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Nuclear Physics B 919 (2017) 297–314
www.elsevier.com/locate/nuclphysb
Inflation from the finite scale gauged
Nambu–Jona-Lasinio model
Tomohiro Inagaki a,b , Sergei D. Odintsov c,d,e,f , Hiroki Sakamoto g,∗
a Information Media Center, Hiroshima University, Higashi-Hiroshima, 739-8521, Japan
b Core of Research for the Energetic Universe, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan
c ICREA, Passeig Luis Companys, 23, 08010 Barcelona, Spain
d Institute of Space Sciences (IEEC-CSIC), Cr.Can Magrans s/n, 08193 Barcelona, Spain
e Laboratory for Theoretical Cosmology, Tomsk State University of Control Systems and Radioelectronics (TUSUR),
634050 Tomsk, Russia
f Tomsk State Pedagogical University, 634050 Tomsk, Russia
g Department of Physics, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan
Received 3 October 2016; received in revised form 21 March 2017; accepted 24 March 2017
Available online 28 March 2017
Editor: Hong-Jian He
Abstract
The possibility to construct an inflationary universe scenario for the finite-scale gauged Nambu–JonaLasinio model is investigated. This model can be described by the Higgs–Yukawa type interaction model
with the corresponding compositeness scale. Therefore, the one-loop Higgs–Yukawa effective potential is
used with the compositeness condition for the study of inflationary dynamics. We evaluate the fluctuations
in the cosmic microwave background for the model with a finite compositeness scale in the slow-roll approximation. We find the remarkable dependence on the gauge group and the number of fermion flavors. It
is also proved that the model has similar behavior with the φ 4n chaotic inflation and the Starobinsky model
at the flat and steep limits, respectively. It is demonstrated that realistic inflation consistent with Planck data
is possible for a range of theory parameters.
© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
* Corresponding author.
E-mail address: h-sakamoto@hiroshima-u.ac.jp (H. Sakamoto).
http://dx.doi.org/10.1016/j.nuclphysb.2017.03.024
0550-3213/© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
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T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
1. Introduction
The occurrence of the early-time inflation provides eventually the easiest explanation of
the astrophysical data, including that the large-scale universe is approximately isotropic,
homogeneous and spatially-flat. The possible origin of the inflationary era maybe related with
existing particle physics models (for review, see [1–4]). The quantum fluctuations in the particle
physics theory grow up through the inflationary expansion. It can be considered as the origin
of the fluctuations in the cosmic microwave background (CMB). Therefore, the cosmological
predictions of particle physics theory can be inspected by the Planck observational data.
A composite scalar field maybe one of the interesting candidates to provide the potential
energy for the inflationary expansion. The Nambu–Jona-Lasinio (NJL) model [5] describes the
composite scalar field which appears in the low-energy phenomena. The gauged NJL model has
been introduced as a low energy-effective model of QCD with QED gauge interaction. The scale
up model can be used as a prototype model of a composite scalar field theory with a gauge
interaction at high energy. Other composite models have been also studied as the origin to induce
the inflationary expansion (see, for example [6–9]).
In Ref. [10] the CMB fluctuations are evaluated in the gauged NJL model. The gauged NJL
model can be represented as the gauged Higgs–Yukawa theory below the compositeness scale
(see, for example [11–18]). In the previous study the CMB fluctuations are calculated in the
gauged Higgs–Yukawa theory with the compositeness scale. The consistency with the Planck
2015 data has been shown at the infinite compositeness scale limit.
In the standard scenario of the chaotic inflation it is often assumed that the field starts from a
super-Planckian domain. Although the infinite compositeness scale limit [10] reduces the model
parameters, the initial field value is too large to adopt the infinite compositeness scale limit.
A more realistic situation corresponds to the model with a finite compositeness scale. Therefore,
it is quite natural to investigate the inflation induced by the gauged NJL model with a finite compositeness scale and calculate the CMB fluctuations in such a case. Therefore we find alternative
features of the gauged NJL model.
This paper is organized as follows. Section 2 is devoted to the review of the gauged NJL
model. Using the auxiliary field method and applying the renormalization group improvement
technique, the model is rewritten as the gauged Higgs–Yukawa theory. In Sec. 3, we employ the
slow-roll approximation and present the explicit forms of the parameters for CMB fluctuations.
In Sec. 4, we find the analytic solutions of the CMB fluctuations at the flat and steep limits. It is
observed that the gauged NJL model behaves as the φ 4n chaotic inflation and Starobinsky model,
respectively at the flat and steep limits. We also discuss why such a feature can not be observed at
the infinite compositeness scale limit considered in Ref. [10]. In Sec. 5 we numerically calculate
the CMB fluctuations and confirm their consistency with the analytic results. The correspondence
with Planck data is explicitly established. The reheating temperature is roughly estimated in
Sec. 6. Finally, some concluding remarks are given.
2. Gauged NJL model with a finite compositeness scale
Following the procedure developed in Refs. [10,19,20], let us construct the effective potential
for the composite scalar field at inflationary era. The gauged NJL model is employed as a prototype model to generate the composite scalar field. Thus, we start from the Lagrangian of the
gauged NJL model, i.e. an SU (Nc ) gauge theory with four-fermion interactions,
T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
/̂ +
LgNJ L = Lgauge + ψ̄i Dψ
2
2
16π 2 g4
,
ψ̄ψ + ψ̄iγ5 τ a ψ
2
8Nf Nc
299
(2.1)
where Lgauge shows the Lagrangian of the pure SU (Nc ) gauge theory, D̂ is the covariant derivative and τ a are the generators of the SU (Nf ) flavor symmetry. The four-fermion coupling is
composed by the dimensionless parameter g4 and compositeness scale . According to the auxiliary field method, the Lagrangian (2.1) is rewritten as
2Nf Nc 2 2
a2
/̂ − σ − iγ5 τ a π a ψ −
Laux = Lgauge + ψ̄ i D
+
π
σ
,
16π 2 g4
(2.2)
where σ and π a represent the composite scalar and pseudo-scalar fields.
From the other side, the corresponding (renormalizable) gauge-Higgs–Yukawa Lagrangian is
given by
1
1
∂μ σ ∂ μ σ + 2 ∂μ π a ∂ μ π a
2
2y
2y
2
1m
λ
−
(σ 2 + π a π a ) − 4 (σ 2 + π a π a )2
2 y2
4y
1 ξ
/̂ − ψ̄(σ + iγ5 τ a π a )ψ,
−
R(σ 2 + π a π a ) + ψ̄i Dψ
2 y2
LgH Y = Lgauge +
(2.3)
where y indicates the Yukawa coupling. Here we rescale the fields in the ordinary gauge-Higgs–
Yukawa model to σ → σ/y and π a → π a /y. There is a solution of the renormalization group
(RG) equations where the Lagrangian (2.2) has an equivalent form with (2.3) at the compositeness scale . The compositeness of the fields, σ and π a , can be represented by the conditions
[21,22],
λ(t )
m2 (t )
2a
1
1
1
1
2
=
0,
=
0,
=
−
, ξ(t ) = , (2.4)
2
4
2
2
g4
(t )
6
y (t )
y (t )
y (t )
16π
where a = 2Nc Nf and (t ) is a function of t ≡ ln(μ/μ0 ) with the renormalization scale, μ,
and the reference scale, μ0 , at μ = . Below we suppose only the composite scalar σ contributes
to the evolution of the Universe and drop the pseudo-scalar field π a .
The running Yukawa coupling y(t), the quartic scalar coupling λ(t) and the curvature coupling ξ(t) in the low energy effective model of the gauged NJL theory are found by solving the
RG equations with the boundary conditions (2.4). Here we neglect running of the SU (Nc ) gauge
coupling g, for simplicity. Thus, the function (t ) coincides with w = 1 − α/(2αc ), where
α ≡ g 2 /(4π) and αc ≡ 2πNc /(3Nc2 − 3). The solutions of RG equations are given by
2 1−w −1
16π 2 α
μ
y (t) =
,
1−
2a αc
2
2 2−2w
2a αc
μ
λ(t)
=
1
−
,
4
2
y (t) 16π α
2
2
(2.5)
(2.6)
1
(2.7)
ξ(t) = .
6
We introduce the compositeness conditions into the one-loop effective potential of the gaugeHiggs–Yukawa model up to linear in the curvature terms and obtain
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T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
1
1
1
V 1 loop (σ ) = m2 σ 2 + λσ 4 + ξ Rσ 2
2
4
2
ay 4 σ 4
aRy 2 σ 2
y2σ 2 3
y2σ 2
−
−
−1 .
−
ln
ln
2
2 · 16π 2
μ2
12 · 16π 2
μ2
(2.8)
The solution of the RG equation for effective potential satisfies
V (g, y, λ, m2 , ξ, σ, μ) = V (ḡ(t), ȳ(t), λ̄(t), m̄2 (t), ξ̄ (t), σ̄ (t), μet ),
(2.9)
where the barred quantities represent the renormalized ones at the scale μet . This scale is fixed
to drop the logarithmic term in RG invariant effective potential,
yσ 1/(2−w)
et =
.
(2.10)
μ
From the one-loop effective potential (2.8) with compositeness conditions (2.4) and (2.10),
one can evaluate the RG improved effective potential [18,19],
B 2 C1 4/(2−w) C2 4 R D1 2/(2−w) R D2 2
−
−
σ +
σ
σ +
σ
σ ,
2
4
4
2 6
2 6
where the functions B, C1 , C2 , D1 and D2 are given by
1
(μ2 /2 )1−w
1
2
B = 2(1 − w)
,
−
g4 () w 1 − (μ2 /2 )1−w
2/(2−w)
1−w
aμ4 4 − 3w
16π 2
,
C1 =
16π 2 1 − w
aμ2 1 − (μ2 /2 )1−w
V=
(2.11)
(2.12)
(2.13)
2
16π 2
(μ2 /2 )1−w
C2 =
,
(1 − w)
a
1 − (μ2 /2 )1−w
1−w
aμ2 2 − w
16π 2
D1 =
16π 2 1 − w
aμ2 1 − (μ2 /2 )1−w
D2 =
(2.14)
1/(2−w)
(μ2 /2 )1−w
.
1 − (μ2 /2 )1−w
,
(2.15)
(2.16)
It should be noted that the functions, C2 and D2 , disappear at the large limit [10]. Note that
RG improved effective potential of renormalizable gauge theory in curved spacetime is found
long ago [23,24]. Such RG improved potential has been used to study the (elementary scalar)
inflation in refs. [25–42].
Therefore, we obtain the effective action for the composite scalar with Einstein–Hilbert term,
√
1
1
S = d 4 x −g − R + g μν ∂μ σ ∂ν σ − V ,
(2.17)
2
2
where the reduced Planck unit Mp2 = 1/(8πG) = 1 is used.
3. CMB fluctuations in the gauged NJL inflation
Let us evaluate the CMB fluctuations starting from the effective action (2.17). Below the
compositeness scale the composite scalar field couples with the curvature R. The useful way to
T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
301
evaluate the CMB fluctuations is to transform the non-minimal (Jordan frame) description to the
minimal (Einstein frame) description.1
This is achieved by the Weyl transformation for the metric tensor gμν
g̃μν =
2
(3.1)
(x)gμν ,
where the Weyl factor 2 (x) is tuned to cancel out the non-minimal curvature coupling term.
After the Weyl transformation and the redefinition of the field one obtains the action with a
canonical kinetic term in the Einstein frame,
1
1
SE = dx 4 −g̃ − R̃ + g̃ μν ∂μ φ∂ν φ − VE ,
(3.2)
2
2
where the subscript E indicates the quantities in the Einstein frame, and the redefined scalar field
φ satisfies
dφ
f
3 ( 2σ )2
,
f
≡
1
+
.
(3.3)
=
2
2
dσ
2
Here and hereafter we use the subscripts σ and φ to describe the derivative with respect to them.
In the Einstein frame the Weyl factor is found to be
2
= 1 + ζ1 μ2−2n σ 2n − ζ2 σ 2 ,
(3.4)
and the effective potential is given by
VE =
U
4
,
U = λ1 μ2 σ 2 + λ2 μ4(1−n) σ 4n − λ3 σ 4 ,
(3.5)
where we set
1
ζ1 =
6n
n
a
2
16π 1 − n
1−n
n
1
1−n
1 − (μ2 /2 ) n
(3.6)
,
1−n
1
(μ2 /2 ) n
ζ2 =
6 1 − (μ2 /2 ) 1−n
n
(3.7)
,
1−n
1
,
nG4r 1 − (μ2 /2 ) 1−n
n
1−2n
n
3 − 2n
a
1
λ2 =
1−n
4n
16π 2 1 − n
2
1 − (μ /2 ) n
λ1 =
1−n
4π 2 1 − n
(μ2 /2 ) n
λ3 =
1−n
a
n
1 − (μ2 /2 ) n
(3.8)
2n
,
(3.9)
2
,
(3.10)
and 1/G4r shows the renormalized coupling, 1/G4 ≡ 1/g4 () − 1/w with (μ2 /2 )−w , and
n = 1/(2 − w).
1 It is known that in slow-roll approximation the calculation of primordial spectral indices gives effectively the same
numerical prediction for these indices calculated in Jordan or in Einstein frames [43–47].
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T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
The stability of the potential is evaluated by observing the positivity of the first derivative,
∂V
> 0. Below we evaluate the CMB fluctuations starting from σ where the potential satisfies
∂σ
the stability condition. For μ/ 1 the stability condition is given by
n−1
16π 2 (1 − n) (σ 2 ) n
a<
n
μ2
(9 − 6n)n2 G4r
(2n − 1)(1 − n)
1/n
.
(3.11)
At the limit, α → 0 (n → 1), it reduces to a simple form
a<
48π 2
G4r .
μ2
(3.12)
It reproduces the one obtained in Ref. [10], Nf Nc < 24π 2 for μ = 1 and G4r = 1.
In the analysis of the CMB fluctuations the parameter n in the exponent of Eqs. (3.4) and (3.5)
has a decisive role. It is given by
n=
Nc2 1
4πNc
4π
−−−−→
.
2
4π + 3αNc
4πNc + 3α(Nc − 1)
(3.13)
At the limit Nc2 1 the parameter n is described as a function of αNc . On the other hand the
RG equations (2.5), (2.6) and (2.7) are calculated in the perturbative regime 0 < αNc < 1 with
the large Nc limit. It means that a non-perturbative approach is necessary for
n
1
∼ 0.8.
