$T\bar{T}$ deformation: Stress-energy momentum tensor deformed in CFT and in QFT for various $d$-dimensions

The $$T\bar{T}$$ deformation is based on the original work of Zamolochikov [1] explored deformations of two-dimensional conformal field theories (CFT) by an operator that is quadratic in the stress-energy tensor, called the $$T\bar{T}$$ operator.

Zamolodchikov [1] obtained analytic results for the expectation value of the operator, using the complex coordinates $$z=x+it$$ and $$\bar z=x-it$$ to express the 2d CFT coordinates:

$$O\equiv T_{zz} T_{ \bar{z} \bar z} − T_{z\bar z}^2$$

in a two-dimensional (2d) relativistic quantum field theory, where the $$T_{µν}$$ denote the components of the (euclidean) energy-momentum, or stress, tensor in complex coordinates. In particular, Zamolodchikov showed that it is well-defined by point-splitting, and that its vacuum expectation value $$\langle O \rangle$$ is proportional to $$-\langle (T^{µ}_{µ})^2 \rangle$$

My question is that what are some possible generalizations of $$T\bar{T}$$ deformation in higher dimensions?

One possibility is consider this operator defined as a bilocal operator,

$$TT¯(x, y) = T_{ij} (x)T_{ij} (y) − T_{ii}(x)T_{jj}(y)$$

where $$T_{ij}$$ is the stress-energy tensor. In a two-dimensional CFT this operator was shown by Zamoldchikov to have a remarkable OPE structure as $$x \to y$$: $$TT¯(x, y) = T (y) + \sum_α A_{α}(x − y) ∇y O_{α}(x).$$ where $$O_α$$ denote local operators and the function $$A_α(x − y)$$ can be divergent as $$x \to y$$; this relation implies that we can identify $$TT¯$$ as a local operator $$T (y)$$, modulo derivatives of other local operators. The $$TT¯$$ operator can be used to deform the CFT, generating a family of theories characterized by the coupling of this operator. While the deforming operator is irrelevant, its particular properties imply that the resulting theory is more predictive than a generic non-renormalizable quantum field theory (QFT).

My question again: are there other possibilities to motivate the (physical/geometrical) meanings of higher dimensional $$d$$ generalizations of $$T\bar{T}$$ deformation, say $$d$$ is larger than 2?

[1] A. B. Zamolodchikov, Expectation value of composite field T anti-T in two-dimensional quantum field theory, hep-th/0401146.

[2] M. Taylor, TT deformations in general dimensions, arXiv:1805.10287