Low-Energy Effective Field Theory

This chapter constitutes the core of the part of the book devoted to spontaneously broken internal symmetries. It develops two methods for construction of effective actions for Nambu–Goldstone bosons. The first method is based on identification of various contributions to the effective Lagrangian with tensor fields on the coset space of broken symmetry. The requirement of invariance of the action is imposed and solved directly using the tools of differential geometry. However, to understand and utilize the final result requires just elementary linear algebra and group theory. The second approach to construction of effective actions introduces a set of background gauge fields, coupled to the Noether currents of the symmetry. This makes it possible to classify invariant actions using elementary field theory, without heavy use of differential geometry. A comparison of the two approaches sheds light on the conditions under which a global symmetry can be gauged. The last section of the chapter revisits the spectrum of Nambu–Goldstone bosons. It is shown, among others, that type-A and type-B Nambu–Goldstone bosons have a distinct geometric interpretation, associated with a presymplectic structure on the coset space.