1 + 3/(4π)
(3.14)
The fluctuations of cosmological microwave background (CMB) led to the quantum fluctuations of the inflaton. Thus, the CMB fluctuations are considered as eventual evidence of the
inflationary expansion. In the Einstein frame the dynamics of the inflaton φ is described by
∂VE
,
φ̈ + 3H φ̇ = −
∂φ
1 1 2
H2 =
φ̇ + VE ,
3 2
where H is the Hubble parameter and we use the dot for the time derivative.
We assume that the inflaton rolls down slowly from the initial value on its potential,
∂VE
.
φ̇ VE , φ̈
∂φ
(3.15)
(3.16)
(3.17)
and employ the slow-roll approximation. It is more convenient for practical calculations to define
the slow-roll parameters, ε( 1), η( 1) and ξCMB ,
1 1 ∂VE 2
ε≡
,
(3.18)
2 VE ∂φ
η≡
1 ∂ 2 VE
,
VE ∂φ 2
ξCMB ≡
1 ∂VE ∂ 3 VE
.
VE2 ∂φ ∂φ 3
(3.19)
(3.20)
T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
303
Substituting the effective potential (3.5) with the Weyl factor (3.4), we obtain
1 VEφ
2 VE
VEφφ
η=
VE
2
ε=
=
+
ξCMB =
=
2f −
σ
1 2
2 f
=
2f
σ
2f 2
Uσ
−2
U
Uσ σ
Uσ
−
U
U
2
f
2
σ
2
Uσ
−2
U
2
(3.21)
,
2
σ
2
2
−2
2
σσ
+2
2
2
σ
2
2
+
2 U
σ
f
U
−2
2
σ
2
2
,
(3.22)
VEφ VEφφφ
VE2
8
1
2
2
2
σ
2
Uσ
−2
U
2
2
σσ
−
2
2f
fσ σ
fσ2
σ σ
−
+
2
2f
f
f2
f4
2 U
2
4 2
fσ
Uσ 2σ
Uσ
σσ
σσ
σ
σ
−
2
+
6
−
− 2 σ2
−4
+2
2
2
4
2
U
U
U
f
3
2
2
1 Uσ
fσ
f2
σ
−
−
− 2 σ2
2− 4
2
2 U
f
2
4 2
Uσ σ
Uσ 2σ
σσ
σ
+
−
2
+
6
−4
2
4
U
U 2
2
2
2
4
Uσ
Uσ 2σ
f2
Uσ σ
σ
σσ
σ
−
−
2
+
6
−2 2
−4
2
−
2
4
4
U
U
U 2
2
Uσ
Uσ 2σ σ
Uσ 4σ
Uσ σ 2σ
Uσ σ σ
+
−6
+ 18
− 2 σ2
−6
2
2
U
U
U
U
U 4
2
σσσ
+ 18
2
−2
2
σ
2
σσ
− 24
4
6
σ
6
,
(3.23)
where the subscript σ σ and σ σ σ indicates the second and third derivative with respect to σ .
We assume that the volume of the Universe increases eN times during the inflation. Thus, the
e-folding number N is given by
φN
N=
φend
VE
dφ,
∂VE /∂φ
(3.24)
where φN , φend denote the field configuration of the horizon crossing and the end of the inflation,
respectively. Inserting the effective potential (3.5), the e-folding number is written as
σN
N=
σend
f
2U
1
σ /U − 2
2/
σ
2
dσ.
(3.25)
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T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
Planck satellite collaboration has observed the curvature perturbation As , the spectral index ns , the running of the spectral index αs and the tensor-to-scalar ratio r. In the slow-roll
approximation these observables are given by [48]
VE
As =
,
(3.26)
24π 2 ε σ =σN
ns = 1 + 2η − 6ε|σ =σN ,
dns
= −24ε 2 + 16εη − 2ξCMB |σ =σN ,
αs =
d ln k
(3.27)
r = 16ε|σ =σN .
(3.29)
(3.28)
σ =σN
The observational values of these parameters are given as ln(1010 As ) = 3.094 ± 0.034, ns =
0.9645 ± 0.0049, αs = −0.0057 ± 0.0071 (68% CL, Planck TT,TE,EE+lowP) and r0.002 < 0.10
(95% CL, Planck TT,TE,EE+lowP) [49].
4. CMB fluctuations at the flat and steep limits
CMB fluctuations for gauged NJL model are evaluated by substituting the slow-roll parameters at the horizon crossing to (3.26), (3.27), (3.28) and (3.29). In order to analytically evaluate
the CMB fluctuations we assume, only this section, μ/ 1 and ignore the last term in (3.4)
and (3.5). Here, we consider the following two limits: 2 ( 2σ )2 and 2 ( 2σ )2 . The former case is named the flat limit and the latter case is called the steep limit [50]. These limits
characterize the specific behavior of the inflation.
