University of Nebraska - Lincoln
DigitalCommons@University of Nebraska - Lincoln
Faculty Publications, Department of Physics and
Astronomy
Research Papers in Physics and Astronomy
2016
Probing the electroweak phase transition with
Higgs factories and gravitational waves
Peisi Huang
University of Chicago, peisi.huang@unl.edu
Andrew J. Long
University of Chicago, andrewjlong@kicp.uchicago.edu
Lian-Tao Wang
University of Chicago, liantaow@uchicago.edu
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Huang, Peisi; Long, Andrew J.; and Wang, Lian-Tao, "Probing the electroweak phase transition with Higgs factories and gravitational
waves" (2016). Faculty Publications, Department of Physics and Astronomy. 219.
h8p://digitalcommons.unl.edu/physicsfacpub/219
Probing the electroweak phase transition with Higgs factories
and gravitational waves
Peisi Huang,
1,2
,*
Andrew J. Long,
1
,†
and Lian-Tao Wang
1,3
,‡
1
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA
2
High Energy Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
3
Kavli Institute for Cosmological Physics, Univ ersity of Chicago, Chicago, Illinois 60637, USA
(Received 1 September 2016; published 18 October 2016)
After the discovery of the Higgs boson, understanding the nature of electroweak symmetry breaking and
the associated electroweak phase transition has become the most pressing question in particle physics.
Answering this question is a priority for experimental studies. Data from the LHC and future lepton
collider-based Higgs factories may uncover new physics coupled to the Higgs boson, which can induce the
electroweak phase transition to become first order. Such a phase transition generates a stochastic
background of gravitational waves, which could potentially be detected by a space-based gravitational
wave interferometer. In this paper, we survey a few classes of models in which the electroweak phase
transition is strongly first order. We identify the observables that would provide evidence of these models at
the LHC and next-generation lepton colliders, and we assess whether the corresponding gravitational wave
signal could be detected by eLISA. We find that most of the models with first-order electroweak phase
transition can be covered by the precise measurements of Higgs couplings at the proposed Higgs factories.
We also map out the model space that can be probed with gravitational wave detection by eLISA.
DOI:
10.1103/PhysRevD.94.075008
I. INTRODUCTION
The discovery of the Higgs boson completes the list of
particles of the Standard Model. However, there are many
open questions regarding the dynamics of electroweak
symmetry breaking. Addressing these questions has been
a driving force in both theoretical and experimental
explorations of the energy frontier.
One of the most outstanding problem is the nature of the
electroweak phase transition. Currently, we have measured
with precision the size of the Higgs vacuum expectation
value (VEV) and the mass of the Higgs boson. However,
we know very little about the shape of the Higgs potential
beyond that. The Standard Model defines its Higgs poten-
tial with only renormalizable terms—the so-called
“Mexican” hat form. In this case the electroweak symmetry
is smoothly restored via a continuous crossover as the
temperature is raised above the electroweak scale
[1].
However, new physics can modify the nature of the phase
transition, possibly turning it into an abrupt first-order
phase transition. On the one hand, such new physics can be
searched for directly at current and future colliders. On the
other hand, a first-order electroweak phase transition in the
early Universe would generate a stochastic background
of gravitational waves that can be searched for with
interferometers
[2]. Discovering such a gravitational wave
signal would establish the electroweak phase transition as a
new milestone in our understanding of the early Universe.
It will advance our knowledge into an epoch significantly
earlier than nucleosynthesis.
A first-order electroweak phase transition requires a
significant deviation away from the renormalizable
Higgs potential, which implies the presence of new physics
close to the weak scale. We can look for such new particles
directly, such as at the LHC. Even though it can be
powerful in certain cases, the reach of this approach is
limited. The new physics can be weakly coupled, and the
searches at the LHC suffer from large background. At the
same time, the most model-independent effect of such new
physics is the induced deviation in the Higgs couplings
[3,4]. Measuring such couplings precisely, and uncovering
potential new physics, is a major physics goal of proposed
Higgs factories. In this paper, we will focus on the potential
of probing new physics associated with electroweak
symmetry breaking at these facilities.
