arX

iv:1

610.

0224

1v3

[he

p-ph

] 1

Dec

201

6KANAZAWA-16-11

KUNS-2643

Scale genesis and gravitational wave in a classically scale invariant

extension of the standard model

Jisuke Kubo1 and Masatoshi Yamada2

1Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan

2Department of Physics, Kyoto University, Kyoto 606-8502, Japan

Institut fur Theoretische Physik, Universitat Heidelberg,

Philosophenweg 16, 69120 Heidelberg, Germany

Abstract

We assume that the origin of the electroweak (EW) scale is a gauge-invariant scalar-bilinear

condensation in a strongly interacting non-abelian gauge sector, which is connected to the standard

model via a Higgs portal coupling. The dynamical scale genesis appears as a phase transition at

finite temperature, and it can produce a gravitational wave (GW) background in the early Universe.

We find that the critical temperature of the scale phase transition lies above that of the EW phase

transition and below few O(100) GeV and it is strongly first-order. We calculate the spectrum of

the GW background and find the scale phase transition is strong enough that the GW background

can be observed by DECIGO.

1

I. INTRODUCTION

One of the problems of the standard model (SM) is that it cannot explain the origin of the

electroweak (EW) scale. Obviously, if we start with a theory which contains a scale from the

beginning, we have no chance to explain its origin. We recall that the Higgs mass term is the

only term that breaks scale invariance at the classical level in the SM. So, a central question

is: What is the origin of the Higgs mass term? Recently, a number of studies on a scale

invariant extension of the SM have been performed, where two ideas have been considered.

The one [1–43] relies on the Coleman-Weinberg (CW) potential [44], and the other one is

based on non-perturbative effects: Dynamical chiral symmetry breaking [45, 46] is applied

in [47–59] and gauge-invariant scalar-bilinear condensation [60–62] is used in [63, 64].

In the case that a scale is dynamically created, there will be a corresponding scale phase

transition. Chiral phase transition is a scale phase transition in this context. If a scale

phase transition is strongly first-order, the Universe undergoes a strong phase transition at

a certain high temperature, thereby producing gravitational wave (GW) background that

could be observed today [65–67]. There are a number of GW experiments that are on-going

or planned in the near and far future (see [68, 69] and also [70–72] for instance). It is in fact

this year in which LIGO [73] has observed the GW for the first time.

In this paper we focus on the second non-perturbative effect, the gauge-invariant scalar-

bilinear condensation, that produces the GW background in the early Universe. The present

work is a natural extension of our recent works [63, 64], because we have shown there that

the scale phase transition due to the scalar-bilinear condensation is strongly first-order in

a wide region of the parameter space. We have performed the analysis, using an effective

field theory that describes approximately the process of the dynamical scale genesis via

the scalar-bilinear condensation. Since we will be using the same effective field theory to

calculate the spectrum of the corresponding GW background, we devote, for completeness,

first two chapters for briefly elucidating our effective field theory method, where parallels to

the Nambu-Jona-Lasinio (NJL) theory [46] could be read off. In [63, 64] we have used the so-

called self-consisting mean-field approximation [74, 75] to derive the mean-field Lagrangian,

we will employ the path-integral approach [76, 77] (sect. II) to arrive at the same Lagrangian,

a procedure which may be more readable in high-energy physics society.

We will single out a benchmark point in the parameter space in section III. The benchmark

2

point represents the wide region of the parameter space, which is consistent with the dark

matter (DM) phenomenology. (In the model we will consider there exists a DM candidate

due to an unbroken flavor symmetry in the hidden sector.) In this region of the parameter

space the scale phase transition is strongly first-order.

In section IV we will calculate the spectrum of the corresponding GW background. There

are three production mechanisms of GWs at a strong first-oder phase transition, in which

the bubble nucleation grows and the GW is produced; collisions of bubble walls Ωcoll [78–

84], magnetohydrodynamic (MHD) turbulence ΩMHD [85–91] and also sound waves Ωsw after

the bubble wall collisions [92–95]. Using the formulas given in these papers and especially

in [71], we will compute these individual contributions to the GW background spectrum

for a set of the benchmark parameters and find that Ωcoll and ΩMHD are several orders of

magnitude smaller than Ωsw. Finally we will compare our result with the sensitivity of

various GW experiments. We will find that the scale phase transition caused by the scalar-

bilinear condensation can be strong enough to produce the GW background that can be

observed by DECIGO [96].

Sect. V is devoted to a summary, and in the appendix we compute the field renormaliza-

tion factor.

II. THE BASIC IDEA AND THE PATH-INTEGRAL APPROACH

We consider a classical scale invariant extension of the SM, which has been studied in

[63, 64]. The basic assumption there is that the origin of the EW scale is a scalar-bilinear

condensation, which forms due a strong non-abelian gauge interaction in a hidden sector and

triggers the EW symmetry breaking through a Higgs portal coupling. The hidden sector is

described by an SU(Nc) gauge theory with the scalar fields Sai (a = 1, . . . , Nc, i = 1, . . . , Nf)

in the fundamental representation of SU(Nc). Accordingly, the total Lagrangian is given by

LH = −1

2trF 2 + ([DµSi]

†DµSi)− λS(S†iSi)(S

†jSj)

− λ′S(S

†iSj)(S

†jSi) + λHS(S

†iSi)H

†H − λH(H†H)2 + L′

SM, (1)

where DµSi = ∂µSi − igHGµSi, Gµ is the matrix-valued SU(Nc) gauge field, and the SM

Higgs doublet field is denoted by H (the parenthesis stands for SU(Nc) invariant products.).

The last term, L′SM, contains the SM gauge and Yukawa interactions. Note that the Higgs

3

mass term, which is the only scale-invariance violating term in the SM, is absent in (1).