4.1. Flat limit (
2(
2 )2 )
σ
Let us assume that the renormalization scale μ is less than Planck scale, i.e. μ < 1 in our
notations. For 2n < 1 the first term of U in (3.5) dominates the energy density of the inflationary
Universe. The first term induces the φ 2 chaotic inflation. Since the second term has a dominant
contribution for 2n > 1, the φ 4n chaotic inflation takes place. At the massless limit, λ1 = 0 the
first term disappears and the model is identical to the φ 4n chaotic inflation.
At the flat limit 2 ( 2σ )2 , Eqs. (3.21)–(3.25) simplify to
8n2
,
(4.1)
σ2
4n(4n − 1) 16n2 + 4(1 − 2n)n − 4n(1 + 7n)
+
ζ1 μ2−2n σ 2n ,
(4.2)
η=
σ2
σ2
16n2 (4n − 1)(4n − 2)
ξCMB =
σ4
3
2
16n 32n − 4n (17n + 10) − 2n(11n2 − 15n − 4) − n(4n2 − 10n + 1)
+
ζ1 μ2−2n σ 2n ,
σ4
(4.3)
ε=
N=
σN2
,
8n
(4.4)
T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
305
Table 1
The typical values of the CMB fluctuations for λ1 = 0 and Nc Nf ∼ O(1) at the flat limit.
μ
ns
r
αs
N = 50
2n = 1.5
2n = 1
2n = 0.5
O(10−13 )
O(10−7 )
O(10−3 )
0.95
0.96
0.97
0.24
0.16
0.08
−0.001
−0.0008
−0.0006
N = 60
2n = 1.5
2n = 1
2n = 0.5
O(10−13 )
O(10−7 )
O(10−3 )
0.958
0.97
0.975
0.2
0.13
0.07
−0.0007
−0.0006
−0.0004
where all the sub-dominant contributions are ignored. Substituting these parameters into
Eqs. (3.26), (3.27), (3.28) and (3.29), we obtain As , ns , r and αs as a function of the model
parameters and e-folding number, N ,
⎧
λ 1 μ2 N 2
⎪
⎪
⎨
(2n < 1 and λ1 = 0),
3π 2
As =
(4.5)
4−4n (8nN)2n+1
⎪
⎪
⎩ λ2 μ
(2n
>
1
and/or
λ
=
0),
1
192π 2 n2
⎧
⎧
⎧
2
2
8
⎪
⎪
⎪
⎨1 −
⎨− 2
⎨
(2n < 1 and λ1 = 0),
N
N
N
(4.6)
ns =
r=
αs =
⎪
⎪
⎪
⎩1 − 2n + 1
⎩− 2n + 1
⎩ 16n
(2n
>
1
and/or
λ
=
0),
1
N
N
N2
As is known, the e-folding number N should be in the interval, 50 ∼ 60, to solve the flatness
problem of the Universe.
To find the typical CMB fluctuations at the flat limit we tune the scale μ to reproduce the observed value of As with Nc Nf ∼ O(1) and evaluate ns , r and αs at λ1 = 0 for 2n = 1.5, 2n = 1,
and 2n = 0.5. The results are presented in Table 1. The result is consistent with the observed
values of Planck 2015 for 2n = 0.5. Such a small n is beyond the scope of the perturbative expansion with respect to αNc . Thus, we conclude that a consistent value for the tensor-to-scalar
ratio, r(< 0.10), can not be obtained in the perturbative regime at the flat limit.
4.2. Steep limit (
2(
2 )2 )
σ
Let us now consider the case 1 (
2 2
dφ
3
σ
.
∼
2
dσ
2
2 )2 /
σ
2 . In this case (3.3) reduces to
(4.7)
Solving this equation with 2 |φ=0 = 1, one obtains the explicit expression of the Weyl factor,
2
2
= exp ±
φ .
(4.8)
3
Here we confine ourselves for the massless limit of the composite scalar field λ1 = 0. Thus,
the potential in the Jordan frame is rewritten as
U = λ2 μ4(1−n) σ 4n = λ̂(
2
− 1)2 ,
(4.9)
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T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
Table 2
The CMB fluctuations for λ1 = 0 at the steep limit.
N
ns
r
αs
50
60
0.9582
0.9654
0.0048
0.0033
−0.00080
−0.00056
with
λ̂ ≡
λ2 144π 2
=
(3 − 2n)(1 − n).
a
ζ12
Plugging (4.8) to (4.9), the potential in the Einstein frame is represented as
U
VE = 4 = λ̂(1 − exp(− 2/3φ))2 .
(4.10)
(4.11)
It is equivalent to the potential in the Starobinsky inflation [51]. As is known Starobinsky inflation
is consistent with the Planck observational data.
Thus, the slow-roll parameters are given by
4
1
√
,
3 (e 2/3φ − 1)2
(4.12)
4 (e 2/3φ − 2)
√
η=−
,
3 (e 2/3φ − 1)2
(4.13)
16 (e 2/3φ − 4)
√
ξCMB =
,
9 (e 2/3φ − 1)3
(4.14)
ε=
√
√
and the e-folding number is
3
√
3 √2/3φN
2/3φend
N=
−e
e
(φN − φend ).
