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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 secondorder 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 byintroducing counterterms. We demonstrate that the renormalized Lagrangian isunambiguously determined by demanding that it gives the correct $S$-matrix of a$T\bar{T}$-deformed integrable field theory. Remarkably, the renormalizedLagrangian is qualitatively very different from its classical counterpart.

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