We assume that the SU(Nc) gauge theory in the hidden sector is asymptotically free and

gauge symmetry is unbroken for the entire energy scale, such that above a certain energy

the theory is perturbative and there is no mass scale except for renormalization scale, which

is present due to scale anomaly [97]. At a certain low energy scale (say the confinement

scale) the gauge coupling gH becomes so large that the SU(Nc) invariant scalar bilinear

dynamically forms a U(Nf ) invariant condensate [61, 62],

〈(S†iSj)〉 = 〈

Nc∑

a=1

Sa†i Sa

j 〉 ∝ δij , (2)

and the mass term (constituent mass) for Si is dynamically generated. The creation of

the mass term from nothing is possible only by a non-perturbative effect: Scale anomaly

[97] cannot generate a mass term, because the SU(Nc) gauge symmetry is unbroken by

assumption. Of course, scale anomaly does contribute to the mass once it is generated,

and how it logarithmically runs is described, for instance in [98], in a modern language of

renormalization1.

It has been intended in [63, 64] to describe the non-perturbative phenomena of conden-

sation (2) approximately by using an effective theory. That is, the effective theory should

describe the dynamical generation of the mass for Si via the scalar-bilinear condensation

(2). Using the effective theory it should be also possible to approximately describe how

the energy scale transfers from the hidden sector to the SM sector. Inspired by the NJL

theory, which can approximately describe the dynamical chiral symmetry breaking in QCD,

we have assumed that the effective Lagrangian does not contain the SU(Nc) gauge fields,

because they are integrated out, while it contains the “constituent” scalar fields Sai . Fur-

thermore, since the hard breaking of scale invariance by anomaly is only logarithmic, the

non-perturbative breaking may be assumed to be dominant, so that scale anomaly may be

1 There exist proofs within the framework of perturbation theory that conformal anomaly does not generate

mass term of scalar field in the class of massless renormalizable (not super-renormalizable) field theories.

This is rigorously proven in the massless φ4 theory by Loewenstein and Zimmermann [99], and the same

conclusion was made in massless non-abelian gauge theories in [100]. If the regularization (e.g. cut-off

regularization) breaks scale invariance, one needs a counter term for the mass of the scalar field. From

this reason we will employ dimensional regularization.

4

ignored in writing down an effective Lagrangian at the tree level:

Leff = ([∂µSi]†∂µSi)− λS(S

†i Si)(S

†jSj)− λ′

S(S†iSj)(S

†jSi)

+ λHS(S†iSi)H

†H − λH(H†H)2, (3)

where all λ’s are positive. This is the most general form which is consistent with the global

SU(Nc)×U(Nf ) symmetry and the classical scale invariance (we have suppressed the kinetic

term for H in (3) because it does not play any important role for our discussions here). That

is, Leff has the same global symmetry as LH even at the quantum level. We emphasize that

the effective Lagrangian (3) is not a Lagrangian for an effective theory of (1) after the

confinement has taken place and the condensation (2) has appeared: It should describe the

process of the condensation. Note also that the couplings λS, λ′S, and λHS in LH are not the

same as λS, λ′S, and λHS in Leff , because the latter are effective couplings which are dressed

by hidden gluon contributions.

A. Path-integral formalism

In [63, 64] the self-consistent mean-field approximation [74, 75] has been applied to treat

the effective Lagrangian (3). Here we base on the path-integral formalism to obtain the

mean-field Lagrangian LMFA. The method is known as the so-called auxiliary field method or

Hubbard–Stratonovich transformation [76, 77]. The starting path-integral with the effective

Lagrangian (3) is given by

Z =

∫

DS†DS exp

[

i

∫

d4x

([∂µSi]†∂µSi)− λS(S

†iSi)(S

†jSj)− λ′

S(S†iSj)(S

†jSi)

+ λHS(S†i Si)H

†H − λH(H†H)2

]

, (4)

where we focus only on the path-integral for S† and S. We insert the Gaussian integral

1 ∝∫

Df ′Dφ′ exp

[

i

∫

d4xLA

]

with LA = Nf (NfλS + λ′S)f

′2 +λ′S

2φ′aφ′a (5)

to the path-integral (4) and make the change of the integration variables to f and φa0 ac-

cording to

f ′ = f − (S†iSi)/Nf , φ′a = φa

0 − 2(S†i t

aijSj), (6)

5

where ta (a = 1, . . . , N2f − 1) are the SU(Nf ) generators in the fundamental representation.

Then the path-integral can be written as

Z =

∫

DS†DSDfDφ0 exp

[

i

∫

d4x

([∂µSi]†∂µSi) + λHS(S

†iSi)H

†H − λH(H†H)2

+Nf(NfλS + λ′S)f

2 − 2(NfλS + λ′S)f(S

†iSi) +

λ′S

2(φa

0)2 − 2λ′

Sφa0(S

†i t

aijSj)

]

, (7)

where we have used the identity

(S†i t

aijSj)(S

†kt

aklSl) = S†

iSjS†kSl

(

taijtakl

)

= S†iSjS

†kSl ×

1

2

(

δilδjk −1

Nfδijδkl

)

=1

2

(

(S†iSj)(S

†jSi)−

1

Nf

(S†iSi)

2

)

. (8)

Note that the Euler–Lagrange equations for the auxiliary fields f and φa0 become f =

(S†iSi)/Nf and φa

0 = 2(S†i t

aijSj), respectively, and substituting them into (7), we are back to

(4). We thus arrive at the mean-field Lagrangian

LMFA = ([∂µSi]†∂µSi)−M2(S†

i Si)− λH(H†H)2

+Nf (NfλS + λ′S)f

2 +λ′S

2(φa

0)2 − 2λ′

Sφa0(S

†i t

aijSj), (9)

where

M2 = 2(NfλS + λ′S)f − λHSH

†H. (10)

