Criticality and Inflation of the Gauged B-L Model

0.4 MB·English

by  Kiyoharu Kawana

KUNS-2540
5 Criticality and Inflation of the Gauged B-L Model
1
0
2

l
u Kiyoharu Kawana
∗
J

5
1 Department of Physics, Kyoto University, Kyoto 606-8502, Japan

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h
p
- July 16, 2015
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e
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3 Abstract
v
2 We consider the multiple point principle (MPP) and the inflation of the gauged
8
B-L extension of the Standard Model (SM) with a classical conformality. We examine
4
4 whether the scalar couplings and their beta functions can become simultaneously zero
0 at Λ := 1017 GeV by using the two-loop renormalization group equations (RGEs).
. MPP
1 We find that we can actually realize such a situation and that the parameters of the
0
5 model are uniquely determined by the MPP. However, as discussed in [28], if we want
1 to realize the electroweak symmetry breaking by the radiative B-L symmetry breaking,
:
v the self coupling λ of a newly introduced SM singlet complex scalar Ψ must have
Ψ
i
X a non-zero value at Λ , which means the breaking of the MPP. We find that the
MPP
r (100)GeV electroweak symmetry breaking can be achieved even if this breaking is
a O
very small; λ (Λ ) 10−10. Within this situation, the mass of the B-L gauge boson
Ψ MPP
≤
is predicted to be
λ(v )
M = 2√2 h v 696GeV,
B−L h
× 0.10 × ≃
r
where λ is the Higgs self coupling and v is the Higgs expectation value. This is a
h
remarkable prediction of the (slightly broken) MPP. Furthermore, such a small λ
Ψ
opens a new possibility: Ψ plays a roll of the inflaton [30]. Another purpose of this
paper is to investigate the λ Ψ4 inflation scenario with the non-minimal gravitational
Ψ
coupling ξΨ2 based on the two-loop RGEs.
R
∗E-mail: [email protected]
1
1 Introduction
The discovery of the Higgs like particle and its mass [1, 2] is very meaningful for the
Standard Model (SM). The experimental value of the Higgs mass suggests that the
Higgs potential can be stable up to the Planck scale M and also that both of the
pl
Higgs self coupling λ and its beta function β become very small around M . This
λ pl
fact attracts much attention, and there are many works which try to find its physical
meaning [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].
Well before the discovery of the Higgs, it was argued that the Higgs mass can
be predicted to be around 130GeV by the requirement that the minimum of the Higgs
potentialbecomeszeroatM [3,4]. Sucharequirement (notalwaysatM )isgenerally
pl pl
called the multiple point principle (MPP). One of the good points of the MPP is its
predictability: The low energy effective couplings are fixed so that the minimum of the
potential vanishes. See [24, 25, 26] for example.
By taking the fact that the MPP can be realized in the SM into consideration, a
natural question is whether such a criticality can be also realized in the models beyond
the SM. One of the interesting extensions is the gauged B-L model with a classical
conformality [27, 28, 29, 30]. Here, “classical conformality” means there is no mass
term at the classical level without gravity. This model can be obtained by gauging
the global U(1) symmetry of the SM with the three right-handed neutrinos and a
B-L
SM singlet complex scalar Ψ. As discussed in the following, if we neglect the Yukawa
couplings between the Higgs and neutrinos, there are six unknown parameters in this
model. In particular, two of them are new scalar couplings: κ and λ . Therefore, in
Ψ
principle, these six parameters can be uniquely fixed by the MPP conditions:
λ(Λ ) = λ (Λ ) = κ(Λ ) = β (Λ ) = β (Λ ) = β (Λ ) = 0, (1)
MPP Ψ MPP MPP λ MPP λΨ MPP κ MPP
where Λ is the scale at which we impose the MPP. The analyses in this paper are
MPP
based on the following assumptions:
1. We consider the MPP at Λ = 1017 GeV.
MPP
2. Aswell astheanalyses in[27, 28], we donotinclude mass termsintheLagrangian.
As a result, all the low energy scales are radiatively generated.
