Holographic RG flow triggered by a classically marginal operator
FFebruary 4, 2021
Holographic RG flowtriggered by a classically marginal operator
Chanyong Park a ∗ a Department of Physics and Photon Science, Gwangju Institute of Science and Technology,Gwangju 61005, Korea
ABSTRACT
We study the holographic renormalization group (RG) flow triggered by a classically marginaloperator. When a marginal operator deforms a conformal field theory, it does not yield a non-trivial renormalization group flow at the classical level. At the quantum level, however, quantumcorrections modify a marginal operator into one of the truly marginal, marginally relevant, andmarginally irrelevant operators and can generate a nontrivial RG flow. We investigate the holo-graphic description of a RG flow triggered by a marginal operator with quantum corrections.We look into how the physical quantities of a deformed theory, a coupling constant and thevacuum expectation value, rely on the RG scale. We further discuss the holographic descriptionof the trace anomaly caused by the gluon condensation. ∗ e-mail : [email protected] a r X i v : . [ h e p - t h ] F e b ontents For the last decade, much attention has been paid to understanding strongly interacting systemsof Quantum Chromodynamics (QCD) and condensed matter theory by using the anti de-Sitter(AdS)/conformal field theory (CFT) correspondence [1–4]. The AdS/CFT correspondence haspassed many nontrivial checks. When a relevant operator deforms a CFT, the original CFT ismodified relying on the energy scale observing the system and eventually flows a new IR theory[5]. At a low energy scale, systems of nuclear and condensed matter physics usually have a strongcoupling constant. Understanding nonperturbative physics governed by strong interactions isone of the important issues in recent theoretical and experimental researches. To accountfor such nonperturbative phenomena, we need to figure out an exact and nonperturbativerenormalization group (RG) flow. Although the perturbative RG flow was well established in thequantum field theory (QFT), it remains an important issue to understand the nonperturbativeRG flow. The AdS/CFT correspondence or holography can shed light on this issue due tothe relation between the nonperturbative RG flow of the QFT and the equations of motion ofthe dual gravity [6–19]. In this work, we investigate the holographic RG flow of a dual QFTdeformed by a marginal operator and compare the holographic result with the gluon condensateappearing in the lattice QCD [20, 21].After the AdS/CFT conjecture, there were many attempts to understand the dual QFTof an asymptotic AdS geometry. In the holographic setup, an IR energy scale of the gravitytheory maps to a UV energy scale of the dual field theory and generally leads to various UVdivergences [9–11, 13]. In order to get rid of such UV divergences, several distinct methods have1een invented. One of them is the ‘background subtraction’ method, which gets rid of the effectof the background reference spacetime. This subtraction method is not applicable to certaincases where an appropriate reference solution is ambiguous or unknown, e.g., topological blackholes [22–26], Taub-NUT-AdS, and Taub-bolt-AdS [27–29]. Another way to remove the UVdivergence is called the holographic renormalization in which the UV divergences are removedby adding appropriate counterterms. This method is similar to the renormalization process ofa QFT. The holographic renormalization method is useful in that it does not require a back-ground reference spacetime [10]. On the dual field theory side, the holographic renormalizationcorresponds to the momentum space RG flow. There is another description called the RG flowof the entanglement entropy where the RG scale is characterized by the subsystem size ratherthan the boundary position. It has been shown that the thermodynamics-like law of the en-tanglement entropy can reproduce the linearized Einstein equation of the dual AdS geometry.From the field theory point of view, the RG flow of the entanglement entropy corresponds to thereal space RG flow [30–36]. Intriguingly, it was shown by applying the entanglement entropyRG flow that a quantum entanglement entropy in the UV region can flow to a thermal entropyin the IR region [37]. From now on, we concentrate on the momentum space RG flow to findthe connection between the gravity equations and the RG equations.When one applies the holographic renormalization technique to the gravity with somematter fields, it was shown that there can exist an ambiguity in choosing the counterterms[38, 39]. In the holographic Lifshitz theory, for example, there are many different countertermsgiving the same UV divergence. Although the dual field theory is uniquely determined at aUV fixed point, choosing different counterterm leads to a different RG flow. This implies that,although we start with the same UV theory, the low energy physics along the RG flow can showdifferent physics relying on the counterterm