If we expand the composite field f and the Higgs doublet around the vacuum values f0 and

vh, i.e.,

f = f0 + Z1/2σ σ, H =

1√2

χ1 + iχ2

vh + h+ iχ0

, (11)

and redefine φa0 as φa

0 = Z1/2φ φa, the mean-field Lagrangian (9) becomes

L′MFA = ([∂µSi]

†∂µSi)−M20 (S

†iSi) +Nf (NfλS + λ′

S)Zσσ2 +

λ′S

2Zφφ

aφa

− 2(NfλS + λ′S)Z

1/2σ σ(S†

iSi)− 2λ′SZ

1/2φ (S†

i taijφ

aSj)

+λHS

2(S†

i Si)h(2vh + h)− λH

4h2(6v2h + 4vhh+ h2), (12)

6

where Zσ and Zφ are field renormalization constants of the canonical dimension two. In

(12) we have neglected the would-be Goldstone bosons χi in the Higgs field and defined the

constituent scalar mass squared as

M20 = 2(NfλS + λ′

S)f0 −λHS

2v2h. (13)

B. Effective potential and the mean-field vacuum

To determine the mean-field vacuum, we next derive the mean-field effective potential by

integrating out the quantum fluctuations of S. The assumption that the bilinear condensate

(2) is U(Nf ) invariant means for the composite fields that 〈f〉 6= 0 and 〈φa〉 = 0. Therefore,

it is sufficient to consider the path-integral (9) with φa = 0. Since (9) is quadratic in S, the

path-integral for S is Gaussian, and then we obtain2

Z =

∫

Df exp

[

i

∫

d4x

−M2(S†S)− λH(H†H)2

+Nf(NfλS + λ′S)f

2 + iNcNf lnDet[

∂2 +M2]

]

, (14)

where the fluctuation of S has been integrated out around its background field S. The last

term in the right-hand side of (14) is evaluated as

lnDet[

∂2 +M2]

= (V T )M4

2(4π)2

(

1

ǫ− ln

(

M2)

+3

2

)

, (15)

where the dimensional regularization has been used, V T is the space-time volume (V T =∫

d4x), and 1/ǫ = 2/(d − 4) − γE + ln(4π). Then the 1PI effective action at the one-loop

level is given by

Γ[S, f,H ] = V T

[

−M2(S†S)− λH(H†H)2

+Nf (NfλS + λ′S)f

2 +NcNf

2(4π)2M4

(

1

ǫ− ln

(

M2)

+3

2

)]

. (16)

Note that we have not used the large-N approximation to derive the effective action (16).

Finally we obtain the effective potential

VMFA

(

S, f,H)

= −Γ[S, f,H ]

V T

2 The field S is expanded around the homogeneous background, i.e., S → S + χ, and the path-integral of

χ is performed.

7

= M2(S†i Si) + λH(H

†H)2 −Nf (NfλS + λ′S)f

2 +NcNf

32π2M4 ln

M2

Λ2H

, (17)

where the divergence 1/ǫ was removed by renormalization of the coupling constants in the

MS scheme and ΛH = µe3/4 is the scale at which the quantum correction vanishes if M =

ΛH . Note here that the scale ΛH is generated by quantum effect in the classically scale

invariant effective theory (3) and becomes the origin of the electroweak scale as it will be

seen below.

The minima of the effective potential (17) can be obtained from the solution of the gap

equations3

0 =∂

∂Sai

VMFA =∂

∂fVMFA =

∂

∂Hl

VMFA (l = 1, 2). (18)

The first equation (18) yields 〈Sai 〉〈M2〉 = 0, which is satisfied in the following three cases:

(i) 〈Sai 〉 6= 0 and 〈M2〉 = 0; (ii) 〈Sa

i 〉 = 0 and 〈M2〉 = 0; (iii) 〈Sai 〉 = 0 and 〈M2〉 6= 0.

The case (i) corresponds to the end-point solution [104] in which the effective potential has

a flat direction, i.e., Veff = 0 for f = H = 0. The gap equations in the case (i) imply the

relation 〈f〉 = 2λH/NfλHS〈H†H〉, and the effective potential at the minimum vanishes, i.e.,

〈VMFA〉 = 0 for an arbitrary value of S. In the case (ii) 〈VMFA〉 = 0 follows trivially. In the

case (iii), using the other gap equations, we obtain

|〈H〉|2 = v2h2

=NfλHS

GΛ2

H exp

(

32π2λH

NcG− 1

2

)

, 〈f〉 = f0 =2λH

NfλHS

|〈H〉|2, (19)

〈M2〉 = M20 =

G

NfλHS

|〈H〉|2 (20)

at the minimum, where G ≡ 4NfλHλS−Nfλ2HS+4λHλ

′S. The value of the effective potential

at this minimum is given by

〈VMFA〉 = −NcNf

64π2Λ4

H exp

(

64π2λH

NcG− 1

)

< 0. (21)

We therefore conclude that the case (iii) corresponds to the absolute minimum of the effective

potential (17) as far as G > 0 is satisfied. The Higgs mass at this level of approximation is

calculated to be

m2h0 = |〈H〉|2

(

16λ2H(NfλS + λ′

S)

G+

NcNfλ2HS

8π2

)

. (22)

3 A similar potential problem has been studied in [101–104]. But they did not study the classical scale

invariant case in detail, and moreover no coupling to the SM was introduced.

