Pseudo entropy of primary operators in $T\bar{T}$/$J\bar{T}$-deformed CFTs

In this work, we investigate the time evolution of the pseudo-(R\'enyi) entropy after local primary operator quenches in 2D CFTs with $T\bar T/J\bar T$-deformation. Using perturbation theory, we analyze the corrections to the second pseudo-R\'enyi entropy at the late time, which exhibit a universal form, while its early-time behavior is model-dependent. Moreover, we uncover nontrivial time-dependent effects arising from the first-order deformation of the $k^{\rm th}$ pseudo-R\'enyi entropy at the late time. Additionally, drawing inspiration from the gravitational side, specifically the gluing of two cutoff AdS geometries, we investigate the $k^{\rm th}$ pseudo-R\'enyi entropy for vacuum states characterized by distinct $T\bar{T}$-deformation parameters, as well as for primary states acting on different deformed vacuum states. Our findings reveal additional corrections compared to the results of pseudo-R\'enyi entropy for globally deformed vacuum states.

Recently, there has been a surge of interest in 2D CFTs deformed by TT /JT operator [19][20][21]. Though the deformations are irrelevant in the sense of the renormalization group, owing to the integrability [19,21] and holographic duality [22][23][24][25][26], they have been extensively investigated in recent years . The TT /JT -deformation triggers a flow of the original theory along a trajectory in the space of field theory. For the TT -deformation, the trajectory of the deformed action satisfies the following condition: where λ represents the coupling constant of the TT operator. S(λ = 0) corresponds to the action of the undeformed theory. In the flat space, the TT operator can be expressed as with T = T zz andT = Tzz. In this paper, we mainly focus on the perturbation theory of S(λ).
The TT -deformed action, up to the first order of λ, is given by and we refer to (TT ) λ=0 as simply TT . We treat the JT -deformation in the same manner.
Pseudo entropy is a generalization of the entanglement entropy proposed via the AdS/CFT correspondence and post-selection [57]. Specifically, the pseudo entropy is a two-state vector version of the entanglement entropy, defined as follows. Let |ψ⟩ and |φ⟩ be two non-orthogonal states in the Hilbert space H S of a composite quantum system S = A∪B. We initially construct an operator acting on H S which is called the transition matrix [57,58] T ψ|φ ≡ |ψ⟩⟨φ| We then define the so-called reduced transition matrix T The pseudo entropy of A is then defined by the von Neumann entropy of T ψ|φ A , i.e., Furthermore, we can introduce the so-called pseudo-Rényi entropy S

(k)
A , a generalization of the Rényi entropy S (k) utilizing the transition matrix. The pseudo-Rényi entropy is defined by replacing the reduced density matrix in the Rényi entropy (6) with the reduced transition matrix T Note that the pseudo-Rényi entropy (7) gives the pseudo entropy (5) in the limit of k → 1.
It has been shown that pseudo entropy, like entanglement entropy, can be used to quantify the topological contributions of entanglement of excited states in two-dimensional rational CFTs [59][60][61] 5 . The behavior of the entanglement entropy in the deformed CFTs, especially in the context of the TT -and JT -deformations, has been extensively studied in recent years [81][82][83][84][85].
However, little is known about the pseudo-Rényi entropy in TT /JT -deformed CFTs. Another motivation for investigating pseudo-Rényi entropy in TT /JT -deformed CFTs comes from the holography. It is thought that pseudo entropy has its holographic dual, and the holographic dual of the positive sign TT -deformed two-dimensional holographic CFT was proposed to be AdS 3 gravity with a finite radius cut-off. So the properties of the pseudo-Rényi entropy in the deformed CFTs would provide insight into the structure of the deformed theories and their relationship to the undeformed CFTs. Therefore, investigating the pseudo-Rényi entropy in the deformed CFTs is an interesting and important research direction.
This paper aims to investigate the pseudo-Rényi entropy in the TT /JT -deformed CFTs.
Our inquiry can be traced back to the investigations on the Rényi entropy of vacuum states [81] and locally excited states [86] in the TT /JT -deformed theories. It has been observed that, with the first-order perturbation of the deformation, the second Rényi entanglement entropy of locally excited states acquires a nontrivial time dependence. It would be captivating to investigate the correction of the pseudo-Rényi entropy in the deformed CFTs. Besides, inspired by gluing together two cutoff AdS geometries and the cutoff AdS/TT -deformed-CFT(cAdS/dCFT) correspondence, we are interested in exploring the pseudo-Rényi entropy of two states with two different deformation parameters.
The structure of this paper is as follows. In section 2, we briefly overview the TT -deformation of the correlation functions and the replica method for locally excited states in 2D CFTs. In section 3, we concentrate on the second pseudo-Rényi entropy of the TT -deformed theories. In section 4, we investigate the excess of the k th Rényi entropy of locally excited states and the k th pseudo-Rényi entropy of locally excited states in TT -deformed CFTs. In section 5, we compute the k th pseudo-Rényi entropy of two vacuum states and two primary states with different deformation parameters in TT -deformed CFTs. In section 6, we extend the analysis from sections 3 and 4 to the JT -deformed theory. Finally, in section 7, we provide concluding remarks and discuss possible future research directions. Some technical details of the calculations are presented in the appendices.

