Renormalization group ow to eective quantum mechanics at IR in an emergent dual holographic description for spontaneous chiral symmetry breaking Ki-Seok Kimab Mitsuhiro Nishidaa and Yoonseok Chounab

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Renormalization group ﬂow to eﬀective quantum mechanics at IR in an emergent dual

holographic description for spontaneous chiral symmetry breaking

Ki-Seok Kima,b, Mitsuhiro Nishidaa, and Yoonseok Chouna,b

aDepartment of Physics, POSTECH, Pohang, Gyeongbuk 37673, Korea

bAsia Paciﬁc Center for Theoretical Physics (APCTP), Pohang, Gyeongbuk 37673, Korea∗

(Dated: October 21, 2022)

Implementing the Wilsonian renormalization group (RG) transformation in a nonperturbative

way, we construct an eﬀective holographic dual description with an emergent extradimension iden-

tiﬁed with an RG scale. Taking the large−Nlimit, we obtain an equation of motion of an order-

parameter ﬁeld, here the chiral condensate for our explicit demonstration. In particular, an in-

tertwined structure manifests between the ﬁrst-order RG ﬂow equations of renormalized coupling

functions and the second-order diﬀerential equation of the order-parameter ﬁeld, thus referred to as

a nonperturbative RG-improved mean-ﬁeld theory. Assuming translational symmetry as a vacuum

state, we solve these nonlinear coupled mean-ﬁeld equations based on a matching method between

UV- and IR-regional solutions. As a result, we ﬁnd an RG ﬂow from a weakly-coupled chiral-

symmetric UV ﬁxed point to a strongly-correlated chiral-symmetry broken IR ﬁxed point, where

the renormalized velocity of Dirac fermions vanishes most rapidly and eﬀective quantum mechanics

appears at IR. Furthermore, we translate these RG ﬂows of coupling functions into those of emer-

gent metric tensors and extract out geometrical properties of the emergent holographic spacetime

constructed from the UV- and IR-regional solutions. Surprisingly, we obtain the volume law of

entanglement entropy in the Ryu-Takayanagi formula, which implies appearance of a black hole

type solution in the limit of inﬁnite cutoﬀ even at zero temperature. We critically discuss our ﬁeld

theoretic interpretation for this solution in terms of potentially gapless multi-particle excitation

spectra.

I. INTRODUCTION

Renormalization group (RG) improved mean-ﬁeld theory (RG-MFT) is not only a natural but also a general

framework at least in the conceptual aspect. Here, interaction vertices are renormalized in the one-loop level, or

resummed in the context of the Bethe-Salpeter equation [1]. These renormalized coupling functions enter a mean-ﬁeld

equation for the description of a phase transition given by an order-parameter ﬁeld. Unfortunately, this ‘perturbative’

RG framework is not enough to understand dynamics of strongly coupled quantum ﬁeld theories.

To generalize the Wilsonian perturbative RG transformation in a nonperturbative way, we introduce an energy-

scale dependent order-parameter ﬁeld and propose an intertwined RG structure between the RG ﬂow equations of

renormalized coupling functions and an extended mean-ﬁeld equation of the order-parameter ﬁeld at each energy

scale. Suppose the Wilsonian RG transformation for a given quantum ﬁeld theory at an energy scale Λ. Then, we

solve the resulting eﬀective interacting ﬁeld theory in a mean-ﬁeld fashion but with renormalized interaction vertices

at the scale Λ. Usually, this is the end of the procedure, referred to as the RG-MFT mentioned above.

In this study, we extend this procedure in an iterative way. After performing the ﬁrst iteration in the RG-MFT, we

consider the second RG transformation for all dynamical variables including the order-parameter ﬁeld. This second

iteration is supported by the renormalized coupling functions and the mean-ﬁeld ‘background’ value of the order

parameter at the scale Λ in the ﬁrst step of the RG-MFT. Again, we take the large−Nlimit to perform the mean-ﬁeld

analysis with the updated renormalized interaction vertices after the second RG implementation. This completes the

second iteration in the RG-MFT.

