DOI: 10.22661/AAPPSBL.2018.28.4.64
Holographic Renormalization Group Flow
from UV to IR
CHANYONG PARK ^{a,b,c }*
^{a }DEPARTMENT OF PHYSICS AND PHOTON SCIENCE, GWANGJU INSTITUTE OF SCIENCE AND TECHNOLOGY
^{b}ASIA PACIFIC CENTER FOR THEORETICAL PHYSICS
^{c }DEPARTMENT OF PHYSICS, POSTECH
ABSTRACT
We construct a threedimensional geometry interpolating two different AdS spaces. From the dual quantum field theory viewpoint, it corresponds to a nontrivial renormalization group flow from a UV to another IR conformal field theory. On this geometry, we discuss the change of the central charge in a momentumspace and realspace renormalization group flow. The result shows in both renormalization prescriptions that the central charge monotonically decreases along the renormalization group flow.
* cyong21@gist.ac.kr
INTRODUCTION
After Maldecena's AdS/CFT conjecture [14], considerable attention has been paid to holography for understanding a variety of strongly interacting systems. Since a strongly interacting system is a nonperturbative system, there is no appropriate mathematical tool to control it in traditional quantum field theory. In this situation, holography may be helpful to figure out the nonperturbative and quantum features of a strongly interacting system. In this short paper, we construct a dual geometry that enables us to understand holographically a nonperturbative renormalization group (RG) flow from a UV to an IR fixed point.
Though several important features of conformal field theory (CFT) have been well described by a onedimensional higher AdS (antideSitter) geometry [59], it still remains an important issue to construct a dual geometry of a deformed CFT [1012]. In this work, we take into account a twodimensional quantum field theory (QFT), which flows from a UV CFT to another IR CFT, and construct a dual threedimensional gravity representing such a RG flow. Since two fixed points in the UV and IR limit are dual to two different AdS spaces, constructing the dual geometry of a nontrivial RG flow corresponds to finding a geometry interpolating those two AdS spaces. Once such an interpolating geometry is known, we can check whether or not the ctheorem conjectured in Ref. [13] is really working. Moreover, the dual geometry enables us to figure out how the dual QFT changes along the RG flow. After constructing a dual geometry analytically, we will investigate the change of a heavy quark's potential [1418]. The resulting quark's potential indicates that there is no phase transition when a UV CFT smoothly changes to an IR CFT.
We note that there are two different prescriptions for a RG flow. One is a momentumspace RG flow [7] and the other a realspace RG flow [1921]. In the holographic model, the momentumspace RG flow corresponds to taking the holographic renormalization procedure with regards to the radial coordinate of the dual geometry as the energy scale of the QFT. On the other hand, the realspace RG flow is governed by a holographic entanglement entropy in which the energy scale of the dual QFT is characterized by the size of the subsystem. We show that, although the cfunction defined with two different prescriptions are different, they all satisfy the ctheorem. At a UV and IR fixed point, in particular, two different prescriptions reproduce the known CFT result.
Now, let us consider the following threedimensional toy gravity model with λ>1

(1) 
where the scalar potential is given by

(2) 
This model represents a dual gravity of a twodimensional QFT with a nontrivial RG flow from a UV to IR. Despite the complicated form of the scalar potential, the gravity solution is given by a relatively simple and analytic solution, which allows the interpolation of two different AdS geometries.
UV behavior
Before discussing an interpolating geometric solution, we first investigate the property of the dual QFT in the UV regime. Considering the scalar field fluctuation near the UV fixed point (𝜙=0+δ𝜙), the scalar potential is expanded into

(3) 
which leads to a local maximum (∂^{2}V/∂δ𝜙^{2}<0) at 𝜙=0. Therefore, the solution of the Einstein equation at the UV fixed point 𝜙=0 reduces to an AdS space with a negative cosmological constant Λ_{uv}=V/2=1/R^{2}_{uv}

