A warped relaxion
HJE
warped relaxion
Nayara Fonseca 0 1 2
Benedict von Harling 0 1 2
Leonardo de Lima 0 1 2
Camila S. Machado 0 1 2 3
Johannes GutenbergUniversitat Mainz,
0 55099 Mainz , Germany
1 Av. Edmundo Gaievski 1000 , 85770000 Realeza , Brazil
2 Notkestrasse 85 , 22607 Hamburg , Germany
3 PRISMA Cluster of Excellence and Mainz Institute for Theoretical Physics
We construct a UV completion of the relaxion in a warped extra dimension. We identify the relaxion with the zero mode of the fth component of a bulk gauge eld and show how hierarchically di erent decay constants for this eld can be achieved by di erent localizations of anomalous terms in the warped space. This framework may also nd applications for other axionlike elds. The cuto of the relaxion model is identi ed as the scale of the IR brane where the Higgs lives, which can be as high as 106 GeV, while above this scale warping takes over in protecting the Higgs mass.
Cosmology of Theories beyond the SM; Field Theories in Higher Dimensions

A
1 Introduction
2 Hierarchical decay constants from warped space
2.1
Anomalous couplings from the bulk
2.2 Anomalous couplings from the UV brane
3 Generating the relaxion potential
3.1
General setup
3.2 A warped model
4 Conditions for successful relaxation
4.1
General conditions
4.2 Conditions on the warped model
5
Warping the doublescanner mechanism
5.1
A UV completion
5.2 Constraints
6 Conclusions
A An anomalous coupling on the UV brane from two throats
B ChernSimons terms from bulk fermions C Pionlike elds in the relaxion potential 1 4
the cosmological relaxation of the electroweak scale [1] (see also [2{12]). It relies on the
scanning of the Higgs mass parameter by a new
eld, the relaxion, and a backreaction
{ 1 {
mechanism that is triggered when the vacuum expectation value (VEV) of the Higgs has
reached the electroweak scale, making the relaxion evolution stop.1 This is a radical change
of paradigm as it implies that the naturalness problem of the Standard Model ceases to be
a reason to expect new physics close to the TeV scale.
In what follows we review the relaxation mechanism for which an axionlike scalar
is introduced which couples to the Higgs doublet H via the potential
which sets the Higgs mass parameter, f the decay constant of the
HJEP07(218)3
Here
relaxion,
the Higgs quartic coupling, g and g0 are small dimensionless couplings, and
f (H) is a scale which depends on the Higgs VEV. Assuming a classical time evolution
with slowroll conditions, the secondlast term in eq. (1.1) causes the relaxion to move
downwards following its potential. The e ective Higgs mass parameter in the
background,
the rst term in parenthesis in eq. (1.1), then varies accordingly. The relaxion is assumed to
start with a VEV such that this mass parameter is initially positive. Due to the evolution
of the relaxion, the mass parameter then eventually turns tachyonic, triggering electroweak
symmetry breaking. In the presence of a Higgs VEV, the oscillatory barrier from the last
term grows, until its slope matches the slope of the linear term. For technically natural
parameters in the potential, this causes the relaxion to stop once the Higgs VEV has reached
the electroweak scale. There must be some mechanism to dissipate the kinetic energy of
the relaxion during its evolution such that the eld does not overshoot the barriers. If the
dynamics happens during a period of in ation, Hubble friction can provide the dissipation
necessary to slow down the eld [1]. As an alternative to in ation, one can also consider
friction due to particle production as proposed in ref. [14] or nite temperature e ects in
the early universe as in ref. [15].
Note that the linear terms in
are in con ict with the assumption that the relaxion
is a pseudoNambuGoldstone boson as they explicitly break the axion shift symmetry [
5
].
This may be reconciled if the linear terms arise from a second oscillatory potential with a
period much larger than f . This is realized if the potential takes the form [16{18]:2
V ( ; H)
2H2 +
H4 +
4F (H) cos
+
f4 (H) cos
F
f
;
(1.2)
where F
f is another decay constant and
F (H) another scale that depends on the
Higgs in such a way as to reproduce the second and fourth term in eq. (1.1) after
expanding in
=F . An interesting possibility to obtain this type of potential is the clockwork
construction which was rst realized for axionlike elds in refs. [16, 17] and generalized for
applications other than the relaxion in ref. [
26
]. Further developments regarding the 5D
1See also N naturalness [13], where instead of multiple vacua, many copies of the Standard Model are
2See also refs. [19{24] for similar earlier ideas in in ation model building. For the viability of the
relaxation mechanism in string theory in the context of axion monodromy, see ref. [25].
{ 2 {
continuum limit of the clockwork can be found in refs. [27{30]. Besides the clockwork, one
can also generate a potential of the form in eq. (1.2) in realizations inspired by dimensional
deconstruction [31, 32], as in ref. [18].
In this work, we show how the required potential for the relaxation mechanism to work
can be naturally obtained by embedding the relaxion and Higgs into a warped extra
dimension. We consider a slice of AdS5 space which is bounded by two branes, as in the
RandallSundrum model [33]. However, in our setup the IR scale or warpeddown AdS scale is not
of order TeV but can be much larger. We introduce a U(1) gauge eld in the bulk of the
extra dimension and break the gauge symmetry on the two branes. The 5th component A5 of
the gauge eld then gives rise to one massless scalar mode in 4D which we identify with the
laxion as in eq. (1.2) with periods given by the decay constants.3 Due to the warping, these
periods can thus naturally be hierarchically di erent as required. We embed the Higgs at
or near the IR brane. Its mass parameter is then naturally of order the IR scale which we
identify with the cuto
of the relaxion theory. The required Higgsrelaxion couplings can
be obtained by introducing fermions on the IR brane with higherdimensional or Yukawa
couplings to the Higgs. To summarize, the warping does two things:
rstly, it generates
the hierarchy between the decay constants F and f in eq. (1.2) and thereby explains the
smallness of the couplings g and g0 in eq. (1.1). Secondly, it provides a UV completion4 for
the relaxion. The relaxation mechanism protects the Higgs up to the IR scale above which
warping takes over.5 We illustrate this in gure 1. Alternatively, one can think of the
relaxation mechanism in our construction as a solution to the little hierachy problem of
RandallSundrum models.6 As is wellknown, various experimental constraints (the most stringent
ones coming from CP violation in K
Kmixing and the electirc dipole moment of the
neutron) require that the IR scale in these models is of order 10 TeV or above. This means that
a residual tuning in the permille range is necessary to generate the electroweak scale. In our
construction with warping and the relaxion, on the other hand, no such tuning is required.