3. The Higgs mass is fixed at
M = 125.7GeV, (2)
h
and we regard the top mass M as one of the free parameters.
t
4. We assume that the small neutrino masses are produced by the seesaw mechanism
via the radiative breaking of the B-L symmetry. As a result, we can neglect the
2
Yukawacouplingsy betweentheHiggsandneutrinosbecausethetypicalbreaking
ν
scale is very small ( 1013GeV).
≪
In subsection 2.2, we will see that Eq.(1) can be actually realized at Λ = 1017GeV.
MPP
One of the good features of this model is that the electroweak symmetry breaking
can be triggered by the U(1) symmetry breaking via the Coleman-Weinberg (CW)
B-L
mechanism. In [28], it was argued that we can naturally obtain v = (100)GeV by
h
O
imposing λ(M ) = 0 and κ(M ) = 0. Here, the important point is that λ (Λ ) = 0
pl pl Ψ MPP
6
is needed to realize such B-L breaking1. Therefore, if we try to combine this fact and
the MPP, a natural question arises:
Is the (100)GeV electroweak symmetry breaking possible even if λΨ(ΛMPP) is
• O
small ?
In subsection 2.3, we will see that this is actually possible even if λ (Λ ) 10−10.
Ψ MPP
≤
The reason for this is very simple: By tuning the parameters of the model, we can
obtain the favorable scale at which U(1) breaks so that v becomes (100)GeV.
B-L h
O
Therefore, the B-L model is a phenomenologically very interesting model in that it can
explain the natural-scale electroweak symmetry breaking while satisfying the (slightly
broken) MPP. Furthermore, within this situation, we find that the mass of the B-L
gauge boson is predicted to be
λ(v )
M = 2g (v )v = 2√2 h v 696GeV, (3)
B−L B−L B−L B−L h
× 0.10 × ≃
r
wherev istheexpectationvalueofΨandwehaveusedthetypicalvalueλ(v ) 0.1.
B−L h
≃
This is a remarkable prediction of the (slightly broken) MPP, and it is surprising that
the predicted value of M depends only on the SM parameters2.
B−L
On the other hand, there are many observational results from the cosmological side.
One of the reliable possibilities to explain them is the cosmic inflation. As is well
known, Higgs inflation is possible in the SM where the criticality of the Higgs potential
plays an important role to realize the inflation naturally [17]. Of course, such a Higgs
inflation is possible in the B-L model, but, we can also consider the inflation scenario
where Ψ plays a roll ofthe inflaton[30]. In this paper, we study the λ Ψ4 inflationwith
Ψ
the non-minimal gravitational coupling ξΨ2 . Our analysis is based on the following
R
condition:
We consider the inflation under the situation where the minimum of the Higgs
•
potential vanishes at ΛMPP = 1017GeV and the electroweak symmetry breaking
occurs at (100)GeV.
O
1
Realizing the B-L symmetry breaking when λΨ(ΛMPP)=0 is difficult. See Section2.
2Unfortunately, this value of MB−L is already excluded by the experiment of ATLAS [31]. See subsection 2.3.
3
SU(3) SU(2) U(1) U(1)
c L Y B−L
Qi 3 2 +1/6 +1/3
L
ui 3 1 +2/3 +1/3
R
di 3 1 1/3 +1/3
R −
ℓi 1 2 1/2 1
L − −
νi 1 1 0 1
R −
ei 1 1 1 1
R − −
H 1 2 1/2 0
−
Ψ 1 1 0 +2
Table 1: The particle contents of the B-L model and their charges except for the gauge bosons.
Here, i represents the generation.
In the following discussion, we will see that this condition strongly constrains the pa-
rameters, and as a result, we can obtain the unique cosmological predictions3 which
are consistent with the recent observed values by Planck [34] and BICEP2 [35].
