MITP-22-088 Echo of the Dark Gravitational waves from dark SU3 Yang-Mills theory

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MITP-22-088

Echo of the Dark:

Gravitational waves from dark SU(3) Yang-Mills theory

Enrico Morgante,1, ∗Nicklas Ramberg,1, †and Pedro Schwaller1, ‡

1PRISMA+ Cluster of Excellence & Mainz Institute for Theoretical Physics,

Johannes Gutenberg-Universit¨at Mainz, 55099 Mainz, Germany

(Dated: February 16, 2023)

We analyze the phase transition in improved holographic QCD to obtain an estimate of the

gravitational wave signal emitted in the conﬁnement transition of a pure SU(3) Yang-Mills dark

sector. We derive the eﬀective action from holography and show that the energy budget and duration

of the phase transition can be calculated with minor errors. These are used as input to obtain a

prediction of the gravitational wave signal. To our knowledge, this is the ﬁrst computation of the

gravitational wave signal in a holographic model designated to match lattice data on the thermal

properties of pure Yang-Mills.

I. INTRODUCTION

First order phase transitions (FOPT) in the early uni-

verse emitting gravitational waves (GWs) are detectable

in upcoming experiments [1–3], and would be clear hints

for new physics since the electroweak and QCD PTs in

the standard model are cross-overs. These GWs might

allow us to probe the dynamics of otherwise inaccessible

dark or hidden sectors [4–6]. SU(N) Yang-Mills theories

are known to feature a color conﬁnement FOPT [7,8],

and they appear in many extensions of the standard

model [9–17]. These scenarios are minimal in the sense

that the conﬁnement scale is their only free parameter,

and thus ideal as benchmark models. However, due to

the strong coupling, nonperturbative methods are needed

for quantitative studies of the dynamics.

Here we employ the AdS/CFT correspondence [18,19]

in a bottom-up framework. We use the Improved Holo-

graphic QCD model [20,21], which successfully repro-

duces lattice data of SU(3) thermodynamics [22,23], to

calculate the equilibrium and quasiequilibrium quantities

relevant for GWs. Previous attempts to study the GW

signal in such models include Refs. [24–31].

The outline of this work is as follows. We will start

by reviewing Improved Holographic QCD and compute

the equilibrium thermodynamics of the model. We will

then construct an eﬀective action by using the free en-

ergy landscape approach [25,26,32–34]. For the kinetic

term of the eﬀective action, we follow the approach of [34]

regarding its normalization. We use our eﬀective action

to calculate the GW signal by using the LISA Cosmol-

ogy Working Group [35] template for the PT parameters

β, α, vw, κ(α). Finally, we will discuss our results and

some future prospects.

∗Electronic address: emorgant@uni-mainz.de

†Electronic address: nramberg@uni-mainz.de

‡Electronic address: pedro.schwaller@uni-mainz.de

II. REVIEW OF IMPROVED HOLOGRAPHIC

QCD

Improved Holographic QCD (IHQCD) [20–23] is a

bottom-up 5-D theory inspired by noncritical string the-

ory, that describes the gluon sector of Yang-Mills theo-

ries. The model is constructed in such a way to reproduce

various features of QCD, for instance, linear conﬁnement,

a qualitative hadron spectrum, asymptotic freedom in

the UV, and a ﬁnite temperature phase diagram that

matches SU(Nc) Yang-Mills theory. Here we will limit

ourselves to a pure gluonic sector, but the inclusion of

ﬂavor and chiral symmetry breaking is possible by intro-

ducing tachyonic D-branes, as well as an axion. More-

over, we will ﬁt the free parameters of the model com-

paring with lattice calculations of SU(3) Yang-Mills. We

plan to extend our results to other values of Ncin a fu-

ture publication. In our equations, we will not impose

Nc= 3 in the expressions which are valid for every Nc.

The model consists of a metric gµν dual to the energy-

momentum tensor, the dilaton Φ dual to (λY M , T rF 2)

and an axion dual to (θY M , T rF ∧F). The axion part

of the theory can be neglected here, as our interest lies

in the thermodynamics mainly. The action for the axion

is N−2

csuppressed. In the Einstein frame, the 5-D ac-

tion which describes this model both at zero and ﬁnite

temperature is given by

S5=−M3

pN2

cZd5x√gR−4

3(∂Φ)2+V(Φ)

+ 2M3

pZ∂M

d4x√hK,(1)

where Mpis the plank mass, Ncis the number of colors,

Rthe Ricci scalar, gthe metric and V(Φ) is the dilaton

potential. The second term in the action is the Gibbons

Hawking term that depends on the induced metric hon

the boundary, and Kis the extrinsic curvature

Kµν =∇µnν=1

2nρ∂ρhµν ,K=habKab .(2)

arXiv:2210.11821v3 [hep-ph] 15 Feb 2023

Since this is a boundary term it does not aﬀect the solu-

tions to the equations of motion in the zero-Ttheory, but

in the ﬁnite Tcase it plays a pivotal role for providing

with a holographically renormalized action [36,37]. For

the dilaton potential we take the ansatz [23]:

