Correlation functions, entanglement and chaos in the $T\bar{T}$/$J\bar{T}$-deformed CFTs

In this paper, we regard the $T\bar{T}$/$J\bar{T}$-deformed CFTs as perturbation theories and calculate the first order correction of the correlation functions due to the $T\bar{T}$/$J\bar{T}$-deformation. As applications, we study the R\'enyi entanglement entropy of excited state in the $T\bar{T}$/$J\bar{T}$-deformed two-dimensional CFTs. We find, up to the perturbation first order of the deformation, the R\'enyi entanglement entropy of locally excited states will acquire a non-trivial time dependence. The excess of the R\'enyi entanglement entropy of locally excited state will also be dramatically changed up to order ${\cal O}(c)$. Furthermore, the out of time ordered correlation function is investigated to confirm that the $T\bar{T}$/$J\bar{T}$-deformations do not change the maximal chaotic behavior of holographic CFTs up to the first order of the deformations.


Introduction
Recently much attention has been paid to the irrelevant deformations of two-dimensional quantum field theories by bilinear form of conserved currents [1]. This deformation [2,3] has been extensively investigated . Although these deformations are irrelevant in the renormalization group sense, the deformed theory appears to be more predictive than the generic non-renormalizable QFT. Remarkably, such deformation preserves the integrability for the integrable quantum field [1]. Even for the non-integrable theory, some properties, e.g. for a finite size spectrum and the S-matrix, of the TT /JTdeformed theory can be exactly calculated based on the data of the undeformed theory [1,3]. For Non-Lorentz invariant cases were studied in [27][28][29][30][31][32].
Here, we study the correlation functions of primary operators in the TT /JT -deformed two-dimensional CFTs without the effect of the renormalization group flow of the op-erator. We will focus on the 2-and 4-point functions in TT /JT . For simplicity, we regard the TT /JT -deformed CFTs as perturbation theories of CFTs and compute the first order correction of the correlation function due to the TT /JT -deformation. Since TT /JT are conserved quantities, the Ward identities are held in the deformed theory.
Once we implement the Ward identity for deformed correlation functions, we have to deal with the divergences. We apply the dimensional regularization procedure and the deformation of correlation function up to the first order can be obtained explicitly.
As applications, we employ the formula to investigate the Rényi entanglement entropy in two-dimensional CFTs perturbatively. We just put a local primary operator following the procedure in [60][61][62][65][66][67] to obtain a locally excited state. With time evolution, the excess of Rényi entanglement entropy of deformed two-dimensional CFTs has been calculated in this paper. We will show how the TT /JT -deformation changes the excess of Rényi entanglement entropy in this local quenched system.
The TT -deformation of the integrable model is proposed to hold the integrability structure. Alternatively, we would like to employ the out of time order correlation function (OTOC) [84][85][86] to gain some insight into integrability/chaos after the deformation, since the OTOC has been broadly regarded as one of the quantities to capture the chaotic or integrable. We investigate the OTOC in the deformed CFTs to see whether the chaotic property is preserved or not after the TT /JT -deformation in a perturbative sense.
The remainder of this paper is organized as follows. In Section 2, we setup the perturbation of TT -deformed theories. In terms of perturbative CFT technicals, we formulate 2-and 4-point correlation functions of TT -deformed CFTs explicitly, where the dimensional regularization has been implemented. In Section 3, we have studied the excess of the Rényi entanglement entropy of the locally excited states in TT -deformed theory. In Section 4, we work out the out of time ordered correlation function of the TT -deformed theory, up to first order perturbation. In Section 5, we directly extend the investigations in Sections 2, 3 and 4 to the JT -deformed theory. Finally, section 6 is devoted to conclusions and discussions. We also mention some likely future problems.
In the appendices, we would like to list some techniques and notations relevant to our analysis.

