Adler's zero and effective Lagrangians for nonlinearly realized symmetry

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.