we took. Therefore, it is important to clarify thecounterterm for describing the IR physics correctly in terms of the RG flow. In this work, weinvestigate how we can get information about the RG flow after the holographic renormalizationwith specifying the counterterms. To do so, we exploit the Hamilton-Jacobi formalism of thegravity theory [10–12]. After finding the relation between the radial coordinate and the RGscale, we show how the Hamilton-Jacobi formalism reproduces to the RG equation of the dualQFT.When a CFT deforms by a marginal operator, the deformed theory remains marginal atthe classical or action level. However, this not true at the quantum level because quantumcorrections may change the property of the deformation operator. The AdS/CFT correspon-dence claimed that a classical gravity theory is dual to a full quantum field theory defined atthe boundary. Therefore, the holographic description allows us to figure out the nonperturba-tive RG flow containing the quantum effect. One of the important quantities affected by such2uantum corrections is the β -function. The β -function describes how the coupling constantdepends on the energy scale. In general, physical phenomena crucially depends on the strengthof a coupling constant. At the classical level, the β -function can be classified by the conformaldimension of a deformation operator. When the coupling constant λ couples to an operatorwith the conformal dimension ∆ in a d -dimensional CFT, the classical β -function is given by[40] β cl = − ( d − ∆) λ. (1.1)If we further take into account quantum corrections, the β -function generalizes to β = β cl + β q , (1.2)where β q indicates quantum corrections. If we concentrate on a classically marginal operatorwith ∆ = d , the classical β -function automatically vanishes. However, the quantum effectleads to a nonvanishing β -function and generates a nontrivial RG flow. As explained here, thequantum correction plays a crucial role in determining the RG flow triggered by a marginaloperator. In this work, we investigate the holographic description of such quantum correctionand discuss how such a marginal deformation modifies a UV CFT.Related to the marginal deformation in QCD, one of the important quantities is the gluoncondensation, (cid:104) G (cid:105) . Its one-loop correction in the UV region leads to the following trace anomaly[20, 21] (cid:104) T µµ (cid:105) = − N c π β λ λ (cid:104) G (cid:105) , (1.3)where λ and β λ indicate the t’ Hooft coupling and its β -function, respectively. This traceanomaly is crucially associated with the RG flow caused by the gluon condensation. In thepresent work, we investigate the trace anomaly and its RG scale dependence by applying theholographic RG flow technique.The rest of this paper is organized as follows. In Sec. 2, we discuss the β -function of amarginal operator and the RG equation appearing in a QFT. In Sec. 3, we study a gravity theoryinvolving a scalar field which is dual of a marginal operator. In addition, we investigate howto construct the holographic RG flow caused by such a marginal operator with discussing thecorrect counterterms required to renormalize the UV divergences. By applying this holographicRG flow technique, we investigate the trace anomaly caused by the gluon condensation in Sec.4. We finish this work with some concluding remarks in Sec. 5. At a low energy scale, understanding new physics laws of macroscopic systems is one of theimportant issues to be resolved. To understand macroscopic theories, we need to know how a3icroscopic UV theory flows to a new IR theory along the RG flow. A relevant or marginallyrelevant operator causes a nontrivial RG flow and its effect becomes signigicant in the IRregion. In this RG flow process, nonperturbative quantum corrections play a crucial role.Unfortunately, we do not have an analytic method accounting for the nonperturbative RG flowin the traditional QFT. In this situation, the AdS/CFT correspondence may provide a newchance to investigate such a nonperturbative RG flow.Before studying the holographic RG flow, we first discuss general aspects of the RG flowin a d -dimensional QFT for the comparison with the holographic result. Assuming that a CFTdeforms by an operator of a conformal dimension ∆, the deformed theory is described by S QF T = S CF T + (cid:90) d d x √ γ µ d − ∆ λ ¯ O, (2.1)where γ µν corresponds to the background metric and ¯ O is a dimensionful composite operatorof the fundamental field. Here, µ and λ denotes an RG scale and a dimensionless couplingconstant, respectively. Ignoring quantum effects, the deformation can be characterized by theconformal dimension ∆. At the classical level, for example, the dimensionless coupling and thedimensionless operator, O = µ − ∆ ¯ O , under the RG or scale transformation scale as [5, 40] λ → µ − ( d − ∆) λ and O → µ − ∆ O. (2.2)Under the RG transformation, the deformation operator is classified into three differentcategories: relevant for ∆ < d , marginal for ∆ = d , and