# Quark Model

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Quark Model
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## Condensed Matter Physics

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2 14. Quark model  14.2. Mesons Mesons have baryon number B  = 0. In the quark model, they are qq  ′ bound states of quarks q  and antiquarks q  ′ (the ﬂavors of  q  and q  ′ may be diﬀerent). If the orbital angular momentum of the qq  ′ state is ℓ , then the parity P  is ( − 1) ℓ +1 . The meson spin J  is given by the usual relation | ℓ − s |≤ J  ≤| ℓ + s | , where s is 0 (antiparallel quark spins) or 1 (parallel quark spins). The chargeconjugation, or C  -parity C  = ( − 1) ℓ + s , is deﬁned only for the q  ¯ q  states made of quarks and theirown antiquarks. The C  -parity can be generalized to the G -parity G = ( − 1) I  + ℓ + s for mesonsmade of quarks and their own antiquarks (isospin I z = 0), and for the charged u ¯ d and d ¯ u states(isospin I = 1).The mesons are classiﬁed in J  PC  multiplets. The ℓ = 0 states are the pseudoscalars (0 − + )and the vectors (1 −− ). The orbital excitations ℓ = 1 are the scalars (0 ++ ), the axial vectors(1 ++ ) and (1 + − ), and the tensors (2 ++ ). Assignments for many of the known mesons are givenin Tables 14.2 and 14.3. Radial excitations are denoted by the principal quantum number n . Thevery short lifetime of the t quark makes it likely that bound-state hadrons containing t quarksand/or antiquarks do not exist.States in the natural spin-parity series P  = ( − 1) J  must, according to the above, have s = 1and hence, CP  = +1. Thus, mesons with natural spin-parity and CP  = − 1 (0 + − , 1 − + , 2 + − ,3 − + , etc. ) are forbidden in the q  ¯ q  ′ model. The J  PC  = 0 −− state is forbidden as well. Mesonswith such exotic  quantum numbers may exist, but would lie outside the q  ¯ q  ′ model (see sectionbelow on exotic mesons).Following SU(3), the nine possible q  ¯ q  ′ combinations containing the light u, d, and s quarks aregrouped into an octet and a singlet of light quark mesons: 3 ⊗ 3 = 8 ⊕ 1 . (14 . 2)A fourth quark such as charm c can be included by extending SU(3) to SU(4). However, SU(4)is badly broken owing to the much heavier c quark. Nevertheless, in an SU(4) classiﬁcation, thesixteen mesons are grouped into a 15-plet and a singlet: 4 ⊗ 4 = 15 ⊕ 1 . (14 . 3)The weight diagrams  for the ground-state pseudoscalar (0 − + ) and vector (1 −− ) mesons aredepicted in Fig. 14.1. The light quark mesons are members of nonets building the middle plane inFig. 14.1(a) and (b).Isoscalar states with the same J  PC  will mix, but mixing between the two light quark isoscalarmesons, and the much heavier charmonium or bottomonium states, are generally assumed to benegligible. In the following, we shall use the generic names a for the I  = 1, K  for the I  = 1 / 2,and f  and f  ′ for the I  = 0 members of the light quark nonets. Thus, the physical isoscalars aremixtures of the SU(3) wave function ψ 8 and ψ 1 : f  ′ = ψ 8 cos θ − ψ 1 sin θ , (14 . 4) f  = ψ 8 sin θ + ψ 1 cos θ , (14 . 5)where θ is the nonet mixing angle and ψ 8 =1 √  6( u ¯ u + d ¯ d − 2 s ¯ s ) , (14 . 6) ψ 1 =1 √  3( u ¯ u + d ¯ d + s ¯ s ) . (14 . 7) February 16, 2012 14:08
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