Relaxing the electroweak scale: the role of broken dS symmetry
HJE
Relaxing the electroweak scale: the role of broken dS symmetry
Subodh P. Patil 0 1 3
Pedro Schwaller 0 1 2
0 CH-1211 Geneva-23 , Switzerland
1 24 Quai Ansermet , CH-1211 Geneva-4 , Switzerland
2 Theory Division , PH-TH Case C01600, CERN
3 Department of Theoretical Physics, University of Geneva
Recently, a novel mechanism to address the hierarchy problem has been proposed [1], where the hierarchy between weak scale physics and any putative `cuto ' M is translated into a parametrically large eld excursion for the so-called relaxion eld, driving the Higgs mass to values much less than M through cosmological dynamics. In its simplest incarnation, the relaxion mechanism requires nothing beyond the standard model other than an axion (the relaxion eld) and an in aton. In this note, we critically re-examine the requirements for successfully realizing the relaxion mechanism and point out that parametrically larger eld excursions can be obtained for a given number of e-folds by simply requiring that the background break exact de Sitter invariance. We discuss several corollaries of this observation, including the interplay between the upper bound on the scale M
Cosmology of Theories beyond the SM; Beyond Standard Model
-
and the order parameter
associated with the breaking of dS symmetry, and entertain the
possibility that the relaxion could play the role of a curvaton. We
nd that a successful
realization of the mechanism is possible with as few as O(103) e-foldings, albeit with a
reduced cuto
M
106 GeV for a dark QCD axion and outline a minimal scenario that
can be made consistent with CMB observations.
1 Introductory remarks 2
Cosmological relaxation revisited
2.1
2.2
2.3
2.4
Field excursions and the breaking of dS symmetry
Cosmological bounds on the relaxion
Observational constraints
Concrete models
3
Concluding remarks
A Field rede nitions and higher order corrections B The relaxion as a curvaton 1 5
Since the discovery of the Higgs boson with a relatively light mass of 125 GeV [2, 3],
explanations that dynamically account for the apparent hierarchy between the electroweak
(EW) scale and any new physics that is presumed to complete the weak sector of the
standard model1 appear to be in tension with atomic physics and collider constraints
excepting rather tuned regions of parameter space. This has led to anthropic arguments
gaining currency as a plausible alternative, although as of yet no convincing solution to the
problem of how to de ne probabilities for observers and observables is available. Evidently,
novel solutions to the hierarchy problem that circumvent current low energy constraints
need no further justi cation.
Recently, the authors [
1
] have proposed a mechanism where the hierarchy between weak
scale physics and the new physics scale M is paraphrased into requiring a parametrically
large
eld excursion for a
eld that couples to the Higgs.2 In order to keep any new
hierarchies introduced by this new sector to be technically natural [4, 5], an obvious choice
would be for this eld to be axion-like, hence a relaxion. The potential for the relaxion
coupled to the singlet component of the Higgs h := (HyH)1=2 is given by
V ( ; h) =
1Such as low energy supersymmetry and large/ warped extra dimensions for completions that incorporate
gravity, and composite models for completions which become relevant at lower energies.
2For precursors in this direction, see [6{8].
{ 1 {
gM 2f:
gM 2f
f 2m2 = hvi
h
:
{ 2 {
where the ellipses denote higher order terms3 in g , and hhi is the ( dependent) vacuum
expectation value of the Higgs. The shape of the potential is shown schematically in gure 1.
We presume the relaxion to begin at very large
eld values
M 2=g wherein the
Higgs has a naturally large (and positive) mass squared. The relaxion evolves under the
in uence of the background cosmology which has to last long enough for
to scan a
su cient range in eld space to eventually break EW symmetry at
M 2=g. Primordial
in ation provides a natural context for this evolution to take place. As soon as the relaxion
expectation value drops below
= M 2=g, the Higgs starts to acquire a non-zero expectation
value and a periodic potential for
is generated by instanton e ects whose scale in the
EW vacuum is set by
where f is the (non-perturbatively generated) pion decay constant and m is the pion
mass. Since m2 grows linearly with the quark masses, this term grows in proportion to
hhi. Under the approximation that
is slow rolling, it will get trapped in a local minimum
once the barriers induced by the instanton potential are large enough to compensate the
slope of the potential, which occurs when
Parameterising the prefactor of the periodic potential as
246 GeV, it follows that
4(hhi) =
4hhi=v, where v =
Since small values of g are technically natural, hhvi of order one can be obtained for very
large values of the cut-o
M
v, by adjusting g accordingly.
Thus far we have taken the cosmological history of (...truncated)