This paper is organized as follows. In Section 2, we study the MPP and the B-L
symmetry breaking from the point of view of the slightly broken MPP. In Section 3,
we investigate the inflation scenario where the SM singlet complex scalar Ψ plays a roll
of the inflaton. In Section 4, we give summary.
2 MPP of the B-L Model and Symmetry Breaking
The flow of this section is as follows. In subsection 2.1, we shortly review the gauged
B-L model. In subsection 2.2, we consider the MPP of this model. In subsection 2.3,
we study whether the (100)GeV electroweak symmetry breaking can be realized even
O
if λ (Λ ) is very small.
Ψ MPP
2.1 Short Review of the B-L Model
In this subsection, we briefly review the B-L extension of the SM. Here, our discussion
is mainly based on [32]. As mentioned in the introduction, this model can be obtained
by gauging the global U(1) symmetry. The kinetic terms of the two U(1) gauge
B-L
fields are given as follows:
1 1 ω
= FµνF Fµν F Fµν F , (4)
Lkin −4 µν − 4 B-L B-Lµν − 4 B-L µν
3
Here,weuse“unique”inthesensethatourpredictionsdonotsodependontheparametersofthemodelexcept
for λΨ, ξ and the initial value of Ψ.
4
where ω( R) represents the kinetic mixing. The U(1) part of the covariant derivative
∈
of a matter field φ is given by
k
2 2
= ∂ +i Yig Aj, (5)
Dµ µ k ij µ
i=1 j=1
XX
where A1 and A2 are the gauge fields of U(1) and U(1) respectively, Yi are the
µ µ Y B-L k
U(1) charges and g represent the U(1) gauge couplings. We can remove the mixing
ij
term by changing A1 and A2 to the new fields AY and AB−L:
µ µ µ µ
1 1 1 1
A1 = AY + AB−L , A2 = AY AB−L.
µ 2(1+ω) µ 2(1 2ω) µ µ 2(1+ω) µ − 2(1 2ω) µ
− −
(6)
p p p p
We simply express Eq.(6) as Ai = Ri Aα. By this transformation, the new gauge
µ α α µ
couplings are
P
g′ := g Rj. (7)
iα ij α
j
X
We denote g′ as g , g , g and g without a prime in the following discussion.
iα YY YE EY EE
Only three of them are meaningful because we can further rotate the gauge fields
without producing the mixing term:
AY cosθ sinθ A˜Y
= − .
AB−L sinθ cosθ A˜B−L
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
Thus, we can choose the angle θ so that one of g vanishes. For convenience, we take
αβ
the following bases:
g AB−L +g AY g AB−L +g AY
B := EE µ EY µ , E := − EY µ EE µ . (8)
µ µ
g2 +g2 g2 +g2
EE EY EE EY
p p
In this bases, the second term of Eq.(5) becomes
g YYB +(g YB−L +g YY)E , (9)
Y k µ B−L k mix k µ
where
g g g g g g +g g
g := EE YY − EY YE , g := g2 +g2 , g := YE EE EY YY . (10)
Y g2 +g2 B−L EE EY mix g2 +g2
EE EY q EE EY
p p
As a result, B plays a roll of the ordinary U(1) gauge field, and E is a new gauge
µ Y µ
field which can have a mass if the B-L symmetry is broken. We use Eq.(9) for the
calculations of the RGEs in Appendix A.
