Physics Letters B 747 (2015) 169–172
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Criticality in the scale invariant standard model (squared)
Robert Foot
a
, Archil Kobakhidze
b,∗
, Alexander Spencer-Smith
b
a
ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Melbourne, VIC 3010, Australia
b
ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Sydney, NSW 2006, Australia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
5 January 2015
Received
in revised form 31 March 2015
Accepted
26 May 2015
Available
online 29 May 2015
Editor:
A. Ringwald
We consider first the standard model Lagrangian with μ
2
h
Higgs potential term set to zero. We point
out that this classically scale invariant theory potentially exhibits radiative electroweak/scale symmetry
breaking with very high vacuum expectation value (VEV) for the Higgs field, φ ≈ 10
17–18
GeV.
Furthermore, if such a vacuum were realized then cancellation of vacuum energy automatically implies
that this nontrivial vacuum is degenerate with the trivial unbroken vacuum. Such a theory would
therefore be critical with the Higgs self-coupling and its beta function nearly vanishing at the symmetry
breaking minimum, λ(μ =φ) ≈ β
λ
(μ =φ) ≈ 0. A phenomenologically viable model that predicts this
criticality property arises if we consider two copies of the standard model Lagrangian, with exact Z
2
symmetry swapping each ordinary particle with a partner. The spontaneously broken vacuum can then
arise where one sector gains the high scale VEV, while the other gains the electroweak scale VEV. The
low scale VEV is perturbed away from zero due to a Higgs portal coupling, or via the usual small Higgs
mass terms μ
2
h
, which softly break the scale invariance. In either case, the cancellation of vacuum energy
requires M
t
= (171.53 ±0.42) GeV, which is close to its measured value of (173.34 ±0.76) GeV.
© 2015 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 SCOAP
3
.
The discovery of a Higgs-like particle [1,2] with mass around
125 GeV confirms the standard picture of electroweak symmetry
breaking via the nonzero vacuum expectation value of a scalar
field [3]. This nontrivial vacuum arises from a Higgs potential, of
the form:
V = λφ
†
φφ
†
φ − μ
2
h
φ
†
φ +hμ
4
h
, (1)
where the hμ
4
h
part is the cosmological constant (CC) term, usually
neglected as it only affects gravitational physics, e.g. [4,5]. In mod-
els
with classical scale invariance, however, the CC term is absent,
as required by this symmetry. The physical cosmological constant
still arises radiatively, but is a calculable function of the other pa-
rameters
of the theory [6].
Interestingly,
a possible hint of a deeper structure beyond
the standard model has emerged in a rather unexpected man-
ner.
The Higgs quartic coupling when evolved up to a high scale
∼ 10
17–18
GeV, appears to approximately satisfy: λ =
˙
λ(≡ β
λ
) = 0
(for
recent calculations, see [7,8]). This condition seems to be ac-
cidental,
since it involves cancellation among numerically large
*
Corresponding author.
E-mail
addresses: rfoot@unimelb.edu.au (R. Foot), archilk@physics.usyd.edu.au
(A. Kobakhidze),
alexss@physics.usyd.edu.au (A. Spencer-Smith).
quantities.
1
In this short note, we show that such a relation can
naturally arise in scale invariant models as a consequence of set-
ting
the physical cosmological constant to its measured small
value. This then automatically implies that two distinct vacua with
broken and unbroken symmetries coexists in the theory, that is,
the theory exhibits criticality. We consider cases of exact classical
scale invariance and also the case where scale invariance is con-
sidered
to be softly broken by the familiar μ
2
h
term in the Higgs
potential.
Let
us define the scale invariant standard model Lagrangian,
L
SI
SM
, to be the same as the standard model Lagrangian except with
the μ
2
h
term set to zero. The Higgs potential is then particularly
simple:
V = λφ
†
φφ
†
φ. (2)
A Coleman–Weinberg analysis [12,13], reveals that such a po-
tential,
radiatively corrected, can exhibit spontaneous symmetry
breaking. This requires λ(μ) to be small at some particular scale
μ = μ
1
, and μ
1
sets the scale of the VEV of φ. If we addition-
ally
require that the CC vanish, then we have much more stringent
1
Previous attempts to justify these conditions were based on somewhat obscure
principle of multiple criticality [9] and still controversial proposal of asymptotic
safety of gravity [10]. See also [11].
http://dx.doi.org/10.1016/j.physletb.2015.05.064
0370-2693/
© 2015 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
SCOAP
3
.