative corrections matter because photon emission can move events from an elastic scattering peak into lower outgoing-energy bins. The analyses subtracted the elastic radiative tail before unfolding inelastic radiation. Independent MIT and SLAC procedures provided a check. An uncorrected low-$E^{\prime }$ tail alone therefore cannot establish deep inelastic scattering.^[ref]

Work in natural units with $c=1$, neglect the electron mass, and put the initial proton at rest with mass $M$. Let $k$ and $k^{\prime }$ be the initial and final electron four-momenta, $q=k-k^{\prime }$, and $Q^2=-q^2>0$. Define the laboratory energy transfer ${\nu }_{\mathrm{lab}}=E-E^{\prime }=P\cdot q/M$. This ${\nu }_{\mathrm{lab}}$ has units of energy; the invariant $P\cdot q$ has units of energy squared. The Bjorken scaling variable and the final hadron invariant mass obeyThese follow by squaring $k-k^{\prime }$ and $P+q$, using $k^2=k^{\prime 2}=0$ and $P=(M,0)$. Solving the Bjorken x definition gives the outgoing energy and then the scattering angle:Alternatively, substituting $Q^2=4E{E}^{\prime }{\sin }^2(\theta /2)$ before solving gives $E^{\prime }=E/[1+2E{\sin }^2(\theta /2)/(Mx)]$. A chosen $x,E,Q^2$ must satisfy $E^{\prime }>0$ and $0\le {\sin }^2(\theta /2)\le 1$. In addition, $W$ must exceed the lightest allowed inelastic hadron threshold; positive $W^2$ alone is insufficient.

For an original numerical example, take $M=0.938\mathrm{GeV}$, $x=0.3$, $E=15\mathrm{GeV}$ and $Q^2=3{\mathrm{GeV}}^2$. Then ${\nu }_{\mathrm{lab}}\simeq 5.330\mathrm{GeV}$, $E^{\prime }\simeq 9.670\mathrm{GeV}$, $\theta \simeq 8.25^{\circ }$, and $W\simeq 2.807\mathrm{GeV}$. Varying $2\le Q^2/{\mathrm{GeV}}^2\le 5$ and $10\le E/\mathrm{GeV}\le 20$ at this same Bjorken x keeps every setting physical and gives $W^2>5.54{\mathrm{GeV}}^2$, above the proton-pion threshold and the stricter chosen continuum selection $W>2\mathrm{GeV}$. These are calculated settings, not historical measurements.

An ideal Bjorken scaling model has $F_2(x,Q^2)=f(x)$: varying $Q^2$ at fixed Bjorken x leaves this dimensionless deep-inelastic structure function unchanged. The comparison requires extracting a deep-inelastic structure function, rather than plotting raw event counts. Quantum chromodynamics predicts departures from exact Bjorken scaling; finite hadron masses also limit the idealization.

For an original shape comparison, choose $f(x)=x(1-x{)}^3$ and define $S(Q^2)=F_2(0.3,Q^2)/F_2(0.3,Q_0^2)$ with $Q_0^2=2{\mathrm{GeV}}^2$. The ideal model has $S=1$. To display a contrasting size-dependent suppression, define an illustrative elastic-form-factor shape $G(Q^2)=(1+Q^2/{\Lambda }^2{)}^{-2}$ with the freely chosen ${\Lambda }^2=1{\mathrm{GeV}}^2$. Its normalized squared magnitude isThus $A(5{\mathrm{GeV}}^2)=1/16$, whereas $S(5{\mathrm{GeV}}^2)=1$. The elastic-form-factor shape is a comparison of suppression mechanisms, not an elastic differential scattering cross-section assigned to $x=0.3$: actual elastic scattering from a proton has $W=M$ and $x=1$. Neither curve is measured data or a fit to the SLAC-MIT experiment.