If a first-order electroweak phase transition occurred in
the early Universe, the collision of bubbles and damping of
plasma inhomogeneities would have generated a stochastic
background of gravitational waves. The frequency of these
waves is relatively model independent, being related to the
scale of the cosmological horizon at the time of the
phase transition. Therefore today we expect the waves to
have redshifted into the millihertz range. This potential
signal is impossible to probe with ground-based gravita-
tional wave interferometers like AdvLIGO due to seismic
noise. However, the signal is ideal for a space-based
interferometer like eLISA
[2] with arm lengths of order
millions of kilometers. In this paper, we assess the
*
peisi@uchicago.edu
†
andrewjlong@kicp.uchicago.edu
‡
liantaow@uchicago.edu
PHYSICAL REVIEW D 94, 075008 (2016)
2470-0010=2016=94(7)=075008(17) 075008-1 © 2016 American Physical Society
possibility of using gravitational waves as probes of a first-
order electroweak phase transition and the complementarity
of this technique with the collider searches.
There are a number of ways in which new particles may
cause the electroweak phase transition to become first
order. In general a first-order phase transition can occur if
the Higgs effective potential is modified from its Standard
Model form so as to develop a potential energy barrier
separating the phases of broken and unbroken electroweak
symmetry. As discussed in
[5], there are three general
model classes in which the barrier can arise. First, if the
new degrees of freedom are scalar fields that participate
in the electroweak phase transition (their VEV changes at
the same time as the Higgs), then tree-level interactions
with the Higgs field can make some regions of field space
energetically disfavorable and lead to a barrier. Second, the
presence of new particles coupled to the Higgs boson
affects the running of the Higgs mass parameter and self-
coupling. Then the barrier can arise by virtue of quantum
effects. Third, if the new particles are present in the early
Universe plasma and acquire their mass (at least partially)
from the Higgs field, then a barrier can arise via thermal
effects. This can be understood as a trade-off between
minimizing the energy, represented by the tree-level Higgs
potential, and maximizing the entropy, which prefers the
Higgs field to take on values where particles in the plasma
are light. Then, this third case can be further divided into
two categories: a barrier arising from light scalars via the
thermal cubic term ∼ðm
2
Þ
3=2
T and a barrier arising from
heavy particles that get their mass predominantly from a
large coupling with the Higgs field.
In this paper, we survey a number of simplified models
that demonstrate the basic ingredients necessary for a
first-order electroweak phase transition. We focus on four
models in which the Standard Model (SM) is extended,
respectively, to include a real scalar singlet, a scalar
doublet, heavy chiral fermions, and varying Yukawa
couplings. This set of models exemplifies all of the
different phase transition model classes, enumerated above.
II. MODELS
In each of the models discussed here, the Higgs field is
represented by ΦðxÞ, and the Standard Model Lagrangian
contains
L
SM
⊃ ðD
μ
ΦÞ
†
ðD
μ
ΦÞ − m
2
0
Φ
†
Φ − λ
h
ðΦ
†
ΦÞ
2
: ð2:1Þ
In calculating the scalar effective potential we write
hΦðxÞi ¼ ð0; ϕ
h
=
ffiffiffi
2
p
Þ with ϕ
h
real. The vacuum sponta-
neously breaks the electroweak symmetry, ϕ
h
¼ v with
v ≃ 246 GeV. The Higgs mass is denoted as M
h
, and it
takes the value M
h
≃ 125 GeV.
A. Real scalar singlet
First, we add to the SM a real scalar field SðxÞ, which is a
singlet under the SM gauge group. This is probably the
simplest extension of the Higgs sector of the SM. At the
same time, due the lack of other interactions, it gives rise to
the most independent signal.
The most general renormalizable Lagrangian is
written as
L ¼ L
SM
þ
1
2
ð∂
μ
SÞð∂
μ
SÞ − t
s
S −
m
2
s
2
S
2
−
a
s
3
S
3
−
λ
s
4
S
4
− λ
hs
Φ
†
ΦS
2
− 2a
hs
Φ
†
ΦS: ð2:2Þ
Without loss of generality, we can set t
s
¼ 0. Since the new
scalar is a singlet, it only interacts with the Standard Model
via the Higgs portal, Φ
†
ΦS
2
and Φ
†
ΦS. The electroweak
phase transition and collider phenomenology in this model,
sometimes called the xSM, have been studied extensively;
see e.g.
[6] and references therein. The gravitational wave
signal in related models has been studied recently by
Refs.
[7–13].
There is no single reason why this model admits a first-
order electroweak phase transition. In fact different limits
of this simple model exhibit each of the phase transition
model classes that were identified in
[5]. Most notably, the
tree-level interactions play a significant role in most of the
parameter space. During the electroweak phase transition,
the singlet v
s
need not remain fixed. If v
s
changes along
with v, then the Higgs portal terms λ
hs
Φ
†
ΦS
2
and a
hs
Φ
†
ΦS
can give rise to a barrier in the effective potential, and the
phase transition is first order.
