Non-perturbative aspects of two-dimensional T ¯ T -deformed scalar theory from functional renormalization group

The T ¯ T -deformed scalar field theory in two-dimensional spacetime is studied by using the functional renormalization group. We drive the beta functions for the couplings in the system and explore the fixed points. In addition to the Gaussian (trivial) fixed point, we find a non-trivial fixed point at which a new universality class exists and especially the deformation parameter becomes relevant at the non-trivial fixed point. Therefore, the T ¯ T -deformed scalar field theory in two-dimensional spacetime could be defined as a non-perturbatively renormalizable theory.


I. INTRODUCTION
Quantum field theory (QFT) is the critical mathematical language for describing the dynamics of quantum particles.In general, however, most QFT models are not solvable even in small spacetime dimensions.Recently, the T T deformation of two-dimensional QFT [1,2] has attracted attention as an integrable deformation at the quantum level, in the sense that the energy spectra of the deformed theory are exactly obtained.See, e.g., Ref. [3] for a review.The T T -deformed action of the massive O(N ) vector model is given at the lowest order of the deformation parameter by with ⃗ ϕ = (ϕ 1 , • • • , ϕ N ) and the energy-momentum tensor Here, η µν = diag(−1, 1) is the flat metric and α is called the deformation parameter with mass dimension −2.Note that det(T µν ) is the determinant of the tensor T µν defined as det(T µν ) = 1/2ϵ µρ ϵ νσ T µν T ρσ , where ϵ µρ is the Levi-Civita tensor in two dimensions.The deformation parameter is a canonically irrelevant coupling in the infrared (IR) regime.Therefore, the theory (1) is perturbatively nonrenormalizable.In this sense, the T T deformation is also called the "irrelevant" deformation.
The T T -deformed theories have several attractive features.One is a relation with the string action.It has been shown in Ref. [4] that with an appropriate change of variables and large α, the deformed massless O(N ) vector model (1) can be written in the form of the Nambu-Goto action in a N + 2-dimensional target space in the static gauge.The inverse of the deformation parameter α −1 is identified with string tension.
Another noteworthy fact is that the deformed action can be written as a scalar theory coupled to gravity in twodimensional spacetime.To see this, we first rewrite the determinant term in Eq. ( 1) by introducing an auxiliary symmetric tensor field C µν such that, within the path integral formalism, where det(C µν ) is defined in the same way as det(T µν ).Thus, the determinant term is decomposed into the interactions between the scalar field ⃗ ϕ and the auxiliary tensor field C µν .Here, the tensor field is decomposed as C µν = γ µν + Cδ µν /2 with the trace mode C = δ µν C µν and the traceless mode γ µν (which satisfies δ µν γ µν = 0).Defining a new tensor field g µν ≡ (δ µν − γ µν )/(1 + C), the action (1) can be rewritten as where √ −g = [− det(g µν )] − 1 2 .In the classical action (1) or (4), there is no kinetic term of the tensor field, i.e., C µν (or, equivalently, g µν ) is the nondynamical field.From the equations of motion for C and γ µν , these fields are regarded as composite operators, Thus, the scalar field dynamics becomes the leading effects and induces an infinite number of effective interactions and makes C µν dynamical.
The deformation parameter plays a crucial role in these aspects of the deformed action (1).In the limit of α → 0 (corresponding to infinite string tension), Eq. ( 1) becomes a simple free scalar theory as a QFT model.When α is large, the degrees of freedom of C µν are expected to become dynamical, as mentioned above, and the system tends to describe a stringlike object.Therefore, the change of α may connect QFT and string theory.This picture is widely inferred from the fact that α is canonically irrelevant and shrinks to zero in the low-energy regime, while it grows in the high-energy regime.
However, in deformed theories, there is an issue of negative norm states for C µν .The large-N analysis for the action (1) has been carried out in Refs.[5,6] and has shown that the quantum loop effects of the scalar field induce the kinetic term of C µν with a negative sign.This fact implies that the T T -deformed theories are ill defined in the large-N limit.
Understanding the features of T T -deformed theory is expected to lead to deep insides both QFT and string theory.T T deformation has been initially proposed in the context of studies on quantum integrable systems.In addition to the methods for integrable systems, such as the Bethe ansatz [1,2] and S-matrix bootstrap [7], earlier studies on T T deformation have mainly relied on perturbation theory [8][9][10][11], the methods of large-N expansion [5,6], and holography [12][13][14][15].Also, several attempts [16][17][18][19] have been made to understand the renormalization group flow of the T T -deformed theories.In this paper, we intend to perform a nonperturbative analysis for the T T -deformed O(N ) scalar theory (1) using the functional renormalization group [20].Our aim is to investigate the impact of the nonperturbative dynamics of C µν , which cannot be captured by the above-mentioned methods.We derive the renormalization group (RG) equations for an effective theory of Eq. ( 1) and then analyze their fixed-point structure.