TT -deformation and correlation functions
In this section, we give a lightning review of the TT -deformation and calculate the correlation function in the TT -deformed CFTs, which are useful in the later parts.
The TT -deformed action is the trajectory on the space of field theory satisfying where λ is the coupling constant of the TT -operator. S(λ = 0) is the action of the un-deformed CFT on the flat metric ds 2 = dzdz. Since the theory is on the flat space, the TT operator can be written as with T = T zz andT = Tzz. In this paper, we will focus on the perturbation theory of where (TT ) λ=0 = TT plays the role of the perturbation operator in the CFT. Without confusion we will denote (TT ) λ=0 as TT from now on.
In this perturbation theory, the first order correction to the n-point correlation function of primary operators O 1 (z 1 ,z 1 )O 2 (z 2 ,z 2 ) · · · O n (z n ,z n ) becomes By using the Ward identity, this correction can be written as where we have used the fact that any correlation function including T zz vanish i.e.
T zz · · · = 0. As examples, we will study the corrections to the 2-and 4-point functions up to the first order perturbation in λ.
Let us first consider the two-point function of primary operator: where z ij = z i − z j ,z ij =z i −z j . The Ward identity leads to Note that due to the effect of ∂ z i , the terms such as (z −z i ) −1 will provide a delta function. One can see that the final two terms will contribute to two non-dynamics terms which are UV divergence after the space time integration. For simplicity, we drop out these terms in later analysis. One can show that the first term in above equation is consistent with the two-point function given by [47,83]. Using the formula (87) in appendix, we obtain the first order correction of the two-point function due to the TT -deformation where ǫ is the dimensional regularization parameter. See (79) for the notation of I 2 .
This result reproduces the one obtained in [47,83]. In the following, we will use the same prescription to handle the divergent integrals.
For the single primary operator O, we have with the cross ratios The first order correction for this four-point function due to TT -deformation is The delta functions presented in the 4-th and 5-th rows of the above equation will not contribute to the dynamics of the four-point function after the proper regularization, which is similar as the situation in two-point correlation function. We thus drop these terms. Moreover, after the space time integration associated with the deformation, the terms with delta function and η∂η G G will also vanish. Using the notations of the integrals introduced in (79), we express the first order correction to four-point function as We then could use the formula (93), (98), (99) and (104) in appendix to express this integral in terms of I 1 , I 2 and I 3 . More precisely, the integral like I 11 (z 1 ,z 1 ) will also appear. The dimensional reduction parameter ǫ in I 1 and I 2 is positive, while the parameterǫ in I 3 is negative. The integral I 11 (z 1 ,z 1 ) contains both ǫ andǫ. In our calculation, the contribution due to integral I 11 (z 1 ,z 1 ) will only replace theǫ in I 3 to ǫ.
So we can just ignore I 11 (z 1 ,z 1 ), and regard the parameterǫ in paired I 3 as positive 4 .
Then we use (84), (87) and (90) to obtain the dimensional regulated result. Since the final result is quite complicated, we will not show the detail here.
However, these works are mainly focused on the Rényi entanglement entropy of vacuum states in deformed CFTs. In this section, we first review the Rényi entanglement entropy of excited state in the un-deformed CFT and then consider their TT -deformation.
Let us consider an excited state defined by acting a primary operator O a on the vacuum state |0 in the two-dimensional CFT. We introduce the complex coordinate (z,z) = (x + iτ, x − iτ ), such that x and τ are the Euclidean space and Euclidean time respectively. We insert the primary operator O a at x = −l < 0 and consider the real time-evolution from τ = 0 to τ = t with the Hamiltonian H [59,60]. The corresponding density matrix is where N is the normalization factor, ǫ is an ultraviolet regularization. Moreover, w 1 and w 2 are defined by where ǫ ± it are treated as the purely real numbers [60]. In other words, we regarded t as the pure imaginary number until the end of the calculation.
We then employ the replica method in the path integral formulation to compute the Rényi entanglement entropy. Let us choose the subsystem A to be an interval 0 ≤ x ≤ L at τ = 0. This leads to a n-sheet Riemann surface Σ n with 2n-operators We are interested in the difference of S

(n)
A between the excited state and the vacuum state: These quantities measure the effective quantum mechanical degrees of freedom of the operator [59, 60] 5 .
Let us consider the n = 2 case. We apply the conformal transformation such that Σ 2 is mapped to Σ 1 . For this case, the coordinates z i are given by 5 We call the difference of the ∆S