V=12

`21 + V0λ+V1λ4

3(log[1 + V2λ4

3+V3λ2]1

2),

(3)

where λ= exp(Φ), and `is the AdS length that sets the

scale of the ﬁfth dimensional coordinate. The parameter

V0and V2are related to the coeﬃcients of the SU(3) YM

β-function

V0=8

9b0, V2=b4

0 23 + 36b1

81V1!2

,(4)

with b0= 22/(3(4π)2) and b1/b2

0= 51/121. These values

depend on setting λequal to the ’t Hooft coupling of the

YM theory in the UV. Other normalizations are possible,

but do not inﬂuence the physical results [23]. The free

parameters V1, V3are set in order to ﬁt lattice results for

the thermodynamical properties of the model: [23]

V1= 14 , V3= 170 .(5)

At ﬁnite temperature, after going into imaginary time

and compactifying time on a circle β= 1/T , we identify

two types of solutions. The ﬁrst reads

ds2=b2

0(r)(dr2−dt2+dxmdxm),(6)

and corresponds to a thermal gas at a temperature T.

Here ris the coordinate of the ﬁfth dimension, and b2

0(r)

is a scale factor. This is the same metric as in the case of

the zero T solution [20,21] except for the identiﬁcation of

time being compactiﬁeld t∼t+iβ. The second solution

is a AdS black hole (BH) metric

ds2=b2(r)dr2

f(r)−f(r)dt2+dxmdxm,(7)

where the “blackening” factor f(r) goes to 0 at the hori-

zon position rh. Regularity of the solution at the horizon

implies that

Th≡|˙

f(rh)|

4π=T , (8)

where This the Hawking temperature of the BH. In

the UV (r→0) the two solutions asymptotically co-

incide, and AdS metric is recovered: b0(r)≈`/r. In

this setup, the AdS BH metric represents the deconﬁned

phase, and the thermal gas solution corresponds to the

conﬁned phase [19].

For a given temperature T, there are either zero or

two values of rhwhich give a solution with Th=T.

These two values identify two separate BH branches, one

for small rh, and correspondingly large b(rh) (big black

0.1 0.2 0.5 1

0.9

1.0

1.1

1.2

1.3

0.95 1.00 1.05 1.10 1.15

1.20

-0.005

0.000

0.005

0.010

FIG. 1. Left: temperature as a function of the horizon posi-

tion λh. Right: free energy as function of temperature, along

the two BH branches.

holes branch), and one at larger rhand smaller b(rh)

(small black hole branch), which is thermodynamically

unstable. Below Tmin there is no BH solution, and the

conﬁnement phase transition must complete. The dilaton

proﬁle λ(r) grows monotonically from 0 at r→0 to λ→

∞at large r, and the horizon position rhcorresponds

to a ﬁnite value λh. A convenient choice is the dilaton

frame [38], in which λis used as the radial coordinate

along the ﬁfth dimension.

The thermodynamical quantity which controls the

phase transition is the free energy diﬀerence between the

BH solution and the thermal gas one, which is deﬁned as

the diﬀerence between the actions of the two solutions:

F=β

(Sdec.− Sconf.).(9)

The action is regularized with a cutoﬀ at r=→0,

and the diﬀerence avoids the need for computing coun-

terterms. The sign of Findicates the energetically fa-

vorable phase, with F<0 corresponding to the decon-

ﬁned phase. The critical temperature Tcis deﬁned at

F= 0. In practice, the free energy can be computed by

integrating the thermodynamic relation dF=−dS/dT

along both black hole branches [22]:

F=−Zλh

∞

b(˜

λh)3dT

d˜

λh

d˜

λh.(10)

Figure 1shows the temperature and the free energy of

the BH solutions.

The entropy is given by the Hawking-Beckenstein for-

mula

S=Area

4G5

= 4πM3

pN2

cV3b(rh)3,(11)

where G5= 1/(16πM3

pN2

c) is the 5D Newton constant

and V3is the volume of 3D space.

More details about the geometric properties of the so-

lution can be obtained as discussed in Sec. 7 of Ref. [22]

and summarized in the appendix.

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MITP-22-088EchooftheDark:GravitationalwavesfromdarkSU(3)Yang-MillstheoryEnricoMorgante,1,NicklasRamberg,1,yandPedroSchwaller1,z1PRISMA+ClusterofExcellence&MainzInstituteforTheoreticalPhysics,JohannesGutenberg-UniversitatMainz,55099Mainz,Germany(Dated:February16,2023)Weanalyzethephasetransitioninim...

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MITP-22-088 Echo of the Dark Gravitational waves from dark SU3 Yang-Mills theory

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