irrelevant for ∆ > d . The deformationeffect of a relevant operator is negligible in the UV regime, while it becomes important in theIR regime and changes the UV theory into another IR theory. A marginal operator at theclassical level does not break the conformal symmetry, so that the deformed theory remainsconformal in the entire energy range. The irrelevant operator gives rise to a serious effect onthe UV theory, so that the resulting theory even in the UV limit differs from the original UVCFT. This means that the deformed theory is UV incomplete. These features become manifestwhen we take into account a classical β -function [40] β cl ≡ ∂λ∂ log µ = − ( d − ∆) λ. (2.3)The coupling constant determined by the classical β -function automatically satisfies the scalingbehavior in (2.2). For a marginal operator with ∆ = d , the classical β -function automaticallyvanishes and the coupling constant does not change along the RG flow. The coupling constant ofan irrelevant operator has a positive β -function, so that the coupling constant becomes large as µ increases. This implies that the effect of the irrelevant deformation becomes more importantin the UV region. As a consequence, the deformed theory is not the same as the undeformed4ne even in the UV limit. On the other hand, a relevant operator has a negative β -function andits coupling constant decreases as µ increases. Therefore, the effect of a relevant operator isnegligible in the UV region. Note that this is the story at the classical level. If we further takeinto account quantum effects, they can modify behaviors of the classical β -function. From nowon, we consider a QFT defined at a flat spacetime to ignore the effect of a conformal anomaly.Quantum effects in the renormalization procedure usually cause UV divergences. Sincephysical quantities must be finite, we need to get rid of such UV divergences by adding appro-priate counterterms. After an appropriate renormalization, the renormalized partition functionreduces to Z = (cid:90) D φ e − ( S QFT + S ct ) = e − Γ[ γ µν ( µ ) ,λ ( µ ); µ ] , (2.4)where Γ is called the generating functional. The background metric γ µν in a QFT usuallydoes not vary along the RG flow. In this work, however, we take into account the backgroundmetric as another coupling constant depending the RG scale as done in Ref. [41, 42]. Since therenormalized partition function is independent of the RG scale, the variation of the partitionfunction gives rise to the following RG equation0 = µ √ γ ∂ Γ ∂µ + γ µν (cid:104) T µν (cid:105) + β λ (cid:104) O (cid:105) , (2.5)where β λ = dλd log µ , (2.6) (cid:104) T µν (cid:105) = − √ γ ∂ Γ ∂γ µν , (2.7) (cid:104) O (cid:105) = 1 √ γ ∂ Γ ∂λ . (2.8)When the background metric is independent of the RG scale, the above RG equation reducesto 0 = µ ∂ Γ ∂µ + β λ (cid:104) O (cid:105) . (2.9)This is the typical RG equation of a QFT. However, assuming that the metric scales underthe RG transformation, the generalized RG equation (2.5) gives us more information about thestress tensor of a system which plays an important role in understanding the connection to thedual gravity, as will be seen later.For the undeformed CFT with β λ = 0, the RG equation reduces to (cid:104) T µµ (cid:105) = − µ √ γ ∂ Γ ∂µ . (2.10)5he stress tensor of a CFT must be traceless up to a conformal anomaly [43]. We may associatethe right hand side with a conformal anomaly depending on the topology of the backgroundspacetime. Hereafter, we concentrate on a flat spacetime which has no conformal anomaly.When a CFT deforms by an operator, a nontrivial trace of the stress tensor newly appears andgenerally breaks the conformal symmetry. Ignoring the RG flow of the generating functionalwith fixed coupling constants ( ∂ Γ /∂µ = 0), the trace of the stress tensor is determined by thedeformation (cid:104) T µµ (cid:105) ∼ − β λ (cid:104) O (cid:105) . (2.11)To avoid the confusion with the conformal anomaly, from now on, we call a nonvanishing (cid:104) T µµ (cid:105) caused by a deformation a trace anomaly.The quantum effect involved in the renormalization procedure modifies the classical β -function, as mentioned before. For a marginal and a relevant deformation, the β -function inthe UV region can be written as β λ = β cl + β q = − ( d − ∆) λ + β q . (2.12)where β q means all quantum corrections. For a relevant deformation, the quantum effectbecomes subdominant in the UV region. In general, the β -function is given by a function ofthe coupling constant relying on the RG scale. Then, the vev of the operator is derived fromthe variation of the generating functional with respect to the coupling constant, as shown in(2.8). For a marginal deformation with ∆ = d , the classical β -function vanishes and the actionstill preserves the scale invariance. Therefore, the classical RG flow becomes trivial and thetrace of the stress tensor vanishes at the classical level. However, this is not the case at thequantum level. The quantum effect allows a nonvanishing β -function. If β q = 0, for example,the corresponding operator is called a truly marginal operator. In this case, the conformalsymmetry of the UV theory preserves under the RG transformation even at the quantum level.Therefore, the deformed theory remains as the CFT in the entire RG scale. For β q < β q > According to the AdS/CFT correspondence, a d -dimensional QFT is dual to a classical ( d + 1)-dimensional gravity theory. Therefore, we may expect that a gravity theory realizes the RG flow6f a dual QFT. If possible, the AdS/CFT correspondence would be helpful to understand thenonperturbative RG flow because the AdS/CFT correspondence claims that the dual gravitymaps to a QFT involving all quantum effects. Now, we investigate how we can describe theRG flow including quantum effects on the dual gravity side. In this work, we concentrateon a gravity theory which is dual to a UV CFT deformed by a classically marginal operator.Although the microscopic details of the deformed theory are obscure, studying the dual gravitytheory can give us important information about the RG flow of the deformed QFT.From now on, we focus on the case of d = 4. According to the AdS/CFT correspondence,the mass of the bulk field is related to the conformal dimension of the dual operator∆ = 2 + (cid:114) m R . (3.1)If we take into account a massless scalar field with m = 0, the dual operator corresponds toa marginal operator with a conformal dimension ∆ = 4. To consider a classically marginaldeformation, we start with the following Euclidean Einstein-scalar gravity S = − κ (cid:90) d X √ g (cid:18) R − − g MN ∂ M φ∂ N φ (cid:19) + 1 κ (cid:90) ∂ M d x √ γ K, (3.2)where Λ = − /R is a cosmological constant with an AdS radius R . Here, g MN and γ µν indicatea bulk metric and an induced metric on the boundary respectively. Since the variation of thegravity action usually has a radial derivative of the metric at the boundary, the last term calledthe Gibbons-Hawking term is required to get rid of such a radial derivative term. Assumingthat the scalar field depends only on the radial coordinate and that the boundary space is flat,the most general metric ansatz preserving the boundary’s planar symmetry is expressed as ds = e A ( y ) δ µν dx µ dx ν + dy . (3.3)The detail of this geometry is governed by φ ( y ) and A ( y ) satisfying the following equationsof motion 0 = 24 ˙ A − ˙ φ + 4Λ , (3.4)0 = 12 ¨ A + 24 ˙ A + ˙ φ + 4Λ , (3.5)0 = ¨ φ + 4 ˙ A ˙ φ, (3.6)where the dot means a derivative with respect to y . Here the first equation is a constraint,while the others describe dynamics of A and φ . Note that only two of the above three equationsare independent. These equations allow the following analytic solution [44, 45] φ = φ + η (cid:114)
32 log (cid:32) √ − φ z /R √ φ z /R (cid:33) , (3.7) e A ( y ) = R z (cid:114) − η φ z R , (3.8)7ith z = Re − y/R , (3.9)where φ and φ are two integral constants. Assuming that two integral constants are positive, φ maps to a coupling constant of the dual UV CFT defined at the boundary ( y = ∞ ). Theinvariance of the gravity action under the parity, φ → − φ , allows two different profiles with η = ±
1. According to the AdS/CFT correspondence, a classical gravity theory is matched to afull quantum theory defined at the boundary. Therefore, the classical geometric solution (3.7)can give us information about the quantum effect of the deformation. Since the inner geometryof (3.7) differs from the AdS space, the conformal symmetry is broken in the IR regime. Thisfact indicates that the deformation is not truly marginal. In other words, the deformation in(3.7) is either marginally relevant or marginally irrelevant due to the quantum corrections.There exists another solution which yields a truly marginal deformation. When the scalarfield has a constant profile which corresponds to the case with η = 0 in (3.7), its gravitationalbackreaction is absent. Thus, the AdS metric of the undeformed theory still becomes thesolution of the deformed theory φ = φ , (3.10) A ( y ) = yR . (3.11)Since the resulting geometry is still the AdS space, the deformed theory remains as a CFT,on the dual field theory side. In this case, the undeformed and deformed theories are differentbecause their dual CFTs have different coupling constants. This is a typical feature of atruly marginal deformation which shifts the value of the coupling constant with a vanishing β -function.We have several important remarks. The classical conformal dimension of the dual operatorcan be easily read from the profile of the bulk scalar field. Near the boundary, for example, ageneral massive bulk scalar field expands into φ = c z − ∆ + c z ∆ + · · · . (3.12)Recalling that the radial coordinate maps to the energy scale of the dual field theory, the scal-ing behavior of the scalar field, without an appropriate renormalization procedure, allows us toidentify c and c with a source (or coupling constant) and the vev of a dual operator, respec-tively. This identification is consistent with the classical scaling