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Generalized TT¯-like flows for scalar theories in two dimensions

H. Babaei-Aghbolagh1,2,*, Song He1,2,3,4,†, and Hao Ouyang3,‡

  • *Contact author: hosseinbabaei@nbu.edu.cn
  • †Contact author: hesong@nbu.edu.cn
  • ‡Contact author: haoouyang@jlu.edu.cn
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Phys. Rev. D 112, 066005 – Published 10 September, 2025

DOI: https://doi.org/10.1103/58fz-mkxx

Abstract

We demonstrate that the necessary condition for SO(N)×SO(N) duality invariance manifests as a partial differential equation in two-dimensional scalar theories. This condition, expressed as a partial differential equation, corresponds precisely to the integrability condition. We derive a general perturbation solution to this partial differential equation, which includes both a root TT¯ flow equation and an irrelevant TT¯-like flow equation. Additionally, we identify a general form for these flow equations that commute with each other. Our results establish a general integrable theory characterized by theory-dependent coefficients at each order in the λ expansion. This unified framework systematically classifies all integrable theories possessing two Lorentz-invariant variables (P1, P2) while accommodating arbitrary orders of the coupling constants (λ, γ). The theory provides a comprehensive classification scheme that encompasses both known and novel integrable systems within this class.

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Phys. Rev. D 112, 066005– Published 10 September, 2025
  • Received 7 May 2025
  • Accepted 24 July 2025
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