Physics Letters B 789 (2019) 154–159
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Revisiting electroweak phase transition in the standard model with
a real singlet scalar
Cheng-Wei Chiang
a,b,c
, Yen-Ting Li
a
, Eibun Senaha
d,∗
a
Department of Physics, National Taiwan University, Taipei 10617, Taiwan
b
Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
c
Kavli IPMU, University of Tokyo, Kashiwa, 277-8583, Japan
d
Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea
a r t i c l e i n f o a b s t r a c t
Article history:
Received
11 August 2018
Received
in revised form 10 December 2018
Accepted
10 December 2018
Available
online 11 December 2018
Editor:
M. Trodden
We revisit the electroweak phase transition in the standard model with a real scalar, utilizing several
calculation methods to investigate scheme dependences. We quantify the numerical impacts of Nambu–
Goldstone
resummation, required in one of the schemes, on the strength of the first-order electroweak
phase transition. We also employ a gauge-independent scheme to make a comparison with the standard
gauge-dependent results. It is found that the effect of the Nambu–Goldstone resummation is typically
∼ 1%. Our analysis shows that both gauge-dependent and -independent methods give qualitatively the
same result within theoretical uncertainties. In either methods, the scale uncertainties in the ratio of
critical temperature and the corresponding Higgs vacuum expectation value are more than 10%, which
signifies the importance of higher-order corrections.
© 2018 The Author(s). 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 SCOAP
3
.
1. Introduction
Cosmic baryon asymmetry [1]is one of the longstanding
problems in particle physics and cosmology. Though the stan-
dard
model (SM) can satisfy the so-called the Sakharov crite-
ria [2]in
principle, the discovered Higgs boson with a mass of
125 GeV [3]is incompatible with successful electroweak baryoge-
nesis
(EWBG) [4]since the electroweak phase transition (EWPT) is
a smooth crossover [5] rather than first order with expanding bub-
bles.
It is known that this drawback can be easily circumvented by
augmenting the minimal Higgs sector. The simplest extension is to
add an SU(2)
L
singlet scalar, which provides not only a strong first-
order
EWPT but also a dark matter candidate if a Z
2
symmetry is
imposed [6,8–13].
A
thorny problem in investigating EWPT using a perturbative
effective potential is the dependence on a gauge fixing parame-
ter
ξ [14,15](for recent studies, see, e.g., Refs. [16,17]). For in-
stance,
the Higgs vacuum expectation value (VEV) obtained by the
effective potential can change with a varying ξ . Such an unwanted
ξ dependence eventually contaminate a baryon-number preserv-
ing
criterion: v
C
/T
C
1, where T
C
denotes the critical tempera-
*
Corresponding author.
E-mail
addresses: chengwei@phys.ntu.edu.tw (C.-W. Chiang),
R04222022@ntu.edu.tw (Y.-T. Li), senaha@ibs.re.kr (E. Senaha).
ture associated with the phase transition and v
C
is the doublet
Higgs VEV at T
C
. As a result, any phenomenological consequences
derived from this criteria suffer from the ξ dependence and are
therefore unreliable unless the dependence can be kept under con-
trol.
Common
lore is that if the EWPT is driven by scalar thermal
loops or a tree-potential barrier, the ξ dependence is expected to
be small. As found in the Abelian–Higgs model with an additional
scalar [18], however, such an expectation is not always correct. It is
concluded that the ξ dependence can be pronounced even when
the tree-potential barrier exists. Nevertheless, this point is often
overlooked in previous studies on the EWPT in the SM with a real
singlet scalar.
Another
issue is the occurrence of IR divergences in the ef-
fective
potential in the R
ξ
gauge with ξ = 0. For example, if the
Higgs boson mass is renormalized using the one-loop effective po-
tential
in such a way that the loop corrections do not modify the
tree-level mass relations,
1
the second derivative of the one-loop
effective potential is ill-defined due to the IR divergences coming
from the Nambu–Goldstone (NG) boson loops. One of the prescrip-
tions
for the problem is to resum higher-order corrections to the
1
This is called “on-shell” renormalization in Ref. [8]. Since it is not the genuine
on-shell renormalization, we refer to it as “on-shell-like” renormalization in the cur-
rent
paper.
https://doi.org/10.1016/j.physletb.2018.12.017
0370-2693/
© 2018 The Author(s). 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
SCOAP
3
.