dimension of the dual QFT. Ifwe further take into account quantum corrections through an appropriate renormalization, theclassical scaling dimension modifies along the RG flow. In the above gravity action, the bulkscalar field is dimensionless because the gravitational constant has the dimension of mass − .Therefore, the bulk field is matched to a dimensionless coupling constant of the dual QFT.8 .1 Holographic description of the RG flow The RG equation is usually represented as first-order differential equations, while the equationsof motion determining the dual geometry are governed by the second-order differential equa-tions. To derive the RG flow from the bulk equations, we reformulate the bulk equations byapplying the Hamilton-Jacobi formalism in which the bulk equations are rewritten as the first-order differential equations [10–13, 15, 16]. To do so, we first notice that the metric solution in(3.3) is a specific case of the following general metric ds = N dy + γ µν ( x, y ) dx µ dx ν , (3.13)where N is a lapse function and γ µν = e A ( y ) δ µν . Note that the lapse function is non-dynamicaland that varying the action with respect to the lapse function gives rise to a constraint equation.Therefore, we can set N = 1, without loss of generality, after all calculation.Regarding the radial coordinate y as a Euclidean time, we can rewrite the previous Einstein-scalar gravity action as a functional form of the extrinsic curvature [10–12] S = (cid:90) d xdy √ g L , (3.14)with L = 12 κ (cid:20) N (cid:0) −R (4) + K µν K µν − K + 2Λ (cid:1) + 12 N ˙ φ (cid:21) , (3.15)where the extrinsic curvature tensor is defined as K µν = 12 N ∂γ µν ∂y . (3.16)Above R (4) denotes an intrinsic curvature of the boundary spacetime. Since the boundary isflat in our setup, R (4) automatically vanishes. The canonical momenta of the boundary metric γ µν and scalar field φ are given by π µν ≡ ∂S∂ ˙ γ µν = − κ ( K µν − γ µν K ) ,π φ ≡ ∂S∂ ˙ φ = 12 κ ˙ φ. (3.17)These canonical momenta enable us to reexpress the action as the first-order form S = (cid:90) d xdy √ g (cid:16) π µν ˙ γ µν + π φ ˙ φ − N H (cid:17) , (3.18)where the Hamiltonian density is given by H = 2 κ (cid:18) γ µρ γ νσ π µν π ρσ − π + 12 π φ (cid:19) − Λ κ . (3.19)9ith π = γ µν π µν . The variation of this action with respect to the lapse function leads to theHamiltonian constraint, H = 0. This Hamiltonian is a generator of the translation in the y -direction. All solutions connected by this transformation are gauge equivalent.After imposing the Hamiltonian constraint, the variation of the action finally results in thevariation of the boundary action δS B = (cid:90) ∂ M d x √ γ ( π µν δγ µν + π φ δφ ) , (3.20)where all variables are defined at the boundary. In the Hamilton-Jacobi formalism, the canonicalmomenta are defined as π µν = 1 √ γ δS B δγ µν and π φ = 1 √ γ δS B δφ . (3.21)According to the AdS/CFT correspondence, we identify the above boundary action with thegenerating functional of the dual QFT. We also identify the radial coordinate of the bulkgeometry with the energy scale of the dual QFT. Then, we can investigate the RG flow of thedual QFT by changing the boundary position on the gravity side. To map the above equationsto the RG equations correctly, we need to resolve two issues. First, we have to clarify how theradial position of the boundary is related to the RG scale [46–48]. Second, the above boundaryaction is an unrenormalized generating functional because we did not get rid of UV divergencesyet. Therefore, we have to renormalize the above boundary action by adding appropriatecounterterms.Although the RG flow can vary the value of coupling constants including the metric, itdoes not change the background geometry where the QFT is defined. On the dual gravityside, this implies that the boundary spacetime described by ds = γ µν dx µ dx ν must be invariantunder the scale transformation. Since the metric component in (3.13) scales by e A (¯ y ) → e σ e A (¯ y ) under x → e − σ x or µ → e σ µ , we can associate the RG scale of the dual QFT with the metriccomponent at the boundary µ = e A (¯ y ) R , (3.22)where A (¯ y ) indicates the value of A at the boundary. Above, we introduced the AdS radius R for the correct dimension count. This relation represents how the RG scale changes when theboundary moves in the radial direction. From now on, we concentrate on the dual QFT definedat the boundary, so we drop out the bar symbol. For η = ±