The TT -deformation of the correlation functions
In this subsection, we briefly overview the TT -deformation in 2D CFTs and its effect on the correlation functions in [86].
primary operators in TT -deformed CFTs can be expressed as ⟨O 1 (z 1 ,z 1 ) · · · O k (z k ,z k )⟩ λ , where λ is the coupling constant of the TT -deformation. The expression for the first-order correction is as follows: To obtain the result, we take advantage of the fact that any correlation function involving T zz vanishes in a 2D CFT. Specifically, we begin with the two-point correlation function of primary where z ij = z i − z j ,z ij =z i −z j . The Ward identity of the two-point function gives us the following relation: The appearance of the delta function arises from the action of ∂ z i on terms such as (z − z i ) −1 , which leads to a non-dynamical contribution. After the integration over spacetime, this contribution gives rise to a UV divergence. To simplify the following analysis, we omit these terms. We can obtain the first-order correction of the two-point function resulting from the TT -deformation using dimension regularization Next, we compute the four-point function within the framework of TT -deformed CFTs. Specifically, the four-point function of primary operators can be written as follows: with the cross ratios and G(η,η) is the conformal block which is a function of cross ratios.
The first-order correction to this four-point function due to the TT -deformation is given by:

The pseudo entropy and the replica method
This subsection presents a concise overview of the replica method for constructing the pseudo-Rényi entropy with locally excited states in rational conformal field theories (RCFTs). We start by considering a RCFT defined on a plane, where the vacuum state is denoted as |Ω⟩. By applying the primary operators O(x 1 ) and O † (x 2 ) to the vacuum state, we create two locally excited states where the infinitesimal parameter ϵ is introduced to suppress the high energy modes [87].
Subsequently, we utilize these states to construct a real-time evolved transition matrix denoted as T 1|2 (t): To obtain the reduced transition matrix of subsystem A at time t, we trace out the degrees of freedom of A c (the complement of A) from the transition matrix T 1|2 (t), giving us T 1|2 It turns out that the excess of the k th pseudo-Rényi entropy of A with respect to the ground state, defined as ∆S (k) (T 1|2 A (t)) := S (k) (T 1|2 A (t)) − S (k) tr A c |Ω⟩⟨Ω| , is of the form [60] according to the replica method. In Eq.(17), Σ k denotes an k-sheet Riemann surface with cuts on each copy corresponding to A, and (w 2j−1 ,w 2j−1 ) and (w 2j ,w 2j ) are coordinates on the j th -sheet surface. The term in the first line of Eq.(17) is given by a 2k-point correlation function on Σ k , and the one in the second line is calculated from a two-point function on Σ 1 .
The coordinates are chosen as The 2k-point correlation function on Σ k in Eq. (17) can be evaluated with the help of a conformal mapping of Σ k to the complex plane Σ 1 . For convenience, the subsystem is chosen such that A = [0, ∞) hereafter. We can then map Σ k to Σ 1 using the simple conformal mapping We subsequently utilize the mapping given by Eq. (19) to express the 2k-point function on Σ k , 3 Second pseudo-Rényi entropy in TT -deformed CFTs The late-time excess of pseudo-Rényi entropy of locally excited operators has a universal behavior: the logarithmic quantum dimension of the operator, while the early-time one does not.
Investigating the late-time behavior of the excess of pseudo-Rényi entropy and its behavior in the early time in TT -deformed CFTs is intriguing.
In [81], the authors discussed Rényi entropy in the TT -deformed CFTs for vacuum theory.
It is shown that up to the first order of the deformation parameter λ, where we denote ∆S (k) A,λ as the correction to the undeformed Rényi entropy. Now let us consider second pseudo-Rényi entropy in TT -deformed CFTs − ∆S (2) A,0 + ∆S The derivation of (23) is a simplified version of Appendix B where we set λ 1 = λ 2 and k = 2 6 .
Under the conformal mapping (19) for k = 2, the stress tensors transform as where is the Schwarzian derivative, and c is the central charge. The TT operator thus transforms as By utilizing this transformation formula and performing an expansion around λ = 0, we obtain expressions for the zero th and first-order terms in λ: Let us focus on the large c case. At the leading order, that is the O(c 2 ) order, Eq. (26) is To regularize this formula, we introduce the IR cutoff and UV cutoff by replacing (0, ∞) with ( 1 Λ , Λ) and we obtain 2λ 6 Notice that there are some typos in [86], the exact coupling in its section 3 and section 5.1 should be λ At the next order, which is of order O(c), Eq. (26) can be evaluated as The derivation of this equation is shown in Appendix A. When we take the late-time limit t → ∞, the conformal block in the four-point correlation function is where Substitute (30) into the (29) and we find the correction to the second pseudo-Rényi entropy in large c limit at the late time: A,λ = 6λ This result is universal in all RCFTs. However, if the two insertion coordinates coincide, Eq.
(31) does not yield the same result as the second Rényi entropy derived in [86]. This discrepancy arises primarily because in [86], we initially set x 1 → x 2 and perform an expansion in ϵ, whereas in the case of the second pseudo-Rényi entropy, we first expand in ϵ and then take the limit Notably, the two limits, ϵ → 0 and x 1 → x 2 , do not commute, as demonstrated in [61].
Since the two insertion coordinates are different, the conformal block G(η,η) may diverge for some models at the early time. The rest of this subsection will discuss the early-time correction to the second pseudo-Rényi entropy in TT -deformed CFTs for the free scalar and Ising model as examples.

Example 1. The free scalar
The free scalar can be described by the operator O = 1 √ 2 (e i 2 ϕ + e − i 2 ϕ ), whose conformal dimension is h =h = 1 8 . The conformal block of free scalar can be written as Substituting (32) into (29), we find that the corrections to the second pseudo-Rényi entropy of free scalar at the early time in large c limit is −∆S Notice that the results (31) and (33) are highly different from that in [86] because the insertion coordinates of two states generating the transition matrix are different.