II. EFFECTIVE ACTION FOR T T -DEFORMED SCALAR THEORY
For the study of RG flows of the T T -deformed scalar field theory in two dimensions, the central method is the Wetterich equation [21], which is formulated as a functional partial differential equation for the scale-dependent (one-particle irreducible) effective action Γ k , Here, k is the ultraviolet (UV) cutoff scale, and k is the full twopoint function obtained by the second-order functional derivative with respect to superfields Φ, namely, Γ Tr acts on all spaces on which Φ is defined, such as momentum and O(N ) space, and R k (p) is the regulator function realizing the Wilsonian coarsegraining procedure.In this work, we use the Litim cutoff function [22] for the regulator function.See Eq. (A4) for its explicit form.Now, we make an appropriate ansatz for effective action.In this work, we are mainly interested in the "dynamicalization" of C µν and the RG flow of the deformation parameter.In this work, we focus on the infinitesimal, that is, first order in the deformation parameter α, T T deformation of the massive O(N ) scalar model as a first step. 1Hence, the effective action in two-dimensional Euclidean spacetime is given by Here, the energy-momentum tensor T µν is the same form as given in Eq. ( 2) with the mass parameter m k .The parameters Λ k (corresponding to the cosmological constant) and λ k are induced by quantum effects, but do not contribute to the dynamics.Note that the invariance of the vacuum |Ω⟩ under the translations and the Lorentz transformations results in ⟨Ω|γ µν |Ω⟩ = 0 and thus no linear term in γ µν appears in the effective action (7).The (dimensionless) field renormalization factor Z C,k describes the dynamicalization of C µν .For Z C,k = 0, C µν has no propagating degrees of freedom, while the use of the local potential approximation (LPA) [23,24], Z C,k = 1, implies that C µν is a priori the dynamical field.The rescaling of the field k that contributes to the β functions for interactions involving C µν , such as α k and β k .
Let us here briefly summarize the behavior of the flow equation around a fixed point.Using Eq. ( 6) for the effective action, we obtain the flow equations for the couplings that we denote here symbolically by g i,k .To analyze the structure of fixed points, we need to define dimensionless couplings gi,k = k −di g i,k , with k as the RG scale and d i as the canonical mass dimension of g i,k .Then, we obtain ∂ t gi,k = β i ({g k }), where {g k } denotes a set of dimensionless couplings and β i is the β function of gi,k .The β function typically takes the following form: where B i,k ({g k }) denotes quantum loop corrections to the β functions of the coupling gi,k .The fixed points g * i,k are obtained by looking for zero points in the β functions: Once a fixed point is found, one can analyze the flows of couplings around the fixed point.Performing the Taylor expansion for the β function up to the linear order, i.e., where V j i is the matrix diagonalizing the stability matrix T ij , and C j are constant coefficients given at a reference scale Λ.The critical exponents θ j are the eigenvalues of T ij and play a crucial role in the energy scaling of the coupling constants gi around the fixed point.The coupling constant with a positive critical exponent grows for k → 0 and is called relevant.On the other hand, the irrelevant coupling constant with the negative critical exponent shrinks toward the fixed point for k → 0. On the contrary, in the continuum limit k → ∞, relevant couplings converge to the fixed point, while irrelevant couplings diverge.To avoid such a divergence, we need fine-tuning for irrelevant couplings so that they do not deviate from the fixed point.This behavior means that relevant couplings are free parameters in the continuum limit; thus, a continuous and renormalizable theory can be constructed at a fixed point with a finite number of relevant couplings.
In particular, at the Gaussian fixed point g * i,k = 0 that characterizes the perturbation theory, we have V j i = δ j i and θ i = d i for gi,k .Hence, from the dimensional analysis of couplings, one can judge the renormalizability of a system as usual.In the system (7) at the Gaussian fixed point (g * i,k = 0), one has Note that κk has a zero critical exponent and is called a marginal coupling.If we expand the β function of κk around the Gaussian fixed point, we find that ∂ t κk is given by −κ 2 k / m2 k multiplied by some positive constant.Therefore, κk is marginally relevant/irrelevant depending on whether m2 k is negative/positive.If we consider higher-order quantum corrections, the relevance of κk may further change.Next, we study the possibility of the nontrivial fixed point in the system (7) and the critical exponents.