1, the RG scale of the dual QFTis related to the radial position of the boundary by µ = e y/R R (cid:18) − φ e − y/R (cid:19) / . (3.23)10ven though we know how to relate the radial coordinate to the RG scale, we need toadd appropriate counterterms to remove the UV divergences appearing in the boundary action.In this renormalization procedure, there can exist several different counterterms which havethe same UV divergent terms [38]. Although the regular terms do not give any effect on therenormalization, they can provide a nontrivial effect on the RG flow in the intermediate energyscale. The regular part of the counterterms is associated with the quantum corrections andseriously modify the IR physics. Therefore, it is an important to fix the counterterms correctlyto understand the RG flow and its IR physics. The UV divergence of the above boundaryaction appears at a UV cutoff ¯ y → ∞ due to the invariant integral measure (cid:90) d x √ γ ∼ (cid:90) d x e y/R . (3.24)On the other hand, the integrands of the boundary action in the aymptotic region behave as π µν γ µν ∼ ˙ A ∼ R ,π φ φ ∼ e − y/R . (3.25)As shown in these asymptotic behaviors, the gravity part, (cid:82) √ γ π µν γ µν , gives rise to a UVdivergence proportional to e y/R , whereas the scalar field part, (cid:82) √ γ π φ φ , does not make anyadditional UV divergence. Therefore, we need the counterterm which cancels only the di-vergence of the gravity part. In other words, since the marginal deformation does not anyadditional UV divergence, we exploit the same counterterm used in the undeformed CFT [43] S ct = − κ (cid:90) d x √ γ L ct . (3.26)with L ct = 6 R . (3.27)Then, the renormalized action is given byΓ[ γ µν , φ ; ¯ y ] = S B − S ct . (3.28)The resulting renormalized action is finite and corresponds to the renormalized generatingfunctional of the dual QFT. The UV cutoff we introduced is artificial, so the the physicalrenormalized action must be independent of this artificial UV cutoff. The scale independenceof the renormalized action leads to the RG equation0 = µ ∂ Γ ∂µ + ∂γ µν ∂ log µ ∂ Γ ∂γ µν + ∂φ∂ log µ ∂ Γ ∂φ , (3.29)11here µ is the RG scale satisfying (3.22). Here φ indicates the boundary value of the bulkscalar field. Identifying φ with the dimensionless coupling of the dual QFT, the vev of a scalaroperator is derived from the generating functional, as it should do. Due to the following relation dγ µν d log µ = − γ µν . (3.30)we further rewrite the above RG equation as the usual form0 = µ √ γ ∂ Γ ∂µ + γ µν (cid:104) T µν (cid:105) + β φ (cid:104) O (cid:105) , (3.31)with the following definitions β φ ≡ ∂φ∂ log µ , (3.32) (cid:104) T µν (cid:105) ≡ − √ γ ∂ Γ ∂γ µν = − (cid:18) π µν − κ γ µν L ct (cid:19) , (3.33) (cid:104) O (cid:105) ≡ √ γ δ Γ δφ = π φ + 12 κ ∂ L ct ∂φ . (3.34)This is the same as the generalized RG equation (2.5) of a QFT, where we take into accountthe metric as an additional coupling constant. When we describe the holographic RG flow, it is more convenient to introduce a superpotential.To do so, we return to the bulk equations of motion. Since the bulk equations of motion,(3.4),(3.5), and (3.6), are depending only on ˙ A and ¨ A , we can introduce a superpotential whichsatisfies [13, 44, 49–53] W ( φ ) = 6 ˙ A. (3.35)Then, the bulk equations reduce to two first-order differential equations2Λ = 12 (cid:18) ∂W∂φ (cid:19) − W , (3.36)˙ φ = − ∂W∂φ . (3.37)Here the first equation is just the Hamiltonian constraint which determines the superpotentialas a function of φ . Then, we rewrite the RG equation in terms of the superpotential0 = µ √ γ ∂ Γ ∂µ + γ µν (cid:104) T µν (cid:105) + β φ (cid:104) O (cid:105) , (3.38)12ith β φ = − W ∂W∂φ , (3.39) (cid:104) T µν (cid:105) = 1 κ ( K µν − γ µν K ) − κ R γ µν , (3.40) (cid:104) O (cid:105) = 12 κ ∂W∂φ . (3.41)For a marginal deformation, ∂ L ct /∂φ automatically vanishes because L ct is given by a constant,as shown in (3.27). The Hamiltonian constraint (3.36) allows two different types of solutions. The first one is givenby [15, 16, 44, 45] W = 6 R cosh (cid:32)(cid:114)
23 ( φ − φ ) (cid:33) . (3.42)After substituting the superpotential into (3.37) and solving (3.37), this superpotential repro-duces the solution in (3.7) and (3.8) with η = ±
1. From the profile of the bulk scalar field(3.7), we can easily read off the corresponding β -function in the UV region β φ = ηφ µ + O (cid:18) µ (cid:19) . (3.43)Recalling that the β -function of the marginal operator vanishes at the classical level, thisnonvanishing β -function comes thoroughly from the quantum effect. If we identify the bulkscalar field with the coupling constant of the dual QFT, the nonvanishing β -function implies thatthe deformation operator is marginally relevant ( β φ <
0) for η = − β φ >
0) for η = 1, see Fig. 1. On the other hand, if we take the bulk scalar as the inverse ofthe coupling constant as will be seen in the next section, the deformation becomes marginallyrelevant for η = 1 and margiallry irrelevant for η = −
1. We see from (3.34) that the vev of thedeformation operator reduces to (cid:104) O (cid:105) = √ κ R sinh (cid:32)(cid:114)
23 ( φ − φ ) (cid:33) . (3.44)These nonvanishing β -function and vev of the operator give rise to a nontrivial trace anomalyalong the RG flow (cid:104) T µµ (cid:105) = − µ √ γ ∂ Γ ∂µ − β φ (cid:104) O (cid:105) . (3.45)This is the expected RG flow when the dual QFT deforms by a marginally relevant or irrelevantoperator. 13 arginally irrelevantmarginally relevantUV fixed point Figure 1 . The RG flows caused by marginally relevant and irrelevant operators.