Repeating these RG iterations with the scale-dependent mean-ﬁeld analysis in the large−Nlimit and expressing

the discrete variable of the RG iteration with a continuous ‘coordinate’ z, we construct an eﬀective ﬁeld theory, where

the RG transformation manifests with an emergent extradimension denoted by z[2–12]. Interestingly, we observe

that this nonperturbative RG-MFT not only shares some characteristic features of the holographic dual eﬀective

ﬁeld theory [13–19] but also modiﬁes them in two ways. First, the background geometry in the dual holographic

description corresponds to the RG ﬂow equations of renormalized coupling functions in the nonperturbative RG-

MFT [17–19]. Second, other ﬁelds besides the metric tensor describe the dynamics of ‘order parameters’ or collective

ﬁelds in a dual fashion [13–16]. For example, the scalar ﬁeld in the Einstein-scalar ﬁeld theory corresponds to the

∗Ki-Seok Kim: tkfkd@postech.ac.kr; Mitsuhiro Nishida: mnishida124@gmail.com; Yoonseok Choun: ychoun@gmail.com

arXiv:2210.10277v2 [hep-th] 20 Oct 2022

chiral condensate in the present description. These two aspects reinterpret the holographic dual eﬀective ﬁeld theory

in terms of nonperturbative Wilsonian RG transformations. Third, the nonperturbative RG-MFT introduces RG

β−functions of interaction vertices beyond the holographic dual eﬀective ﬁeld theory with gravity, where the RG

ﬂows for all dynamical variables appear naturally to show conformal invariance only in the low-energy limit [5–12, 20–

23]. Fourth, there is essential diﬀerence on how to assign UV and IR boundary conditions in the present description,

where an eﬀective on-shell (boundary) action determines both boundary conditions [5–12]. Here, the eﬀective on-shell

(boundary) action may be regarded as a solution of the Hamilton-Jacobi equation [5, 6].

For concreteness, we consider spontaneous chiral symmetry breaking [1]. Here, we have three types of coupling

functions, corresponding to the wave-function renormalization constant for Dirac fermions, the velocity of Dirac

fermions, and their eﬀective interactions for chiral symmetry breaking, respectively. In addition, we have one order

parameter ﬁeld to describe the chiral symmetry breaking. Taking the large−Nlimit in the nonperturbative RG-MFT,

where Nis the ﬂavor number of Dirac fermions, we obtain four coupled diﬀerential equations, where three of them are

given by the ﬁrst order to describe the RG ﬂows of the coupling functions and the last is the second-order diﬀerential

equation for the order-parameter ﬁeld, supported by renormalized coupling functions at a given energy scale. We

verify that the wave-function renormalization constant (the velocity renormalization constant) in the nonperturbative

RG-MFT corresponds to the space (time) component of the metric tensor in the holographic dual eﬀective ﬁeld theory.

We believe that it would not be an easy task to solve these four heavily intertwined diﬀerential equations. Here, we

apply a matching method to solving these coupled diﬀerential equations [24]. First, we solve such coupled diﬀerential

equations near both UV and IR boundary regions, where these equations become simpliﬁed. Second, we apply the

UV (IR) boundary condition to the UV-regional (IR-) solution. Since the number of boundary conditions would

be less than that of integration constants, some of the integration constants remain undetermined in both UV- and

IR-regional solutions. Third, we require that the UV-regional solution should be smoothly connected to the IR-

regional solution in the extradimensional space. Of course, there must be a certain condition for the existence of this

matching solution, which will be clariﬁed later. Based on this matching method, we ﬁnd an RG ﬂow from a weakly-

coupled chiral-symmetric UV ﬁxed point to a strongly-correlated chiral-symmetry broken IR ﬁxed point, where the

renormalized velocity of Dirac fermions vanishes most rapidly and eﬀective quantum mechanics appears at IR. It is

a feature of the nonperturbative RG-MFT the appearance of this local strong-coupling ﬁxed point. Furthermore,

we investigate several geometrical objects in the holographic spacetime constructed from the UV- and IR-regional

solutions we found. In particular, we calculate the minimal surface for a single interval, which is essential for the

holographic entanglement entropy formula [25, 26]. We uncovered the volume law of entanglement entropy at zero

temperature, which implies appearance of a black hole type geometry at Λ → ∞ to describe the strongly-correlated

chiral-symmetry broken IR ﬁxed point. We critically discuss our ﬁeld theoretic interpretation that strong correlations

may allow gapless multi-particle spectra between the single-particle excitation gap due to spontaneous chiral symmetry

breaking.

This paper is organized as follows. First, we discuss a general formulation of emergent dual holography in section

II. Here, we introduce a general framework for the RG transformation and clarify how the present construction

implements the RG transformation in a nonperturbaive way. In addition, we introduce an emergent dual holographic

description given by gravity with an extradimension and argue that our nonperturbative RG framework may be

regarded as an eﬀective holographic dual description. Second, we take the large−Nlimit in this eﬀective ﬁeld theory

and obtain the equation of motion for an order-parameter ﬁeld, here the chiral condensate, in section II. Here, we ﬁnd

an intertwined structure for the RG ﬂows between the renormalized coupling vertices and the order-parameter ﬁeld.

Third, we try to solve these coupled mean-ﬁeld equations based on a matching method between UV- and IR-regional

solutions in section III, assuming translational symmetry as a vacuum solution. In particular, we could reveal some

analytic behaviors in the low-energy limit and ﬁnd a local strong-coupling ﬁxed point. In section IV, we compute the

Ricci scalar curvature and the minimal surface on the three-dimensional holographic spacetime. Then, we summarize

our main results and discuss several unresolved issues here in the last section.

II. A NONPERTURBATIVE APPROACH OF THE WILSONIAN RENORMALIZATION GROUP

THEORY FOR SPONTANEOUS CHIRAL SYMMETRY BREAKING

A. A general framework for the renormalization group transformation

We start our discussions, reviewing a general framework of the RG transformation [1]. Here, we consider interacting

Dirac fermions, described by

Z=ZDψbσ(x) exp h−ZdDxn¯

ψbσ(x)γµ∂µψbσ(x) + λbχ

2N¯

ψbσ(x)ψbσ(x)¯

ψbσ0(x)ψbσ0(x)oi.(1)

ψbσ(x) is a Dirac spinor at xin Dspacetime dimensions, where σis a ﬂavor index from 1 to Nand bis ‘bare’ or

‘unrenormalized’. ¯

ψbσ(x) = ψ†

bσ(x)γ0is the canonical conjugate variable to ψbσ(x). γµwith µ= 0, ..., D −1 is the

Dirac matrix to satisfy the Cliﬀord algebra in Dspacetime dimensions. λbχ is an interaction coeﬃcient for chiral

symmetry breaking, where balso denotes ‘bare’ or ‘unrenormalized’.

Performing the Hubbard-Stratonovich transformation for the chiral symmetry breaking channel, we introduce an

order-parameter ﬁeld ϕb(x) as follows

Z=ZDψbσ(x)Dϕb(x) exp h−ZdDxn¯

ψbσ(x)γτ∂τ−ivbψγi∂iψbσ (x)−iλbχϕb(x)¯

ψbσ(x)ψbσ(x) + Nλbχ

2ϕ2

b(x)oi.