After electroweak symmetry breaking hΦi¼ð0;v=
ffiffiffi
2
p
Þ,
and generically we expect the singlet field to acquire a
vacuum expectation value as well, hSi¼v
s
. Then, the
Higgs portal operators allow the Higgs and singlet fields to
mix. The mixing angle −π=4 ≤ θ ≤ π=4 satisfies
sin 2θ ¼
4vða
hs
þ λ
hs
v
s
Þ
M
2
h
− M
2
s
; ð2:3Þ
where M
h
≃ 125 GeV is the physical Higgs boson mass
and M
s
is the physical mass of the singlet.
Interactions between the Higgs boson and the singlet
scalar affect the coupling of the Higgs to the Z boson.
Writing the effective hZZ coupling as g
hZZ
, we define the
fractional deviation from the SM value as
δZ
h
≡ 1 −
g
hZZ
g
hZZ;SM
s¼ð250 GeVÞ
2
; ð2:4Þ
where the couplings are evaluated at a center of mass
energy s ¼ð250 GeVÞ
2
. We calculate δZ
h
as
HUANG, LONG, and WANG PHYSICAL REVIEW D 94, 075008 (2016)
075008-2
δZ
h
≈ ð1 − cos θÞ −
1
2
ja
hs
þ λ
hs
v
s
j
2
16π
2
I
B
ðM
2
h
; M
2
h
;M
2
s
Þ
−
1
2
jλ
hs
j
2
v
2
16π
2
I
B
ðM
2
h
; M
2
s
;M
2
s
Þ − 0.006
λ
3
λ
3;SM
− 1
:
ð2:5Þ
The first term arises from the tree-level Higgs-singlet
mixing
(2.3). This is typically the dominant contribution
to δZ
h
. At the one-loop order, the singlet contributes to the
wave function renormalization of the Higgs. This gives rise
to the second and third terms in
(2.5). We have generalized
the calculation in Refs.
[14,15], to allow for the cases
without a Z
2
symmetry. The bosonic loop function is given
by
[16]
I
B
ðp
2
; m
2
1
;m
2
2
Þ¼
Z
1
0
dx
xð1 − xÞ
xð1 − xÞp
2
− xm
2
1
− ð1 − xÞm
2
2
:
ð2:6Þ
The wave function renormalization terms are typically
subdominant, except for the Z
2
limit (discussed in
II A 1) where θ ¼ a
hs
¼ v
s
¼ 0 and the jλ
hs
j
2
v
2
term is
dominant.
The fourth term in
(2.4) also arises at the one-loop order.
As recognized in [17] this term appears when the Higgs
trilinear coupling λ
3
deviates from its SM value λ
3;SM
. The
effect on δZ
h
depends on the center of mass energy, and for
ffiffiffi
s
p
¼ 250 GeV the prefactor evaluates to 0.006
[17]. The
cubic self-coupling of the mass eigenstate Higgs (hhh)is
calculated as
λ
3
¼ð6λ
h
vÞcos
3
θ þð6a
hs
þ 6λ
hs
v
s
Þsin θ cos
2
θ
þð6λ
hs
vÞsin
2
θ cos θ þð2a
s
þ 6λ
s
v
s
Þsin
3
θ: ð2:7Þ
In the Standard Model we have λ
3
¼ λ
3;SM
≡ 3M
2
h
=v≃
191 GeV. The last term in λ
3
arises from a three-vertex,
one-loop graph. As we will see, models exhibiting a
strongly first-order phase transition typically have an
Oð1Þ deviation in λ
3
, and, therefore, this effect on δZ
h
can be sizable.
If the singlet is sufficiently light, M
s
<M
h
=2 ≃ 62.5 GeV,
the Higgs decay channel h → SS opens. This decay con-
tributes to the Higgs invisible width. The invisible width is
calculated as
[18]
Γ
inv
¼ Γðh → SSÞ¼
λ
2
211
32πM
h
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 −
4M
2
s
M
2
h
s
; ð2:8Þ
where
λ
211
¼ð2a
hs
þ 2λ
hs
v
s
Þcos
3
θ þð4λ
hs
v − 6λ
h
vÞsin θ cos
2
θ
þð6λ
s
v
s
þ 2a
s
− 4λ
hs
v
s
− 4a
hs
Þsin
2
θ cos θ
þð−2λ
hs
vÞsin
3
θ ð2:9Þ
is the effectiv e trilinear coupling of the mass eigenstates. If
the invisible channel is open, the branching fraction is
typically so large as to be excluded already by LHC limits,
BR
inv
≲ 30%
[19,20]. Therefore we require M
s
>M
h
=2.