5
The particle contents (except for the gauge bosons) and their charges are presented
in Table1. In addition to the SM particles, there are three right-handed neutrinos and
a SM singlet complex scalar whose U(1) charge is +2. The relevant terms of the
B-L
renormalizable Lagrangian are
λ H†H 2 λ Ψ†Ψ 2 κ H†H Ψ†Ψ
Ψ
L ⊃ − − −
1
(cid:0) yij(cid:1)ν¯i H†ℓj(cid:0) (cid:1) Yijν(cid:0)¯ci νjΨ(cid:1)(cid:0)+h.c(cid:1). (11)
− ν R L − 2 R R R
ij ij
X X
In the following discussion, we use the bases such that yij and Yij are real and diag-
ν R
onalized, and assume that they are equal respectively for the three generations. As
a result, by including the top mass M , there are seven unknown parameters in this
t
model:
M , g , g , λ , κ , y , Y . (12)
t B−L mix Ψ ν R
If we assume that the small neutrino masses (. 1eV) are generated by the ordinary
seesaw mechanism triggered by the U(1) symmetry breaking at a low energy scale
B-L
( 1013 GeV), y should be very small, and its effects to the RGEs can be negligible.
ν
≪
In this paper, we assume such a situation.
2.2 Multiple Point Principle
To understand how these couplings behave at a high energy scale, we need to know the
RGEs. The two-loop RGEs of this model are presented in Appendix A. Furthermore,
the one-loop effective potentials in the Landau gauge are as follows4:
λ(µ) λ (µ)
VH(µ,φ) = φ4 +VH (µ,φ) , VΨ(µ,Ψ) = Ψ Ψ4 +VΨ (µ,Ψ), (13)
eff 4 1loop eff 4 1loop
M (φ)4 M (φ)2 3
VH (µ,φ) := e4Γ(µ) 12 t log t +2Γ(µ)
1loop − · 64π2 µ2 − 2
(
(cid:20) (cid:18) (cid:19) (cid:21)
M (φ)4 M (φ)2 5 M (φ)4 M (φ)2 5
W W Z Z
+6 log +2Γ(µ) +3 log +2Γ(µ) ,
· 64π2 µ2 − 6 · 64π2 µ2 − 6
)
(cid:20) (cid:18) (cid:19) (cid:21) (cid:20) (cid:18) (cid:19) (cid:21)
M (Ψ)4 M (Ψ)2 3
VΨ (µ,Ψ) := e4ΓΨ(µ) 6 R log R +2Γ (µ)
1loop − · 64π2 µ2 − 2 Ψ
(
(cid:20) (cid:18) (cid:19) (cid:21)
M (Ψ)4 M (Ψ)2 5
B−L B−L
+3 log +2Γ (µ) ,
· 64π2 µ2 − 6 Ψ
)
(cid:20) (cid:18) (cid:19) (cid:21)
(14)
4
Here,weneglecttheone-loopcontributionswhichincludeλ,λΨ andκbecausetheireffectsareverysmallwhen
we consider the MPP.
6
where
y (µ) g (µ) g2(µ)+g2(µ)
M (φ) = t φ , M (φ) = 2 φ , M (φ) = 2 Y φ,
t W Z
√2 2 2
p
Y (µ)
M (Ψ) = R Ψ , M (Ψ)2 = 22g (µ)2Ψ2. (15)
R B−L B−L
√2
Here, µ is the renormalization scale and Γ, Γ are the wave function renormalizations.
Ψ
To minimize the one-loopcontributions, we take µ = φ(Ψ) in the following discussion5.
From these results, we can define the effective self couplings and their effective beta
functions as follows:
4VH(φ) dλeff(φ)
λeff(φ) := eff , βeff := , (16)
φ4 λ dlnφ
4VΨ(Ψ) dλeff(Ψ)
λeff(Ψ) := eff , βeff := Ψ . (17)
Ψ Ψ4 λΨ dlnΨ
Fig.1 shows the typical behaviors of λeff(φ) and its parameter dependences. Here,
for the later convenience, the initial values of λ , κ, g , g and Y are given at
Ψ B−L mix R
Λ = 1017GeV, and their typical values are chosen to be 0.1 respectively. One can
MPP
see that λeff(φ) depends weakly on g and Y because they appear in β at two-loop
B−L R λ
level.
