Perturbative renormalization of the $T \bar{T}$-deformed free massive Dirac fermion

In this paper we explicitly carry out the perturbative renormalization of the $T\bar{T}$-deformed free massive Dirac fermion in two dimensions up to second order in the coupling constant. This is done by computing the two-to-two $S$-matrix using the LSZ reduction formula and canceling out the divergences by introducing counterterms. We demonstrate that the renormalized Lagrangian is unambiguously determined by demanding that it gives the correct $S$-matrix of a $T\bar{T}$-deformed integrable field theory. Remarkably, the renormalized Lagrangian is qualitatively very different from its classical counterpart.