−
4
8
At the steep limit the spectral index and tensor-to-scalar ratio are approximated as
(4.15)
2
12
2
9
, r ∼ 2 , αs ∼ − 2 .
(4.16)
−
2
N
2N
N
N
These equations reproduce the results of the Starobinsky model. We present the values, ns , r and
αs for N = 50 and 60 in Table 2.
In the present case the curvature perturbation is given by
λ̂N 2
3 4
As ∼
.
(4.17)
1
−
4N
18π 2
ns ∼ 1 −
To generate the observed curvature perturbation As , the effective coupling λ̂ should be of the
order O(10−10 ).
It should be noticed that the solution (4.16) coincides with the universal attractor discussed in
Ref. [52]. At the steep limit the solution converges on the universal attractor with no dependence
on the gauge coupling, α. In Ref. [53] the solution (4.16) is also found for large curvature coupling cases, ξ 1 in the NJL model. If we take the limit α → +0 in the gauged NJL model, the
tensor-to-scalar ratio converges to the twice as large as the Starobinsky model. We conjecture the
existence of new attractor in our model.
T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
307
Fig. 1. Curvature perturbation, As , as a function of t = ln(μ/) for α = 0, G4r = 1010 , a = 4 and n = 1. The dashed
line shows the observed value by Planck 2015.
4.3. Small gauge coupling limit
The large compositeness scale limit of the CMB fluctuations obtained in this section do not
reproduce the ones in Ref. [10]. To see the difference we evaluate the effective potential at the
α → 0 limit. Taking the small coupling limit α → 0 of the effective potential (3.5), it reduces to
μ2
π2
−at μ2
σ 2 + 2 3 + 2 ln
− 4t σ 4 + O(α),
2t G4r
8π 2 σ 2
at
1
1
−at μ2
2
=1+
1−
1 + ln
σ 2 + O(α),
6
2t
8π 2 σ 2
U =−
(4.18)
(4.19)
where we write t = ln(μ/) < 0. In Fig. 1 we plot typical behavior of the curvature perturbation, As , from the effective potential (4.18) with (4.19). We should fine tune the ration between
7
the renormalization scale and the compositeness scale, μ/ ∼ O(e−10 ), to acquire a observed
value of the curvature perturbation, As . To obtain the observed value of the curvature perturbation
without fine tuning we employ a larger gauge coupling, α, here.
At the large compositeness scale limit, → ∞ the α-independent terms in Eq. (4.18) disappear. The finite corrections appear from the next to leading terms,
α
μ2 2 4π 2 4
σ +
σ + O(α 2 )
2αc G4r
a
1
α
aαc μ2
2
=1+
1+
1 − ln
6
2αc
8π 2 ασ 2
U=
(4.20)
σ 2 + O(α 2 ).
(4.21)
Thus the α-dependence of the effective potential has an decisive role for the CMB fluctuations.
From this effective potential it is found that the observed value of the curvature perturbation, As ,
is generated for α ∼ O(10−12 ) and μ ∼ O(1). In Ref. [10] the CMB fluctuations are evaluated
only for the small gauge coupling α ∼ O(10−12 ) and the larger gauge coupling case has not been
discussed to avoid tuning of the renormalization scale, μ.
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T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
Fig. 2. Behavior of the tensor-to-scalar ratio, r, and the running of the spectral index, αs , as a function of the spectral
index, ns , for N = 50, 60 at α = 0.5, = 20 and Nf = 1. The dashed lines is written to indicate the points with the
equal Nc .
5. Numerical results
In this section we numerically calculate the CMB fluctuations, Eqs. (3.21)–(3.25). To confirm
the arguments in the previous section we set the parameters (α, ) to (0.5, 20) as a typical case.
In Fig. 2 we plot the tensor-to-scalar ratio, r, and the running of the spectral index, αs , as a
function of the spectral index, ns , for Nf = 1 with Nc varies. The behavior in Fig. 2 is consistent
with the analysis in the previous section. The second term of U in (3.5) has a dominant contribution between Nc = 2 and Nc = 9. In this interval the parameter 2n is grater than one, 2n > 1, for
α = 0.5. The tensor-to-scalar ratio, r, and the running of the spectral index, αs , monotonically
decreases and increases as a function of ns , respectively. These results reproduce the one in the
φ 4n chaotic inflation. For Nc 9 the first term of U in (3.5) gives a dominant contribution. The
lines curve and the results approach to the ones for Nc ∼ 9 (2n = 1) at the large Nc limit. We
indicate the point with Nc ∼ 9 by the star.
In these calculations we fix the renormalization scale, μ, to generate the observed curvature
perturbation, As . We show the scale in Table 3 at α = 0.5, = 20 and G4r = 1010 for Nc =
2, 3, 5, 8 and 25. Similar behavior is obtained with the one at the flat limit. The parameter, n, is
defined as a function of Nc and α in (3.13). The α-dependence of the renormalization scale is
also observed in the Table 3. The parameter, n, is monotonically decreasing as a function of α.
