Chiral Perturbation Theory (CHPT)

Abstract

Let us, for the sake of completeness, briefly repeat some facts which have been already stressed in this book in different contexts. The eight light hadrons (π, K, η) are pseudoscalar mesons. They are believed to be composed of the quarks (u, d, s) and their antiquarks. For yet unknown reasons, their masses (m u , m d , m s ) happen to be small. If these masses were strictly zero, the Lagrangian of QCD would exhibit an exact SU(3) R ΧSU(3) L symmetry. For the standard model to be consistent with the facts of life, it is crucial that the ground state of the theory is not symmetric under this group, such that this RL symmetry breaks down to SU(3) R+L • Pseudoscalar mesons are identified with the Goldstone bosons generated by the symmetry breakdown. If m u , m d and m s were zero, we would have the chiral limit, and the pseudoscalar mesons would be massless. In real world, their masses are proportional to square roots of quark masses, e.g. $$ M_\pi ^2 = \left( {{m_u} + {m_d}} \right){B_0}\left[ {1 + O\left( m \right)} \right]$$ (7.1.1) where the constant B0 depends on the quark condensate. The hidden symmetry reveals itself in the low energy properties of the pseudoscalar mesons. Weinberg (1979) showed that these properties can be analyzed on the basis of an effective Lagrangian, replacing the quark and gluon fields of QCD by a meson field, represented by an element in the SU(3) flavour gauge group. The effective Lagrangian is expanded in powers of derivatives of the meson field and in powers of mass matrix: