We study Wilson loops in the exact renormalization group ow at zero tempera- ture via the 5d holographic model that reproduces the behavior of the QCD running coupling. Calculating the time-like rectangular Wilson loop we show that the model includes a conning phase at T = 0 in the IR region

Golubtsova, Anastasia A.

Vu, Nguyen Hoang

English

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 29, No. 3SI (2019), pp. 371-376
DOI:10.15625/0868-3166/29/3SI/14248
HOLOGRAPHIC WILSON LOOPS IN THE CONFINING BACKGROUNDS AT
ZERO TEMPERATURE
ANASTASIA A. GOLUBTSOVAa,†AND NGUYEN HOANG VUb
aBogoliubov Laboratory of Theoretical Physics, JINR,
141980 Dubna, Moscow region, Russia
bInstitute of Physics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
†E-mail: golubtsova@theor.jinr.ru
Received 19 August 2019
Accepted for publication 8 October 2019
Published 22 October 2019
Abstract. We study Wilson loops in the exact renormalization group flow at zero temperature via
the 5D holographic model that reproduces the behavior of the QCD running coupling. Calculating
the time-like rectangular Wilson loop we show that the model includes a confining phase at T = 0
in the IR region.
Keywords: holographic gauge; Wilson loops.
Classification numbers: 11.10.Gh; 12.38.-t; 24.85.+p.
I. INTRODUCTION
In the recent years the holographic gauge/gravity duality has been shown to be a very use-
ful tool to get theoretical insight into systems in the strongly coupling regime [1]. Some of holo-
graphic models represent a class of useful toy models, because they can be studied analytically
while having features resembling the real QCD.
The description of RG flow via holography corresponds to a gravitational domain-wall
solution interpolating between boundaries. These boundaries of the solutions are supposed to be
related to fixed points of a dual theory and should have particular properties. The holographic
direction corresponds to the energy scale, while the dilaton is related to the running coupling.
In [2] it was constructed the holographic exact RG flows for the 5D dilaton gravity. For a particular
case, it was shown that the behaviour of the running coupling mimics the dependence of the QCD
running coupling on the energy scale. Thus it is of interest to perform a check if the confinement
takes place for the backgrounds found in [2].
c©2019 Vietnam Academy of Science and Technology
372 HOLOGRAPHIC WILSON LOOPS IN THE CONFINING BACKGROUNDS AT ZERO TEMPERATURE
The Wilson loop represents one of criteria for studies of the IR region and manifests an
area law behaviour in the case of the confinement. Using the holographic prescription of Wilson
loops [3] one can explore a holographic dual for possible phase transitions. The aim of this work
is to calculate the expectation value of a rectangular temporal Wilson loop for the holographic RG
flow constructed in [2].
Recently, Wilson loops with different orientations in holographic RG flows with anisotropy
were studied in [5, 6]. Using a so-called effective potential [1] it was shown that the model has
a confinement-deconfinement phase transition. In [7, 8] it was also explored holographic Wilson
loops in RG flow backgrounds estimating a position of the turning point. In this work we will
follow the method of the effective potential [1,5,6] and show that for the holographic RG flow [2]
at T = 0 one can observe a confinement. The work is organized as follows. In section 2 we
give a brief description of the holographic model and corresponding zero-temperature solutions.
In section 3 we calculate the holographic temporal Wilson loop and discuss the related effective
potential. In section 4 we conclude.
II. THE HOLOGRAPHIC MODEL
The 5-dimensional holographic model [2] is given by an action of the form
S =
1
16piG5
∫
d5x
√
|g|
(
R− 4
3
(∂φ)2−C1e2k1φ −C2e2k2φ
)
+G.H., (1)
where Ci, ki with i = 1,2 are constants. We choose the constants as C1 < 0, and C2 > 0. We note
that the choice of the dilaton potential is motivated by a relation to a bosonic sector of supergravity
[4]. Taking the dilatonic constants as k1 = k, k2 = 169k with 0 < k <
4
3 , one can reduce the model
(1) to the Toda chains and integrate this [2]. The solutions for the metric and the dilaton at zero
temperature for the model (1) are
ds2 = F
8
9k2−16
1 F
9k2
2(16−9k2)
2
(−dt2 +d~x 2)+F 329k2−161 F 18k216−9k22 du2, (2)
where~x = (x1,x2,x3) and the corresponding dilaton
φ =− 9k
9k2−16 logF1 +
9k
9k2−16 logF2. (3)
The functions F1 and F2 in (2)-(3) are given by
F1 =
√∣∣∣∣ C12E1
∣∣∣∣sinh(µ u), F2 =
√∣∣∣∣ C22E2
∣∣∣∣sinh( 43kµ (u−u0)), µ =
√∣∣∣∣3E12
(
k2− 16
9
)∣∣∣∣. (4)
The parameters u0, E1 and E2 are the constants of integration and one has the zero energy
constraint E1 +E2 = 0, with E1 < 0, E2 > 0.
The solutions possess the Poincare invariance. It’s convenient to come to the domain wall
coordinates, thus the metric takes the form
ds2 = e2A (w)
(−dt2 +dx21 +dx22 +dx23)+dw2. (5)
ANASTASIA A. GOLUBTSOVA AND NGUYEN HOANG VU 373
We note that the change of the coordinates is given by dw = F
16
9k2−16
1 F
9k2
16−9k2
2 du, so the scale factor
is
A =
4
9k2−16 logF1 +
9k2
4(16−9k2) logF2. (6)
The quantity eA is supposed to be associated with the energy scale of the dual theory, so it should
be monotonic. The constant of integration u0 in (4) splits our solution into the following branches:
a) u < u0; b) u0 < u < 0, c) u > 0, d) u0 = 01. In [2, 9] it was discussed in details the features
of this solutions. It was shown that the solution a) has a non-monotonic behaviour of the scale
factor that precludes to give an interpretation of this solutions as a holographic RG flow. The
solution b) obeys the condition of the monotonicity of the scale factor and moreover corresponds
to a confining theory in the IR limit and a free theory in the UV limit. The solutions c) and d) have
monotonic scale factors but describe theories that are free both in the UV and IR limits. In what
follows we will focus on the RG flow related to the solution and omit the discussion of others
solution.
We note that for the holographic RG flow corresponding to the solution b) the boundary
with u ∼ 0 corresponds to the UV limit and the boundary near u0 point is related to the IR limit.
Both boundaries are spacetimes with a hyperscaling violation.
III. THE RECTANGULAR TIME-LIKE WILSON LOOP
In the holographic prescription the expectation value of the rectangular temporal Wilson
loop of size T×` can be defined through the Nambu-Goto action SNG of a string stretched between
two quarks in a considered background
〈W 〉 ∼ e−SNG . (7)
In this section we discuss the Nambu-Goto action for the general solution (2) in the domain
wall coordinates (5). This corresponds to consideration of Wilson loops in the holographic RG
For the string configuration we choose the following gauge [3]
σ0 = t, σ1 = x1, w = w(x1) (8)
with w(0) = w(`) = 0, w
(
`
2
)
= wmin.
Then the induced metric has the following form
ds2 =−e2A dt2 +(e2A +w′2)dx21. (9)
The Nambu-Goto action in the string frame can be presented in the following way
SNG =
1
2piα ′
∫
dx1dte
2φ
3
√
|g|= T
∫
dx1
(
F2
F1
) 6k
9k2−16 √
e2A (e2A +w′2). (10)
Coming to the u-variable the distance between quarks is
`
2
=
∫ 0
u∗
du
ce3A√
e4A+
4
3φ − c2
. (11)
1In what follows we suppose that u0 < 0.
374 HOLOGRAPHIC WILSON LOOPS IN THE CONFINING BACKGROUNDS AT ZERO TEMPERATURE
For the Nambu-Goto action we have
SNG
2
=
T
2piα ′
∫ 0
u∗
du
e7A+
4φ
3√
e4A+
4φ
3 − c2
. (12)
As expected the Nambu-Goto action is divergent near 0 (the UV boundary). Doing the renormal-
ization one can write
SNG
2
=
T
2piα ′
∫ 0
u∗
du
 e7A+ 4φ3√
e4A+
4φ
3 − c2
− e5A+ 23φ
+∫ u(Λ)
u∗
due5A+
2
3φ , (13)
where we introduced the cutoff parameter u(Λ).
One can define the so-called effective potential with u′ = 0 as
Ve f f = e2A+
2
3φ = F
8−6k
9k2−16
1 F
9k2−12k
2(16−9k2)
2 . (14)
Then in terms of Ve f f the distance between two endpoints of the string can be represented as
`
2
=
∫ 0
u∗
due−φ
Ve f f (u)
√
Ve f f (u)√
V 2e f f (u)
V 2e f f (u∗)
−1
(15)
and the Nambu-Goto action is
SNG
2
=
T
2piα ′
∫ 
u∗
du
e−φV 3e f f (u)
√
Ve f f (u)√
V 2e f f (u)−V 2e f f (u∗)
, (16)
where  is a cutoff parameter. One can see that (15)-(16) make sense if Ve f f is a decreasing function
for the interval 0< u< u∗.
Let us suppose that
V ′e f f (u)|u=umin = 0, V
′′
e f f (umin)> 0. (17)
Near the point u = umin we have V
′′
e f f (umin)> 0 and√√√√ V 2e f f (u)
V 2e f f (umin)
−1 =
√
V ′′e f f (umin)
Ve f f (umin)
(u−umin)+O(u−umin)2. (18)
ANASTASIA A. GOLUBTSOVA AND NGUYEN HOANG VU 375
Inserting (18) into (15)-(16) one gets for the distance between quarks
`
2
=
∫ 0
umin
due−φ
V
5
4
e f f (umin)
(
Ve f f (umin)+V ′′e f f (umin)(u−umin)2
) 3
4√
V ′′e f f (umin)(u−umin)
=
2V 5/4e f f (umin)e
−φ(umin)
3V ′′3/2e f f (umin)
(
Ve f f (umin)+V ′′e f f (umin)(u−umin)2
) 3
4
×
(
V ′′e f f (umin)−
3Ve f f (umin)
(u−umin)2 2F1(1,1,
5
4
,− Ve f f (umin)
V ′′e f f (umin)(u−umin)2
)
)
(19)
and for the string action
S
2
=
T
2piα ′
∫ 
umin
due−φ
V
3
4
e f f (umin)
(
Ve f f (umin)+V ′′e f f (umin)(u−umin)2
) 7
4√
V ′′e f f (umin)(u−umin)
=
2V 3/4e f f (umin)e
−φ(umin)
7
√
V ′′e f f (umin)
(
Ve f f (umin)+V ′′e f f (umin)(u−umin)2
) 3
4
×
(
(u−umin)2V ′′e f f +
10
21
Ve f f (umin)
− 7V
2
e f f (umin)
(u−umin)2V ′′e f f (umin)2
F1(1,1,
5
4
,− Ve f f (umin)
V ′′e f f (umin)(u−umin)2
)
)
,
(20)
k=0.4
k=1
k=1.2
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 u
1.0
1.5
2.0
2.5
3.0
Veff
Fig. 1. The ”potential” Ve f f as a function of u for the
holographic RG flows defined on (u0;0) with u0 = −1,
E1 = E2 =−1, C1 =−C2 =−1, k = 0.4,1,1.2.
where 2F1 is a Gauss hypergeomet-
ric function and ` → +∞ as u →
umin−0 and S→ ∞ as u→ umin−0.
Comparing (19)-(20) one can see the
area law for the quark-antiquark po-
tential. We note that the station-
ary point, V ′e f f |u=umin = 0 is usu-
ally called a dynamical domain wall
point and obeys the equation
V ′e f f = 0. (21)
In Fig. 1 we present the plot
of Ve f f (14) as a function of u for the
holographic RG flow defined on the
interval (u0;0) with u0 = −1. From
Fig. 1 we observe that Ve f f has a minimum umin in (u0;0). It also confirms that we observe a
confined phase in the background (2) corresponding to a theory with T = 0. We also see that for
smaller values of k the minimum of the potential is closer to the IR boundary u0.
376 HOLOGRAPHIC WILSON LOOPS IN THE CONFINING BACKGROUNDS AT ZERO TEMPERATURE
IV. DISCUSSION
In this paper we have discussed the 5D background of the holographic RG flow at T = 0
constructed in [2]. The model can resemble some QCD features, particularly, the dependence
of the running coupling on the energy scale. Following the holographic prescription we have
discussed a rectangular time-like Wilson loop in the RG flow background. For this purpose we
have considered a string stretched between two quarks in the RG flow geometry. Knowing the
action of the string we have calculated the effective potential and performed some analysis of the
Nambu-Goto action and the distance between quarks near a supposed minimum of the effective
potential. We have shown that the effective potential indeed has a point of minimum which tends
to the IR boundary for small values of k, that defines the shape of the dilaton potential. Thus we
have demonstrated that the holographic RG flow defined for the region (u0,0) has a confinement
phase.
A forthcoming problem is to study more carefully a string tension corresponding to the
temporal Wilson loops. We are also going to explore the Wilson loops for holographic RG flows
at finite temperature from [2] and consider a confinement-deconfinement phase transition as it was
done in [6]. It would be interesting to study the Regge spectrum of mesons in these holographic
models.
ACKNOWLEDGEMENTS
We are grateful to Irina Ya. Aref’eva for useful comments and discussions. AG is supported
by RFBR according to the research project 18-02-40069 mega.
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Holographic Wilson Loops in the Confining Backgrounds at Zero Temperature

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Holographic Wilson Loops in the Confining Backgrounds at Zero Temperature

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