3A potential for A5 can be generated perturbatively if the underlying gauge eld is coupled to charged
bulk states. In the nonabelian case (see e.g. [34]), this includes the gauge elds themselves due to the
nonlinear interactions, while the abelian case requires charged scalars or fermions in the bulk (see e.g. [37]).
Here we consider a U(1) gauge eld and do not add charged bulk states as we are interested in generating
a nonperturbative potential for A5.
4As a caveat, we should stress that the RandallSundrum model itself requires a UV completion. In
particular, near the IR brane gravity becomes strongly coupled at energies not far above the IR scale. Near
that brane, the UV completion therefore needs to kick in at correspondingly low scales. There are known
UV completions to the RandallSundrum model in string theory [38, 39].
5See [6, 10, 12] for how the relaxation mechanism can protect the Higgs up to some high
supersymmetrybreaking scale instead.
6See [40] for an alternative solution where an accidental form of supersymmetry protects a little hierarchy
between the electroweak scale and the IR scale of the RandallSundrum model.
{ 3 {
the Higgs into a warped extra dimension. The hierarchy problem is then solved in two steps: the
relaxation mechanism protects the Higgs mass up to the IR scale (which can be much larger than
the electroweak scale) and from there warping provides protection till the Planck scale.
We nd that for an e ective anomalous coupling localized on the UV brane, the decay
constant is of order MP2L= IR with MPL and
IR being the Planck and IR scale. For an
anomalous coupling in the bulk, we instead
nd a decay constant of order
IR. We then
identify F = MP2L= IR and f =
IR. Generating a suitable barrier f4 (H) cos( =f ) for the
relaxion requires some additional structure. The reason is that this term generically contains
a contribution which is independent of the Higgs and which could stop the relaxion before
the Higgs VEV has reached the electroweak scale. To avoid this problem, we consider two
di erent options. One employs a construction from ref. [1] for which new fermions are
introduced which couple to the Higgs. If the masses of these fermions are near the electroweak
scale, the Higgsindependent barrier can be su ciently small. The drawback of this
construction is a coincidence problem as it requires to introduce the fermions at a scale which
is dynamically generated by the relaxation mechanism and thus a priori determined by
completely di erent parameters. An interesting alternative is the socalled doublescanner
mechanism of ref. [2] (see also [10]). To this end, one introduces another axionlike scalar
which dynamically cancels o
the Higgsindependent barrier. We identify this axionlike
scalar with the 5th component of another U(1) gauge eld in the bulk of the extra
dimension. We then show how the potential which is required for the doublescanner mechanism
can be obtained. This construction is largely independent of the embedding into warped
space and can therefore also be useful for other UV completions of the relaxion. For both
options to generate the barrier, we discuss the relevant theoretical and phenomenological
constraints for successful relaxation. The highest cuto and IR scale consistent with these
constraints in our warped implementation of the relaxation mechanism is
=
IR . 106 GeV.
The plan of this work is as follows. In section 2, we discuss the properties of the A5
and show how hierarchical decay constants can be obtained. In section 3, we generate
the desired potential for the relaxation mechanism. We analyse the relevant constraints
to guarantee a successful relaxation of the electroweak scale in section 4. In section 5, we
present our implementation of the doublescanner mechanism and we conclude in section 6.
Additional details are given in three appendices.
2
Hierarchical decay constants from warped space
We will now show how hierarchical decay constants can be obtained from warped space.
These will be used in later sections to generate the relaxion potential. We consider a slice
{ 4 {
HJEP07(218)3
where FMN is the U(1) eld strength, g5 the 5D gauge coupling and pg = a5(z). In order
to eliminate the mixing between A and A5, we add the gauge xing term (see e.g. [34, 43])
S5D
Z d4x dz pg
1
2g52
The bulk equations of motion for the 4D component A and the 5th component A5 then
read
S5D
Z
d4x dz pg
1
4g52 FMN F MN
;
= 0
= 0 :
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
HJEP07(218)3
of AdS5 space with metric in conformal coordinates given by
where a(z) = (kz) 1 is the warp factor with k being the AdS curvature scale (see e.g. [41]
for a review). The slice is bounded by the UV brane at zUV = 1=k and the IR brane at
zIR = ekL=k. The length L of the extra dimension can be stabilized for example by means
of the GoldbergerWise mechanism [42]. The e ective 4D Planck scale for this space is
given by MP2L ' M 3=k, where M is the 5D Planck scale. We will assume that the Planck
scale and the AdS scale are of the same order of magnitude (and will later often equate
them). For later convenience, let us also de ne the IR scale IR
k e kL.
Let us consider a U(1) gauge boson in the bulk. Its action is given by
A5(x; z) = h(z) (x) ;
{ 5 {
We are interested in obtaining a massless scalar mode from the bulk gauge boson.
