Linear response of entanglement entropy to $T\bar{T}$ in massive QFTs

We compute the leading correction to entanglement entropy in $T\bar{T}$ deformed massive QFTs. We show that both for massive scalar and Dirac fermion, the leading order correction to the entanglement entropy of half-space comes from the boundary of the entangling surface. For the case of massive scalar, the boundary term is finite while for massive fermion, it diverges logarithmically giving rise to an additional log-square divergence in the entanglement entropy.


Introduction
Over the past few years, there have been interesting developments in solvable deformations of two-dimensional quantum field theories (QFT) by a class of irrelevant operators quadratic in the conserved currents [1,2]. Although such deformations involve flowing up the renormalization group (RG), the ultraviolet (UV) divergences are very much under control and the deformed theory turns out to be well-defined. One such solvable irrelevant deformation is the so-called TT deformation (where T andT are respectively the holomorphic and anti-holomorphic components of the stress tensor) where the seed theory is deformed in such a way that the tangent to the RG trajectory in the space of theories, at all points along the flow, is given by the determinant of the stress-energy tensor of the deformed theory at that point on the RG flow. It was shown in [1,2] that the spectrum of the deformed theory satisfies a flow equation popularly known as the Burgers equation. Given the spectrum of the undeformed theory, the spectrum of the deformed theory can be uniquely determined in terms of the spectrum of the undeformed theory and the TT coupling. The high energy limit of the theory is nonlocal in the sense that the short-distance behavior is not governed by a fixed point. For one particular sign of the TT coupling, the high energy density of states is Hagedorn, while for the other sign of the TT coupling, the deformed energies of the excited states (above some threshold) become complex. For an elaborate review of the subject, we will refer to the reader to [3] and for generalizations to [4][5][6][7][8].
The TT operator is a composite operator and needs to be defined carefully so that it's free from divergences. It was shown in [9] that in a generic Lorentz invariant local QFT 2 on flat space, the TT operator defined by point splitting, is unambiguous and non-singular (up to total derivative terms) in the collision limit. Correlation functions in a generic CFT 2 deformed by TT have been studied in [10] to all orders in perturbation theory. In particular, it was shown that the UV divergences can be absorbed into nonlocal field renormalization to all orders in perturbation theory. Correlation functions of stress tensor operators in a TT deformed CFT in a 't Hooft-like limit have been studied in [11], where it was argued that, in this limit, the non-local features of TT deformation disappears and the theory can be treated as a local QFT.
Perturbative renormalization of TT deformed QFTs was first studied in [12] where the renormalized Lagrangian was derived from the S-matrix in the case of massive field theories. Renormalization of composite operators in TT deformed QFTs has been studied in [13,14].
There have been a few approaches to understanding entanglement entropy in TT deformed CFTs [15][16][17][18]. Most of the results available in the literature are holographic. Field theory computation of entanglement entropy of a TT deformed CFT is, in general, difficult and not known beyond leading order in perturbation theory. At linear order in the TT coupling, the response of entanglement entropy to TT vanishes in a CFT 1 , and to the best of our knowledge, the next to leading order correction is not known. However, the leading order correction to entanglement entropy in a TT deformed QFT with a mass scale is non-vanishing. In this paper, we investigate this problem in the case of a free massive scalar field and Dirac fermion. We show that the linear response of entanglement entropy of half space comes from the boundary of the entangling surface. In the case of massive scalar the leading correction has a logarithmic divergence while for massive Dirac fermion we find a log-square divergence along with a log divergence. The presence of the log-square term renders the coefficient of the log term non-universal.
The paper is organized as follows. In section 2, we give a lightning review of the entanglement entropy of half space in a Lorentz invariant QFT. In section 3, we compute the renormalized TT operator and the necessary correlation functions needed to compute entanglement entropy. In section 4, we evaluate the leading order correction to entanglement entropy and finally in section 5 we discuss our finding with avenues to future research.

