Physics Letters B 771 (2017) 281–287
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
On the gauge invariance of the decay rate of false vacuum
Motoi Endo
a,b,c
, Takeo Moroi
d,c,∗
, Mihoko M. Nojiri
a,b,c
, Yutaro Shoji
e
a
KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan
b
The Graduate University of Advanced Studies (Sokendai), Tsukuba, Ibaraki 305-0801, Japan
c
Kavli IPMU (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan
d
Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
e
Institute for Cosmic Ray Research, The University of Tokyo, Kashiwa 277-8582, Japan
a r t i c l e i n f o a b s t r a c t
Article history:
Received
3 April 2017
Received
in revised form 9 May 2017
Accepted
15 May 2017
Available
online 25 May 2017
Editor:
J. Hisano
We study the gauge invariance of the decay rate of the false vacuum for the model in which the scalar
field responsible for the false vacuum decay has gauge quantum number. In order to calculate the decay
rate, one should integrate out the field fluctuations around the classical path connecting the false and
true vacua (i.e., so-called bounce). Concentrating on the case where the gauge symmetry is broken in
the false vacuum, we show a systematic way to perform such an integration and present a manifestly
gauge-invariant formula of the decay rate of the false vacuum.
© 2017 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
.
There have been continuous interest in the theoretically correct
calculation of the decay rate of the false vacuum. One of the recent
motivations has been provided by the discovery of the Higgs bo-
son
at the LHC [1] and the precision measurement of the top quark
mass at the LHC and Tevatron [2]; in the standard model, we are
facing the possibility to live in a metastable electroweak vacuum
with lifetime much longer than the age of the universe [3–9]. Fur-
thermore,
the false and true vacua may show up in various models
of physics beyond the standard model. One important example is
supersymmetric standard model in which the electroweak symme-
try
breaking vacuum may become unstable with the existence of
the color or charge breaking vacuum at which colored or charged
sfermion fields acquire vacuum expectation values; the condition
that the electroweak vacuum has sufficiently large lifetime con-
strains
the parameters in supersymmetric models [10–19]. Thus,
detailed understanding of the decay of the false vacuum is impor-
tant
in particle physics and cosmology.
In
[20–22], the calculation of the decay rate of the false vacuum
was formulated with the so-called bounce configuration which is
a solution of the 4-dimensional (4D) Euclidean equation of mo-
tion
connecting false vacuum and true vacuum (more rigorously,
the other side of the potential wall). The decay rate of the false
vacuum per unit volume is given in the following form:
γ = Ae
−B
, (1)
*
Corresponding author.
E-mail
address: moroi@phys.s.u-tokyo.ac.jp (T. Moroi).
where B is the bounce action, while the prefactor A is obtained by
integrating out field fluctuations around the bounce configuration
as well as those around the false vacuum.
In
gauge theories, if a scalar field with gauge quantum num-
ber
acquires non-vanishing amplitude at the true or false vacuum,
the gauge, Higgs and the ghost sectors contribute to A. The decay
rate should be calculated with the gauge-fixed Lagrangian which
contains the gauge parameter ξ . In the present study, we con-
centrate
on the gauge dependence (i.e., the ξ-dependence) of the
decay rate of the false vacuum. Formally, the ξ-dependence of A
should cancel out exactly. This is due to the fact that the decay
rate is derived from the effective action of the bounce configu-
ration,
and also that the effective action for any solution of the
equation of motion is assured to be gauge invariant [23,24]. In
the actual calculation, however, the gauge independence is not
manifest because the ξ -dependence should cancel out among the
contributions of gauge field, Nambu–Goldstone (NG) boson, and
Faddeev–Popov (FP) ghosts.
1
In particular, the gauge boson and the
NG mode, whose fluctuation operator is ξ -dependent, mix with
each other around the bounce configuration. This makes the study
of the decay rate complicated. Furthermore, it is difficult to check
1
The gauge invariance of the effective potential of the model we consider was
discussed in [25]; however, the scalar configuration was assumed to be space–time
independent, and hence the result is not applicable to the present case. The gauge
independence of the sphaleron transition rate was studied in [26] using functional
determinant method which is also adopted in our analysis.
http://dx.doi.org/10.1016/j.physletb.2017.05.057
0370-2693/
© 2017 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
.
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