(4) 
where the UV region corresponds to z → 0. The dual QFT of this AdS geometry becomes a twodimensional CFT with a central charge, c_{uv}=12πR_{uv}/k^{2} [6].
If we further consider the next order correction of the scalar field fluctuation, we see that it represents a relevant scalar deformation to the UV CFT. To see this perspective, let us take into account a scalar field fluctuation δ𝜙 at the UV fixed point 𝜙=0, In order to understand the asymptotic behavior of the scalar fluctuation, it is sufficient to consider the probe limit because the gravitational backreaction of the scalar fluctuation is negligible. In the UV AdS space, the scalar field fluctuation is governed by

(5) 
and its solution reads

(6) 
This result shows that the scalar field fluctuation corresponds to a relevant scalar operator with the conformal dimension 3/2. If we further consider the gravitational backreaction of the scalar fluctuation in the asymptotic region, it changes only the values of two integral constants, c_{1} and c_{2}, but does not modify the zdependence. This becomes manifest when we consider an exact solution including all gravitational backreactions later. On the dual QFT side, c_{1} and c_{2} are reinterpreted as a source and vacuum expectation value (vev) of the scalar operator [24].
IR behavior
For λ>1, the above scalar potential also allows an additional local minimum at 𝜙=𝜙_{ir}, which corresponds to an IR fixed point. If we focus only on the case with 𝜙&≥0, the local minimum appears at

(7) 
and near the local minimal the scalar potential is expanded into

(8) 
which satisfies ∂^{2}V/∂δ𝜙^{2}>0. In this case, the IR fixed point is again represented by another AdS space with an IR cosmological constant Λ_{ir}=1/R^{2}_{ir} =λ^{2}/R^{2}_{uv}

(9) 
As shown in this IR solution, the parameter λ in the scalar potential plays a crucial role in determining the IR fixed point. From the IR AdS geometry, we expect that the dual IR CFT has a central charge, c_{ir}=12πR_{ir}/k^{2}. Comparing the central charges of UV and IR CFTs, the RG flow yields

(10) 
which is consistent with Zamolodchikov's ctheorem [13].
Above, we showed that the scalar field plays a role in the relevant deformation in the UV region. Relying on the observationalenergy scale, its conformal dimension can vary to a different value. Near the IR fixed point, the scalar field can be represented as 𝜙=𝜙_{ir}+δ𝜙 and the scalar fluctuation in the probe limit is governed by

(11) 
and is given by

(12) 
In the IR limit (z → ∞), the gravitational backreaction of the first term cannot be ignored and, furthermore, can modify the background AdS geometry. As a result, c_{3} must vanish in order to get an IR fixed point. As will be seen, a solution interpolating two UV and IR AdS geometries satisfies the constraints discussed above.
Renormalization group flow from UV to IR regimes
Let us try to find a solution interpolating two UV and IR AdS geometries. To do so, we first take the following metric ansatz

(13) 
If the unkown function f(z) smoothly interpolates f(0)=1 in the UV region and f(∞)=R^{2}_{ir}/R^{2}_{uv} in the IR region, a UV CFT continuously changes to another IR CFT by a relevant deformation. To find such a function, we look into the equations of motion derived from (1)

(14) 
The first equation is a constraint coming from the Einstein equation and the others describe the dynamics of 𝜙(z) and f(z). Since we can reconstruct the third equations by combining the other two equations, only two of them are independent.
Now, let us consider a kinktype solution that interpolates the previous UV and IR AdS spaces. We check that the following kinktype solution really satisfies two independent equations for λ>1

(15) 
This analytic solution allows for the existence of an asymptotic AdS geometry with the AdS radius R_{uv}, whereas another AdS space with the AdS radius, R_{ir}=R_{uv}/λ, also appears in the IR limit (z → ∞). At these two fixed points, the analytic solution reproduces the behaviors of the dual CFT mentioned earlier. For example, the scalar field in the UV region has the following expansion