To this end, we break the gauge symmetry on both branes by imposing Dirichlet
boundary conditions on A . For consistency, this then requires to impose Neumann boundary
conditions for A5. Together the boundary conditions read
A jUV;IR = 0 ;
Alternatively we could break the gauge symmetry with Higgs elds on the two branes (see
e.g. [44, 45]). The above boundary conditions are then obtained in the limit of their VEVs
going to in nity. In unitary gauge,
! 1, the bulk equation of motion for A5 gives
= 0 :
Notice that this equation is consistent with the boundary conditions and there is thus one
massless mode from A5. Its other KaluzaKlein modes are all eaten by A . In particular,
there is no massless mode from A , consistent with the fact that the gauge symmetry is
broken. As usual, the A5 massless mode can be parameterized as
with the 5th component of a U(1) gauge eld in the bulk. Its wavefunction is then localized towards
the IR brane. The Higgs is localized on (or near) the IR brane. The UV brane corresponds to the
Planck scale. We draw the IR brane with a dashed contour as a reminder that the IR scale in our
model can be much larger than the usual TeV scale of the RandallSundrum model.
where h(z) is its pro le along the extra dimension. From eqs. (2.6) and (2.7), we then
see that h(z) = N a(z) 1. Demanding canonically normalized kinetic terms for (x), the
normalization constant N of the wavefunction is determined by
(2.9)
(2.10)
(2.11)
(2.12)
N
g
2
5
2 Z zIR dz
zUV a(z)
= 1 :
h(z) ' g4p2kL e kLkz :
= B z2 + C
!
+
2B
N k
:
At this point, the relaxion is thus an exact NambuGoldstone boson which nonlinearly
realizes a remnant global U(1). By virtue of the 5D gauge invariance, no 5D local,
higherdimensional operators can break this shift symmetry (see [46] for a detailed discussion). A
{ 6 {
For kL
g4
p
1, this gives N ' g4p2kL e kL, where we de ne the dimensionless coupling
g5= L. Altogether, the wavefunction of the massless mode then reads
The wavefunction is thus peaked towards the IR brane (see
gure 2 for a sketch of the
wavefunction pro le in the extra dimension).
Furthermore, the fact that N
! 0 for
zIR ! 1 shows that the A5 massless mode is indeed localized in the IR.
Performing a 5D gauge transformation, AM (x; z) ! AM (x; z) + @M (x; z), we see that
the boundary conditions in eq. (2.6) and the bulk equation of motion in eq. (2.7) remain
invariant only for the subset of transformations
with B and C being independent of x and z. The remaining symmetry in 4D is thus
global, again consistent with the fact that the gauge symmetry is broken. Under this
remnant symmetry, the massless mode transforms as
potential for the relaxion could be generated by nonlocal e ects in the presence of bulk
states which are charged under the U(1) but we assume such states to be absent from the
theory.7 Instead we introduce anomalous couplings of the relaxion to con ning nonabelian
gauge groups. A potential then arises from instantons, similar to what happens for the
axion in QCD. We localize these anomalous couplings in the bulk or on the UV brane.
In what follows, we show that these possibilities, thanks to the warp factor, can naturally
explain the required hierarchy between the decay constants in the relaxion potential.
Anomalous couplings from the bulk
Let us add a nonabelian gauge group in the bulk, whose eld strength and coupling we
denote respectively as GNP and g5c. We choose boundary conditions for the gauge eld such
that the 4D gauge symmetry remains unbroken on the branes. Its tower of KaluzaKlein
modes then contains one massless mode which is the 4D gauge boson. We next introduce
a ChernSimons coupling of the U(1) gauge eld to this gauge group. Including the kinetic
term, the action reads
where cB is a dimensionless constant and the normalization is chosen for later convenience.8
Under a U(1) gauge transformation AM (x; z) ! AM (x; z) + @M (x; z), the action
trans(2.13)
forms as
S5D ! S5D
Z
d4x dz (x; z)
c
B
are thus no gauge anomalies.
In the 4D e ective theory, this gives rise to an anomalous coupling for . Let us
restrict ourselves to the massless mode of the nonabelian gauge eld, whose eld strength
we denote as G . Integrating over the extra dimension, eq. (2.13) then in particular gives
S4D
Z
d4x
1
2(g4c)2 Tr [G
G ] +
(x)
16 2fB
where g4c = g5c=pL is the gauge coupling of the massless mode. The decay constant is given
7Alternatively, for example for bulk fermions charged under the U(1) it is su cient if their masses are
somewhat larger than the AdS scale in which case any perturbative contribution to the potential is highly
8Note that a factor of 2 arises from the normalization Tr[T aT b] = 12 a;b of the generators of the
non{ 7 {
HJEP07(218)3
which is of order the IR scale
IR and thus warpeddown.
From eqs. (2.12), (2.14)
and (2.15), we see that
reproduces the anomaly under a transformation
= Bz2. In
appendix B, we brie y review how ChernSimons terms can arise from charged bulk fermions.
As we also discuss there, any perturbative contribution from such a fermion to the
potential for A5 can be su ciently suppressed. Nevertheless, in the remainder of this paper we
will never assume any charged bulk states and will instead include the ChernSimons terms
directly into our e ective 5D theory.
Note that eq. (2.13) also yields couplings of
to the higher KaluzaKlein modes of
the nonabelian gauge eld. As eq. (2.15) for the massless mode, these couplings are total
derivatives (see e.g. ref. [48]) and therefore do not contribute perturbatively to the potential
for . We will later assume that the nonabelian gauge group con nes in order to generate
a nonperturbative potential for . But we will choose the con nement scale below the IR
scale and thus below the KaluzaKlein masses. The KaluzaKlein modes of the nonabelian
gauge group therefore do not contribute nonperturbatively to the potential either.
2.2
Anomalous couplings from the UV brane
We now discuss how a decay constant which is much larger than
IR can be obtained. To
this end, we consider an e ective anomalous coupling of A5 which is localized on the UV
brane [43],
S5D
Z
d4x dz (z
zUV) 16 2 k
cUV A5
where cUV is a dimensionless constant and GMN is the eld strength of a nonabelian gauge
eld in the bulk. As we outline in appendix A, this interaction can for example arise as
an e ective coupling from a ChernSimons term in a twothroat geometry. Under a U(1)
gauge transformation, the action transforms similar to eq. (2.14) but restricted to the UV
brane and with @5 (x; z) instead of (x; z). Let us again restrict ourselves to the massless
mode of the gauge eld, whose eld strength we denote as G . Using the wavefunction of
the massless mode of A5 from eqs. (2.8) and (2.10), this gives
with decay constant given by [43]
S4D =
Z
d4x
1
(x)
16 2 fUV
p2kL
(2.18)
(2.19)
or fUV
MP2L= IR. We see that a warpedup decay constant, much larger than the cuto ,
appears naturally in this case. This large decay constant can be intuitively understood as
being of order the natural scale MPL on the UV brane times an inverse suppression factor
from the wavefunction overlap of A5 with the UV brane.
Note that superPlanckian decay constants may be constrained by the weak gravity
conjecture in theories of quantum gravity [49] (see also [50{53]). Given that the relaxion
is an axionlike
eld, the conjecture necessarily restricts its eld excursion (
=g0)
to be subPlanckian, setting a lower bound on the coupling g0 in the potential (1.1). The
{ 8 {
weak gravity conjecture is then at odds with any relaxion model with transPlanckian
eld excursions, including our proposal. On the other hand, there are known loopholes
to the conjecture [54{58]. For instance, the application of the conjecture to e ective eld
theories may result in a much weaker bound on the coupling g0 [57]. Furthermore, in [58],
a better understanding of the conclusions of [57] is achieved by considering a string theory
embedding. There it is shown that if a clockwork model is successfully embedded in string
theory, one may in principle obtain a large cuto , avoiding the naive bound from the weak
gravity conjecture, as long as the number of sites in the construction is large.
We conclude that two hierarchically di erent decay constants can be obtained,
depending on the localization of the anomalous interactions in the warped space. For the relaxion,
portional to the warp factor, the potential in eq. (1.2) does not respect a discrete shift
symmetry since, in general, F=f is a noninteger number. This is a consequence of the nonlocal
nature of the residual symmetry transformation
= Bz2 + C in eq. (2.11) which explicitly
depends on the localization. In the following, we build an explicit model that makes use
of this toolkit to generate a phenomenologically viable potential in the form of eq. (1.2).
3
3.1
Generating the relaxion potential
General setup
Let us next discuss the relaxion parameters in more detail and how they can be understood
in terms of our UV model. Provided that electroweak symmetry remains unbroken in the
con nement phase transition which generates the periodic potentials in eq. (1.2),
F;f (H)
both depend quadratically on the Higgs (plus generically higher even powers of the Higgs
which are, however, not important in the following).9 We can then parametrize
4F;f (H) =
4
F;f
1 +
H2 !
MF2;f
;
g =
4
F ;
F 3
g0 =
4
F
F MF2
9As proposed in [1], one can also use the QCD axion as the relaxion. The last term in eq. (1.1) is
then the usual QCD axion potential which depends linearly on the Higgs (see e.g. [59]). However, barring
additional model building, this spoils the axion solution to the strong CP problem. See also [60{62] for
solutions to the strong CP problem in the context of the relaxion.
{ 9 {
where
F;f and MF;f can be understood as the scales where the periodic terms and
higherdimensional couplings to the Higgs are generated, respectively. The potential in eq. (1.2)
then reads
V ( ; H) =
2H2 +
H4 +
4
F
1 +
H2
MF2
cos
F
+
4
f
1 +
f
cos
: (3.2)
H2 !
Mf2
For simplicity, we have dropped terms which may be generated at higher looporder. We
will discuss these terms later in section 4. Assuming that
is in the linear regime of the
F=2 mod 2 , we can expand it for
F=2 . F . After the
, this gives the linear part of the relaxion potential in eq. (1.1)
lowfrequency cosine,
rede nition
F=2 !
with the identi cations
(3.1)
(3.3)
The last term in eq. (3.2) stops the relaxion once the Higgs VEV has reached the
electroweak scale. For this to work, we need to ensure that Mf . vEW, otherwise the
Higgsindependent barrier proportional to cos( =f ) would stop the relaxion already before
the Higgs VEV has obtained the right value. Note also that the Higgsindependent barrier
receives corrections from closing the Higgs loop in the Higgsdependent one and will thus
generically be present. We discuss radiative corrections to the potential in more detail in
section 4. But to get a sense of the scales involved, we already note here that radiative
stability of the potential demands that
f2 . 4 vEWMf and
F . 4 MF .
To obtain Mf . vEW requires that the higherdimensional coupling of the Higgs to
the periodic potential is generated near the electroweak scale. In the next section, we
make use of a construction from ref. [1] which introduces light fermions for this purpose.
The drawback of this scenario is of course a coincidence problem: one has to assume new
particles at a scale which is dynamically generated by the relaxation mechanism and is thus
determined by a priori completely unrelated parameters. One way around this problem is
the doublescanner mechanism of ref. [2]. To this end, one introduces another axionlike
eld which dynamically cancels o the Higgsindependent barrier in eq. (3.2). This allows
HJEP07(218)3
vEW.10 We discuss a UV completion of
the relaxation mechanism to work even for Mf
this scenario in section 5.
3.2
A warped model
We now build a simple explicit model that successfully generates the needed terms in the
Higgsrelaxion potential at a phenomenologically viable scale, making use of the results
of section 2.
We assume that the Higgs is localized on or near the IR brane, so that
its mass is warped down to the IR scale (see
gure 2). We note that it may also be
possible to implement the relaxation mechanism in a model where the Higgs is instead
localized on the UV brane. As usual, the relaxion can only protect the Higgs up to some
cuto
signi cantly below the Planck scale. Such a model would therefore require a UV
completion above this cuto on the UV brane. We leave a study of this possibility to future
work. As we
nd later, the highest IR scale that we can achieve in our implementation
of the relaxation mechanism (while still solving the hierarchy problem) is below the GUT
scale. If the remaining Standard Model elds are then also localized on the IR brane,
higherdimensional operators violating baryon number lead to too fast proton decay [63].