The experimental interpretation required more than a flat curve. The angular dependence and small longitudinal response supported spin one-half partons, expressed at leading order by the Callan-Gross relation $F_2=2x{F}_1$. Identification with fractionally charged quarks also drew on later electron, muon and neutrino comparisons. A scaling plot alone does not determine the individual electric charges of constituents.^[ref]

The leading parton model formula already shows the ambiguity: $F_2=x{\sum }_f{Q}_f^2(q_f+{\bar{q}}_f)$ combines squared electric charges with unknown parton distribution functions. One measured function cannot independently fix all these quantities. Confinement also explains why an inclusive final hadron state need not contain an isolated outgoing quark.

In deep inelastic scattering, the reference frame where the exchanged virtual photon has zero energy and spatial momentum magnitude $Q$.

The inelasticity in deep inelastic scattering is $y=P\cdot q/(P\cdot p)$, where $p$ is the incoming lepton momentum, $P$ the target momentum and $q$ the transfer. In the target rest frame it is the fraction of the incoming lepton energy transferred to the target. It is unchanged by replacing $P$ by a collinear parton momentum $\xi P$, and $Q^2=xyS$ in massless kinematics.

In deep inelastic scattering, the scaling variable is $x=Q^2/(2P\cdot q)$ with $Q^2=-q^2$. In the massless collinear parton model, the outgoing on-shell condition $(xP+q{)}^2=0$ identifies $x$ with the struck parton's incoming longitudinal momentum fraction.

Here $Q^2=-q^2>0$. In the leading massless collinear parton approximation, this kinematic variable equals the momentum fraction of the struck parton.

The inclusive product of hadronic-current matrix elements, summed over final states with a Lorentz-invariant phase-space measure. Electromagnetic current conservation makes both indices transverse to the exchanged four-momentum.

For an absorbed spacelike photon, use $q^2=-Q^2<0$, $\nu =P\cdot q>0$, $x=Q^2/(2\nu )$ and the physical momentum delta function ${\delta }^{(4)}(P+q-p_X)$. The displayed invariant final mass is compatible with deep inelastic scattering for $0<x<1$. Using $P-q$ with the same positive-$x$ definition instead gives $M^2-Q^2(1+1/x)$, which becomes negative in the Bjorken scaling limit and has no physical hadronic spectral support. Reversing the photon convention requires reversing the associated scalar-product and scaling-variable conventions too.

For electromagnetic deep inelastic scattering with an unpolarized target, Lorentz invariance, parity and current conservation leave two independent deep-inelastic structure functions. With $T^{\mu \nu }=g^{\mu \nu }-q^{\mu }{q}^{\nu }/q^2$ and $\tilde{P}=P-(P\cdot q/q^2)q$, both tensor structures are transverse to $q$. Terms with a free $q$ index disappear on contraction with a conserved leptonic tensor, although such terms can still be necessary to make the full hadronic tensor transverse.

For an on-shell massless quark with incoming momentum $k$ and outgoing momentum $k+q$, average the initial spin and sum the final spin. The gamma matrix trace identities give $S^{\mu \nu }=\frac{1}{2}\mathrm{Tr}[/k{\gamma }^{\mu }(/k+/q){\gamma }^{\nu }]=2[k^{\mu }(k+q{)}^{\nu }+k^{\nu }(k+q{)}^{\mu }-g^{\mu \nu }k\cdot q]$. On shell $2k\cdot q=Q^2=-q^2$, and defining $T^{\mu \nu }=-g^{\mu \nu }+q^{\mu }{q}^{\nu }/q^2$ and $\tilde{k}=k-(k\cdot q/q^2)q$ yields the displayed transverse decomposition. With the hadronic-tensor prefactor $1/(4\pi )$, one-particle Lorentz-invariant phase space gives $w_f^{\mu \nu }=(Q_f^2/2){\delta }_{+}((k+q{)}^2)S^{\mu \nu }$. The antiquark has the same unpolarized tensor and squared electric charge.

The two scalar coefficients in the unpolarized electromagnetic hadronic tensor. Extra polarization or parity-violating currents can require further coefficients.

In the massless-target convention, the Callan-Gross relation sets this coefficient to zero at leading order in the parton model.

The product of lepton-current matrix elements with a specified spin sum or spin average. Its normalization must be stated when quoting a differential scattering cross-section.