Thus, the scale μ decreases as the gauge coupling, α, increases.
Next we evaluate the Nf dependence on the tensor-to-scalar ratio, r, and the running of the
spectral index, αs . In Fig. 3 we draw r and αs as a function of ns for Nc = 3, 5 and 10 with
Nf varies. The behavior depends on the gauge group SU (Nc ). A larger spectral index, ns , is
T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
309
Table 3
The renormalization scale μ corresponding to the observable of curvature perturbation for Nc = 2, 3, 5, 8, 25 and the e-folding number N = 50, 60 at α = 0.5,
Nf = 1 and G4r = 1010 . The mass scale is normalized by Mp = 1.
2n = 1.70 (Nc = 2)
2n = 1.52 (Nc = 3)
2n = 1.38 (Nc = 5)
2n = 1.03 (Nc = 8)
2n = 0.50 (Nc = 25)
N = 50
N = 60
1.75 × 10−21
9.08 × 10−22
2.11 × 10−13
2.67 × 10−10
2.58 × 10−6
6.15 × 10−5
3.37 × 10−13
1.65 × 10−8
3.12 × 10−6
6.74 × 10−5
Fig. 3. Behavior of the tensor-to-scalar ratio, r, and the running of the spectral index, αs , as a function of the spectral
index, ns , for N = 50 at α = 0.5, = 20. The dashed lines is written to indicate the flat limit. The star indicates the
steep limit.
obtained as Nc increases. For Nf = 1 the results can be well described by Eq. (4.6) at the flat
limit. It is observed that the results converge to the attractor point (4.16) at the large Nf limit.
The results are consistent with the observation by Planck 2015 for a large number of fermion
flavors.
In Table 4 we show the CMB fluctuations for Nc = 2, N = 50, α = 0.5 and G4r = 1010 as
and Nf vary. All the cases are indistinguishable in the accuracy of calculations under discussion.
The CMB fluctuations have no large dependence on the compositeness scale, , and the number
of the fermion species, Nf . In the original NJL model the composite scale corresponds to the
QCD scale. It is at most 10–100 times larger than the renormalization scale, i.e. meson scale.
Here we regard the model as a scale up of QCD like model. However, the different dynamics may
induce a larger compositeness scale. Thus we consider the large compositeness scale. Though the
scale, ∼ 2000, is too large, it is also useful to verify that the results do not converge to the ones
obtained at the large compositeness scale limit in Ref. [10].
The inflationary parameters values approach to the universal attractor for an extremely
large Nf . Although the case, Nf = 4.5 × 1011 , is unnatural and not consistent with the constant SU (Nc ) gauge coupling assumption, we include the case to observe the universal attractor
at the steep limit.
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T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
Table 4
CMB fluctuations for Nc = 2, N = 50, α = 0.5 and G4r = 1010 .
Nf
ns
r
αs
Nf = 1
Nf = 1
Nf = 1
Nf = 11
Nf = 21
Nf = 31
= 20
= 200
= 2000
= 20
= 20
= 20
0.947
0.947
0.947
0.947
0.947
0.947
0.27
0.27
0.27
0.27
0.27
0.27
−0.0010
−0.0010
−0.0010
−0.0010
−0.0010
−0.0010
Nf = 4.5 × 1011
= 20
0.964
0.0048
−0.00076
Table 5
Reheating temperature for Nc = 2, 3, 5, 8, 25 and the e-folding number N =
50, 60 at α = 0.5 and G4r = 1010 .
2n = 1.70 (Nc = 2)
2n = 1.52 (Nc = 3)
2n = 1.38 (Nc = 5)
2n = 1.03 (Nc = 8)
2n = 0.50 (Nc = 25)
N = 50
N = 60
4.9yh × 1013 GeV
5.9yh × 1013 GeV
2.1yh × 1014 GeV
9.4yh × 1013 GeV
1.8yh × 1012 GeV
4.4yh × 1013 GeV
5.3yh × 1013 GeV
6.0yh × 1013 GeV
8.6yh × 1013 GeV
1.8yh × 1012 GeV
6. Reheating temperature
After the inflationary expansion cooled down the Universe, the potential energy of the inflaton
is released through the decay process of the inflaton into light particles. The reheating tempera¯
between
ture depends on the decay modes. Here we assume a Yukawa-type interaction yhσ
the composite scalar σ and a light fermion field, with the coupling constant yh . In a standard
scenario of the chaotic inflation the reheating temperature is roughly estimated by [54]
Tr ∼ 0.2
yh2
MMp ,
8π
(6.1)
where M represents the effective σ mass at the end of the inflation. The composite field σ oscillates around the origin of the effective potential after the inflation era. The typical scale of the
effective mass is given by
∂ 2 VE
M =
,
∂σ 2 σ =σ∗
2
(6.2)
where σ∗ is the amplitude of oscillations.