III. RG FLOWS AND FIXED-POINT STRUCTURE
β functions of system (7) can be derived by using the Wetterich equation (6).Their explicit forms are too long to be shown here, so we display their explicit forms in Eq. (A26) in the Appendix.Instead, we discuss the structure of the β functions and a mechanism to obtain a nontrivial fixed point.
The coupling κ k becomes a crucial interaction that transmits the dynamics of the scalar field to the tensor field.Switching off κ k decouples the scalar sector from the tensor sector and makes the system a free theory.Therefore, we start by looking at the β function of κk (= Z −1/2 C,k κ k ).The canonical dimension of κk is zero, so that quantum corrections give a nonzero β function.Within the effective action (7), all quantum corrections are proportional to κ3 k .Therefore, a nontrivial fixed-point value of κk is not obtained from its β function.However, since the operator T µν C µν includes the kinetic term and the mass term of ⃗ ϕ, the We first explore nontrivial fixed points in the case of an LPA, that is, Z C,k = 1 for which η C = 0. Table I shows the fixed points for N = 1, 2, 3. 2 For N > 3, no reliable nontrivial fixed point was found.The value of the couplings at these fixed points is observed not to diverge as N is increased.The reason is speculated as follows.The β functions of the couplings receive contributions from fluctuations of both scalar and tensor fields.As N is increased, loop effects of the scalar field enhance, while those of the tensor field do not.Because the fixed-point value is determined as a point where the contributions from scalar and tensor fields cancel each other out, there is no fixed point within the real-valued couplings for N to be large.This fact implies that such a fixed point is inaccessible in the large-N analysis.Including the finite anomalous dimension η C slightly modifies the fixedpoint value from the LPA.The value of η C at the fixed point is sufficiently smaller than 1, indicating that the validity of the derivative expansion is guaranteed as an approximation for the effective action (7).
The critical exponents at the fixed points in Table I are summarized in Table II.Note here that the imaginary parts of θ 3 and θ 4 imply the strong mixing between m2 k 2 The appearance of the pair of λ * k and κ * k with ones that have the sign reversed simultaneously results from the redundancy of defining the fields Cµν and C, that is, even if we flip the sign of these tensor fields (Cµν , C → −Cµν , −C) in Eq. ( 7), the RG flow should not be changed.and κk .Indeed, such an imaginary part of critical exponents is often observed in asymptotically safe gravity; see, e.g., Ref. [25].Although, in general, critical exponents at a nontrivial fixed point are eigenvalues of linear combinations of the original basis, it is convenient to investigate the diagonal parts of the stability matrix T ij on the coupling basis {g i } = { Λk , λk , m2 k , κk , αk , βk } in order to roughly identify the critical exponents with the original basis.For example, for N = 1 and with finite η C , we have diag(T ) ≈ (2, 1.88, −6.83, −3.39, 1.75, 1.75).From this fact, the critical exponents (θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , θ 6 ) correspond to approximately (θ Λ , θ λ , θ m 2 , θ κ , θ α , θ β ), respectively.
It turns out that the couplings with the scalar field m2 k and κk become irrelevant, while those with the tensor field αk and βk become relevant.Therefore, the tensor field C µν (or γ µν and C) are effective degrees of freedom in low energy.The flow diagram in the N = 1 case with finite η C on the ( βk , αk ) plane is displayed in Fig. 1, where the arrows indicate flows from the UV to the IR direction and the purple and red points are the nontrivial and Gaussian fixed points, respectively.A separatrix is shown as the green line.To plot it, we have used the fixed-point value for κk and m 2 k for which the Gaussian fixed point is shifted from β * k = α * k = 0 to β * k = −0.239and α * k = 0.In other words, Fig. 1 displays the two-dimensional subspace of αk and βk with the fixed value of κk and m2 k within four-dimensional theory space.
It can be seen from Fig. 1 that there are at least two different phases in the βk -α k plane.If we start from a value of couplings at a UV scale on the green line, its IR physics is described by the Gaussian fixed point.Otherwise, the theory does not flow into the Gaussian fixed point and may converge to other IR fixed points.As for the nontrivial UV fixed point, depending on the boundary condition for those couplings, the deformation parameter grows toward the IR direction.This behavior contracts to flow around the Gaussian fixed point.