In order to suppress these operators, we assume that the Standard Model instead lives
in the bulk. As usual, the light quarks are localized towards the UV brane, while the
topbottom doublet and the righthanded top live nearer to the IR brane. This has the
added advantage that the hierarchy of Yukawa couplings can then be generated from the
warping too. The IR scale in our model can be high enough, on the other hand, to ensure
that oblique corrections and avour and CP violating processes are su ciently suppressed
without imposing custodial or avour symmetries.
10Another proposal for the relaxion that does not require new physics close to the electroweak scale is
the particleproduction mechanism of ref. [14].
We identify the relaxion with the 5th component of a U(1) gauge eld in the bulk. In
order to generate a potential for this eld, we add two nonabelian gauge groups Gf and
GF which also live in the bulk. We assume that these gauge groups con ne at the scales
GF , respectively. In order to ensure that con nement can be discussed using only
the zeromodes of the bulk gauge elds, we take
GF to be below the IR scale.
This can always be arranged by choosing the 5D gauge couplings and ranks of the gauge
groups appropriately.
We assume anomalous couplings of the relaxion
to the eld strengths G
f and GF
of the massless 4D gauge elds corresponding to Gf and GF , respectively:
S4D
Z
d4x
(x)
As we have discussed in section 2, these can arise from a ChernSimons coupling in the
bulk and an e ective anomalous coupling of A5 on the UV brane. But for now, we only
assume that F
f and postpone a concrete choice for the decay constants to section 4.
On the IR brane, we add a pair of chiral fermions
and
c in the fundamental and
antifundamental representation of GF , respectively. These fermions transform under a
chiral symmetry which we assume to be broken only by a Dirac mass m . This allows for
the terms in the action
S5D
Z
where gIR is the induced metric determinant on the IR brane. We have included a
higherdimensional coupling to the Higgs which is generically present and which we expect to be
suppressed by a scale near the Planck scale. Note that we will use the symbol H for both
the SU(2)doublet Higgs eld, writing the singlet combination jHj2 as H2 for simplicity,
and its VEV. It will be clear from context which one is meant. For simplicity, we also
ignore any numerical prefactors for now and set k = MPL. Similarly, we assume that all
parameters are real.
We will reinstate prefactors and phases later on. Performing the
integral over the extra dimension and canonically normalizing the elds gives
S4D
d4x m
1 +
c + h.c. ;
Z
.
Z
H2
2
IR
H2
2
IR
! ei =F ;
where we have rede ned e kLm
c. Note in particular that m
rede nition
! m , e kLH ! H, e 3kL=2
!
and similarly for
IR after the rede nition. Let us next perform the eld
while c is left invariant. Due to the nontrivial transformation of the path integral measure,
this chiral rotation removes the coupling of
in eq. (3.4) and transforms
eq. (3.6) to
{
{
{
If m is below the con nement scale of GF (which in turn is below
to the Higgsrelaxion potential after con nement. Parametrizing11 h
IR), this term contributes
ci =
3GF , this gives
HJEP07(218)3
cos
F
:
This has the same form as the potential with period F in eq. (3.2), including the coupling
to the Higgs. We can then make the identi cations
4F = m
3GF ;
MF2 =
I2R :
Next we need to generate the potentials with smaller period f . To this end, we use
a construction from ref. [1] and add fermions L and N on the IR brane with the same
Standard Model charges as the lepton doublet and the righthanded neutrino, respectively.
In addition, these fermions are in the fundamental representation of the gauge group Gf .
We also include fermions Lc and N c in the conjugate representations. Together they allow
for the terms in the action
Notice that we have not included a higherdimensional coupling to the Higgs. It could be
present but will be subdominant as we will see momentarily. Performing the integral over
the extra dimension and canonically normalizing the elds gives
Z
d4x
mL LLc + mN N N c + y HLN c + y~ HyLcN
+ h.c. ;
(3.12)
where we have rede ned e kLmL ! mL, e kLH ! H, e 3kL=2L ! L and similarly for mN ,
N and the conjugated elds. Note in particular that mL; mN .
IR after the rede nition.
Assuming that mN
satis es yy~H2
mL and restricting to a region in eld space where the Higgs VEV
m2L, we can integrate out L and Lc. This gives
S4D
Z
d4x
mN
11This is thus our de nition of the scale GF .
(3.14)
(3.15)
(3.16)
(3.17)
We can then perform the chiral rotation
N
! ei =f N ;
while N c is left invariant. This removes the coupling of
transforms eq. (3.13) to
to Tr hGf G
f i in eq. (3.4) and
relaxion potential after con nement. Parametrizing hN N ci =
3Gf , this gives
Provided that mN is below the con nement scale of Gf , this term contributes to the
Higgscos
f
This has the form of the potential with period f in eq. (3.2), including the coupling to the
Higgs. We can then make the identi cations
f4 = mN
3Gf ;
Mf2 =
mN mL
yy~
:
For su ciently small mN and mL, this allows for Mf . vEW as required in a technically
natural way. Notice that if we had instead relied on the higherdimensional operator in
eq. (3.5) to generate the barrier, we would have obtained Mf
IR
vEW. We discuss
constraints on the parameters of this construction in more detail in section 4. A summary
of the matter content on the IR brane is given in table 1.
We next reinstate the numerical prefactors and the phases of the parameters which
we have ignored so far. Let us denote the prefactor of the Higgs coupling in eq. (3.5) as
c H . We absorb possible phases in the fermionic condensates h
(relaxionindependent)
terms for GF and Gf into the mass parameters m
and mN ; mL,
respectively. Redoing the derivation above then gives
ci and hN N ci and any
cos
F
+ b
+ jc H j 2 cos
+ b H
H2
IR
F
jyy~j H2
jmN mLj
3
+ 2jmN j Gf cos
f
+ bN
f
cos
+ bNH
; (3.18)
where the complex phases are given by b = arg(m ), b H = arg(m c H ), bN = arg(mN )
and bNH = arg(yy~=mL). Note that this does generically not match the form of the potential
in eq. (3.2). Nevertheless the relaxation mechanism can still work. Indeed expanding the
rst two terms in the linear part of the cosines again gives the sliding term for the relaxion
and its linear coupling to the Higgs. In order to ensure that these terms have the same
sign as required, we need to demand that b
b H . As before, the Higgsindependent
barrier in the third term should be too small to stop the relaxion by itself. It is then
negligible for the dynamics and the phase bN has no consequences. The phase bNH in the
Higgsdependent barrier in the fourth term, on the other hand, slightly shifts the minimum
where the relaxion eventually stops but has no other consequences either.
