Explicit derivation of the completeness condition in pseudoscalar meson photoproduction

By exploiting the underlying symmetries of the relative phases of the pseudoscalar meson photoproduction amplitude, we provide a consistent and explicit mathematical derivation of the completeness condition for the observables in this reaction. In particular, we determine all the possible sets of four double-spin observables that resolve the phase ambiguity of the amplitude in transversity basis up to an overall phase. The present work substantiates and corroborates the original findings of Chiang and Tabakin [Phys. Rev. C 55, 2054 (1997)]. It is found, however, that the completeness condition of four double-spin observables to resolve the phase ambiguity holds only when the relative phases do not meet the condition of equal magnitudes. In situations where this condition occurs, it is shown that one needs extra chosen observables, resulting in the minimum number of observables required to resolve the phase ambiguity reaching up to eight, depending on the particular set of four double-spin observables considered. Furthermore, a way of gauging when the condition of equal magnitudes occurs is provided.