Adler's Zero and Effective Lagrangians for Nonlinearly Realized Symmetry

TL;DR

This paper extends the shift symmetry method to derive effective Lagrangians for multiple Nambu-Goldstone bosons, showing it reproduces CCWZ results without needing the full broken group G, relying only on the unbroken subgroup H.

Contribution

It generalizes the shift symmetry approach to multiple NGBs in symmetric cosets, deriving CCWZ Lagrangians using only H generators and the Adler's zero condition.

Findings

Derivation of NGB covariant derivatives to all orders in decay constant f.

Reproduction of CCWZ Lagrangian without explicit G knowledge.

Extension of shift symmetry method to multiple NGBs in symmetric cosets.

Abstract

Long ago Coleman, Callan, Wess and Zumino (CCWZ) constructed the general effective lagrangian for nonlinearly realized symmetry by finding all possible nonlinear representations of the broken group G which become linear when restricted to the unbroken group H. However, in the case of a single Nambu-Goldstone boson (NGB), which corresponds to a broken U(1), the effective lagrangian can also be obtained by imposing a constant shift symmetry. In this work we generalize the shift symmetry approach to multiple NGBs and show that, when they furnish a linear representation of H that can be embedded in a symmetric coset, it is possible to derive the CCWZ lagrangian by imposing 1) the "Adler's zero condition," which requires scattering amplitudes to vanish when emitting a single soft NGB, and 2) closure of shift symmetry with the linearly realized symmetry; knowledge of the broken group G is not required at all. Using only generators of H, the NGB covariant derivative and the associated gauge field can be computed to all orders in the NGB decay constant f.