(2)

Here, we introduced vbψ as the ‘bare’ velocity of Dirac fermions. To describe quantum ﬂuctuations of chiral symmetry

breaking, we integrate over short-distance ﬂuctuations of Dirac fermions and obtain

Z=ZDψbσ(x)Dϕb(x) exp h−ZdDxn¯

ψbσ(x)γτ∂τ−ivbψγi∂iψbσ (x)−iλbχϕb(x)¯

ψbσ(x)ψbσ(x)

+Nϕb(x)−∂2

τ−v2

bϕ∂2

i+m2

bϕϕb(x)oi.(3)

Here, the kinetic energy of the order-parameter ﬁeld results from the polarization bubble of high-energy quantum

ﬂuctuations of Dirac fermions, denoted by Π(x−x0) = λ2

bχD¯

ψbσ(x)ψbσ(x)¯

ψbσ0(x0)ψbσ0(x0)E, where h...iis an ensemble

average. Performing the Fourier transformation and expanding it up to the second order with respect to the frequency

and momentum, we obtain Π(iΩ, q) = c1Nλ2

bχ

Λ2

f(Ω2+c2v2

bψq2) + Π(0,0), where c1and c2are numerical constants. Λf

is the short-distance cutoﬀ to control high-energy ﬂuctuations. Rescaling coordinates, ﬁelds, and coupling constants

appropriately, we obtain the above expression for the kinetic energy of the order-parameter ﬁeld, where vbϕ and mbϕ

are the ‘bare’ velocity and ‘bare’ mass of the collective ﬁeld ϕb(x).

Now, we consider the RG transformation. First, the coordinate transforms as

x→µx, (4)

where µis the scaling parameter. To make the time-derivative term be invariant under this coordinate transformation,

both fermion and boson ﬁelds have to transform as follows

ψbσ(x) = µ−D−1

2Z1

ψψrσ (x), ϕb=µ−D−2

2Z1

ϕϕr.(5)

Here, Zψand Zϕare wave-function renormalization constants to take divergent contributions from quantum correc-

tions and to relate fermion and boson bare ﬁelds with their renormalized ones denoted by the subscript r. Accordingly,

fermion and boson velocities transform as

vbψ =Z−1

ψZvψvrψ , v2

bϕ =Z−1

ϕZv2

ϕv2

rϕ,(6)

which lead the space-derivative term to be invariant under this scaling transformation. Zvψand Zv2

ϕare velocity

renormalization constants for fermions and bosons, respectively. The boson mass term remains invariant if the mass

parameter transforms as

bϕ =µ2Z−1

ϕZm2

ϕm2

rϕ,(7)

where Zm2

ϕis the mass renormalization constant. Finally, the Yukawa coupling term is invariant if the Yukawa

interaction vertex transforms as

λbχ =µ−D−4

2Z−1

ψZ−1

ϕZλχλrχ,(8)

where Zλχis the interaction renormalization constant.

Replacing all bare ﬁelds and coupling constants with renormalized ones, we obtain the following eﬀective ﬁeld theory

Z= (µ−(2D−3)ZN

ψZϕ)VDZDψrσ(x)Dϕr(x) exp h−ZdDxn¯

ψrσ (x)Zψγτ∂τ−iZvψvrψ γi∂iψrσ(x)

−iZλχλrχϕr(x)¯

ψrσ (x)ψrσ (x) + Nϕr(x)−Zϕ∂2

τ−Zv2

ϕv2

rϕ∂2

i+Zm2

ϕm2

rϕϕr(x)oi,(9)

where VDis the volume factor of the D−dimensional spacetime. Below, we ﬁnd all these renormalization constants

in the nonperturbative RG-MFT.

B. Nonperturbative implementation of the Wilsonian renormalization group transformation

To implement the Wilsonian RG transformation in a nonperturbative way, we introduce wψwith an order-parameter

ﬁeld as follows

Z=ZDψσ(x)Dϕ(x) exp h−ZdDxn¯

ψσ(x)wψγτ∂τ−ivψγi∂iψσ(x)−iλχϕ(x)¯

ψσ(x)ψσ(x) + Nλχ

2ϕ2(x)oi.

(10)

As expected, wψwill play the role of the wave-function renormalization constant for Dirac fermions.