To ensure that our calculation of the potentials is applicable, the masses of the fermion
pairs ; c and N; N c need to be below their respective condensation scales. This means
that the chiral symmetries under which these fermion pairs transform are only weakly
broken at the con nement scales. We then expect corresponding pseudoNambuGoldstone
bosons in the spectrum of composite states. As we discuss in appendix C, their contribution
to the potential factorizes from the remaining potential and they can be trivially integrated
out if the spectrum of fermions is doubled.
4
Conditions for successful relaxation
We now discuss various conditions that need to be ful lled for the relaxation mechanism
to be viable. In section 4.1, we derive general conditions on the parameters in the relaxion
potential in eq. (3.2). In section 4.2, we then discuss additional conditions that arise in
our warped model with a barrier at the electroweak scale.
4.1
General conditions
We begin our discussion of the evolution of the Higgs and relaxion with the Higgs
masssquared being positive and of order
2. In order to allow the relaxion to subsequently turn
the Higgs mass tachyonic, its average VEV ~ during this stage of the evolution needs to
satisfy
Since the lefthand side is bounded by 1, this in particular implies the condition
The relaxion stops rolling down its potential when the derivatives of the periodic
terms balance each other. We will nd below that MF
vEW and the term proportional to
cos( =F ) is thus dominated by the Higgsindependent part. On the other hand, the term
proportional to cos( =f ) needs to be dominated by the Higgsdependent part as discussed
in section 3. The relaxion then stops once the Higgs VEV becomes
where we have set sin( ~=F )
1. This is a good approximation as long as cos( ~=F ) is not
very close to its extrema. The parameters need to be chosen such that the combination
on the righthand side gives the electroweak scale vEW. In the following, we will use this
relation to trade f for vEW.
Notice that the Higgsdependent barrier H2 cos( =f ) in the potential contributes to
the Higgs mass. Imposing that this contribution be less than the electroweak scale (see
~
F
cos
&
2 MF2 :
4
F
2F &
MF :
H2
Mf2 Ff
4
F4 ;
f
(4.1)
(4.2)
(4.3)
e.g. ref. [64])12 gives the constraint f2 . Mf vEW which using eq. (4.3) leads to
F . vEW
F
f
1=4
Together with eq. (4.2), this gives a constraint on the cuto in our model as we discuss
in section 4.2. In order to ensure that the Higgs mass is scanned with su cient precision,
we need to demand that the change of the Higgsdependent term proportional to cos( =F )
over one period of the barrier,
f , is less than the electroweak scale. This gives the
constraint F . (MF vEW)1=2(F=f )1=4 which is weaker than eq. (4.4).
Furthermore, there are several requirements on the in ation sector for the relaxation
mechanism to be viable. If the relaxion is not the in aton, its energy density should be
subdominant compared to the in aton. The energy density in the minimum where the
relaxion eventually settles needs to be (close to) zero. This requires an additional constant
contribution that is added to the potential and chosen such that the energy density at
the minimum (nearly) vanishes. The tuning that is necessary to achieve this is just a
manifestation of the cosmological constant problem. The contribution of the relaxion to
the energy density relevant for in ation is then determined by how much it changes during
its evolution. Using eq. (4.1) in the potential of eq. (3.2) gives the condition
where HI is the Hubble rate during in ation. In addition, to ensure that our classical
analysis of the eld evolution is applicable, quantum
uctuations of the relaxion while it
roles down the potential should be su ciently small. Over one Hubble time, the relaxion
changes classically by (
)class:
are ( )quant:
HI . This leads to the condition
HI 2dV =d . Its quantum
uctuations, on the other hand,
Combining the last two inequalities, we get
condition Ne(
)class: &
using eq. (4.1) this gives
Finally, the number of efolds of in ation must be su ciently large to ensure that the
relaxion scans the required
eld range. Denoting the latter by
, this leads to the
. Provided that the relaxion is in the linear part of cos( =F ),
12This constraint can be slightly relaxed if one includes the barrier term in the scanning of the Higgs
mass [65]. One then still needs to impose that
potential are small. This gives a similar condition as eq. (4.4) but with an additional factor p
4 on the
f2 . 4 Mf vEW to ensure that loop corrections to the
righthand side.
HI &
MF
MPL
;
HI .
2F &
p
F
4=3
F F1=3 :
MF
MPL
Ne &
HI F MF
4
F
3=2
:
2
:
(4.5)
(4.6)
(4.7)
(4.8)
The resulting required number of efolds can be very large. We will not specify the
ination sector but will simply assume that it can be arranged to ful ll the conditions in
eqs. (4.5), (4.6) and (4.8). Possible complications in achieving this are discussed e.g. in
ref. [9]. Note also that the above conditions are somewhat alleviated if the e ect of the
time evolution of the Hubble rate during in ation is taken into accout [3].
We also need to ensure that the potential is radiatively stable. The potential is an
e ective theory with a cuto
determined by the con nement scales
gauge groups that give rise to the periodic terms (assuming they are smaller than the
cuto s associated with generating the H2terms in the potential). In the region of the
potential where the Higgs mass parameter13
m2H ( )
4
MFF2 cos
F
is smaller than these cuto s, the Higgs can give important corrections to the potential.
From the oneloop e ective potential, we nd
f
f
2
Gf
#
log
m2H ( )
!
