Physics Letters B 786 (2018) 272–277
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
The fundamental need for a SM Higgs and the weak gravity conjecture
Eduardo Gonzalo
∗
, Luis E. Ibáñez
Departamento de Física Teórica and Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
a r t i c l e i n f o a b s t r a c t
Article history:
Received
18 July 2018
Received
in revised form 11 September
2018
Accepted
17 September 2018
Available
online 19 September 2018
Editor:
M. Cveti
ˇ
c
Compactifying the SM down to 3D or 2D one may obtain AdS vacua depending on the neutrino mass
spectrum. It has been recently shown that, by insisting in the absence of these vacua, as suggested by
Weak Gravity Conjecture (WGC) arguments, intriguing constraints on the value of the lightest neutrino
mass and the 4D cosmological constant are obtained. For fixed Yukawa coupling one also obtains an
upper bound on the EW scale
H
1/4
4
/Y
ν
i
, where
4
is the 4D cosmological constant and Y
ν
i
the
Yukawa coupling of the lightest (Dirac) neutrino. This bound may lead to a reassessment of the gauge
hierarchy problem. In this letter, following the same line of arguments, we point out that the SM without
a Higgs field would give rise to new AdS lower dimensional vacua. Absence of latter would require the
very existence of the SM Higgs. Furthermore one can derive a lower bound on the Higgs vev
H
QCD
which is required by the absence of AdS vacua in lower dimensions. The lowest number of quark/lepton
generations in which this need for a Higgs applies is three, giving a justification for family replication. We
also reassess the connection between the EW scale, neutrino masses and the c.c. in this approach. The EW
fine-tuning is here related to the proximity between the c.c. scale
1/4
4
and the lightest neutrino mass
m
ν
i
by the expression
H
H
(a
1/4
4
−m
ν
i
)
m
ν
i
. We emphasize that all the above results rely on the assumption
of the stability of the AdS SM vacua found.
© 2018 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
.
1. Introduction
It is a frustrating fact how poor our present understanding
of the origin of the different fundamental mass scales in Particle
Physics is. Simplifying a bit, there are essentially three regions of
scales in fundamental physics. There is a deep-infrared region in
which there are only two fundamental massless particles, photon
and graviton with the three neutrinos with masses in the region
m
ν
i
10
−3
–10
−1
eV, where one of the neutrinos could even be
massless. Interestingly, this is also very close to the scale of the ob-
served
cosmological constant
4
= (2.25 ×10
−3
eV)
4
.
1
The second
region is that of the masses of most elementary particles which
are around 10
−3
–10
2
GeV. These masses are dictated both by the
value of the QCD condensate
QCD
10
−1
GeV and the Higgs vev
H
0
=
246 GeV. Finally there is the Planck scale and presumably a
unification/string scale somewhat below. We would like, of course,
*
Corresponding author.
E-mail
addresses: eduardo.gonzalo@uam.es (E. Gonzalo), luis.ibannez@uam.es
(L.E. Ibáñez).
1
We are assuming here that the origin of dark energy is a 4D cosmological con-
stant.
to understand why the scales are what they are and what is the
information that this distribution of scales is giving us concerning
the fundamental theory. In particular it is difficult to understand
why
4
and the EW scale are so small compared to the fundamen-
tal
scales of gravity and unification. Also, the proximity of neutrino
masses to
1/4
as well as the (relative) proximity of
QCD
to the
EW scale could be just coincidences or could be telling us some-
thing
about the underlying theory.
A
natural question is whether all these scales are independent
or whether they are related or constrained within a more fun-
damental
theory including quantum gravity coupled to the SM
physics. Recently it has been pointed out that quantum gravity
constraints could have an impact on Particle Physics [1–3]. The ori-
gin
of these constraints is based on the Weak Gravity Conjecture
(WGC) [4,5], see [6]for a review and [7–9]for some recent refer-
ences.
A sharpened variation of the WGC was proposed by Ooguri
and Vafa in [1] which states that a non-SUSY Anti-de Sitter stable
vacuum cannot be embedded into a consistent theory of quantum
gravity (see also [10]). This general statement, together with an
assumption of background independence, may be applied to the
Standard Model (SM) itself [1]implying that no compactification
of the SM to lower dimensions should lead to a stable AdS vac-
https://doi.org/10.1016/j.physletb.2018.09.034
0370-2693/
© 2018 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
.