(16) 
while in the IR region it is expanded into

(17) 
where the ellipsis indicates higher order corrections. These two asymptotic behaviors near the UV and IR fixed points are perfectly matched with the previous results obtained in the probe limit.
From the metric form in (13), we may reinterpret R_{eff}=R_{uv}√f as an effective radius corresponding to the observation energy scale of the dual QFT. This is called the holographic renormalization representing the momentumspace renormalization. The effective central charge of the dual QFT then can be written as

(18) 
which reproduces the known central charges of UV and IR CFTs. At an intermediate energy scale, however, R_{eff} is given by a nontrivial function of z, so that c_{eff} crucially relies on the observation energy scale because the dual QFT is not a CFT at an intermediate energy scale. Despite this fact, c_{eff} can have an important implication to the dual QFT. When a CFT at UV scale runs to another CFT at IR scale by a relevant deformation, then it is well known that there exists a cfunction that monotonically decreases along the RG flow and that it reduces to the central charges at two fixed points. Since c_{eff} reduces to the central charges of two CFTs at the fixed points, it would be interesting to check whether it monotonically decreases along the RG flow or not. To do so, let us calculate the change of c_{eff} along the radial coordinate

(19) 
For λ>1, this is always negative. Noting that dz is the RG flow direction, this result indicates that c_{eff} monotonically decreases along the RG flow. Consequently, c_{eff} can be taken into account as a candidate of a cfunction. Expanding c_{eff} near the UV region gives rise to

(20) 
whereas c_{eff} in the IR regime becomes

(21) 
Since the cfunction describes the degrees of freedom of the dual QFT, the ctheorem implies that the degrees of freedom of the dual QFT monotonically decreases along the RG flow.
Heavy quark potential with a nontrivial RG flow
For a CFT without a nontrivial RG flow, it has been known that the interaction of heavy quarks and antiquarks is described by the Coulomb potential inversely proportional to their distance [14, 15]. However, since the geometry we constructed describes a nontrivial RG flow from a UV CFT to another IR CFT, it would be interesting to investigate how the heavy quark potential is modified along the nontrivial RG flow. In string theory, a heavy quark is described as an end of an open string extended to the dual geometry. When two ends of an open string are attached to the AdS boundary, it was known that its worldsheet action represents the potential energy between two heavy quarks [14]. In this case, the heavy quark's potential energy plays an important role in indicating the deconfinement phase transition of QCD. Using the holographic prescription, we will investigate the quark potential when the underlying theory changes from UV to IR.
Let us first assume that the position of the open string in the zdirection depends on the boundary coordinate x. Then, the induced metric on the open string is reduced to

(22) 
and the string action becomes

(23) 
where T indicates an appropriate time interval. Using the fact that this string action has no explicit dependence on x, the distance between two quarks is given by

(24) 
where z_{0} corresponds to the turning point and f_{0} indicates the value of f(z) at the turning point. Due to the existence of the turning point, the open string is extended only in the range of 0≤z≤z_{0}. On the other hand, the string action in terms of the turning point is given by

(25) 
where ε is introduced to regularize the UV divergence. In the UV limit where the distance between quarks is very short, we can expand and perform the above integrals perturbatively.
After integrating (24), the turning point can be perturbatively represented in terms of the quark's distance

(26) 
The integration of the string action usually leads to a UV divergence. In order to remove such a UV divergence, we can consider an action for free quarks which also gives rise to the same UV divergence.
Fig. 1: Quark's potential for = 1, 1.5, and 2.
The action of two free quarks corresponding to the free quark's mass is given by

(27) 
Subtracting the free quark's mass from the string action in (25), the resulting renormalized quark potential is given by

(28) 
For the AdS case where λ=1, the quark potential reduces to a Coulomb potential, V ~ 1/d, as expected by the conformal symmetry [14]. For a large quark's distance, we plot the quark potential with several different λ in Fig. 1. This shows that the strength of the interaction between quarks becomes weaker as λ increases. Even for λ → ∞, the numerical result shows that the quark's potential approaches 0. This fact implies that there always exists a bound state of two quarks, which is more stable than two free quarks. As a consequence, there is no phase transition representing the dissociation of heavy quarkonium, when the underlying theory changes continuously from a UV CFT to another IR CFT.
Entanglement entropy
Now, let us investigate the change of the entanglement entropy along the RG flow [2225]. The RG flow governed by the entanglement entropy is related to realspace renormalization [26, 27]. The entanglement entropy of the dual field is represented as the area of the minimal surface extended to the dual geometry [19, 20]. When we take a subsystem at l/2 ≤ x ≤ l/2, the holographic entanglement entropy is described by