(n)
A as the excess of the Rényi entanglement entropy.
On the Σ 1 , the four-point function can be expressed as where We then apply the map (17) to express the four-point function on Σ 2 : Here the Rényi entanglement entropy in 0 < t < l or t > L + l for O a |0 will be vanishing [60]. When l < t < L + l, Rényi entanglement entropy will be the logarithm of the quantum dimension of the corresponding operator. Therefore, the excess of the Rényi entanglement entropy of locally excited states between the early time and late time is logarithmic quantum dimension of the local operator. In twodimensional rational CFTs, the authors of [60-62, 66, 67] obtain that the excess of Rényi entanglement entropy is a logarithmic quantum dimension of corresponding local operator which excites the space time 6 .

TT -deformation
In two-dimensional rational CFTs, we have investigated the excess of the Rényi entanglement entropy of locally excited states between the early time and late time is the logarithmic quantum dimension of the local operator, which have been proved to be universal. In this section, we consider the Rényi entanglement entropy of excited state in the TT -deformed CFT. Let us focus on the TT -deformation of ∆S (2) A in (16). The vacuum state |0 will be deformed by the TT deformation. To take the deformation of the vacuum state, we also expand the vacuum state in deformation theory up to the first order. For the sake of simplicity, we start from the density matrix of the locally excited states and focus on the first order correction of Rényi entropy in TT -deformation by using the CFT perturbation theory. Therefore, we assume that the conformal transformation is still an approximated symmetry and make use of replica trick to obtain the Rényi entropy. We have to insert the TT -deformation operator in the n-sheeted manifold. Since we only focus on the first order of the REE, combining the vacuum deformation and replica effects, it will give the 3n of the deformation on one sheet totally.
To evaluate the correlator on the Σ 2 , we use the conformal map (17) to map w in Σ 2 to z in Σ. Under the conformal map (17), the stress tensors transform as z ′2 is the Schwarzian derivative. The TT -operator thus transforms as Using this transformation formula and expanding around λ = 0, we find − ∆S A,0 + ∆S Let us focus on the large c case. The leading order is evaluated as To take the regularization, we introduce the cutoff by replacing (0, ∞) to ( 1 Λ , Λ): In the leading order O(c 2 ), the Rényi entanglement entropy depends on the UV and IR cut off introduced by regularization. By using the Ward identity and the integrals in Appendix B, the order c of eq. (25) can be written as Then our task is to substitute the conformal block and evaluate the next order correction (28). In generic CFTs, the function G(η,η) in (19), can be expressed by using the conformal blocks [87] G a (η, where b runs over all the primary operators.
When we take the early time in our setup 0 < t < l or t > L + l, one finds the cross In this limit, i.e. (η,η) → (0, 0), the dominant contribution arise from the identity operator. We thus get Plugging this into (28), we obtain the next order correction As we take the late time l < t < l + L, the cross ratios behaves as [60] η The conformal block at the limit (η,η) → (1, 0) can be written as where F bc [a] is a constant called as Fusion matrix [88,89]. Substituting this into (28), By using the Ward identity, at order O(c 0 ), the correction to the Rényi entanglement entropy of excited state can also be written in terms of integrals, which appear in a complicated form. We will not consider the correction in order O(c 0 ) in the present paper.
Together with the result at leading order (27), we obtain the TT -deformed ∆S (2) A,λ at large c limit:

−∆S
A,λ = 6λ The first term is associated with UV and IR cutoff. The second term is related to the nontrivial time dependence at the linear order of the central charge c. Comparing the Rényi entanglement entropy of excited state at the early time and late time, we find where t e and t l label the early time and late time respectively. We thus find that at the leading order of λ the excess of the Rényi entanglement entropy change dramatically in the order of O(c), which depends on the details of the CFT.