8

In the small λHS limit, this mass can be expanded as

m2h0 = 2NfλHS〈f〉+Nfλ

2HS

(

Nc

8π2+

1

NfλS + λ′S

)

|〈H〉|2 + · · · . (23)

We see that the Higgs mass is generated by the scalar-bilinear condensation f and the second

term in the right-hand side of (23) comes from the back-reaction due to the finite vacuum

expectation value of the Higgs field. In other words, the first term is, due to (19), equal to

4λH |〈H〉|2, which is the tree level expression in the SM model, so that the second term must

be a correction due to the back-reaction. The correction coming from the SM sector to the

Higgs mass (22) will be calculated below.

C. Corrections from the SM sector

We calculate the one-loop contribution from the SM sector to the effective potential

(17) and evaluate the correction to the Higgs mass (22). The one-loop contribution to the

effective potential can be calculated from

VCM(h) =∑

I=W,Z,h

nI

2

∫

d4k

(2π)4ln(

k2 +m2I(h)

)

− nt

2

∫

d4k

(2π)4ln(

k2 +m2t (h)

)

+ c.t., (24)

where nI (I = W,Z, t, h) is the degrees of freedom of the corresponding particle, i.e., nW = 6,

nZ = 3, nt = 12 and nh = 1, and c.t. stands for the counter terms. The contributions from

the would-be Goldstone bosons in the Higgs field have been neglected4. As before we use the

dimensional regularization to respect scale invariance, and the counter terms are so chosen

that the normalization conditions

VCW(h = vh) = 0,dVCW(h)

dh

∣

∣

∣

∣

h=vh

= 0 (25)

with vh = 246 GeV are satisfied. In this way we obtain the Coleman–Weinberg potential [44]

VCW(h) = C0(h4 − v4h) +

1

64π2

[

6m4W ln(m2

W/m2W ) + 3m4

Z ln(m2Z/m

2Z)

+m4h ln(m

2h/m

2h)− 12m4

t ln(m2t/m

2t )]

, (26)

where

C0 ≃ − 1

64π2v4h

(

3m4W + (3/2)m4

Z + (3/4)m4h − 6m4

t

)

, (27)

4 We work in the Landau gauge.

9

m2W = (mW/vh)

2h2, m2Z = (mZ/vh)

2h2, m2t = (mt/vh)

2h2,

m2h = 3λHh

2 +λHS

64π2

7NcNfλHSh2 − 4fNcNf(NfλS + λ′

S)

− 2NcNf

[

−3λHSh2 + 4f(NfλS + λ′

S)]

ln4f(NfλS + λ′

S)− λHSh2

2Λ2H

, (28)

and mI ( masses given at the vacuum vh = 246 GeV) are

mW = 80.4 GeV, mZ = 91.2 GeV, mt = 174 GeV, mh = 125 GeV. (29)

We find that the Coleman–Weinberg potential (26) yields a one-loop correction to the Higgs

mass squared (22)

δm2h =

d2VCW

dh2

∣

∣

∣

∣

h=vh

≃ −16C0v2h, (30)

which gives about −6% correction to (22).

III. BENCHMARK POINTS AND SCALE PHASE TRANSITION

Thanks to the unbroken U(Nf ) flavor symmetry the excitation field φa can be a DM

candidate. How to evaluate its relic abundance ΩDMh2 and its spin-independent elastic

cross section off the nucleon σSI is explained in [63], where h is the dimensionless Hubble

constant today. Since we would like to perform a benchmark point analysis, let us explain

the parameter space which we are interested in. To obtain the DM relic abundance and its

spin-independent elastic cross section off the nucleon, the annihilation process, in which a

pair of φa annihilates into the SM particles through the effective interaction (φa)2h2, has

to be considered. The effective interaction is generated by the loop effects of S as shown

in Fig. 2, where the vertices in the diagrams are S†i t

aijφ

aSj and (S†i Si)h

2 in the mean-field

Lagrangian (12). The s-channel describes the DM annihilation process, while the t-channel

is used for the DM interaction with the nucleon.

The Higgs portal coupling λHS plays a triple role. First, as we see from (19), the smaller

λHS is, the larger ΛH has to become, because |〈H〉| is fixed at vh/√2 = 246/

√2 GeV.

Secondly, since the coupling constant for the effective interaction (φa)2h2 is proportional to

λHS, the DM relic abundance decreases as λHS increases, while σSI increases. That is, to

satisfy the experimental constraints on ΩDMh2 [105] and σSI [? ] at the same time, λHS

10

FIG. 1: The spin-independent elastic cross section σSI of DM off the nucleon as a function of

mDM for Nf = 2, Nc = 6. The black solid line stands for the central value of the LUX upper bound

[106] with one (green) and two (yellow) σ bands, and the black dotted line indicates the sensitivity

of XENON1T [107, 108].

has to lie in an interval. Equivalently, too small or too large ΛH is inconsistent with the

DM constraints. The interval depends strongly on Nf , because ΩDMh2 is proportional to

N2f − 1. This implies that the weakest constraint on λHS is given for Nf = 2. Furthermore,

the color degrees of freedom in the hidden sector is not completely free within our effective

field theory approach. This is due to the inverse propagator Γ(p2) for the DM, which can

have a zero for a positive p2 only if λ′SNc is large enough (the zero of Γ(p2) defines the DM

mass). If we restrict ourselves to λ′S<∼ 2, we find that Nc > 4. On the other hand, the

results (at least for the DM phenomenology) for Nc = 5− 8 are very similar. The predicted

region for Nf = 2 and Nc = 6 is shown in Fig. 1. As we can see from Fig. 1 this result could

be tested by XENON1T [107, 108], whose sensitivity is indicated by the dotted line.