(29) 
where the prime means a derivative with respect to x and G=k^{2}/(8π). Denoting the turning point of the minimal surface as z_{*}, z', vanishes at this turning point and the range of z extended by the minimal surface is restricted to 0 ≤ z ≤ z_{*}. Solving the equations of motion derived from S_{E}, the subsystem size and the entanglement entropy can be parameterized by the turning point

(30) 

(31) 
where f_{*} indicates the value of ε at the turning point and is introduced as a UV cutoff.
In the UV region satisfying z_{*}/R_{uv} ≪ 1, performing the integral in (30) perturbatively leads to

(32) 
The inverse of this relation determines the turning point in terms of the subsystem size

(33) 
The integral in (31) leads to

(34) 
Here, the first term is exactly the result of a twodimensional CFT and the last term represents the nontrivial contribution caused by the scalar deformation. This result shows the UV fixed point (l → 0) is not stationary because dS_{E}/dl ≠ 0. The central charge of the dual CFT is determined by [2830]

(35) 
Fig. 2: The RG ow of (a) the renormalized entanglement entropy, S_{RE} ≡ S_{E} (log , and (b) the cfunction for λ= 1, 1.5, and 2 where we take G =1 and R_{uv} = 1 for simplicity.
c_{E} defined here is another candidate of a cfunction represented in the realspace RG flow. In the UV limit (l → 0), as expected, c reduces to the known CFT's central charge, c_{uv }= 3R_{uv}/(2G).
In Fig. 2, we depict the change of the entanglement entropy and the cfunction with several λs at the intermediate energy scale. The results show that the cfunction defined by the entanglement entropy also monotonically decreases along the RG flow, which is again consistent with the Zamolodchikov ctheorem. Moreover, the result in Fig. 2(b) shows that the cfunction approaches to the central charge of the IR CFT, c_{ir }= 3R_{ir}/(2G), when the subsystem size becomes large (l → ∞). At the fixed points, we have obtained the exact same central charges consistent with the CFT results regardless of the RG flows defined in momentum and real spaces. This is because of the restoration of the conformal symmetry, under which the momentum scales as the inverse of the coordinate, p ~ 1/l. Therefore, there exists a direct relation between the momentum and coordinate at the fixed points. However, since the conformal symmetry is broken at the intermediate energy scale, there is no simple relation between p and l. Despite this fact, the holographic RG flows in the momentum and realspace consistently show that the cfunction defined in a different way perfectly satisfies the Zamolodchikov ctheorem.
In this work, we have discussed the cfunction of a twodimensional QFT along the nontrivial RG flow. In spite of the importance of the nonperturbative RG flow to understand the IR features, it is not easy to know the nonperturbative RG flow of a higher dimensional QFT in a traditional QFT. In this situation, the holographic method we investigated here would be useful to account for the nonperturbative IR phenomena of nuclear and condensed matter physics. We hope to report more results on higher dimensional theories in future works.
Acknowledgements: C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF2016R1D1A1B03932371) and also by the Korea Ministry of Education, the Ministry of Science and ICT, GyeongsangbukDo and Pohang City.
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Chanyong Park is a Professor of Gwangju Institute of Science and Technology (GIST). After receiving the Ph. D from the Hanyang University, he worked at Center for Quantum SpaceTime (CQUeST), National Institute for Mathematical Science (NIMS), Institute for Early Universe (IEU) and Asia Pacific Center for Theoretical Physics (APCTP) before joining GIST. His research field is theoretical high energy physics related to string theory, string cosmology, and gauge/gravity duality. 