;
(4.10)
GF
where we have neglected some subdominant terms. In the opposite region m2H ( )
or
2 , on the other hand, the corrections are strongly suppressed.14 This ensures that
2
Gf
the term proportional to m2H ( ) cos( =f ) gives only a small contribution to the
Higgsindependent barrier. In order to guarantee that the other term proportional to cos( =f )
is suppressed too, we require that
Provided that ; GF ; F . 4 MF the rst two terms in eq. (4.10) give small corrections to
the sliding term for the relaxion and do not a ect the dynamics. Finally if
f2 . 4 Mf vEW,
the cos2( =f )term is negligible compared to the Higgsdependent barrier when the Higgs
reaches the electroweak scale. Using eq. (4.3), this translates to the constraint
Gf . 4 Mf :
p
4 vEW
F
f
1=4
:
(4.9)
This is less stringent than eq. (4.4).
Conditions on the warped model
The Higgs is localized on or near the IR brane in our warped model. Its mass parameter
is then naturally of order I2R. We therefore identify the cuto of our relaxion model with
13Note that the Higgs mass parameter has an additional contribution from the cos( =f )term. Since it
is subdominant except in a small region of , we de ne eq. (4.9) without this contribution.
14See the oneloop e ective potential e.g. in eq. (2.64) of ref. [66] in the limit U 00
2
.
the IR scale:
IR :
(4.13)
As we have discussed in section 2, we can obtain the decay constants fB
IR from a
ChernSimons term in the bulk and fUV
on the UV brane. Since F
MP2L= IR from an e ective anomalous coupling
f is required, we identify F = MP2L= IR and f =
From the conditions in eqs. (4.2) and (4.4) and using that MF
IR, we obtain an
upper bound on the IR scale in our warped model:
IR .
vE2WMPL
1=3
4 104 TeV :
(4.14)
HJEP07(218)3
Note that this is slightly lower than the maximal cuto found in ref. [1]. The reason is
that there the bound on the cuto
is partly determined by the requirement of a
nite
viable window for the Hubble rate. In our warped model, the corresponding contraint in
eq. (4.7) is always trivially satis ed as we discuss below. The dominant bound on the cuto
instead involves the constraint in eq. (4.2) that the H2 cos( =F )term in the potential can
compensate for a Higgs mass near the cuto . This di erence arises because g is a free
parameter in the e ective description of ref. [1], whereas in our warped model g / 1=F is
determined in terms of other parameters.
We need to ensure that collider and avour bounds on the KK modes in our warped
model are ful lled. We have assumed that the Standard Model elds live in the bulk. The
dominant constraints then arise from CP violation in K
Kmixing and the electric dipole moment of the neutron. This requires [67, 68]:
IR & 10 TeV :
(4.15)
This also satis es constraints from electroweak precision tests without imposing a custodial
symmetry [69, 70] and on the radion (for a typical stabilization mechanism).
The potential leads to mixing between the Higgs and the relaxion.
This further
constrains the IR scale. We use results from ref. [64], where bounds on the parameter
2br =
f2 vEW=Mf controlling the mixing have been derived from several experiments ( fth
force, astrophysical and cosmological probes, beam dump, avor, and collider searches).
Using eq. (4.3), this translates to limits on
F and thereby on IR. For our case F = MP2L= IR
and f =
IR, the most stringent bound comes from the distortion of the di use
extragalactic background light spectrum due to relaxion late decays. This gives the constraint
which is more stringent than eq. (4.14).
We have discussed the con nement of Gf and GF in terms of only the massless modes
of the gauge elds in our extradimensional model. This is a good approximation provided
that the con nement scales are smaller than the KK mass scale:15
15It may be possible to alleviate this condition by including some of the KK modes in the e ective theory.
(4.16)
(4.17)
IR . 4 103 TeV
Gf ; GF . IR :
IR
IR
IR
IR
f
IR
f
3=2
IR
MP1L=2
vEW
10 TeV . IR . 4 103 TeV
barrier. The range for the IR scale is allowed by all phenomenological constraints considered in this
section.
Since
F
F .
GF and MF
IR according to eq. (3.10), it then follows from eq. (4.2) that
IR is required for successful relaxation. This in turn means that m ; GF
Since the fermions ; c are localized on the IR brane, the former condition can be
naturally ful lled. In order to discuss the latter condition, let us focus on GF = SU(N ) for
de niteness. If we estimate the con nement scale as the scale where the 4D gauge coupling
diverges, we nd (see e.g. ref. [71])16
GF
MPL
IR
MPL
24 2
11N(g5c)2k
;
(4.18)
GF is close to the IR scale if 24 2=(11N (g5c)2k)
where g5c is the 5D gauge coupling of GF . From this we see that the con nement scale of
1. This can be achieved for a wide range
of values for g5c and N but clearly requires a coincidence between two parameters which
are a priori not related. It may be possible to instead trigger the con nement of GF by
adding states on the IR brane and thereby achieve
We leave a detailed study of this question to future work.
GF
IR without such a coincidence.
We next consider constraints related to the fermions N; N c and L; Lc on the IR brane.
The last two terms in eq. (3.12) break the chiral symmetry of N; N c, in addition to their
Dirac mass. Loop corrections then contribute to the Dirac mass (see gure 3), leading to
the constraint
The Higgsdependent barrier can only stop the relaxion if Mf . vEW. Using eq. (3.17), the
loop contribution to mN then implies that
mN &
y1y~6m2L log( IR=mL) :
mL .
4 vEW
plog( IR=mL)
:
(4.19)
(4.20)
The electroweak doublets L; Lc can thus not be much heavier than the electroweak scale.
On the other hand, due to collider constraints on such particles, they cannot be much
lighter either. This limits their mass to a region near the electroweak scale. The question
why their mass should be near the scale that is dynamically generated via the relaxation
mechanism is the coincidence problem that we have mentioned in section 3. This problem
does not appear in the doublescanner scenario that we discuss in section 5.
