Correlation functions in $$ \textrm{T}\overline{\textrm{T}} $$-deformed theories on the torus

A bstract We study the correlation functions of local operators in unitary $$ \textrm{T}\overline{\textrm{T}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> -deformed field theories defined on a torus, using their formulation in terms of Jackiw-Teitelboim gravity. We focus on the two-point correlation functions in momentum space when the undeformed theory is a conformal field theory. The large momentum behavior of the correlation functions is computed and compared to that of $$ \textrm{T}\overline{\textrm{T}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> -deformed field theories defined on a plane. For the latter, the behavior found was $$ {\left(\frac{\sqrt{t}\left|q\right|}{\pi e}\right)}^{-\frac{tq^2}{\pi }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mfenced> <mml:mfrac> <mml:mrow> <mml:msqrt> <mml:mi>t</mml:mi> </mml:msqrt> <mml:mfenced> <mml:mi>q</mml:mi> </mml:mfenced> </mml:mrow> <mml:mi>πe</mml:mi> </mml:mfrac> </mml:mfenced> <mml:mrow> <mml:mo>−</mml:mo> <mml:mfrac> <mml:msup> <mml:mi>tq</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>π</mml:mi> </mml:mfrac> </mml:mrow> </mml:msup> </mml:math> , where q is the momentum and t is the deformation parameter. For a torus, the same behavior is found for | q | ≪ L / t , where L is the torus’ length scale. However, for | q | ≫ L / t , a different behavior is found: $$ {\left(\frac{2{\sqrt{t}}^5{q}^2}{\pi e{L}^3{\left|T\right|}^2}\right)}^{\frac{tq^2}{\pi }} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mfenced> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:msup> <mml:msqrt> <mml:mi>t</mml:mi> </mml:msqrt> <mml:mn>5</mml:mn> </mml:msup> <mml:msup> <mml:mi>q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mrow> <mml:mi>πe</mml:mi> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:msup> <mml:mfenced> <mml:mi>T</mml:mi> </mml:mfenced> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> </mml:mfenced> <mml:mfrac> <mml:msup> <mml:mi>tq</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>π</mml:mi> </mml:mfrac> </mml:msup> </mml:math> , where T is the complex structure of the torus. Hence, at large momentum, the correlator decays and then grows. This behavior suggests that operators carrying momentum q are smeared on a distance scale t | q |. The difference from the plane’s result illustrates the non-locality of the theory and the UV-IR mixing.