# Quark Model - Mesons

Mesons

The Eightfold Way classification is named after the following fact. If we take three flavours of quarks, then the quarks lie in the fundamental representation, 3 (called the triplet) of flavour SU(3). The antiquarks lie in the complex conjugate representation 3. The nine states (nonet) made out of a pair can be decomposed into the trivial representation, 1 (called the singlet), and the adjoint representation, 8 (called the octet). The notation for this decomposition is

.

Figure 1 shows the application of this decomposition to the mesons. If the flavour symmetry were exact, then all nine mesons would have the same mass. The physical content of the theory includes consideration of the symmetry breaking induced by the quark mass differences, and considerations of mixing between various multiplets (such as the octet and the singlet). The splitting between the η and the η′ is larger than the quark model can accommodate. This "η–η′ puzzle" is resolved by instantons.

Mesons are hadrons with zero baryon number. If the quark–antiquark pair are in an orbital angular momentum L state, and have spin S, then

• |LS| ≤ JL + S, where S = 0 or 1,
• P = (−1)L + 1, where the 1 in the exponent arises from the intrinsic parity of the quark–antiquark pair.
• C = (−1)L + S for mesons which have no flavour. Flavoured mesons have indefinite value of C.
• For isospin I = 1 and 0 states, one can define a new multiplicative quantum number called the G-parity such that G = (−1)I + L + S.

If P = (−1)J, then it follows that S = 1, thus PC= 1. States with these quantum numbers are called natural parity states while all other quantum numbers are called exotic (for example the state JPC = 0−−).