Goldstone bosons in soft limit

This statement is called in the literature Adler zero. Nambu-Goldstone boson couples to the associated Noether current with a strength parametrized by the decay constant $f$ : $$ langle 0 | J_mu (x)| phi(p) rangle = -i p_mu f e^{-i p x} $$ The matrix element between physical states has a pole for $p^2 rightarrow 0$ and the residue corresponds to the scattering amplitude for NGB emission. For the $S$-matrix element between states $alpha$ and $beta$ : $$ langle alpha | J_mu (x)| beta rangle = f frac{p_mu}{p^2} A_n (alpha + phi(p), beta ) + R_{mu}(p) $$ where $A_n$ is on-shell amplitude for emission of the $phi$ with $p = P_alpha - P_beta$, and $R_mu (p)$ is regular in limit $p^2 rightarrow 0$. Due to the conservation of the current $p_mu langle alpha | J^mu (x)| beta rangle = 0$, so: $$ A_n (alpha + phi(p), beta ) = -frac{p_mu R^mu (p)}{f} $$ Further assuming that $R_mu (p)$ is regular in the limit $p rightarrow 0$ (it is an additional assumption), one has the vanishing of the amplitude in the soft limit: $$ lim_{p rightarrow 0}A_n (alpha + phi(p), beta ) = 0 $$

I've taken this extract from the paper- https://www.researchgate.net/publication/340008599_New_Soft_Theorems_for_Goldstone-Boson_Amplitudes.