Smirnov and Zamolodchikov recently introduced a new class of two-dimensional quantum field theories, defined through a differential change of any existing theory by the determinant of the energy-momentum tensor. From this TT¯ flow equation one can find a simple expression for both the energy spectrum and the S-matrix of the TT¯ deformed theories. Our goal is to find the renormalized Lagrangian of the TT¯ deformed theories. In the context of the TT¯ deformation of an integrable theory, the deformed theory is also integrable and, correspondingly, the S-matrix factorizes into two-to-two S-matrices. One may thus hope to be able to extract the renormalized Lagrangian from the S-matrix. We do this explicitly for the TT¯ deformation of a free massive scalar, to second order in the deformation parameter. Once one has the renormalized Lagrangian one can, in principle, compute all other observables, such as correlation functions. We briefly discuss this, as well as the relation between the renormalized Lagrangian, the TT¯ flow equation, and the S-matrix. We also mention a more general class of integrability-preserving deformations of a free scalar field theory.
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We thank A. Cavaglià, S. Dubovsky, D. Gross, S. Komatsu, P. Kraus, J. Maldacena, S. Minwalla, D. Simmons-Duffin, and H. Verlinde for helpful discussions. The work of V. R. is supported by NSF Grant No. PHY-1911298. V. R. thanks the Aspen Center for Physics (NSF Grant No. PHY-1607611) for hospitality while this work was being completed. The work of M. S. is supported by the Binational Science Foundation (Grant No. 2016186) and by the Quantum Universe I-CORE program of the Israel Planning and Budgeting Committee (Grant No. 1937/12).
© 2020 authors. Published by the American Physical Society.