In general the model has seven parameters corresponding
to the potential terms in
(2.2) with t
s
¼ 0. Two parameters
can be exchanged for the Higgs mass and VEV, leaving five
free parameters. In
IV we present the main result for this
model, which entails a scan over the five-dimensional
parameter space. In the following subsections we discuss
a couple of special limiting cases of this model. Even
though they do not represent generic models with first-
order electroweak phase transition, they give rise to differ-
ent predictions. We include them in our discussion for
completeness.
1. Z
2
-symmetric limit
We impose the Z
2
discrete symmetry under which the
singlet is odd, ϕ
s
→ −ϕ
s
, and the Standard Model fields are
even. In terms of the singlet Lagrangian
(2.2) the symmetry
enforces
t
s
¼ 0;a
s
¼ 0; and a
hs
¼ 0: ð2:10Þ
We also require that the Z
2
symmetry is not broken
spontaneously, and thus v
s
¼ 0. The only interaction
between the Standard Model and the singlet is through
the Higgs portal λ
hs
Φ
†
ΦS
2
.
The Z
2
symmetry forbids a mixing between the Higgs
and singlet fields. In the absence of mixing, modifications
to the hZZ coupling
(2.5) first arise at the one-loop level.
The tree-level modifications to the trilinear coupling
(2.7)
are also suppressed in this limit, and, therefore, only the
jλ
hs
j
2
term contributes to δZ
h
. Thus we expect that this
corner of parameter space can evade constraints on δZ
h
from future colliders as discussed later. As such, this model
has been identified as a “worst-case scenario” for finding
evidence of a first-order electroweak phase transition at
colliders
[15,21].
Despite the vanishing mixing, the electroweak phase
transition may still be first order. Morally speaking, the
Higgs-singlet mixing is a “local” property of the theory,
related to the behavior of small fluctuations about the
vacuum, but the nature of the phase transition depends also
upon “global” properties of the theory, e.g. whether the
theory admits other metastable vacua. The presence or
absence of such metastable vacua is not directly related
to the mixing at the true vacuum. Two specific scenarios
have been studied. If the Higgs portal coupling is suffi-
ciently large, the singlet can affect the running of the Higgs
PROBING THE ELECTROWEAK PHASE TRANSITION WITH … PHYSICAL REVIEW D 94, 075008 (2016)
075008-3
self-coupling, which may induce a barrier in the effective
potential
[22]. Alternatively, the Higgs portal interaction,
λ
hs
ϕ
2
h
ϕ
2
s
with λ
hs
> 0, may give rise to a barrier in the
effective potential at tree level when the phase transition
passes from a vacuum with ϕ
h
¼ 0 and ϕ
s
¼ v
s
ðTÞ to a
vacuum with ϕ
h
¼ vðTÞ and ϕ
s
¼ 0.
2. Unmixed limit
The tree-level Higgs-singlet mixing
(2.3) vanishes when
we take
a
hs
þ λ
hs
v
s
¼ 0: ð2:11Þ
Unlike the Z
2
symmetric limit of (2.10), the choice of
parameters in
(2.11) is not associated with any enhanced
symmetry. Significant fine-tuning among tree-level param-
eters is necessary to reach this limit. Furthermore, such a
tuning is not technically natural; the mixing is induced
radiatively. At the one-loop order, the induced mixing is
∝ λ
hs
vða
s
þ λ
hs
v
s
Þ. Otherwise, the first-order electroweak
phase transition and collider phenomenology is similar to
the Z
2
case.
In the unmixed limit, the singlet can be pair produced
through an off-shell Higgs via the hSS coupling λ
hs
v and
can decay to hh, WW, and ZZ final states via the radiatively
generated mixing. Then by the Goldstone equivalence
theorem, the singlet decay branching ratios for the hh,
WW, and ZZ channels are 25%, 50%, and 25%, respec-
tively. Those final states contribute to a multilepton,
multijet signature, which can be probed at the LHC.
