Quantum chromodynamics

Quantum chromodynamics (QCD) is the theory describing one of the fundamental forces, the strong interaction. It describes the interactions of quarks and gluons and takes the form of a quantum field theory of a special kind called a non-abelian gauge theory. QCD forms an important part of the standard model of particle physics. A huge body of experimental evidence for QCD has been gathered over the years.

The theory

Some definitions

Every field theory of particle physics is based on certain symmetries of nature whose existence is deduced from observations. These can be

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• local symmetries, that is the symmetry acts independently at each point in space-time. Each such symmetry is the basis of a gauge theory and requires the introduction of its own gauge bosons.
• global symmetries, which are symmetries whose operations must be simultaneously applied to all points of space-time.

QCD is a gauge theory of the SU(3) gauge group obtained by taking the color charge to define a local symmetry.

Related Topics:
SU(3) - Color charge

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Since the strong interaction does not discriminate between different flavors of quark, QCD has approximate flavor symmetry, which is broken by the differing masses of the quarks.

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There are additional global symmetries whose definitions require the notion of chirality, discrimination between left and right-handed. If the spin of a particle has a positive projection on its direction of motion then it is called left-handed; otherwise, it is right-handed.

Related Topics:
Chirality - Spin - Projection

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• Chiral symmetries involve independent transformations of these two types of particle.
• Vector symmetries (also called diagonal symmetries) mean the same transformation is applied on the two chiralities.
• Axial symmetries are those in which one transformation is applied on left-handed particles and the inverse on the right-handed particles.

The symmetry groups

The color group SU(3) corresponds to the local symmetry whose gauging gives rise to QCD. The electric charge labels a representation of the local symmetry group U(1) which is gauged to give QED: this is an Abelian group. If one considers a version of QCD with Nf flavors of massless quarks, then there is a global (chiral) flavor symmetry group SU_L(N_f) imes SU_R(N_f) imes U_B(1) imes U_A(1). The chiral symmetry is spontaneously broken by the QCD vacuum to the vector (L+R) SU_V(N_f) with the formation of a chiral condensate. The vector symmetry, U_B(1) corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry U_A(1) is exact in the classical theory, but broken in the quantum theory, an occurrence called an anomaly. Gluon field configurations called instantons are closely related to this anomaly.

Related Topics:
QED - Abelian - Chiral - Spontaneously broken - QCD vacuum - Chiral condensate - Anomaly - Instanton

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Cautionary note

In many applications of QCD one can ignore the heavy flavors (charm, bottom and top). In this case the effective flavor group is often SU(3), which should not be confused with the color group. In QCD the color group belongs to a local symmetry and hence is gauged. The flavor group is not gauged. The Eightfold way is based on the flavor group and ignores the local symmetry which gives QCD.

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The fields

Quarks are massive spin-1/2 fermions which carry a color charge whose gauging is the content of QCD. Quarks are represented by Dirac fields in the fundamental representation 3 of the gauge group SU(3). They also carry electric charge (either -1/3 or 2/3) and participate in weak interactions as part of isospin doublets. They carry global quantum numbers including the baryon number, which is 1/3 for each quark, hypercharge and one of the flavor quantum numbers.

Related Topics:
Quark - Fermion - Color charge - Dirac field - Fundamental representation - Gauge group - SU(3) - Weak interactions - Baryon number - Hypercharge - Flavor quantum numbers

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Gluons are spin-1 bosons which also carry color charges, since they lie in the adjoint representation 8 of SU(3). They have no electric charge, do not participate in the weak interactions, and have no flavor. They lie in the singlet representation 1 of all these symmetry groups.

Related Topics:
Gluon - Boson - Color charge - Adjoint representation - SU(3) - Singlet representation

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Every quark has its own antiquark. The charge of each antiquark is exactly the opposite of the corresponding quark.

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QCD

The Lagrangian of QCD (with color, flavor and spin indices suppressed) looks exactly like that of QED:

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:L = -rac{1}{4} F_{mu u} F^{mu u} + overline{psi}(igamma_mu D^mu - m)psi

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where F denotes the gluon field tensor, ψ the quark field and D the covariant derivative. Part of its content lies in the Feynman rules which state that all processes which occur in the theory can be resolved into the elementary interactions (called vertices): qqg, ggg and gggg. A quark may emit (or absorb) a gluon, a gluon may emit (or absorb) a gluon, and two gluons may directly interact. In QED, only the first kind of vertex occurs, since photons have no charge.

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