For the current with $(1-{\gamma }^5)$ rather than $P_L$, the spin-summed symmetric leptonic tensor is $8(q_{1\mu }{q}_{2\nu }+q_{1\nu }{q}_{2\mu }-g_{\mu \nu }{q}_1\cdot q_2)$. Integrating the standard two-body Lorentz-invariant phase space of two massless particles givesIts antisymmetric Levi-Civita symbol term integrates to zero. The normalization changes by a factor of four if the current uses $P_L$ instead. The result separates the leptonic integration from hadronic pseudoscalar-to-pseudoscalar form factors in a three-body decay rate.

A gluon is a spin-one quantum of the quantum chromodynamics gauge field. There are eight color-adjoint gluon components, and the gauge description has zero gluon mass. Unlike the neutral photon of an Abelian theory, gluons themselves carry the charge associated with their interaction, producing three- and four-gluon vertices.

A gluon is not observed as an isolated color-charged final particle because of confinement. An energetic gluon can instead initiate a jet after radiation and hadronization. Gluons also contribute to a proton's momentum distribution and energy even though its net valence flavor content is described as $uud$. Color labels are internal gauge labels, not ordinary visible colors or eight independently identifiable classical substances.

A QCD branching in which a gluon produces a quark-antiquark pair or two gluons. Quark-pair production supplies a perturbative source of sea quarks. Its rates depend on scale and on the allowed daughter quark masses.

Confinement is the absence of isolated color-charged asymptotic states in the physical low-energy spectrum of quantum chromodynamics. Ordinary detected particles are color-singlet hadrons, even when a short-distance reaction is calculated in terms of quarks and gluons.

Pulling a quark and antiquark apart increases the energy stored in their color field. With dynamical light quarks, producing additional quark–antiquark pairs can break the growing flux configuration into separate hadrons rather than yielding two free colored particles. This gives physical intuition for hadronization. It does not supply a complete nonperturbative derivation from the QCD action. High-energy scattering can still resolve partonic structure, so evidence for constituents inside a hadron is consistent with confinement of isolated final states.

The QCD scale is the renormalization-group invariant energy at which the perturbative running coupling becomes strong. It sets the characteristic scale of confinement and most hadron masses.

It is a characteristic energy scale of quantum chromodynamics.

Flavor symmetry rotates quark species whose strong interactions are identical. Quark masses break it explicitly, so it is a useful approximate symmetry only among quarks light compared with the hadronic scale.

For a colour singlet hadron built entirely from up quarks, down quarks, strange quarks, antiquarks and gluons, the net number of quarks is a multiple of three. The central SU(3) colour element multiplies the state by ${\omega }^{n_q-n_{\bar{q}}}$, while a singlet must be unchanged. The central flavour element supplies the same phase because all three light flavours form the fundamental representation. Every resulting flavour multiplet therefore has zero SU(3) triality. Heavy quarks are singlets of light-flavour SU(3), so hadrons containing them do not obey this argument. The zero-triality restriction allows higher light-quark multiplets as well as the familiar singlet, octet and decuplet.

The $SU(2)$ flavor symmetry rotating down and strange quarks, with the up quark a singlet. Since down and strange quarks have equal electric charge, the charge operator commutes with all U-spin generators, and every irreducible U-spin multiplet has a common charge. Unequal down- and strange-quark masses break this symmetry.

For flavor symmetry acting on $(u,d,s)$, trace-orthonormal Hermitian diagonal generators are $h_1=\mathrm{diag}(1,-1,0)/\sqrt{2}$ and $h_2=\mathrm{diag}(1,1,-2)/\sqrt{6}$. The isospin and flavor hypercharge operators are $I_3=h_1/\sqrt{2}$ and $Y=\sqrt{2/3}{h}_2$. Their multiples by $i$ lie in the compact SU(3) Lie algebra.

An additive flavor symmetry quantum number: a strange quark has $S=-1$, its antiquark has $S=+1$, and up and down quarks have $S=0$. The strong interaction conserves net strangeness. Together with baryon number, it determines flavor hypercharge $Y=B+S$.