OTOC in TT -deformed CFTs
The out of time order correlation function (OTOC) has been identified as a diagnostic of quantum chaos [84][85][86]. Remarkably, the field theory with Einstein gravity dual is proposed to exhibit the maximal Lyapunov exponent, which measures the growth rate of the OTOC. In this section, we investigate the OTOC between pairs of operators: in the deformed CFTs to see whether the chaotic property is preserved or not after the TT −deformation perturbatively. Since the OTOC can be broadly regarded as one of the quantities to capture the chaotic or integrable behavior, our study will shed light on the integrability/chaos after the TT -deformation.
The thermal four-point correlator O(x, t) · · · β , x, t are the coordinates of the spatially infinite thermal system, can be computed by the vacuum expectation values through the conformal transformation: where z i ,z i are We may deform the thermal system, i.e. two-dimensional CFT at finite temperature 1/β, by inserting the TT operator The first order correction to the thermal correlator where w = x + t andw = x − t. Taking account the transformation of the stress tensor transform under the conformal transformation, we find Under the TT -deformation, the four-point function is deformed to Expanding around λ = 0 and performing the coordinate transformation (40), we obtain The term of order O(c 2 ) thus can be written as Note that this divergence only depends on the cutoff. Since no dynamics appear, this is not interested for us.
We then consider the order O(c) Note that the four-point function in un-deformed CFT is given by The two-point function in un-deformed CFT behaves as The two-point functions for operator V also has the similar construction. Using the Ward identity and the integrals in Appendix C, we evaluate the next order correction Then our task is to evaluate (50). In the two-dimensional CFT, G(η,η) can expand in terms of global conformal blocks [90]: where F is the Gauss hypergeometric function. The summation is over the global SL(2) primary operator. The coefficient p is related to operator expansion coefficient For the two-dimensional CFT corresponding to the Einstein gravity theory, all the desired propagation can be expressed by using the identity operator, and the conformal block can be replaced by where F is the Virasoro conformal block whose dimension is zero in the intermediate channel.
Here we will use a slight different notation compared with previous section.
The function F is not known in generic cases. However, at large c with small h w /c fixed and large h v fixed, the formula reads [92] F where the function has a branch cut at η = 1. For the contour around η = 1 and small η, one finds Since the path ofη does not cross the brach cut atη = 1, one findF(η) = 1 at smallη.
To apply the TT -deformed correlation function to the OTOC, we follow the steps in [86,91] to evaluate the OTOC by using the analytic of the Euclideans of the four-point function by writing as the function of the continuation parameter t. Substituting the coordinates (55) and (54) to (50), we find where we have located the operators in pairs: ǫ 2 = ǫ 1 + β/2 and ǫ 4 = ǫ 3 + β/2, and set ǫ 1 = 0 without loss generality [91].
Let us now consider the order O(c 0 ) correction. By using the Ward identity, we find where ǫ is the cutoff denoted by |z i | 2 = z izi + ǫ 2 . Taking together with (46), (56) and (57), we find the TT -deformed OTOC at late time behaves as where C 1 (x) and C 2 (x) are the terms independent of t. Therefore, the Lyapunov exponent, which measures the time growth rate, is not affected. Further, the choices of the sign of λ do not affect the late time behavior e − 2π β t in the above equation. We thus expect the TT -deformation does not affect the maximal chaos found by OTOC up to the perturbation first order of the deformation 7 . Moreover, since the bound of the Lyapunov exponent found in OTOC is un-affect, the gravity dual of the TT -deformed holographic CFT is expected to saturate the bound of the chaos. Here we focus on the late time behavior of the OTOC in the TT deformed large central charge CFT which is expected to have holographic dual. Although we have not investigated the integral model directly, it is also natural to expect that the TT -deformed integrable model is still integrable up to the perturbation first order of the deformation.

Correlation functions in JT -deformed CFTs
It is also interesting to consider the deformation of JT , which is defined by adding an operator constructed from a chiral U(1) current J and stress tensorT in the action In a similar way as in TT -deformation, we regard the deformation as a perturbative theory, in which case the action can be written as where we denoted (JT ) λ=0 = JT . The first order correction to the correlation function is O 1 (z 1 ,z 1 ) · · · O n (z n ,z n ) λ = λ d 2 z JT (z,z)O 1 (z 1 ,z 1 ) · · · O n (z n ,z n ) .
By using the Ward identity, this correction becomes where O i is the primary operator with dimension (h,h) and charge q. Therefore, the first order correction of two-point correlator due to JT -deformation is By using the integral (92), we can express this correction in terms of I 3 , which is evaluated by using the dimensional regularization.
Since the final result is quite complicated, we will not show the details here.

