Adler's Zero and Effective Lagrangians for Nonlinearly Realized Symmetry

TL;DR

This paper shows that the CCWZ effective Lagrangian for Nambu-Goldstone bosons can be derived from infrared data alone by imposing Adler's zero and a consistent shift symmetry, without reference to the UV symmetry-breaking group G. By generalizing to multiple NGBs in a linear representation of a simple unbroken group H and requiring the shift to close with H, a Closure Condition is obtained, equivalent to embedding in a symmetric coset. The authors derive universal all-order expressions for the NGB covariant derivative and gauge fields purely in terms of H generators, illustrating that NGB interactions are largely independent of the UV breaking pattern. They also develop a bootstrapping approach to extend the shift symmetry to finite field excursions and present compact all-order formulas that reproduce CCWZ results from the IR perspective, up to a single normalization parameter f. The work highlights the universality of NGB dynamics in the IR and lays groundwork for models with pNGB Higgs and non-compact cosets, while outlining future directions such as topological terms and spacetime-symmetry breaking.

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.