Example 2. Ising model
The 2n-point correlation function of the spin operator in the Ising model can be written as The first-order corrections to the second pseudo-Rényi entropy of the spin operator at the early Since all correlation functions in the Ising model are known, one can easily calculate the correction to the pseudo-Rényi entropy in TT -deformed CFTs at the early time. As Eq. (35) shows, the earlier time behavior of the pseudo-Rényi entropy depends on the initial setup and details of the theories.
The correction to the undeformed Rényi entropy seen in appendix B is where O(z i ,z i ) is denoted by O i . Notice that if we take all primary operators as identity operators, (37) becomes the Rényi entropy for vacuum states in TT -deformed CFTs, and it matches the result in [81] precisely.
Using the conformal transformation (19) between Σ k and Σ 1 , we can rewrite this formula We still focus on the large c case. At the leading order O(c 2 ), by introducing the IR cutoff 1 Λ , the correction to the k th Rényi entropy is The derivation of this formula is shown in Appendix A. At the next-to-leading order, the integral in the first line of (38) can be divided into two parts at the late time. The first part, due to T (z) acting on the 2k-point correlation function, can be written as where in the last equality, we use the fact and the fact that the holomorphic part of the 2k-point correlation function can be written as The second part due toT (z) acting on the 2k-point correlation function can be written as where in the last equality, we use the fact and the fact that the anti-holomorphic part of the 2k-point correlation function can be written as Now we replace all coordinates in (40) and (43) with We get the correction to the k th Rényi entropy in the TT -deformed CFT at the late time We next consider the integral (38) at the early time. The coordinates and the 2k-point correlation function have the following property: We can get the correction to the k th Rényi entropy at the early time.

∆S (k)
A,λ = 3kλ Results (47) and (49) are precisely the results we find in [86] if we take the interval of A in [86] to infinity and set k = 2. Taking the limit of k → 1 for (47) and (49) Combining (40) and (43), and replacing all the coordinates with (21), we get the correction to the k th pseudo-Rényi entropy in the TT -deformed CFT, The result (51) precisely matches the result we find in the previous section 3 if we take k = 2.
Taking the limit of k → 1 for (51), we obtain the late-time correction of the pseudo entropy under the TT -deformation up to the first-order which is In the limit of late time, it is evident that ∆S A,λ ∼ 1 t 2 . Therefore, the first-order TT deformation does not alter the late-time behavior of the pseudo-entropy, which is consistent with the late-time behavior of the excess entanglement entropy derived in (50). Moreover, there exist singularities at t = x 1 or t = x 2 , which emerge due to the non-smooth behavior of the correlation function at those specific points. These singularities exhibit similarities to the finding discussed in the previous subsection.