IV. SUMMARY AND DISCUSSION
In this paper, we have performed the functional renormalization group study on the two-dimensional T Tdeformed scalar field theory.As seen from Eq. ( 10) and the flow diagram in Fig. 1, the T T -deformed term det T µν is irrelevant around the Gaussian fixed point, so that we cannot define the continuum quantum field theory with T T interactions around the Gaussian fixed point.This result means that the ordinary perturbative analysis is no longer valid for T T -deformed theory.
The novel finding in this work is the existence of the nontrivial UV fixed point.This finding may lead to defining the T T -deformed theory in a nonperturbative and renormalizable way as an asymptotically safe theory around the nontrivial fixed point.In addition, it may provide a new picture of the T T -deformed theory.In particular, the fact that the deformation parameter α k becomes relevant at the nontrivial fixed point may imply the existence of different phases.In the strong coupling phase α k > α * k , α k becomes large along the RG flow toward the IR regime, while the flow of α k in the weak coupling phase α k < α * k converges to the Gaussian fixed point in the IR limit.In other words, depending on the value of the deformation parameter, the theory could show different behaviors in the IR regime.This result contrasts the naive picture from the perturbation theory where the flow of α k around the Gaussian fixed point gives a connection between a free scalar field theory (α k → 0 in the IR regime) and the Nambu-Goto action (α k → ∞ in the UV regime).
Once the theory is scale invariant at the fixed point, it involves conformal invariance thanks to the c theorem [26].Simultaneously, this theory cannot describe the dynamics of the Nambu-Goldstone bosons accompanied by spontaneous breaking of the global O(N ) symmetry, which is prohibited by the Coleman-Hohenberg-Mermin-Wagner theorem [27][28][29].Therefore, a conformal field theory (CFT) with global O(N ) symmetry should describe this UV fixed point.Specifying this CFT in more detail is left for future work.
Another future direction is to study the stability of our results when increasing the truncation level, especially considering higher-order terms of the T T deformation.In this study, we consider the lowest-order term of the T T -deformed massive vector model with respect to the deformation parameter α.Naively, since the higher-order terms have negative and large canonical scaling, they are expected to significantly affect the UV fixed point.However, the finite T T deformation of the free massless O(N ) vector model is the Nambu-Goto action.Thus, the relation between this UV fixed point and string theory is worth further investigating.
Here, we introduce the cutoff function such that for which the numerator of the flow equation ( 6) is computed as In this work, we employ the Litim-type cutoff function [22] The diagonal parts in the Hessian with the regulator function are where we have used ϵ µρ ϵ νσ = δ µν δ ρσ − δ µσ δ ρν and have introduced

Regulated propagator
To obtain the β functions, we need to evaluate the inverse form of the regulated Hessian (Γ k + R k ).To this end, we first compute the inverse forms of Eqs.(A5) and (A6): where we have defined Then, the inverse form of the regulated Hessian reads Here, the off-diagonal parts are irrelevant for deriving the β functions in the case of the regulator (A3), so we do not specify them.We have from which the (1, 1) component of Eq. (A10) is computed as whose inverse form is given by CC −1

Flow generator
Now, we are in the position to compute the flow generator, i.e., the right-hand side of the Wetterich equation ( 6).From Eqs. (A3) and (A10), we have Tr Γ (2) First, we evaluate the ϕ-loop contribution denoted by, Here, the first term is computed as while the second term is Let us next evaluate the C-loop contribution denoted by, Here, we obtain the first term as The second term is (A21)

Flow equations
The left-hand side of the Wetterich equation ( 6) for the effective action ( 7) is given by We obtain the flow equations by projecting onto each field operator from the flow generators (A17), (A18), (A20), (A21) obtained in the previous subsection such that From Eqs. (A23e) and (A23f), we obtain the flow equation for κ k as (A24) To study the fixed-point structure, we introduce the dimensionless couplings such that Then, the flow equations for the dimensionless couplings are obtained as ) 2 . (A26f) Here, we have defined the anomalous dimension of C µν as This quantity is obtained in the next subsection.In Section III, we have defined the threshold function I m 2 in the β function for m2 k such that .

5 FIG. 1 .
FIG. 1. Flow diagram on βk -αk plane in the N = 1 case with finite ηC .The arrows show flows from the UV to IR direction, and the green line is a separatrix.For κk and m2 k , we used the fixed-point value κ * k = 0.462 and m2 * k = −1.26.The nontrivial fixed point (purple point) is located at β * k = −0.239and α * k = 0.323 (see Table I), while the Gaussian fixed point (red point) is shifted to α * k = 0 and βk = −0.239due to the use of the fixed values for κk and mk .
m 2 denotes the threshold function given in Eq. (A28) in the Appendix.For a finite value of m 2 * k , there exists a nonvanishing value of κk such that β m 2 = 0 due to cancellation between −2 m2 * k I m 2 ( m2 k , αk , βk ), where I k and κ * 2 k I m 2 ( m * 2 k , α * k , β * k ).Once a finite value κ * k is found, a nontrivial fixed point for α k and β k is obtained in a similar way.Note that threshold functions give finite values for fixed values of couplings.

TABLE I .
Nontrivial fixed-point values for several values of N .

TABLE II .
Critical exponents at the nontrivial fixed points listed in TableIfor several values of N .