SU(3)

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SU(3)

The symmetry group SU(3) figures prominently in elementary particle physics. There are two important and distinct SU(3) symmetries that are relevant for the strong interactions: SU(3) color symmetry of the quark and gluon dynamics and SU(3) flavor symmetry of light quarks. Each of these symmetries refers to an underlying threefold symmetry in strong interaction physics.

Mathematically, SU(3) is the group of special unitary 3 × 3 matrices U . The SU(3) group consists of all symmetry transformations that preserve the unit magnitude of 3 vectors: where the ψ_i are complex numbers satisfying In quantum mechanics, the 3-vector ψ is called the probability amplitude or wavefunction for finding a particle in any one of three possible states, whereas the dot product ψ^*·ψ is the probability for measuring the particle in any one of the three possible states. The total probability ψ^*·ψ is the sum of the probabilities for finding the particle in each of the three states: ψ^*₁·ψ₁ = |ψ₁ |² is the probability that the particle is found in state 1; |ψ₂|² is the probability that the particle is found in state 2; and |ψ₃|² is the probability that the particle is found in state 3. The sum of these three probabilities is equal to 1, since any measurement is guaranteed to find the particle in one of the three possible states. An arbitrary symmetry transformation U maps the wavefunction ψ into a new wavefunction, ψ →U ψ. The dot products of all complex 3-vectors ψ are left invariant under this mapping if U is special (its determinant is equal to one) and unitary (its Hermitian conjugate U^† is equal to its inverse U^-1). Thus, SU(3) is the group symmetry transformations of the 3-vector wavefunction ψ that maintain the physical constraint that the total probability for finding the particle in one of the three possible states equals 1.

Any arbitrary SU(3) matrix can be written in the form where the α^a are arbitrary real numbers, and the eight 3 × 3 matrices T^a, a = 1, ..., 8, are all traceless:

Tr T^a = 0 and Hermitian: T^a† = T^a.

The T^a are called the generators of SU(3) since all SU(3) group transformations can be written as exponentials of linear combinations of these eight generators. Because the number of SU(3) group transformations U is infinite, it is a great simplification to express them in terms of a finite number of group generators.

It is conventional to define the generators of SU(3) in terms of the eight Gell-Mann matrices λ^a: where The generators T^a of SU(3) satisfy the commutation relations where the ƒ^abc, the structure constants of SU(3), are real numbers. The product of any two SU(3) group transformations can be determined from the commutation relations of the generators, so they determine the structure of the group. For SU(3) the maximal set of generators that commute with each other is given by the two diagonal matrices T³ and T⁸. In quantum mechanics, commuting operators correspond to physical quantities that can be known with certainty at the same time. Thus, SU(3) charges T³ and T⁸ of a physical system can be measured simultaneously because the generators T³ and T⁸ commute.

The fundamental representation of SU(3) is the three-dimensional representation, which is referred to as the 3 of SU(3). The generators T³ and T⁸ are both diagonal, so the three states of the 3 each have definite values of the charges T³ and T⁸. The three independent states of the 3-vector ψ correspond to the (T³, T⁸) states It is useful to plot the (T³, T⁸) quantum numbers of any given SU(3) representation in a plane with coordinate axes labeled by the charges T³ and T⁸. The fundamental representation 3 is plotted in Figure 1. Note the threefold symmetry of the 3 . SU(3) generators acting on the 3 transform the three states into one another. It is interesting to note that the group SU(3) contains three SU(2) subgroups that transform any two states of the 3 into one another, while leaving the third state invariant.

The 3 of SU(3) is not equivalent to its conjugate representation, which is obtained by reversing the signs of all (T³, T⁸) quantum numbers. The conjugate representation of the 3 is called the 3̄ (3-bar) and is shown in Figure 2.

All higher-dimensional representations of SU(3) can be obtained as products of the fundamental 3 and antifundamental 3̄ representations. The product of 3 and 3̄ representations yields the eight-dimensional representation displayed in Figure 3. The 8 of

FIGURE 1

FIGURE 2

SU(3) is called the adjoint representation of SU(3). The SU(3) generators or charges T^a, a = 1, . . . , 8, form an eight-dimensional adjoint representation of SU(3). In general, every SU(3) representation exhibits threefold symmetry in the (T³, T⁸) plane.

The SU(3) color group is the exact gauge symmetry of the Standard Model, which accounts for the strong interactions of quarks and gluons. The theory of the strong interactions is called quantum chromodynamics (QCD). Quarks occur in the fundamental three-dimensional representation of SU(3) color. The three complex components of the quark color wavefunction denote the probability amplitudes for finding a quark with one of three different colors, where color is a charge that comes in three varieties: red, green, and blue. Antiquarks, the antiparticle of quarks, occur in the conjugate 3̄ representation and carry an anticolor charge. The gauge boson mediators of the strong interactions are massless gluons that occur in the eight-dimensional adjoint representation of SU(3) color. The number of gluons corresponds to the number of SU(3) generators T^a. A different-colored

FIGURE 3

gluon couples to each of the eight color charges. In QCD color charge is conserved in the interactions of quarks, antiquarks, and gluons. A quark and an antiquark couple to a colored gluon, so a gluon in the eight-dimensional adjoint representation carries both color in the fundamental 3 representation and anticolor in the 3̄ representation.

The SU(3) flavor group is an approximate symmetry of QCD resulting from the universality of quark-gluon couplings. All quark flavors with a given color couple to gluons in precisely the same manner, that is, gluons are flavor-blind. The light quarks, up, down, and strange, occur in the fundamental three-dimensional representation of SU(3) flavor. The three complex components of the light quark flavor wavefunction denote the probability amplitudes for finding a light quark with one of the three different flavors, where light quark flavor is a charge that comes in three varieties: up, down, and strange. The antiquarks ū , ū , and d̄ occur in the conjugate 3̄ flavor representation and carry antiflavor charge. SU(3) flavor symmetry is not

an exact symmetry because the masses of the u , d , and s quarks are not the same, and so the quark flavors are distinguishable. Nevertheless, the mass difference of the u , d , and s quarks are all small compared to the scale at which the QCD coupling constant becomes large, so neglecting the mass splittings of the three light quarks is a good approximation.

SU(3) flavor symmetry is a useful approximate symmetry in QCD because hadrons containing light quarks and antiquarks of different flavors have similar properties. Colorless hadrons, either mesons or baryons, can be organized into SU(3) flavor multiplets. The lowest-lying meson and baryon multiplets are both in the eight-dimensional representation of SU(3) flavor. For SU(3) flavor multiplets, it is conventional to refer to the charges isospin I³ and hypercharge Y, which are related to the charges T³ and T⁸ by

Isospin refers to the SU(2) flavor subgroup for the two lightest quark flavors u and d, whereas hypercharge is proportional to the net number of strange antiquarks minus strange quarks, or strangeness, of a hadron. Isospin and hypercharge are both approximately conserved in decays and scattering processes resulting from strong interactions.