Entanglement entropy in JT -deformed CFTs
In this section, we consider the Rényi entanglement entropy of excited state in the JT -deformed CFT. In the parallel with the TT -deformation, we consider the JT -deformation of ∆S (2) A in (16).

−(∆S (2)
A,0 + ∆S To evaluate the correlator on the Σ 2 , we use the conformal map (17) to map w in Σ 2 to z in Σ 1 . Under the conformal map, the current and the stress tensors transform as Expanding around λ = 0, we find A,0 + ∆S We still focus on the large c case, ∆S A,λ can be written as where we used the Ward identity. This integrals can be evaluated in the similar way as in Section 3, and we find we find ∆S (2) A,λ = 6λ Plugging the coordinates (18) in, we obtain ∆S (2) A,λ = 6λ for t > 0. From the above equation, up to the λ leading order of the JT -deformation and the leading order of the large c limit, the Rényi entanglement entropy will obtain the corrections, where the first two terms associated with non trivial time dependence and the other term is about the UV cutoff due to the regularization. We also find that the excess of Rényi entanglement entropy will be dramatically changed.

OTOC in JT -deformed CFTs
We then consider the OTOC under the JT -deformation. Under the coordinate transformation, the correlation function transform as The JT -deformation of the function becomes where we have expanded around λ = 0. Under the conformal transformation (40), we By using the Ward identity, we find This integral can be evaluated by using the similar method as in the Section 3. Using the coordinates (55) and conformal block (54), we obtain where we have located the operators in pairs: ǫ 2 = ǫ 1 + β/2 and ǫ 4 = ǫ 3 + β/2, and set ǫ 1 = 0 without loss generality [91]. At late time, the JT -defomed OTOC behaves as where C 3 (x) is a coefficient independent of t. We thus find the JT -deformation does not affect the maximal chaos found by OTOC up to the perturbation first order of the deformation, which is the similar as the result found in TT -deformation.

Conclusions and discussions
In this paper, we study the TT /JT -deformation of two-dimensional CFTs perturba- The excess of Rényi entanglement entropy between early and late times is significantly changed up to the order O(c), see (37) and (71).
In [1], it claimed that the integrability structure is still held in integrable models with TT -deformation. We read out the signals of integrability by calculating the OTOC in the TT /JT -deformed field theory. To this end, the OTOC of deformed theory has been given explicitly and it shows that the TT /JT -deformation does not change the maximal chaotic property of holographic CFTs in our calculation. Although we do not explicitly exhibit the integrability structure of TT /JT -deformed integrable CFTs, up to the first order of deformation, we expect that such deformations do not change the integrability structure of un-deformed theory which is an interesting direction in the future work.
One can directly extend the perturbation to the higher order of these deformations to calculate the higher-point correlation functions, which will give us some highly nontrivial insights into the renormalization flow structure of the correlation function. One can compare the correlation function in the deformed theory with the non-perturbative correlation functions proposed by [83], and check the non-local effect in the UV limit.
Further, one can exactly check the crossing symmetry of four-point function in a perturbative sense, as we have done in this paper, or non-perturbative sense [83] as elsewhere.
To exactly match these two methods is a very interesting direction for future research.
As applications, one can apply these higher order corrected correlation functions to study the Rényi entanglement entropy and the OTOC to see the chaotic signals of the deformed theory in a perturbative sense.
By using the Feynman parameter We then replace two-dimension to d-dimension: whereǫ < 0.

By using
The integral thus becomes Note that this integral divergent. We introduce the cutoff on ρ as ( 1 Λ , Λ).
Comparing with the result obtained by using dimensional regularization (90), we find these two prescriptions of regularization are equivalent.