When discussing the scale phase transition at finite temperature and the GW background,

we will consider a benchmark point in our parameter space:

Nf = 2, Nc = 6, λS = 0.145, λ′S = 2.045, λH = 0.15, λHS = 0.032, (31)

which yields

ΛH = 0.0621 TeV, mDM = 0.856 TeV, ΩDMh2 = 0.122,

mh = 0.126 TeV, σSI = 5.12× 10−46 cm2. (32)

11

φα

φβ

h

h

S

+ cross

φα

φβ

h

h

+

φα

φβ

h

h

+ crosses

FIG. 2: The effective interaction φ2h2 generated by the S loop effect.

As we can see from Fig. 1, most of the predicted points in the mDM − σSI plane, except for

those with smaller mDM, are close to the benchmark point. The third role of λHS will be

discussed when considering phase transition at finite temperature below.

It is expected that at high temperature the thermal effects restore the electroweak sym-

metry and scale symmetry. We assume that even at finite temperature the mean-field ap-

proximation is still a good approximation to the original strongly interacting gauge theory

(1). The effective potential consists of four components,

Veff(f, h, T ) = VMFA(f, h) + VCW(h) + VFT(f, h, T ) + Vring(h, T ) , (33)

where VMFA(f, h) and VCW(h) are given in (17) and (26), respectively. Since the absolute

minimum of VMFA is located at 〈S〉 = 0, we suppress the S dependence of VMFA. The main

thermal effects are included in VFT(f, h, T ), which is

VFT(f, h, T ) =∑

I=S,W,Z,h

nIT

∫

d3k

(2π)3ln(

1 + eωI/T)

− ntT

∫

d3k

(2π)3ln(

1− eωt/T)

=T 4

2π2

[

2NcNfJB

(

M2(T ) /T 2)

+ JB

(

m2h(T ) /T

2)

+ 6JB

(

m2W/T 2

)

+ 3JB

(

m2Z/T

2)

− 12JF

(

m2t/T

2)

]

, (34)

12

where ωI =

√

~k2 +m2I(h), and the thermal masses are given by

M2(T ) = M2 +T 2

6

(

(NcNf + 1)λS + (Nf +Nc)λ′S − λHS

)

, (35)

m2h(T ) = m2

h +T 4

12

(

9

4g22 +

3

4g21 + 3y2t + 6λH −NcNfλHS

)

. (36)

Here g2 = 0.65 and g1 = 0.36 are the SU(2)L and U(1)Y gauge coupling constants, re-

spectively, while yt = 1.0 stands for the top-Yukawa coupling constant. Further, M2 and

mI(I = W,Z, t, h) are given in (10) (with H†H = h2/2) and (28), respectively. The thermal

functions and their high temperature expansions are

JB

(

r2)

=

∫ ∞

0

dx x2 ln(

1− e−√x2+r2

)

≃ −π4

45+

π2

12r2 − π

6r3 − r4

32

[

ln(r2/16π2) + 2γE − 3

2

]

for r2 <∼ 2, (37)

JF

(

r2)

=

∫ ∞

0

dx x2 ln(

1 + e−√x2+r2

)

≃ 7π4

360− π2

24r2 − r4

32

[

ln(r2/π2) + 2γE − 3

2

]

for r2 <∼ 2. (38)

Although the high temperature expansions are useful, they are not suitable for large r > 2.

Therefore, the following fitting functions [109] are used5:

JB(F )(r2) ≃ e−r

40∑

n=0

cB(F )n rn. (39)

The contribution from the ring (daisy) diagrams of gauge boson is [111]

Vring(h, T ) = − T

12π

(

2a3/2g +1

2√2

(

ag + cg − [(ag − cg)2 + 4b2g]

1/2)3/2

+1

2√2

(

ag + cg + [(ag − cg)2 + 4b2g]

1/2)3/2 − 1

4[g22h

2]3/2 − 1

8[(g22 + g21)h

2]3/2)

,

(40)

where

ag =1

4g22h

2 +11

6g22T

2, bg = −1

4g2g1h

2, cg =1

4g21h

2 +11

6g21T

2. (41)

Note that the ring contributions from the scalar S and the Higgs field are included in (34).

5 The validness of the approximations for the thermal functions is discussed in [110].

13

Using the thermal effective potential (33), it is possible to consider phase transitions at

finite temperature: There exist two phase transitions in this model, the scale and EW phase

transitions (whose critical temperatures are denoted by TS and TEW, respectively). The

scale phase transition in hidden sector can be strongly first-order for a wide range in the

parameter space [64]. In the case with Nf = 1 (no DM) we see that both phase transitions

can be strongly first-order and occur at the same temperature, TS = TEW. On the other

hand, if DM is consistently included (Nf > 1), the EW phase transition becomes weak,

although the scale phase transition is still strongly first-order, and TEW < TS. The crucial

difference between the two cases comes from the value of the Higgs portal coupling constant

λHS, which controls the strength of the connection between the SM and hidden sector. As

discussed in section III, smaller λHS means larger ΛH , which in turn implies larger TS, while

TEW stays around O(100) GeV. This is the third role of λHS. Therefore, within the minimal

model with a DM candidate we are considering here, the scale phase transition occurs at a

higher temperature than the EW phase transition and is strongly first-order. Since we are

especially interested in the possibility to explain the origin of the EW scale and DM at the

same time, we focus in the following discussions on the model with Nf = 2, in which, as

explained in section III, the weakest constraint on λHS has to be satisfied.

The EW phase transition is shown in Fig. 3 for the set of the benchmark parameters

(31). We see from Fig. 3 that a weak transition appears around TEW ≃ 0.161 TeV. Fig. 4

(left) presents 〈f〉1/2/T as a function of T , while Fig. 4 (right) shows the effective potential

Veff given in (33) with h = 0 at T = TS = 0.323 TeV as a function of f 1/2, showing that the

scale phase transition is strongly first-order.

