1. Effective Theory for Mesons
Matthew Musgrave
Dec. 5, 2008

2. Two light quarks: u and d.
Approximate SU(2)xSU(2) symmetry.
Three pions acting as Goldstone bosons.
Three light quarks: u, d, and s.
Approximate SU(3)xSU(3) symmetry.
Eight mesons acting as Goldstone bosons.

3. There are eight Goldstone boson fields for the eight broken symmetries.

4. SU(3)xSU(3) transformations can be written as a product of a SSB transformation and a non-SSB transformation.

5. If the u, d, and s quarks are assumed to be massless, we obtain the SU(3)xSU(3) invariant Lagrangian density.

6. The quark masses break chiral symmetry, and a new mass term is added to the Lagangian density.
If q~ is a Goldstone boson free field, we can make the following substitution.

7. The third term in the Lagrangian density is purely bosonic and represents the Goldstone boson mass term. The quark masses cause the Goldstone bosons in this spontaneous symmetry breaking to be massive.

8. The Goldstone boson mass term is solved for the vacuum expectation value of the quark field.

9. The meson masses as a function of the quark masses.
And a mixing term.