Entanglement entropy of half space
In this section, we review the results which are essential for evaluating the entanglement entropy associated with a semi-infinite spatial cut A = {(x 1 , x 2 ) ∈ R 2 : x 1 ∈ (0, ∞) , x 2 = 0}. It was argued in [20] that the modular Hamiltonian for this geometry tremendously simplifies. For any QFT in the vacuum state it can be identified with the generator of rotations around the origin 2 where ρ = TrĀ|0 0| is the vacuum state reduced to A, and c is a real normalization constant, such that Tr A ρ = 1. By definition, entanglement entropy is given by Representing the vacuum expectation value of the modular Hamiltonian in terms of the Euclidean path integral, one can derive a closed form expression for the variation of entanglement entropy with respect to any coupling λ with other parameters fixed [21][22][23] where the subscript c indicates connected correlator, whereas I E is the full Euclidean action including counterterns which are necessary to generate finite correlation functions. If the normalization condition Tr A ρ = 1 is not imposed for some reason, the right hand side should be supplemented with an additional term 0| ∂K ∂λ |0 . For us this term is absent since the reduced state in the definition of entanglement entropy is normalized. 3 Furthermore, since K generates rotations around the origin and can be evaluated on any semi-infinite cut not necessarily with θ = 0, it follows that the correlation function of K with any scalar operator is independent of the polar angle θ. It is only a function of the radial distance. In particular, (2.3) can be rewritten as follows In higher dimensions K corresponds to the generator of rotations in the plane orthogonal to the entangling surface. The minus sign on the right hand side ensures K is a positive definite operator for our choice of the stress tensor T µν = 2 where O(r, θ) is the operator that couples to λ, whereas θ can take any value, the final answer is independent of a particular choice. This general formula will be used in section 4 to evaluate entanglement entropy associated with the so-called TT deformation.

Free massive fields and TT
In this section, we evaluate the correlation function K TT c for free massive scalar and Dirac fields. which will be used in the following section to calculate the linear response of entanglement entropy to the TT deformation. Free massive field theories are gaussian, and therefore the stress tensor in the definition of K is quadratic in the fields, whereas TT is quartic in the fields. In particular, to calculate K TT c for free fields one has to evaluate the only diagram shown in Fig.1.
In what follows we consider the massive scalar and Dirac fields separately.

Massive scalar field theory
The Euclidean action for a 2D massive scalar field is The energy-momentum tensor of the theory and the TT operator, denoted by TT φ , are given by To get the expression for the TT φ operator we manipulated (3.2) as a classical object. While this approach sounds too naïve, as it may easily overlook contact terms etc., for free fields we expect it to be precise [12]. In fact, we could use the definition with points split apart and take the limit of coincident points at the level of the correlation functions. The final result will be exactly the same as one obtained based on (3.3). As shown in Fig.1, to calculate the correlator K TT φ , we have to contract two fields in (3.3), where various vevs can be calculated using the standard two point function of the massive scalar field e.g., where µ 0 is a spherical UV cutoff. Hence, 4 where we dropped the non-universal power-law divergent terms (∼ µ 2 0 ). Using (2.1) and (3.9), the correlation function of the TT φ with the modular Hamiltonian takes the form where r is the distance of the insertion point of TT φ from the entangling point. Expression in the second line represents K φ 2 in the massive free field theory. In the next section we use this formula along with (2.3) to evaluate the linear response of entanglement entropy to the TT deformation. 4 We used the following identity to simplify the right hand side of (3.9) Note that the last term is proportional to the equations of motion, and therefore can be dropped in the calculation of K TT , because the insertion point of TT does not hit the support of K.