In Table 5 we set σ∗ = 1(< σend = O(1)) and evaluate the reheating temperature. Since we
evaluate the decay process of the inflaton at tree level with respect to the coupling yh , the results
is valid for yh < 1. In the perturbative regime the upper limit of the reheating temperature is
below the order 1015 GeV for α = 0.5 and G4r = 1010 .
T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
311
7. Conclusion
In this paper we investigated the gauged NJL model with a finite compositeness scale as
an alternative scenario of Higgs inflation. We assume that the potential energy of a composite
scalar field causes the exponential expansion of the Universe at inflation era. Hence the CMB
fluctuations are produced from the quantum fluctuation of the composite scalar.
There is a solution of the RG equations where the gauged NJL model coincides with the
gauge-Higgs–Yukawa theory at the compositeness scale. The effective potential is calculated by
solving the renormalization group equations below the compositeness scale. In our model the
potential depends on the gauge coupling, the four-fermion coupling, the number of the gauge
group, the number of the fermion species, the renormalization scale and the compositeness scale.
We evaluated the CMB fluctuations starting from the obtained potential.
Analytic results are obtained at the specific limits. Taking the flat limit, the gauged NJL model
behaves as the φ 4n chaotic inflation. The exponent n ∈ (0, 1) is given by the gauge coupling, α,
and the rank of the gauge group, SU (Nc ). It is found that the tensor-to-scalar ratio generated in
the model with a large n > 1/2 is larger than the upper limit by Planck 2015. At the steep limit
the effective potential of the model coincides with the Starobinsky inflation which is equivalent
to Higgs inflation by Bezrukov and Shaposhnikov. Hence, the solution reaches the universal
attractor with no dependence on the gauge coupling. It is quite remarkable that finite scale gauged
NJL model inflation interpolates between above two well-known inflationary universe models.
We numerically evaluated the CMB fluctuations as the model parameters vary and found large
dependence on the gauge group SU (Nc ) and the number of the fermion flavors Nf . A smaller
tensor-to-scalar ratio is generated at larger Nf . Therefore we found a parameter range of the
gauged NJL model to generate CMB fluctuations consistent with the observation by Planck 2015.
In the possible parameter range the product αNc is larger than unity. On the other hand we employ
the one-loop effective potential which is perturbatively calculated with the expansion parameter,
αNc . Fermion loop corrections in the gauge boson vacuum polarization proportional to αNc . It
contributes the screening of the gauge interaction. Since we drop the running effect of the gauge
coupling here, we expect that the fermion loop corrections have no large contribution to the CMB
fluctuations even for αNc > 1. It is one of our remaining problems to include the running of the
gauge coupling with the fermion loop corrections. It may modify the gauge coupling dependence
of the results. However, we expect that the results show some characteristic features of the gauged
NJL inflation. To study the further detail of NJL inflation, we need to develop a procedure beyond
the perturbation for αNc . This will be discussed elsewhere.
The important remark is in order. Even we consider the compositeness scale of the model,
though finite, to be much larger than the Planck scale, we cannot avoid to comment on the quantum gravity effects beyond the Planck scale. However, this depends very much from the quantum
gravity theory under discussion. For instance, if we consider stringy quantum gravity, one should
extend the number of gravitational partners. From other side, it is pointed out that the scalar type
quantum fluctuations dominate much more than the tensor type ones in a two-loop conformal
gravity [55]. Although it is the result in a specific renormalizable model of the quantum gravity,
we expect that the purely quantum gravity effects do not enhance the tensor-to-scalar ratio. In
higher-derivative multiplicative quantum gravity the effect will be in the inducing of R 2 term in
the action. However, its quantum-induced coefficient will be very small so it plays no essential
role in our considerations. In the gauged NJL model under consideration the quantum gravity
effects also may slightly modify the RG flow and then the corresponding effective potential.
However, we consider the particle physics model with a large number of fermion species. At the
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T. Inagaki et al. / Nuclear Physics B 919 (2017) 297–314
large Nf Nc limit the perturbative quantum gravity effects can be neglected compared with the
fermion loop corrections at the equal order of the loop expansion. The corresponding effects are
higher order corrections and perturbatively suppressed in a sense of the loop expansion. This is
similar to conformal anomaly which is proportional to number of fields (i.e. the fermion fields
number, Nf Nc , proportional sector plus the gauge fields number proportional sector). It introduces R 2 terms in the effective action [56] and contributes to approach the fluctuation to the
steep limit for large Nf Nc . Note that this is customary in the inflationary cosmology induced
by matter quantum effects which are always considered to be dominant compare with quantum
gravity effects even if the inflation scale is much higher than Planck scale.
Acknowledgements
The work by TI is supported in part by JSPS KAKENHI Grant Number 26400250 and that by
SDO is supported in part by MINECO (Spain), project FIS2013-44881 and FIS2016-76363-P.
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