Equivalence of three-particle scattering formalisms

In recent years, different on-shell 3 -> 3 scattering formalisms have been proposed to be applied to both lattice QCD and infinite-volume scattering processes. We prove that the formulation in the infinite volume presented by Hansen and Sharpe in [M. T. Hansen and S. R. Sharpe, Phys. Rev. D 92, 114509 (2015).] and subsequently Briceno et al. in [R. A. Briceno, M. T. Hansen, and S. R. Sharpe, Phys. Rev. D 95, 074510 (2017).] can be recovered from the B-matrix representation, derived on the basis of S-matrix unitarity, presented by Mai et al. in [M. Mai, B. Hu, M. Doring, A. Pilloni, and A. Szczepaniak, Eur. Phys. J. A 53, 177 (2017).] and Jackura et al. in [A. Jackura, C. Fernandez-Ramirez, V. Mathieu, M. Mikhasenko, J. Nys, A. Pilloni, K. Saldana, N. Sherrill, and A. P. Szczepaniak (JPAC Collaboration), Eur. Phys. J. C 79, 56 (2019).] Therefore, both formalisms in the infinite volume are equivalent and the physical content is identical. Additionally, the Faddeev equations are recovered in the nonrelativistic limit of both representations.