Subatomic Physics

478 Quark Models of Mesons and Baryons
their quark constituents are
ρ + = ud
ρ 0 =
ρ − = du.
dd − uu
√ 2
(15.7)
These quark combinations also apply to the pions; the difference between the rho
and the pion lies in the ordinary spin. The rho is a vector meson (J π =1 − ), while
the pion is a pseudoscalar meson (J π =0 − ). The other neutral mesons will be
discussed in Section 15.6.
If masses beyond 1 GeV/c 2 are considered, mesons of orbital angular momentum
l =1� begin to appear, with J π =0 + , 1 + ,and2 + ; they correspond to qq states
1 P1, 3 P0, 3 P1, and 3 P2; the isospin can be zero or one, as for the lower mass mesons.
15.5 Baryons as Bound Quark States
Three quarks form a baryon. Since quarks are fermions, the overall wave function
of the three quarks must be antisymmetric; the wave function must change sign
under any interchange of two quarks:
|q1q2q3〉 = −|q2q1q3〉. (15.8)
To explain why the wave function of the three quarks must be antisymmetric, the
ideas expounded in Chapter 8 are generalized. There, with the introduction of
isospin, proton and neutron were considered to be two states of the same particle.
The total wave function, including isospin, of a two-nucleon system then must be
antisymmetric under exchange of the two nucleons. Here it is assumed that the three
quarks are three states of the same particle, and Eq. (15.8) is then the expression
of the Pauli principle. The simplest situation arises when the three quarks have no
orbital angular momentum between any pairs and have their spins parallel. The
resultant baryon then has spin 3/2 and positive parity. As in the case of the mesons,
it is straightforward to find the quantum numbers of the various quark combinations.
Consider, for instance, the combination uuu.
uuu : A =1,S =0,I3 = 3
,q =2e, J =
2
� �
3
,
2
where S is the strangeness. These are just the quantum numbers of the ∆ ++ ,
the doubly charged member of the ∆(1232). For a ∆ ++ , however, with all spin
components parallel (J =3/2,Jz =3/2) and all quarks in S + states, or no orbital
angular momentum, the wavefunction is symmetric under interchange of any pair
of quarks.