IV. GRAVITATIONAL WAVE FROM SCALE PHASE TRANSITION

As we have seen in the previous section, the scale phase transition occurs at T <∼ few

hundred GeV. This is larger than TEW, and moreover the phase transition is strongly first-

order6. This strong first-order phase transition produces the EW scale and at the same

time dark matter in the early Universe. Fortunately, this phase transition could be observed

indirectly, because it can generates stochastic GW background, which can be detected in

6 The GW background produced by a strong EW phase transition has been studied in [112–119]. Similar

studies have been made in [120, 121].

14

FIG. 3: The EW phase transition. The transition occurs around T = TEW ≃ 0.161 TeV.

FIG. 4: Left: The scale phase transition. The strong first-order transition occurs at T = TS =

0.323 TeV. Right: The effective potential Veff(f, 0, T )/Λ4H against f1/2 at the critical temperature

T = TS = 0.323 TeV. The potential energy density at the origin is subtracted from Veff .

GW experiments [65–67]. The most important quantity in studying the GW relics is the GW

background spectrum (see for instance [68, 69]) ΩGW(ν) ∼ dρGW

d log ν, where ν is the frequency of

the GW, and ρGW is the energy density of the GW. The energy density ρGW is proportional to

〈hijhij〉, where hij is the transverse, traceless, spatial tensor perturbation of the Robertson–

Walker metric, and the dot on hij denotes the derivative with respect to time. The evolution

of hij is described by the Einstein equation with the energy-momentum tensor which contains

the information about the phase transitions.

The characteristics of a first-oder phase transition in discussing the GW background

spectrum are the duration of the phase transition and the latent heat released. The duration

15

time has to be sufficiently short compared with the expansion of the Universe, and it is clear

that the more latent heat is released, the larger ΩGW can become. In the following two

subsections we will discuss these issues.

A. Latent heat

Given the effective potential Veff in (33), it is straightforward to compute the latent heat

ǫ(T ) at T < TS:

ǫ(T ) = −Veff(fB(T ) , T ) + T∂Veff(fB(T ) , T )

∂T, (42)

where fB(T ) is 〈f〉 at T , and we have set h equal to zero, because 〈h〉 = 0 for T > TEW .

The ratio of the released vacuum density ǫ(Tt) to the radiation energy density is

α =ǫ(Tt)

ρtrad, (43)

which is one of the basic parameters entering into ΩGW, where Tt (defined in (49)) is the

temperature just below TS, and ρtrad = g∗(Tt)π2T 4

t /30 with g∗ = 106.75.

Since the latent heat in the lattice SU(3)c gauge theory has been calculated [123], it may

be worthwhile to compare our result with that of [123]. To this end, we calculate the latent

heat in the mean-field approximation with Nc = 3, Nf = 1 and λHS = λH = 0. In this case

the dynamical scale symmetry breaking (at T = 0) occurs for λS + λ′S ≥ 2.8, and therefore,

we calculate the latent heat ǫ(T ) just below the critical temperature for λS + λ′S = 3, 4 and

5, respectively. We find:

ǫ(T )

T 4=

0.70

0.55

0.43

for λS + λ′S =

3

4

5

. (44)

These results should be compared with ǫ(T ) /T 4 = 0.75± 0.17 of [123]. Though this lattice

result is obtained in the theory without matter field, we see that the values (44) are com-

parable in size to the lattice result. This is a good news, because the scale phase transition

and the deconfinement phase transition appear at the same time, and the latent heat is

proportional to the change of entropy which we do not expect to change a lot if one scalar

field is included or not.

16

B. Duration time

To estimate the duration time of a first-order phase transition, we have to consider the

underlying physical process, the tunneling process from the false vacuum to the true vacuum,

because the duration time is the inverse of the decay rate of the false vacuum. The decay

(the tunneling probability) per unit time per unit volume Γ(t) can be written as

Γ(t) ∼ e−SE(t), (45)

where SE is the Euclidean action in the full theory (1). At finite temperature T the theory is

equivalent to the Euclidean theory, which is periodic in the Euclidean time with the period

of T−1. Above a certain high temperature the typical size of the bubbles generated by the

phase transition may become much larger than the period T−1 [122]. Then we may assume

[122] that SE = S3/T , where S3 is the corresponding three-dimensional action.

There are various complications when computing S3 in the effective theory in the mean-

field approximation, where they are related with each other. First, the canonical dimension

of the mean field f is two. This itself is not a big problem, and we redefine f as

f = γ χ2, (46)

where γ is dimensionless so that the canonical dimension of χ is one. Second, the kinetic

term for f is absent at the tree-level and is generated in the one-loop order. Consequently,

the kinetic term of f in the action involves the field renormalization factor Z. Since the

effective potential is computed at zero external momenta, Z may be computed also at zero

external momenta, which is done in the appendix. As we can see from (A8), Z depends

on f as well as on T . However, we have a problem here, because how to include Z in

the kinetic term is not unique: Z−1(∂if)(∂if) and (∂iZ−1/2f)(∂iZ

−1/2f), for instance, give

different equations of motion for f . The third complication is most serious: In principle

we have to compute SE in the full theory (1). That is, instantons have to be taken into

account. Therefore, we expect that the action S3/T computed in the effective theory in the

mean-field approximation can not be a good approximation to the full quantity, because

the effective theory does not know about instantons and confinement. So, we do not trust

S3/T obtained in the effective theory, although we believe that the effective potential Veff

is a good approximation (in fact the latent heat computed from Veff gives reasonable values