Now, we perform the RG transformation. First, we separate fast and slow degrees of freedom from all dynamical

ﬁelds at a given energy scale Λ, here Dirac fermions ψσ(x) and chiral symmetry breaking ﬂuctuations ϕ(x). Sec-

ond, we integrate over short-distance ﬂuctuations for ϕ(x) and obtain newly-generated eﬀective interactions between

Dirac fermions. Third, we perform the Hubbard-Stratonovich transformation for such RG-transformation generated

interactions and have an additional collective ﬁeld variable, saying ϕ(1)(x) to decompose RG-generated eﬀective in-

teractions. Calling the previous low-energy mode ϕ(0)(x) and shifting ϕ(1)(x) to ϕ(1)(x)−ϕ(0)(x), we ﬁnish the ﬁrst

RG transformation for chiral symmetry breaking ﬂuctuations. Fourth, we perform the path integral for short-distance

ﬂuctuations of Dirac fermions. Actually, this RG transformation gives rise to genuine renormalization eﬀects for

all coupling functions introduced in the above UV eﬀective ﬁeld theory. Although any concrete procedures would

depend on regularization, we have the following structure in the Wilsonian RG transformation. The path integral

for high-energy quantum ﬂuctuations of Dirac fermions generates an eﬀective potential in terms of unrenormalized

coupling functions and ϕ(1)(x). Fifth, taking a functional derivative of the RG-generated eﬀective potential with

respect to each coupling function and ϕ(1)(x), we obtain the so called RG β−function to encode how the coupling

function evolves through this RG transformation. Calling the previous unrenormalized coupling function that with a

superscript (0), we update it to a renormalized one with a superscript (1). This completes the ﬁrst iteration of the

RG transformation.

To proceed the second iteration of the RG transformation, we consider quantum ﬂuctuations of ψσ(x) and ϕ(1)(x).

Here, ϕ(0)(x) is determined by free-energy minimization as usual. In this respect ϕ(0)(x) is an energy-scale dependent

order-parameter ﬁeld as discussed in the introduction. Now, it is straightforward to perform the RG transformation

in a recursive way. Each coupling function with a superscript (k−1) renormalizes into that with a superscript (k) by

its RG−βfunction. Within these background renormalized coupling functions, the energy-dependent order parameter

ﬁeld is determined by minimization of the free energy functional. To make the resulting eﬀective ﬁeld theory be more

tractable, we introduce a continuum variable zto replace the discrete step (k) and rewrite the RG transformation

from (k−1) to (k) in a form of the z−derivative. As a result, the RG ﬂows of the coupling functions and the RG

evolution of the order-parameter ﬁeld become manifested through the emergent extradimensional space denoted by

the coordinate z. This completes our nonperturbative RG-MFT [5–12].

All these discussions can be summarized by the following eﬀective ﬁeld theory

Z=ZDψσ(x)Dϕ(x, z)Dπϕ(x, z)Dwψ(x, z)Dπwψ(x, z)Dvψ(x, z)Dπvψ(x, z)Dλχ(x, z)Dπλχ(x, z)

exp h−ZdDxn¯

ψσ(x)wψ(x, zf)γτ∂τ−ivψ(x, zf)γi∂iψσ(x)−iλχ(x, zf)ϕ(x, zf)¯

ψσ(x)ψσ(x) + Nλχ(x, 0)

2ϕ2(x, 0)o

−NZzf

dz ZdDxnπϕ(x, z)∂zϕ(x, z)−βϕ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)]−1

2λχ(x, z)π2

ϕ(x, z)

+πwψ(x, z)∂zwψ(x, z)−βwψ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)]

+πvψ(x, z)∂zvψ(x, z)−βvψ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)]

+πλχ(x, z)∂zλχ(x, z)−βλχ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)]

+Veff [ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)]oi.(11)

Although this expression looks rather complicated, we explain several characteristic features of this eﬀective ﬁeld

theory, discussed above.

wψ(x, z), vψ(x, z), and λχ(x, z) are renormalized coupling functions. In particular, wψ(x, z) is related with the

ﬁeld renormalization to be clariﬁed in the next subsection. πwψ(x, z), πvψ(x, z), and πλχ(x, z) are their canonical

conjugate ﬁelds, respectively. Integrating over these canonical ﬁelds, we obtain the RG−ﬂow equations for these

coupling functions as follows

∂zwψ(x, z) = βwψ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)],(12)

∂zvψ(x, z) = βvψ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)],(13)

∂zλχ(x, z) = βλχ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)].(14)

Physical meaning of these ﬁrst-order diﬀerential equations is clear. The real question is how to ﬁnd these RG

β−functions in a nonperturbative way, to be clariﬁed below.