Pseudo entropy in TT -deformed CFTs with different deformed states
In the previous section, we have explicitly shown the correction of the k th pseudo-Rényi entropy after the TT -deformation, up to the first order of the deformation coefficient. Recent research about the TT -deformed CFTs has focused on how to glue two different AdS spacetimes [88,89] and its holographic dual. Inspired by the cAdS/dCFT correspondence, in this section, we will consider the pseudo-Rényi entropy for states with different TT -deformation on the CFT side.
We first consider the transition matrix for the vacuum state taking the form where the vacuum states are deformed by TT operators with different coupling constants λ 1 and λ 2 , and this transition matrix is defined in the boundary theory of a glued AdS spacetime with two cutoff AdS spacetimes gluing together along a space-like hypersurface, as shown in Figure   1. In the rest of this section, we mainly consider the case that the deformation parameters, λ 1 cutoff In Eq. (54), LHP denotes the lower half-plane of Σ 1 , while UHP denotes the upper half-plane of Σ 1 .
Using the transformation of the TT operator between Σ k and Σ 1 , the integral becomes whereΛ 1 andΛ 2 are two different IR cutoffs. Since we are dealing with two distinct deformation couplings, there is no requirement for the two cutoffs to be identical. However, in the case of vacuum states, if we exchange the indices (1 ↔ 2), the path integral constructed through the replica method should remain invariant. This symmetry is also satisfied by the result in Eq. (55). In the scenario where the two deformation parameters are equal, we must setΛ 1 =Λ 2 .
In the subsequent part of this section, we shall examine the pseudo-Rényi entropy derived from primary operators acting on distinct vacuum states with two different infinitesimally small constants in TT -deformed CFT. The states used to construct the transition matrix are now expressed as follows: According to appendix B, the deformed k th pseudo-Rényi entropy is now taking the form Through the calculation in appendix C, we have the correction to the k th pseudo-Rényi entropy with different deformed parameters of O(c) under TT deformation at the late time Notice that there are also some possible cutoff terms with ϵ 1 and ϵ 2 corresponding to the LHP and UHP separately and we have regularized them through minimal subtraction seen in (98)-(105). When the two deformation parameters are identical, we require that both cutoffs, ϵ 1 and ϵ 2 , are set to the same value ϵ. In this case, formula (58), including the ignored cutoff terms, will precisely give rise to the result in Eq. (51). However, when λ 1 ̸ = λ 2 , the correction to the k th pseudo-Rényi entropy may involve additional nontrivial terms compared to (51).
When the deformation parameters are large enough, the two deformed theories denoted as CFT 1 and CFT 2 are far from the undeformed theory, so states in CFT 1 and CFT 2 belong to two different Hilbert space H 1 and H 2 . The gluing procedure in figure 1 gives rise to an invalid transition matrix, since ordinarily the transition matrix is defined with states in the same Hilbert space [57] 7 . In order to establish a well-defined transition matrix, we could consider the tensor product theory CFT 1 ⊗CFT 2 which has been extensively investigated in [90]. From the holographic view, such tensor