17

compared with the lattice result [123]). That is, we do not trust the kinetic term obtained

in the effective theory (which is anyhow ambiguous), and instead we make an Ansatz for

the kinetic term. The simplest assumption is that the redefinition (46) gives a correct,

canonically normalized kinetic term for χ, where γ should be regarded as a free parameter

constant independent of χ and T .7 That is, we make an Ansatz for S3:

S3(T ) =

∫

d3x

[

1

2(∇iχ)

2 + Veff

(

γχ2, T)

]

, (47)

where the h dependence of the effective potential is suppressed (because 〈h〉 = 0 for T >

TEW), and it is normalized as Veff(0, T ) = 0. At high enough temperatures we may assume

[122] that, not only SE = S3/T is satisfied, but also the classical solution to the field equation

is O(3) symmetric: χ depends only on r = (x21 + x2

2 + x23)

1/2 and satisfies

d2χ

dr2+

2

r

dχ

dr=

dVeff(γχ2, T )

dχ. (48)

To compute the tunnel provability at T < TS from the false vacuum with χ = 0 to the true

vacuum with χ = χB 6= 0, we look for the classical solution, the so-called bounce solution,

that satisfies the boundary conditions, dχ/dr|r=0 = 0 and χ(r = ∞) = 0. The initial value

χ(0) should be chosen slightly smaller than χB, such that χ(r = ∞) = 0 is satisfied. Then

we insert the solution into (47) to compute S3(T )/T .

At the temperature Tt (or at the time tt) the vacuum is overwhelmed by the bubble of

the broken phase. Since the Universe is expanding, it is the time at which the tunneling

provability per Hubble time per Hubble volume becomes nearly one, because after each

tunneling process we have one bubble nucleation. This defines the transition time tt and

also the transition temperature Tt:

Γ(t)

H(t)4

∣

∣

∣

∣

t=tt

≃ 1, (49)

where H(t) is the Hubble parameter at t. This condition is rewritten as [113]

SE(tt) =S3(Tt)

Tt≃ 140–150. (50)

7 As we see from (A6), Z at T = 0, obtained in the one-loop order in the effective theory, is

16π212Nf(NfλS + λ′

S)f ∼ 8 × 102f for the benchmark parameters (31). Then Z−1∂if∂if = ∂iχ∂iχ

if γ = 4π212Nf(NfλS + λ′

S) ∼ 2 × 102. This implies that, although 〈f1/2〉/TS ∼ 1 for the benchmark

point (31), 〈χ〉/TS ∼ 0.02 for χ. So, in terms of χ the phase transition would no longer be a strong phase

transition. This is also a reason why we do not trust the kinetic term for f obtained in the effective

theory.

18

Since the transition time tt (or the transition temperature Tt) is now defined, we can compute

the duration of the phase transition. To this end, we expand the action around tt:

SE(t) = SE(tt)− β∆t+O((∆t)2), (51)

where ∆t = (t− tt) > 0. Then the tunneling per unit time per volume can be approximated

as

Γ(t) ∼ eβ∆t. (52)

Therefore, β−1 is the duration time and can be computed from

β = −dSE

dt

∣

∣

∣

∣

t=tt

=1

Γ

dΓ

dt

∣

∣

∣

∣

t=tt

= HtTtd

dT

(

S3(T )

T

) ∣

∣

∣

∣

T=Tt

= Htβ, (53)

where dT/dt = −HT is used and Ht is H(tt).

This β and α in (43) are the parameters which determine the characteristics of the first-

order phase transition and enter into the GW background spectrum. Their values for the

set of the benchmark parameters (31) with γ = 0.5, 1 and 2 are shown in Table I.

C. GW background spectrum

It is known at present that there are three production mechanisms of GWs at a strong

first-oder phase transition: The bubble nucleation in a strong first-order phase transition

grows, leading to the contributions to ΩGW from collisions of bubble walls Ωcoll [78–84],

magnetohydrodynamic turbulence ΩMHD [85–91] and also sound waves Ωsw after the bubble

wall collisions [92–95]. Then the total GW background spectrum is given by

ΩGW(ν)h2 = [Ωcoll(ν) + ΩMHD(ν) + Ωsw(ν)] h2, (54)

where h2 is the dimensionless Hubble parameter and ν is the frequency of the GW at present.

The individual contributions to the GW background spectrum can be estimated for given α

and β along with the bubble wall velocity vb and the efficiency factor κ. It is the fraction of

the liberated vacuum energy to the bulk kinetic energy of the fluid, which can become the

source of the GW.

19

γ Tt [TeV] S3(Tt)/Tt α β Ωswh2 νsw [Hz]

0.5 0.300 149 0.070 3.7× 103 1.9 × 10−13 0.37

1.0 0.311 145 0.062 7.0× 103 7.4 × 10−14 0.73

2.0 0.316 146 0.059 13× 103 3.4 × 10−14 1.4

TABLE I: Relevant quantities for the GW background spectrum for the set of the benchmark

parameters (31). The quantities α, β and γ are defined in (43), (53) and (46), respectively.

FIG. 5: The GW background spectrum. The doted lines are the four different sensitivities of

eLISA, where the labels (“C1”, · · · ,“C4”) correspond to the configurations listed in Table 1 in [71].

The data sets of their configurations are taken from [124]. The dashed lines are sensitivities of

three different designs (“Pre-DECIGO”, “FP-DECIGO” and “Correlation”) of DECIGO [96]. The

parameter γ is defined in (46).