Massive Dirac field theory
The Euclidean action for a 2D massive Dirac field has the following form The energy-momentum tensor of the theory and the TT operator, denoted by TT ψ (x), take the following form where the symmetric tensor X µν is given by 5 (3.14) The two point function of the Dirac field can be expressed in terms of its scalar coun- where D(x) is given by (3.5). It can be used to get vevs of various local operators where µ 0 is a spherical cut off. Note that we can use the equations of motion, ( / ∂ + m 0 )ψ = 0, while calculating the correlator K TT ψ , because TT ψ does not hit the support of the modular Hamiltonian. For instance, where ∼ means equality up to terms proportional to the equations of motion.
Furthermore, as illustrated in Fig.1, to calculate K TT ψ we have to contract two out of four fields in the definition of TT ψ operator, e.g., contracting two fields in the last term of the above equation, one gets Similarly, using the definition and contracting two fields, yields where we used the following contraction rules which hold up to terms proportional to equations of motion (3.20) Combining, we finally get Thus, Using (2.1), yields and 25) where r is the radial distance from the origin.

Linear response of entanglement entropy to TT
It was shown in [24] that the entangelment entropy in the case of massive 1 + 1dimensional quantum field theory is given by where c is the central charge, m is mass or inverse correlation length, and δ is the lattice spacing or UV cut off. We expect correction to this formula if the theory is deformed by the TT operator, i.e., the coupling λ in (2.3) is identified with the TT deformation of a quantum field theory. In principle, one can derive the full answer by integrating (2.4) with TT substituted for O and using (4.1) as the boundary condition at λ = 0. However, it is difficult to proceed along this guideline in full generality without additional simplification. For instance, the form of the TT operator is not simple, and far away from a gaussian field theory. Therefore we adopt a perturbative approach. The deformed action is defined at some cut off scale µ 0 . To ensure the model is weakly coupled and perturbation theory is applicable, we assume the following hierarchy of scales where m 0 are masses of the fields in the undeformed theory. Furthermore, the derivative with respect to λ in (2.3) is taken while parameters m 0 , µ 0 etc. are held fixed. The latter is necessary to obey the definition of the TT deformation.
To evaluate the linear response of entanglement entropy to the TT deformation, one should calculate the right hand side of (2.3) to zeroth order in λ. Hence, the stress tensor and TT operator of the undeformed theory should be substituted for T 22 and O in (2.3). The corresponding correlation function was evaluated in the previous section and we use it to calculate the linear response of entanglement entropy for scalar and Dirac fields.

Scalar field
To calculate correction to the leading order result (4.1), one should substitute (3.10) into (2.3) and integrate it over the Euclidean 2D plane, where the last term corresponds to the contribution of the total derivative term in (3.10). Combining with (4.1), yields Using (3.6) and the physical mass of the deformed model, see Appendix A for definitions, one can rewrite it as follows The boundary term does not vanish, it is given by Thus the leading order correction to entanglement entropy takes the form

Dirac fermion
Substituting (3.25) into (2.3) and integrating over the Euclidean 2D plane, yields where the last term corresponds to the contribution of the total derivative term in (3.25). Combining with (4.1) and using (3.16) and the definition of the physical mass, see Appendix A, yields  where γ E is the Eular gamma function. Thus the leading order correction to entanglement entropy takes the form (4.11) Note that there is an intriguing difference in the leading order correction to entanglement entropy between the deformed free scalar and the Dirac Fermion. The leading order correction in the case of free boson is a log term whose coefficient is universal (4.7), whereas in the case of the Dirac fermion, there is a log-square divergence along with a log divergence (4.11). The presence of the log-square terms renders the coefficient of the logarithmically divergent term non-universal.