5Precisely speaking, µ should be determined as a function of φ and Ψ by minimizing the one-loop effective
potential. However, in this paper, we simply choose µ = φ(Ψ) when we focus on λeff (λeff). It is known that this
Ψ
choice is a good approximation [17].
7
ΛYHLMPPL=0.2,ΚHLMPPL=0.1,gB-LHLMPPL=0.1,gmixHLMPPL=0.1,YRHLMPPL=0.1Mt=171.2GeV,ΛDMHLMPPL=0.2,gB-LHLMPPL=0.1,gmixHLMPPL=0.1,YRHLMPPL=0.1
0.10
0.10
0.08 Mt=171.2GeV 0.08 ΚHLMPPL=0
ΚHLMPPL=0.4
0.06 0.06
Λeff 0.04 Λeff 0.04
0.02
0.02
0.00
0.00
-0.02
5 10 15 20 -0.02 5 10 15 20
Φ Φ
log10GeV log10GeV
Mt=171.2GeV,ΛDMHLMPPL=0.2,ΚHLMPPL=0.1,gB-LHLMPPL=0.1,YRHLMPPL=0.1Mt=171.2GeV,ΛYHLMPPL=0.2,ΚHLMPPL=0.1,gmixHLMPPL=0.1,YRHLMPPL=0.1
0.10 0.10
0.08 gmixHLMPPL=0 0.08 gB-LHLMPPL=0
gmixHLMPPL=0.4 gB-LHLMPPL=0.4
0.06 0.06
Λeff 0.04 Λeff 0.04
0.02 0.02
0.00 0.00
-0.02 -0.02
5 10 15 20 5 10 15 20
Φ Φ
log10GeV log10GeV
Mt=171.2GeV,ΛYHLMPPL=0.2,ΚHLMPPL=0.1,gB-LHLMPPL=0.1,gmixHLMPPL=0.1 Mt=171.2GeV,ΚHLMPPL=0.1,gB-LHLMPPL=0.1,gmixHLMPPL=0.1,YRHLMPPL=0.1
0.10 0.10
0.08 YRHLMPPL=0 0.08 ΛYHLMPPL=0
YRHLMPPL=0.6 ΛYHLMPPL=0.4
0.06 0.06
Λeff 0.04 Λeff 0.04
0.02 0.02
0.00 0.00
-0.02 -0.02
5 10 15 20 5 10 15 20
Φ Φ
log10GeV log10GeV
Figure 1: The runnings of the Higgs effective self coupling λ as a function of φ. The
eff
upper left (right) panel shows the M (κ(Λ )) dependence. In the case of M , the
t MPP t
blue band corresponds 95% CL deviation from 171.2GeV. The middle left (right) panel
shows the g (Λ )(g (Λ ))) dependence. The lower left (right) panel shows the
mix MPP B−L MPP
Y (Λ )(λ (Λ )) dependence.
R MPP Ψ MPP 8
8´1061
Mt=171.8GeV,gB-LHLMPPL=0.042,gmixHLMPPL=0.26,YRHLMPPL=0.1 2.0´1058 Mt=171.8GeV,gB-LHLMPPL=0.042,gmixHLMPPL=0.26,YRHLMPPL=0.1
6´1061
1.5´1058
H4@DVGeVeff4´1061 [email protected]´1058
2´1061 5.0´1057
0
016.80 16.85 16.90 16.95 17.00 17.05 17.10 16.80 16.85 16.90 16.95 17.00 17.05
Log10GeΦV Log10GeYV
0.00005
4.´10-6 gB-LHLMPPL=0.017
gB-LHLMPPL=0.030
3.´10-6 gB-LHLMPPL=0.042 0.0000
Y
effΛ 2.´10-6 Κ -0.00005
gB-LHLMPPL=0.017
1.´10-6
-0.0001 gB-LHLMPPL=0.025
gB-LHLMPPL=0.042
0
5 10 15 20 25 5 10 15
Μ Μ
log10GeV log10GeV
Figure 2: The effective potentials (upper) and the runnings of λeff and κ which satisfy the MPP
Ψ
conditions (lower). The upper left (right) panel shows VH (VΨ). They are exactly flat at Λ =
eff eff MPP
1017GeV. In the lower panels, we leave g (Λ ) as a free parameter. The different colors
B−L MPP
correspond to the different values of g (Λ ).