Strong and electromagnetic interactions conserve net strangeness; a decay changing it requires the weak interaction. A strange baryon decay into a less-strange baryon and a pion is an example. The weak interaction does not impose a parity selection rule, so a final orbital partial wave cannot by itself determine the parent's parity. Qualitative lifetime classes do not provide an exact lifetime without an amplitude calculation or measurement.

The additive flavor symmetry quantum number $Y=B+S$ combines baryon number with strangeness. For the $(u,d,s)$ quark triplet its values are $(1/3,1/3,-2/3)$. The Gell-Mann--Nishijima formula is $Q=I_3+Y/2$. This is distinct from the normalization of electroweak hypercharge.

Some formulas write $Q=I_3+Y$ rather than $Q=I_3+Y_{\mathrm{usual}}/2$. Then $Y=(B+S)/2$, where $B$ is baryon number and $S$ is strangeness. The light-quark weights are $u:(1/2,1/6)$, $d:(-1/2,1/6)$, $s:(0,-1/3)$. Baryon weight rows therefore have half the usual vertical coordinates. This is a flavour convention, distinct from weak hypercharge.

The baryon octet has rows $Y=1/2,0,-1/2$ with isospins $1/2$, $1\oplus 0$, $1/2$. The central weight has two states, ${\Sigma }^0$ and ${\Lambda }^0$. The baryon decuplet has rows $Y=1/2,0,-1/2,-1$ with isospins $3/2,1,1/2,0$. The three-quark flavour singlet has one state at the origin. Both abstract octet copies in $3^{\otimes 3}$ share the same weights.

For the light quark flavors, electric charge is $Q=I_3+Y/2$, with flavor hypercharge $Y=B+S$. On $(u,d,s)$ it gives $(2/3,-1/3,-1/3)$, and additivity extends it to the baryon octet and meson octet.

Chiral symmetry allows independent flavor rotations of left-handed and right-handed fields. For two massless quark flavors, the non-Abelian chiral transformations form $SU(2{)}_L\times SU(2{)}_R$. The additional vector $U(1)$ symmetry of baryon number acts trivially on the pion field $U$. The low-energy pion field $U\in SU(2)$ transforms as $U\mapsto LU{R}^{†}$. Its derivative Skyrme model terms are invariant under constant $L,R$. A chosen vacuum $U=1$ preserves the diagonal subgroup $L=R$, which acts by conjugation and is the isospin symmetry. An ordinary common nonzero pion mass explicitly breaks the independent left/right symmetry to that vector subgroup; vacuum selection in the massless theory is instead spontaneous symmetry breaking. The singlet axial anomaly is a separate issue and is not part of this two-flavor non-anomalous group.

A nonzero vacuum value of the scalar quark density breaks the massless nonsinglet chiral symmetry to its diagonal vector subgroup. Indeed, the nonsinglet axial charge algebra of quark densities gives $⟨[Q_{5,a},P_b]⟩=-i{\delta }_{ab}⟨\bar{q}q⟩$, whereas invariance of the vacuum under every conserved axial charge would make these expectations vanish. The three broken generators produce three massless Goldstone bosons in the exact chiral limit. Small equal quark masses explicitly break that symmetry and turn the pions into pseudo-Goldstone bosons, with squared masses linear in the quark mass at leading order.

Let $S=\bar{q}q$, $P_a=i\bar{q}{\gamma }_5{\tau }_{a}q$ and $Q_{5,a}=\int q^{†}{\gamma }_5{\tau }_{a}q/2d^{3}x$. The canonical anticommutation relations imply $[Q_{5,a},q]=-{\gamma }_5{\tau }_{a}q/2$ and $[Q_{5,a},\bar{q}]=-\bar{q}{\gamma }_5{\tau }_a/2$. The Leibniz rule for the commutator, together with $\{{\tau }_a,{\tau }_b\}=2{\delta }_{ab}$ and ${\gamma }_5^2=1$, gives the displayed transformations. These are equal-time operator relations; common nonzero quark masses remove conservation of the charges without changing their canonical action on the fields.