Discussion
In this paper, we studied perturbative entanglement entropy in field theories with a mass scale deformed by TT . It is remarkable that both for massive free boson and Dirac fermion the leading correction to the entanglement entropy comes from the boundary of the entangling surface. In the case of free massive boson, the boundary contribution (4.6) is finite giving rise to the expected logarithmic divergence in the entanglement entropy (4.7). For free massive Dirac fermion on the other hand, the boundary contribution is logarithmically divergent (4.10) giving rise to an extra logsquare divergence in the entanglement entropy (4.11).
Although the leading order correction to the entanglement entropy, in both cases, are sourced by the boundary terms of the entangling surface, they are not equal when expressed in terms of the physical parameters in the theory and the cutoff scale. At first glance, this is a bit confusing because one would naively expect an equality of the leading order correction term due to the bosonization duality of two-dimensional QFTs [16,24,25]. However, the bosonization duality has been established in local twodimensional QFTs by comparing local observables, whereas the entanglement entropy is a "non-local observable" supported on a region with a boundary. Our finding shows that the bosonization duality holds up to contributions coming from the boundary of the entangling surface.
In an RG flow of a local Lorentz invariant QFT in two dimensions, connecting two local fixed points, the quantity C ≡ µ 0 ∂ µ 0 S EE is independent of the regularization [26]. This follows from the fact that in local QFTs all physical divergent terms are logarithmic. In the case of TT deformed free boson, at leading order, we do see that C is indeed independent of the cutoff scale, but in the case of free Dirac fermions, this is not the case. This is probably due to the fact that TT is an irrelevant operator and its short distance properties are different from local QFTs. It would be interesting to investigate higher order correction to entanglement entropy to see at what order in perturbation theory exotic non-local divergent terms show up in the case of free bosons.
It's well known that the Rényi entropy computed for conformal scalars (where the stress tensor is computed by taking derivative with respect to the metric) and minimally coupled scalars (where the stress tensor is computed by Noether's theorem) differ by contributions from the boundary of the entangling surface [27][28][29]. A resolution to this discrepancy comes from a careful consideration of the boundary conditions near the entangling surface. In our case the leading correction to the entanglement entropy comes entirely from the boundary of the entangling surface. It would be interesting to analyse the boundary term along the line of [27][28][29] to extract universal features of entanglement in a TT deformed massive QFT. This involves various subtleties concerning the boundary terms. We leave it to future research work.
Following our analysis, it would be nice to understand if the boundary term in the entanglement entropy is a universal feature of an arbitrary massive quantum field theory deformed by TT .
The holographic dual of a CFT 2 deformed by λTT is AdS 3 gravity with mixed boundary condition [30]. The holographic entanglement entropy in such a setup (i.e., AdS 3 with mixed boundary condition) has been investigated in [31]. A natural question that arises at this point is how to compute the holographic entanglement entropy in a TT deformed QFT 2 with a mass scale. It is well known that for a CFT with mass scale at (i.e., for λ = 0), the holographic entanglement entropy of half space is computed by introducing an IR cutoff in the radial coordinate [32]. The radial IR cutoff introduces the mass scale in the problem. It would be interesting to revisit the holographic computation in [31] with a hard radial IR cutoff and compare with the perturbative results obtained in this paper.
Finally, there is another integrable irrelevant deformations of two-dimensional CFTs that goes in the name of JT deformation [4,5]. It would be interesting to calculate the linear response to entanglement due to JT deformation. More generally, our analysis can be extended to deformations by a general linear combination of TT , JT and TJ [7,8].
Marie Sk lodowska-Curie grant agreement n°945298. T.S. thanks the Israeli Science Foundation Center of Excellence (grant No. 2289/18) for support of his research.

A Physical masses
In this appendix we give a brief derivation of physical mass of TT deformed massive free boson and Dirac fermion at leading order in perturbation theory. The deformed scalar field theory Lagrangian is given by Its physical mass, m ph , up to a linear order in λ can be evaluated based on the 1-loop inverse propagator. In momentum space it is given by 6 Note that the residue of D(p) at p 2 = m 2 ph is not unity. To get unit residue one needs to do a wave function renormalization. But its contribution will not change the physical mass. 7 To liner order in λ the X µν X µν term does not contribute to the 1-loop effective action, because it is a total derivative.