ϕ(x, z) is an energy-dependent chiral symmetry breaking order-parameter ﬁeld, whose dynamics is determined by

minimization of the bulk eﬀective action in the large−Nlimit, given by

Sbulk =NZzf

dz ZdDxnπϕ(x, z)∂zϕ(x, z)−βϕ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)]−1

2λχ(x, z)π2

ϕ(x, z)

+Veff [ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)]o.(15)

πϕ(x, z) is the canonical conjugate ﬁeld to ϕ(x, z). In this respect this eﬀective bulk action is written in the Hamiltonian

formulation. The essential ingredient is an eﬀective potential Veff [ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)], generated by

the RG transformation for Dirac fermions and given by

Veff [ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)]

=−1

Nln ZΛ(z)

Dψσ(x) exp h−ZdDxn¯

ψσ(x)wψ(x, z)γτ∂τ−ivψ(x, z)γi∂iψσ(x)−iλχ(x, z)ϕ(x, z)¯

ψσ(x)ψσ(x)oi.

(16)

Here, RΛ(z)Dψσ(x) means that the fermion path integral is performed at a given energy scale Λ(z), which will be

clariﬁed in the next section. Accordingly, the RG β−function of the chiral condensate is given by the functional

derivative of this eﬀective potential with respect to the order-parameter ﬁeld [1] as follows

βϕ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)] = −δϕVef f [ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)] = i

NDλχ(x, z)¯

ψσ(x)ψσ(x)E.

(17)

Here, h...iis an ensemble average with respect to the eﬀective action functional

SΛ(z)=ZdDxn¯

ψσ(x)wψ(x, z)γτ∂τ−ivψ(x, z)γi∂iψσ(x)−iλχ(x, z)ϕ(x, z)¯

ψσ(x)ψσ(x)o.(18)

Three other RG β−functions are given by functional derivatives of this eﬀective potential with respect to the

corresponding coupling function in the following way

βwψ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)] = −δwψVef f [ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)] = −1

ND¯

ψσ(x)γτ∂τψσ(x)E,

(19)

βvψ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)] = −δvψVef f [ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)] = i

ND¯

ψσ(x)γx∂xψσ(x)E,

(20)

βλχ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)] = −δλχVef f [ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)] = i

NDϕ(x, z)¯

ψσ(x)ψσ(x)E.

(21)

This procedure is completely consistent with what we learnt in the RG transformation [1].

It is straightforward to ﬁnd the Hamiltonian equation of motion in the large−Nlimit, given by

πϕ(x, z) = λχ(x, z)∂zϕ(x, z)−βϕ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)](22)

and

∂zπϕ(x, z) = −βϕ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)] −πϕ(x, z)δϕβϕ[ϕ(x, z), wψ(x, z), vψ(x, z), λχ(x, z)],(23)

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摘要：

RenormalizationgroupowtoeectivequantummechanicsatIRinanemergentdualholographicdescriptionforspontaneouschiralsymmetrybreakingKi-SeokKima;b,MitsuhiroNishidaa,andYoonseokChouna;baDepartmentofPhysics,POSTECH,Pohang,Gyeongbuk37673,KoreabAsiaPacicCenterforTheoreticalPhysics(APCTP),Pohang,Gyeongbuk37673...

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Renormalization group ow to eective quantum mechanics at IR in an emergent dual holographic description for spontaneous chiral symmetry breaking Ki-Seok Kimab Mitsuhiro Nishidaa and Yoonseok Chounab

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