There exists another solution satisfying the Hamiltonian constraint. The second superpotentialis given by W = 6 R . (3.46)This reproduces the trivial solution, (3.7) with η = 0. Plugging this superpotential into (3.37)and (3.35), we finally reobtain the trivial solution in (3.10). In this case, the coupling constant φ = 0 of the undeformed CFT changes into φ = φ for the deformed CFT. This is a typicalfeature of the truly marginal deformation satisfying β φ = (cid:104) T µµ (cid:105) = 0 . (3.47)Therefore, this truly marginal operator changes a CFT into another CFT with shifting thevalue of the coupling constant and without generating a nontrivial RG flow. Several distinct condensations appear in QCD. They are usually associated with a certainspontaneous symmetry breaking and responsible for the mass of hadrons. Due to the Lorentzinvariance, the condensation should be a Lorentz scalar and have vanishing charges undervarious global and local symmetries. The well-known condensations in QCD are the chiral and14luon condensations. The chiral condensation is the condensate of two fermions with breakingthe chiral symmetry spontaneously and yields a large effective mass to quarks and most hadrons.The other example is the gluon condensation. For a four-dimensional Yang-Mills theory, thekinetic term of the gauge field is given by S Y M = − g Y M (cid:90) d x √ γ Tr F . (4.1)In this case, since the Yang-Mills coupling constant is classically marginal, the classical β -function automatically vanishes. However, the one-loop quantum correction generates a non-vanishing β -function [5, 54, 55].When QCD deforms by the gluon condensation, we are able to regard φ = 1 / (4 g Y M ) as acoupling constant and G = − Tr F as a deformation operator called the gluon condensation,respectively. Although it is not clear whether the gluon condensation is associated with theknown phase change, there are many indications for the nonvanishing gluon condensation inlattice QCD simulations. The gluon condensation may be partly responsible for masses ofhadrons and leads to a nontrivial RG flow closely related to the trace anomaly. The latticeQCD simulations expects that the one-loop quantum correction in the UV region leads to thefollowing trace anomaly [20, 21] (cid:104) T µµ (cid:105) = − N c π β λ λ (cid:104) G (cid:105) , (4.2)where N c indicates the rank of the gauge group and β λ means a β -function of the ’t Hooftcoupling constant λ . In general, the condensation plays a crucial role in nonperturbativephenomena. To understand such nonperturbative features, the AdS/CFT correspondence maybe helpful. The effect of the gluon condensation on the trace anomaly can be understood by theholographic RG flow discussed before. Although the gravity model we considered is too simpleto study IR physics, the present model is still useful to account for the UV physics because itgives rise to the leading behavior of the marginal deformation at least in the UV region.Now, we specify the parameters of the gravity theory in terms of those of the dual QFT. Forapplying the AdS/CFT correspondence, we define the ‘t Hooft coupling constant as λ = N c g Y M and take the double scaling limit where N c → ∞ and g Y M → λ . Despite an infiniterank of the gauge group, the AdS/CFT correspondence still has an advantage in catching theimportant feature of the nonperturbative RG flow. If we identify the vev of the dual operatorwith the gluon condensation, (cid:104) G (cid:105) , the bulk scalar field is associated with the ’t Hooft couplingconstant φ = N c λ . (4.3)15n the large ’t Hooft coupling constant limit ( λ (cid:29) φ is proportionalto 1 /µ in (3.7). This implies that the ’t Hooft coupling constant is related to the RG scale by λ ∼ µ in the UV region ( µ (cid:29) β -function (3.43) derived from the bulk scalar field canbe reexpressed as the β -function of the ’t Hooft coupling constant β φ = − N c β λ λ . (4.4)When we identify the bulk scalar field with the inverse of the ’t Hooft coupling constant in (4.3),we have to take η = 1 rather than η = − N c is sufficiently larger than the number of othermatter fields, β λ has a negative value representing a marginally relevant deformation. In (4.4),the negative β λ enforces a positive β φ which is the same as choosing η = +1 in (3.43). This isalso consistent with our previous prescription that, when the bulk field is identified with theinverse of the coupling constant, η = +1 describes a marginally relevant deformation.From the holographic RG flow description studied before, the β -function and the gluoncondensation in the UV region are given by functions of the RG scale β φ = φ R µ − φ R µ + O (cid:0) µ − (cid:1) , (cid:104) G (cid:105) = − φ κ R µ + O (cid:0) µ − (cid:1) . (4.5)Here the β λ and (cid:104) G (cid:105) are also represented as functions of the ’t Hooft coupling constant, forexample, β λ ∼ − λ and (cid:104) G (cid:105) ∼ − /λ in the UV region. The leading behavior of the