product theory may relate to the gluing of two cutoff spacetimes along their time-like cutoff boundaries (or branes) as shown in figure 2. In the rest of this section, we will concentrate on pseudo entropy constructed from states of the tensor product Let us first consider a sufficiently simple tensor product theory as a warm-up. The Hamiltonian of the theory is given by where H i (i=1,2) is the Hamiltonian of the corresponding deformed theory CFT i and is the identity operator for CFT i . The inner product of |φ⟩ = |φ 1 ⟩ ⊗ |φ 2 ⟩ is defined as We then construct two states of the tensor product theory by acting tensor product operators on the deformed vacuum state |Ω λ 1 ⟩ ⊗ |Ω λ 2 ⟩ Before calculating the time evolution of pseudo-Rényi entropy of |ψ 1 ⟩ and |ψ 2 ⟩ using the replica method, we would like to show specifically the path integral in the tensor product theory. As an explicit example, we first consider the correlation function of O ⊗ O ′ in the tensor product theory, which is The transition matrix of |ψ 1 ⟩ and |ψ 2 ⟩ is and after using replica method (T Therefore, the correction to the k th pseudo-Rényi entropy for tensor product theory can be written as In the above discussion, our focus is on the condition that the two deformed theories remain independent. However, as demonstrated in [88], if we consider the gluing of two AdS spacetimes, we should add an interaction term H int to the Hamiltonian of tensor product theory (59) which probably can be understood in terms of an irrelevant deformation that couples the two deformed CFTs together, where the author in [90] call it sequential flow, and this term arises due to the exchange of gravitons between two AdS spacetimes. Under such flow, the Lagrangian of the tensor product theory with interactions up to the first order of deformation parameters can be expressed as [90] Due to the interaction term, the transition matrix of interest cannot be decomposed as (63) in this case where |Ω⟩ SF denotes the vacuum of the sequential-flowed theory (66). We still can expand the correlation function under sequential flow perturbatively, and the result is where Z SF is the partition function under sequential flow which can be expressed as Using the method in section 4 we can derive the correction to the k th pseudo-Rényi entropy for tensor product theory under sequential flow where the correlation function containing the term such as T 1T2 can be expressed as and the term such as T 1T1 can be expressed as All terms in (70) can be computed similarly using the results in (39), (40) and (43). However, comparing (70) and (65), we find the result in (70) has additional corrections shown in the last two lines of (70), which can not be derived through the renaming of λ 1 and λ 2 in (65), and these corrections are totally due to the interaction between the two CFTs of the tensor product theory.
The above computations are an initial attempt in RCFTs under the sequential flow [90].
In order to gain insight into the connection between the gluing procedure and the sequential flow, we expect to find the holographic pseudo-Rényi entropy of transition matrix (67) in the concept of "holography without boundaries" initiated in [88]. The holographic calculations in glued AdS and the field-theoretic calculations in holographic CFTs under sequential flow are necessary and will be served as our future work.