B−L MPP
Now, let us consider the MPP. By including the top mass M and neglecting y , there
t ν
are six parameters in this model:
M , g , g , λ , κ , Y . (18)
t B−L mix Ψ R
Therefore, in principle, they are uniquely determined by the MPP conditions:
λeff(Λ ) = λeff(Λ ) = κ(Λ ) = βeff(Λ ) = βeff(Λ ) = β (Λ ) = 0.
MPP Ψ MPP MPP λ MPP λΨ MPP κ MPP
(19)
Among them, λeff(Λ ) = κ(Λ ) = 0 are just the initial conditions of λ and κ,
Ψ MPP MPP Ψ
and other conditions give us constraints between the remaining parameters. We can
understand such constraints qualitatively from the one-loop RGEs:
9
βeff(Λ ) = 0 mainly relates M and g because they appear in β at one-
• λ MPP t mix λ
loop level (see Eq.(65) in Appendix A). As a result, we can fix M and g by
t mix
λeff(Λ ) = βeff(Λ ) = 0. They are
MPP λ MPP
171.74GeV M 171.82GeV , 0.21 g (Λ ) 0.27 (20)
t mix MPP
≤ ≤ ≤ ≤
according to 0 g (Λ ) 0.4 6.
B−L MPP
≤ ≤
We can obtain a relation between g (Λ ) and Y (Λ ) by β (Λ ) = 0
• B−L MPP R MPP λΨ MPP
because the one-loop part of β at Λ is
λΨ MPP
1
β (Λ ) = 96g4 3Y4 . (21)
λΨ|1-loop MPP 16π2 B−L − R
(cid:0) (cid:1)
Finally, g (Λ ) (or Y (Λ )) can be fixed at 0 by β (Λ ) = 0 because
B−L MPP R MPP κ MPP
•
the one-loop part of β at Λ is
κ MPP
1 12g2 g2
β (Λ ) = 12g2 g2 12Y2y2 B−L mix. (22)
κ|1-loop MPP 16π2 B−L mix − R ν ≃ 16π2
(cid:0) (cid:1)
In Fig.2, we show the effective potentials (upper) and the runnings (lower) of λeff
Ψ
and κ which satisfy the above MPP conditions. Here, in the lower panels, we leave
g (Λ ) as a free parameter. One can see that the flat potentials can be actually
B−L MPP
realized at Λ .
MPP
Summary : From the MPP at Λ = 1017GeV, the parameters of the gauged B-L
MPP
extension of the SM are fixed at
M 171.8GeV , g (Λ ) 0 , g (Λ ) 0.2 ,
t B−L MPP mix MPP
≃ ≃ ≃
λ (Λ ) 0 , κ(Λ ) 0 , Y (Λ ) 0. (23)
Ψ MPP MPP R MPP
≃ ≃ ≃
2.3 Electroweak Symmetry Breaking by Breaking the MPP
We first explain how the electroweak symmetry breaking is triggered by the B-L sym-
metry breaking. If Ψ has an expectation value Ψ := v /√2, the interaction term
B−L
h i
κ H†H (Ψ†Ψ) produces the mass term of H:
−
κ
(cid:0) (cid:1) v2 H†H. (24)
L ∋ −2 B−L
Thus, if κ is negative at the B-L breaking scale, the electroweak symmetry breaking
occurs, and the corresponding Higgs expectation value v is given by
h
κ
v = v . (25)
h B−L
−2λ × (cid:12)
r
(cid:12)µ=vh
(cid:12)
6YR(ΛMPP) dependence is negligible. (cid:12)(cid:12)
10