Using the formulas given in [71], we have computed these individual contributions to the

GW background spectrum for the set of the benchmark parameters (31) and found that

Ωcoll and ΩMHD are several orders of magnitude smaller than Ωsw. Therefore, we consider

here only the contribution to ΩGW from the sound wave [92–95]:

Ωsw(ν) h2 = Ωswh

2

(

ν

νsw

)3(7

4 + 3 (ν/νsw)2

)7/2

, (55)

where

Ωswh2 = 2.65× 10−6vbβ

−1

(

κα

1 + α

)2(100

g∗

)1/3

(56)

20

with the peak frequency νsw given by

νsw = 1.9× 10−5Hz× β

vb

(

T∗100GeV

)

( g∗100

)1/6

. (57)

Since we are interested in order-of-magnitude estimates of the GW background spectrum,

we consider the case that the wall velocity is the sound velocity cs = 0.577. In this case the

efficiency factor κ becomes [92–95]:

κ =α2/5

0.017 + (0.997 + α)2/5. (58)

The result for the set of the benchmark parameters (31) with γ = 0.5, 1 and 2 is shown in

Table I, and the model prediction of the GW background spectrum is compared with the

experimental sensitivity of present and future experiments in Fig. 5. As we can see from

Fig. 5, the GW background produced by the scale phase transition in the early Universe can

be observed by DECIGO, where γ is introduced in (46) and its double meaning is explained

there.

V. SUMMARY

In this paper we have considered a non-perturbative effect, the gauge-invariant scalar-

bilinear condensation, that generates a scale in a strongly interacting gauge sector, where

the scale is transmitted to the SM sector via a Higgs portal coupling λHS. The dynamical

scale genesis appears as a phase transition at finite temperature, and it can produce the

GW background in an early Universe. Since λHS controls the strength of the connection

between the SM and hidden sector, smaller λHS means larger ΛH , which in turn implies larger

critical temperature TS for the scale phase transition. Therefore, TS ≫ TEW ∼ O(100) GeV

for smaller λHS. The coupling λHS, on the other hand, cannot be too small, because it

is constrained from below by the DM relic abundance, implying that TS < few hundreds

GeV. Interestingly, in this interval of λHS, where we obtain consistent values of the DM relic

abundance, the scale phase transition is strongly first-order.

We have calculated the spectrum of the GW background, using the effective field theory,

which has been developed in [63]. Our intension in the present paper has not been to

describe in detail the process of the generation of the GW background. We instead have

applied the formulas given in [71] to compute the GW background spectrum. We have found

21

that the contributions to the GW spectrum from the collisions of bubble walls and also

from the magnetohydrodynamic turbulence are negligibly small compared with the sound

wave contribution. We have found that the peak frequency νsw of the GW background

is O(10−1) ∼ O(1) Hz with the peak relic energy density Ωswh2 = O(10−14) ∼ O(10−13).

Therefore, the scale phase transition caused by the scalar-bilinear condensation is strong

enough that the corresponding GW background can be observed by DECIGO [96] in future.

Acknowledgements

We would like to thank K. Hashino, M. Kakizaki and S. Kanemura for useful discus-

sions and suggestions. J. K. is partially supported by the Grant-in-Aid for Scientific Re-

search (C) from the Japan Society for Promotion of Science (Grant No.16K05315). M. Y. is

partially supported by the Grant-in-Aid for Encouragement of Young Scientists (B) from

the Japan Society for Promotion of Science (Principal Investigator: Koji Tsumura, Grant

No.16K17697).

Appendix A: Field renormalization factor

We derive the kinetic term of the composite field at the one-loop level. The two-point

function is given by

Π(ωn, ~p, T,M) = T∑

l

∫

d3k

(2π)31

(ωn − ωl)2 + (~p− ~k)2 +M2

1

ω2l +

~k2 +M2, (A1)

where M2 is (10) with λHS = 0. Although at finite temperature the kinetic term between

the time-direction momentum mode and spacial-direction one is anisotropic, hence the field

renormalization factors should be defined as8

Z−1⊥ (f) ≡ dΓ

dω2n

∣

∣

∣

∣

p=0

, Z−1‖ (f) ≡ dΓ

d~p2

∣

∣

∣

∣

p=0

, (A2)

we hereafter assume that Z−1⊥ ≈ Z−1

‖ and write them as simply Z−1.

Expanding the integrant of (A1) into the polynomial of ~p around ωn = ~p = 0,

1

(ωn − ωl)2 + (~p− ~k)2 +M2

1

ω2l +

~k2 +M2

8 More precisely, the field renormalization factor should be defined at on-shell external momentum. However,

we calculate it at p = 0 for simplicity.

22

=1

(ω2l +

~k2 +M2)2− ~p2 − 2~p · ~k

(ω2l +

~k2 +M2)3+

4(~k · ~p)2

(ω2l +

~k2 +M2)4+ · · · , (A3)

and replacing ~p · ~k and its squared by 0 and ~k2~p2/3, respectively, we obtain

Z−1(f) = T∑

l

∫

d3k

(2π)3

(

1

(ω2l +

~k2 +M2)3− 4~k2

3(ω2l +

~k2 +M2)4

)

. (A4)

Using the standard calculation method at finite temperature, we can separate this into the

zero temperature mode and the finite one:

Z−1(f) = Z−1T=0(f) + Z−1

T 6=0(f) , (A5)

with

Z−1T=0(f) =

∫

d4k

(2π)3

(

1

(k2 +M2)3− k2

(k2 +M2)4

)

=1

16π2

1

6M2, (A6)

and

Z−1T 6=0(f) =

∫

d3k

(2π)3

(

3

8

1

(k2 +M2)5/2+

5

12

1

(k2 +M2)7/2

)

1

e√k2+M2/T − 1

=1

16π2T 2

(

3g3/5(M/T ) +10

3g5/7(M/T )

)

. (A7)

The thermal function is defined as

gn/m(y) =

∫ ∞

0

dxxn−1

(x2 + y2)m/2

(

e√

x2+y2 − 1)−1

. (A8)

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