Pseudo-Rényi entropy in JT -deformed CFTs
So far, we have discussed the TT -deformation of the pseudo-Rényi entropy. In this section, we study the JT -deformation of the pseudo entropy, with JT operator constructed from a chiral U (1) current J and stress tensorT .
The JT -deformed action is the trajectory on the space of field theory satisfying Similar to the TT -deformation, we regard the deformation as a perturbative theory, in which case the action can be written as where (JT ) λ=0 is denoted by JT later on. The first-order correction to the correlation function by using the Ward identity is We then calculate the correction to the k th pseudo-Rényi entropy in the JT -deformed CFT.
Under the conformal map (19), the U (1) current and the stress tensor transform as The correction to the undeformed k th pseudo-Rényi entropy can be derived similarly from that in the previous section. Thus we have We still focus on the large c case. At O(c), ∆S

(k)
A,λ at the late time can be written as Substituting (21) into (79), we get the first-order corrections to the k th pseudo-Rényi entropy in JT -deformed CFT, Specifically, we consider the k = 2 case, ∆S A,λ = 2λ cπ 32 If we take the limit x 1 → x 2 , (81) returns to the result we found in [86]. Using the method mentioned in subsection 4.1, one can also calculate the late-time and early-time corrections to the entanglement entropy in JT -deformed CFTs.

Conclusion and prospect
This paper employs perturbation theory to investigate the pseudo-Rényi entropy associated with locally excited states in the two-dimensional TT /JT deformed CFTs. Our primary objective is to derive nontrivial corrections to the pseudo-Rényi entropy of the undeformed theory. Initially, we provide a comprehensive review of the calculation of two-point and four-point correlation functions using the Ward identity in deformed CFTs and the pseudo-Rényi entropy obtained through the replica method in undeformed CFTs. We then derive the leading order corrections to the second pseudo-Rényi entropy and find that it may depend on the introduced infrared (IR) cutoff and may also depend on the ultraviolet (UV) cutoff, depending on whether the subsystem A is finite. Furthermore, at order O(c), we discover that the second pseudo-Rényi entropy may acquire nontrivial corrections in the deformed CFTs at the late time, as demonstrated in Eqs. (31) and (81). However, the early-time correction becomes more intricate, and we illustrate its behavior using the free scalar and Ising models as specific examples.
In addition, we extend our investigation to the k th Rényi entropy, examining its behavior both at the early and late times as expressed in Eqs. (47) and (49) A Details of integrals in TT -deformed pseudo-Rényi entropy In this appendix, we evaluate the integral of (29) in detail. It mainly takes the following forms The first factor of the right-hand side of Eq. (92) corresponds to a two-point function on the complex plane Σ 1 , while the second factor corresponds to a 2k-point function on the k-sheeted A,0 ), respectively. Substituting (93) into (7), we obtain 8 We denote (w,w) by (w) in O and TT for simplicity. the k th pseudo-Rényi entropy of subsystem A under the TT -deformation ∆S (k) + sub-leading order terms.

C Details of integrals in pseudo-Rényi entropy with different deformed states
In this appendix, we evaluate the integral of (57) Similar to the discussion above, we still focus on the large c case. The leading order of the integral (95) is simple, so we do not evaluate it in detail. We will calculate the integral (95) of O(c) in the rest of this appendix. At O(c), using the conformal transformation (19) and Ward Replacing all the coordinates in (106) with (21), we get the correction to the k th pseudo-Rényi entropy with different deformed parameters under TT deformation at the late time ∆S (k) A,λ 1 ,λ 2 = c(k + 1) 24k 2 λ 1