gluoncondensation, (cid:104) G (cid:105) ∼ µ − , is the form expected by the classical dimension counting in (2.2).This result shows that the gluon condensation rapidly suppresses as the ’t Hooft couplingconstant increases. From (3.34), moreover, we show that the trace anomaly has the followingRG scale dependence (cid:104) T µµ (cid:105) = − φ κ R µ + φ κ R µ + O (cid:0) µ − (cid:1) . (4.6)Comparing the results of the holographic RG flow, we find that the gluon condensation satisfiesthe following relation (cid:104) T µµ (cid:105) = − N c β λ λ (cid:104) G (cid:105) + O (cid:0) λ − (cid:1) . (4.7)This is the form of the trace anomaly expected in the lattice QCD [20, 21]. There are remarkablepoints in this holographic RG flow. Intriguingly, the holographic RG flow shows that the traceanomaly (4.2) expected from the lattice QCD is valid up to the λ − order. The holographicdescription of the gluon condensation leads to the expected behavior of the RG flow. The gluoncondensation and the trace anomaly, as expected, rapidly suppress in the UV region, so thatthe conformal symmetry is restored at the UV fixed point.16 Discussion
In this work, we studied the holographic RG flow of a CFT deformed by a marginal operator.We discussed how we can understand the RG flow of a boundary QFT in terms of the Hamilton-Jacobi formulation on the dual gravity side. At the classical level, a marginal operator does notchange the CFT. The quantum effect, however, can lead to the nontrivial modification of theCFT along the RG flow. Using the holographic description, we studied how the quantum effectof the marginal deformation modifies the β -function and the vev of the operator. Furthermore,we compared this result with the result of the gluon condensation known in QCD.There were several distinct prescriptions to realize the nonperturbative RG flow of the QFTon the dual gravity side. In the present work, we exploited the Hamilton-Jacobi formalismwhich allows us to rewrite the gravity equations as the first-order differential equations. TheRG equations are generally given by the first-order differential equations, so that the Hamilton-Jacobi formulation is useful to understand the RG flow of the dual QFT. When we applied theHamilton-Jacobi formulation, it suffers from the UV divergences similar to those appearing inthe QFT renormalization. We discussed the counterterms, which get rid of the UV divergencesof the holographic renormalization, and then find the finite boundary action, which is identifiedwith the generating functional of the dual QFT.After the holographic renormalization, we explicitly showed that the quantum correctionmodifies a classically marginal operator into one of the truly marginal, marginally relevant andmarginally irrelevant operators at the quantum level. More precisely, the quantum correctiongives rise to a nonvanishing β -function for marginally relevant and irrelevant deformations.If we focus on the marginally relevant deformation, the undeformed theory at the UV fixedpoint becomes unstable under this marginally relevant deformation. Thus, the UV CFT flowsto a new IR theory. In this case, other quantities like the stress tensor and the vev of theoperator also vary along the RG flow. We explicitly calculated the RG scale dependenceof these quantities near the UV fixed point. We also showed that the holographic RG flowreproduces the known trace anomaly of the gluon condensation in the lattice QCD [20, 21].Intriguingly, the holographic RG flow indicates that this trace anomaly is valid only up to λ − order in the UV region.In the present work, the Einstein-Scalar gravity we considered has no well defined IRgeometry because of the existence of a singularity at z = 4 √ /φ . This means that the dualQFT of the present model is IR incomplete, so that the end of the RG flow is not manifest.This means that the present model is not valid in the IR region. Nevertheless, the results weobtained are still valid in explaining the quantum correction in the UV region because it givesrise to the leading contribution. To study the IR physics further, we have to consider a moregeneral gravity theory which allows a well-defined IR fixed point. For example, we can take17nto account a scalar field potential with higher-order terms V ( φ ) = (cid:88) n ≥ a n φ n . (5.1)For a relevant deformation, this potential crucially modifies IR physics and can allow an IRfixed point. Despite this fact, its contribution in the UV region is subdominant. Due to thisreason, in the present work we just focus on the UV behavior of the RG flow. Nevertheless,it is still important to take into account higher-order terms to understand the IR phenomena.We hope to report more results on this issue in future works. Acknowledgement
This work was supported by the National Research Foundation of Korea(NRF) grant fundedby the Korea government(MSIT) (No. NRF-2019R1A2C1006639).
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