Flavor Symmetry

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FLAVOR SYMMETRY

The label that distinguishes different types of quarks, u for up, d for down, s for strange, c for charm, b for bottom, and t for top, is called the flavor of the quark. In this context, flavor is a technical term that bears no relation to the experience associated with the sense of taste. The term flavor symmetry refers to relationships between hadrons composed of different flavor quarks. These relationships exist because the strong force, responsible for binding quarks into hadrons, acts with identical strength on all quarks, regardless of their flavor. The relationships are, however, only approximate since the much feebler electroweak interactions do distinguish between flavors, and, in addition, quarks of different flavors have different masses.

Quarks are classified as light or heavy according to whether their masses are small or large compared to the mass of a proton. The up, down, and strange quarks are light, whereas the charm, bottom, and top quarks are heavy, and, moreover, their masses differ from each other by a large multiple of the proton mass. Flavor symmetry is a good approximation for hadrons composed of the light quarks because the differences between light quark masses are small when compared to the proton mass. Conversely, flavor symmetry does not hold at all for heavy quarks. In units of the proton mass, the masses of the up, down, and strange quarks are approximately0.005, 0.010, and 0.100, respectively. Hence, the flavor symmetry relating up and down quarks holds to excellent accuracy, whereas the flavor symmetry relating all three light quarks holds somewhat less accurately. The former is known both as isospin or SU(2) symmetry, while the latter is commonly known as SU(3) symmetry. The theory of the SU(3) symmetry of hadrons was first proposed in 1961 by American physicist Murray Gell-Mann and, independently, by Israeli physicist Yuval Ne'eman as a scheme for classifying and relating properties of a multitude of observed particles. It was not until 1964 that Gell-Mann and American physicist George Zweig advanced the quark hypothesis to explain the observed SU(3) symmetry.

The dynamics of subatomic particles is best accounted for by quantum mechanics. Particles are described by state vectors. Much like the position of an object in space is specified by three real numbers x, y, z, for example, latitude, longitude, and altitude, that form a vector, the state of a light quark can be described by two complex numbers that can be thought of as the degree to which the quark is a u or a d quark, or by three complex numbers that can be thought of as the degree to which the particle is a u, d, or s quark. And just like the laws of physics are invariant under transformations that rotate vectors, the strong interactions are approximately symmetric under transformations that rotate the quark state vectors. SU(2) refers to the group of transformations that rotate state vectors with two components, whereas SU(3) refers to transformations of vectors with three components.

Because hadrons are composed of quarks, their state vectors also transform under SU(2) or SU(3) rotations in specific ways but not necessarily the same way as quarks. Figure 1 shows eight states that comprise a state vector called an octet with components that transform among themselves under SU(3) rotations. Similarly, the ten states in the decouplet of Figure 2 rotate into themselves only. The quark content of the proton and neutron is uud and udd , respectively. Since they differ in their quark content by one light quark, they are described by a two-component state vector, just like the lightest quarks

FIGURE 1

are. These two states, collectively known as the nucleon N, are said to form a doublet of SU(2). Replacing one light quark by a strange quark gives three particles collectively called Σ, to wit the Σ⁻ (dds ), the Σ⁰ (uds ), and the Σ⁺(uus ), and a fourth particle, the Λ (uds ). The Σ is said to form a triplet of SU(2), whereas the Λ is a singlet of SU(2). Replacing one more light quark by a strange quark gives another doublet of SU(2), known as the Ξ with components Ξ⁻(dss ) and Ξ⁰ (uss ). The properties of the two components of the N are related by isospin, as are those of the three components of the Σ and the two components of the Ξ. Thus, for example, the masses of the proton and neutron are 938.3 and 939.6 MeV/c² respectively; those of the Σ⁻,Σ⁰ and Σ⁺ are 1,197.4, 1,192.6, and 1,189.3 MeV/c², respectively; and those

FIGURE 2

of the Ξ⁻ and Ξ⁰ are 1,321.3 and 1,314.8 MeV/c², respectively. The Λ, with a mass of 1,115.7 MeV/c², remains unchanged under the action of SU(2) symmetry transformations.

