S-matrix on effective string and compactified membrane

Expanding Nambu-Goto action near infinitely long string vacuum one can compute scattering amplitudes of 2d massless fields representing transverse string coordinates. As was shown in arXiv:1203.1054, the resulting S-matrix is integrable, in agreement with the known free string spectrum and also with an interpretation of the static-gauge NG action as a $T\bar T$ deformation of a free massless theory. We consider a generalization of this computation to the case of a membrane, expanding its 3d action near an infinite membrane vacuum that has cylindrical $\mathbb R \times S^1$ shape (we refer to such membrane as "compactified"). Representing 3d fields as Fourier series in $S^1$ coordinate we get an effective 2d model in which the massless string modes are coupled to an infinite KK tower of massive 2d modes. We find that the resulting 2d S-matrix is not integrable already at the tree level. We also compute 1-loop scattering amplitude of massless string modes with all compactified membrane modes propagating in the loop. The result is UV finite and is a non-trivial function of the kinematic variables. In the large momentum limit or when the radius of $S^1$ is taken to infinity we recover the expression for the 1-loop scattering amplitude of the uncompactified $\mathbb R^2$ membrane. We also consider a 2d model which is the $T\bar T$ deformation to the free theory with the same massless plus infinite massive tower of modes. The corresponding 2d S-matrix is found, as expected, to be integrable.