With a large WW branching ratio, the 4W channel leads
to a same-sign dilepton with multiple jets (zero b-jet) final
state. The background processes for this channel include t
¯
t,
t
¯
tW, t
¯
tZ, WZ, and same sign WW plus jets. All back-
grounds except same sign WW plus jets can be estimated
from the t
¯
th searches in the same sign dilepton with at least
two b-jets channel, by replacing the b tagging with a b-jet
veto
[23]. We assume the b-tagging efficiency is 70%.
For the same sign WW plus jets, we include both single
parton scattering and double parton scattering
[24] and
assume a 90% acceptance to account for the lepton
efficiency and kinematics. We assume the same 90%
acceptance for signal as well. Then at the high-luminosity
Large Hadron Collider (HL LHC), we expect a 2σ
significance for σðpp → SSÞ ∼ 1.8 fb, which corresponds
to λ
hs
∼ 1, and M
S
∼ 200 GeV
[25].
B. Scalar doublet (top-squark-like)
In this section, we go beyond the singlet to consider
new particles in nontrivial representations of the SM gauge
group. Some of the simplest cases are obtained by
introducing SU2
L
scalar doublets and singlets with U1
Y
charge. Perhaps the most well-known example is the
Minimal Supersymmetric Standard Model (MSSM) top
squark. However, the light top squark scenario is very
restricted and, at least in simple cases, it cannot give rise to
a first-order electroweak phase transition without running
afoul of collider constraints
[26–28]; see also [29]. Many of
these constraints are a consequence of the supersymmetry
(SUSY). For example, the scalar top partner must to be
colored and hence the top squark is subject to stringent
limits from collider searches. To avoid the collider con-
straints, models like folded SUSY have been proposed
[30], in which the top squarks can still solve the hierarchy
problem but are not colored. In the following, we consider
a similar top-squark-like model. The new particles are
taken to have the same electroweak gauge quantum
numbers as the top squark, but they are not colored. In
addition, their couplings are not subject to the constraints of
supersymmetry.
We extend the SM to include n
f
¼ 3 copies (flavors) of
scalar doublets and complex scalar singlets. We will denote
the doublets and singlets as
~
Q
i
¼ð
~
u
i
;
~
d
i
Þ
T
and
~
U
i
, where
the index i runs from 1 to n
f
. In order to mimic the
interactions of colored squarks, we require the Lagrangian
to respect the global SUn
f
symmetry, under which the
~
Q
i
and
~
U
i
transform in the fundamental representation, and the
SM fields are invariant. Notice that we have used a SUSY-
like notation to indicate the electroweak gauge quantum
numbers, but no SUSY relations are implied.
With the new top-squark-like particle content, the scalar
potential can be written as
V ¼
1
2
m
2
0
ϕ
2
h
þ
λ
h
4
ϕ
4
h
þ m
2
Q
ðj
~
uj
2
þj
~
dj
2
Þþm
2
U
j
~
Uj
2
þ λ
Q
ðj
~
uj
2
þj
~
dj
2
Þ
2
þ λ
U
ðj
~
Uj
2
Þ
2
þ λ
QU
ðj
~
uj
2
þj
~
dj
2
Þj
~
Uj
2
þ
λ
hU
2
ϕ
2
h
j
~
Uj
2
þ
λ
hQ
2
ðj
~
uj
2
þj
~
dj
2
Þϕ
2
h
þ
λ
0
hQ
2
j
~
uj
2
ϕ
2
h
þ
λ
00
hQ
2
j
~
dj
2
ϕ
2
h
þ
a
hQU
ffiffiffi
2
p
~
uϕ
h
~
U
þ H:c:
: ð2:12Þ
The sum over i ¼ 1; …;n
f
flavors has been suppressed. In
general the model has 12 parameters, but two of these can
be exchanged for the Higgs mass and VEV, leaving ten free
parameters. Additionally, we will later assume a universal
dimensionless coupling, λ
Q
¼ λ
U
¼ λ
UQ
¼≡ λ, which
reduces the free parameters to four: fm
2
Q
;m
2
U
; λ;a
hQU
g.We
present the results of a parameter-space scan in
IV.
In the well-known light top squark scenario of the
MSSM
[31], the electroweak phase transition can become
first order due to the presence of these scalar particles in the
plasma. Their contribution to the Higgs thermal effective
potential (background-dependent free energy density) goes
as V
eff
∼−N
c
½m
~
t
ðϕ
h
;TÞ
2
3=2
T, where N
c
¼ 3 is the num-
ber of colors and the effective top squark mass m
~
t
ðϕ
h
;TÞ
depends on the background Higgs field ϕ
h
and the plasma
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075008-4