The slightly less accurate SU(3) symmetry relates the properties of the N, Σ, Λ, and Ξ. These eight spin-½ baryons form an octet, a mathematical object that, like a vector, has specific SU(3) transformation properties. Similarly, there exists a spin-3/2 baryon decouplet with ten states: Δ⁺⁺ (uuu ), Δ⁺ (uud ), Δ⁰ (udd ), Δ⁻(ddd ), Σ⁺ (uus ), Σ⁰ (uds ), Σ⁻(dds ), Ξ⁰(uss ), Ξ⁻(dss ), and Ω⁻(sss ). The existence and mass of the Ω⁻ were predicted by SU(3) symmetry three years before its discovery in 1964. Had SU(3) been an exact symmetry, the masses of all states in the decouplet would be the same. Making the assumption that SU(3) fails to be an exact symmetry only because the strange quark is heavier than the up and down quarks, SU(3) symmetry predicts the Ω to be heavier than the Ξ by the same amount that the Ξ is heavier than the Σ and that this must be the same amount by which the Σ is heavier than the Δ. The three mass differences are experimentally determined to be 139, 149, and 152 MeV/c², respectively. Similarly, for the baryon octet the approximate SU(3) symmetry implies that the Λ is heavier than the N by the same amount that the Ξ is heavier than the Λ and that the Λ and Σ have equal masses. The observed mass differences are 177 and 203 MeV/c², respectively. The magnitude of these mass differences in units of the proton mass, about 20 percent, is a measure of how accurate SU(3) symmetry is.

The particle content of the octet of spin-1/2 baryons and the decouplet of spin-3/2 baryons is summarized in Figures 1 and 2. The vertical axis represents the number of strange quarks in a particle, and the oblique axis represents its charge. SU(2) relates particles on a horizontal line, whereas SU(3) transformations relate all particles in a multiplet.

The branch of mathematics known as group theory gives the number of states that must be grouped into an object which has specific SU(2) or SU(3) transformation properties. Since baryons contain three quarks, there are 2 × 2 × 2 = 8 combinations of u and d flavors for a baryon, 2 × 2 × 2 × 2 × 2 + 4. Group theory instructs that these are to be grouped into one object with four components and two objects with two components each (two doublets). The Δ and N are examples of four and two component objects, respectively. Incorporating the s quark, group theory determines that 3 × 3 × 3 = 27 + 1 + 8 + 8 + 10. Examples of decouplet and octet baryons are the spin-3/2 and spin-1/2 multiplets given above. The lightest singlet baryon is the Λ₁, a spin-½ particle of mass 1,406 MeV/c².

Mesons are hadrons composed of a quark and an antiquark. Group theory also determines the size of a meson multiplet. If made out of u and d quarks and antiquarks, the 2 × 2 × 4 combinations are 2 × 2 = 1 + 3, a singlet and a triplet. The π⁺,π⁰, and π⁻spin-0 mesons form a triplet of SU(2), whereas the η meson is a triplet. Including the s quark, the 3 × 3 = 9 combinations are grouped into an octet and a singlet of SU(3). With the π and η mesons, the Κ⁺, Κ⁰, K̄⁰;, and Κ⁻ mesons complete the octet, whereas the η´ meson is an example of a singlet. There are similar examples for spin-1 mesons: The ρ⁺, ρ⁰, and ρ⁻ mesons form a triplet of SU(2), and these with the four π^± mesons and the two π mesons complete the octet and a singlet.

As opposed to the strong force, which preserves flavor, the weak force can change the flavor of a quark. Nuclear beta decay is an example of a process in which a weak force induces flavor change. For example, the neutron can decay into a proton, an electron, and an antineutrino, n → pev̄ . In this process, one of the d quarks in the neutron is transformed into a u quark, so the transformation d → u gives (udd ) → (uud ), that is, n → p . Electromagnetic forces do not change flavor but act differently on the charge +2/3 u quark than on the charge -1/3 d and s quarks. Thus, for example, the electromagnetic force is responsible for the difference in mass between the π mesons and the π⁰ meson.

Flavor symmetry can also be used in the context of hadrons that contain heavy quarks in addition to light quarks. For example, the B⁺ (ub̄ ) and B̄⁰ (db̄ ) mesons form a doublet of SU(2). Together with the B_s(sb̄ ) meson, they form a triplet of SU(3). As such, their properties are related. The B⁰ and B⁺, of almost equal mass, 5,279 MeV/c², are 90 MeV/c² lighter than the B_s. The mass difference is of about the size expected given the approximate nature of SU(3) symmetry.