16Branelocalized kinetic terms for the gauge eld would give another factor multiplying one side of this
relation. This would change the required relation between g5c and N accordingly.
Let us brie y pause to count parameters. The potential in eq. (3.2) has 7 dimensionful
parameters. Of these,
, MF and
F are of order
IR, whereas Mf is of order vEW.
Furthermore, F and f are given in terms of IR and MPL, while
f is xed as a function of
the other parameters via eq. (4.3). We can then express all parameters (up to O(1) factors)
uniquely in terms of IR (plus MPL and vEW). In table 2, we summarize the corresponding
relations and the phenomenologically viable range for the IR scale in our warped model.
Additional loop corrections arise in the e ective
eld theory at energies below
and
Gf as discussed in section 4.1. In particular, eq. (4.11) is an upper bound on the
con nement scale of Gf . An additional constraint arises from the requirement that the
mass of the lightest fermion after diagonalizing eq. (3.12) is smaller than the con nement
GF
scale (cf. the comment above eq. (3.16)). Together this gives
HJEP07(218)3
mN
yy~ vE2W
2mL
.
Gf . 4 vEW ;
where we have used Mf
vEW and that the largest Higgs VEV of interest is the electroweak
scale (as the relaxion stops before the Higgs VEV can grow even further). Using eq. (4.3)
and that
f .
Gf , this upper bound on
Gf gives an upper bound on
F which is less
stringent than eq. (4.4). On the other hand,
Gf can be very low provided that y; y~ and
mN are su ciently small. In order to ensure that Gf does not contribute to dark radiation
during big bang nucleosynthesis, its con nement scale should be larger than a few MeV:
Gf & O(few) MeV :
From eq. (4.3) and since f .
Gf , it follows that such low
Gf is only possible for the IR
scale near its lower bound in eq. (4.15). Furthermore, we need to ensure that the decay
of composite states does not destroy heavy elements during big bang nucleosynthesis. The
resulting limits have been worked out in ref. [72]. For
Gf = 10 MeV, mL = 500 GeV and
y = 2y~, it is found that y; y~ & 0:15 is required. This limit quickly becomes weaker for larger
Gf or smaller mL. On the other hand, the Yukawa couplings must not be too large in order
to satisfy bounds on the invisible decay width of the Higgs. The corresponding limit is
y; y~ . 0:1 for y = y~ and mL = 200 GeV which becomes slightly less stringent for larger mL.
Given that the fermions , c, L, Lc, N and N c are all localized on the IR brane, we
expect higherdimensional terms in the action. These include
S4D
Z
d4x c
m2
4 (
IR
c)2 + cNN
IR
m42N (N N c)2 + c N
m mN
4
IR
c N N c + h.c. :
(4.21)
(4.22)
!
(4.23)
The coe cients c , cNN and c N could be estimated using naive dimensional analysis. For
simplicity, we assume them to be real. After con nement, this gives the additional terms
in the Higgsrelaxion potential. Note that higherdimensional couplings involving LLc
either do not directly contribute to the potential as the pair LLc does not condense or the
contribution is very suppressed.17 The rst term in eq. (4.24) contributes to the sliding
term for the relaxion.
But for c
. 1 as expected from naive dimensional analysis,
this is suppressed compared to the sliding term in eq. (3.2) and can thus be neglected.
The second and third term, on the other hand, give additional contributions to the
Higgsindependent barrier for the relaxion. Again these are suppressed compared to the
barrier in eq. (3.2) and can be neglected. Adding higherdimensional couplings to the
Higgs in eq. (4.23) gives terms which can similarly be neglected.
Finally, we check constraints related to in ation. Due to the temperature and quantum
uctuations in deSitter space, we need to demand that the con nement scales of Gf and
GF are larger than the Hubble rate during in ation:
HI .
Gf ; GF :
(4.25)
For both
GF
IR and
Gf &
f given by eq. (4.3), this is less stringent than eq. (4.6)
from requiring that quantum
uctuations of the relaxion are negligible for the dynamics.
For F = MP2L= IR and since
F
MF
IR, the condition for having a
nite viable
window for the in ation scale in eq. (4.7) is trivially ful lled. Furthermore, the upper
limit on the in ation scale in eq. (4.6) is signi cantly smaller than the IR scale. We will
assume that the in ationary sector, which we do not specify further, is located on the UV
brane. Then HI
IR guarantees that the e ect of in ation on the geometry of the extra
dimension is negligible [73, 74]. Similarly, for a typical stabilization mechanism it ensures
that the extra dimension is safe from destabilization during in ation. In order to ensure
that the barrier for the relaxion is not removed during reheating after in ation, we demand
that the reheating temperature be below
Gf . This may require a relatively low reheating
temperature. As follows from eq. (4.22), it can still be su ciently high to allow for big
bang nucleosynthesis though. Under certain conditions, the reheating temperature may
also be higher than
Gf [1] (see also [75]).
To summarize, after imposing all the constraints the usual parameters of the relaxion
potential (1.1) in the model discussed in section 3.2 can be written just in terms of IR,
vEW and MPL as can be seen from table 2 and using eq. (3.3). The dimensionless couplings
of the relaxion potential and the relaxion mass are now determined as
g = g0 =
m
2
IR ;
MP2L
2
IR :
MPL
(4.26)
17A higherdimensional coupling ( c)yN N c would give a term proportional to cos( =F
=f ) in the
potential.
These couplings can thus be very small, provided that there is a large hierarchy between
the IR scale and the Planck scale. This in turn can be naturally achieved (i.e. without the
input of very small numbers) e.g. by means of the GoldbergerWise mechanism to stabilize
the extra dimension [42].
In addition to IR and MPL, the input parameters of the model discussed in section 3.2
include the con nement scales
couplings y and y~. Of these,
GF and
GF and m
Gf , the fermion masses m , mN and mL and the
are both required to be of order the IR scale.
Since the corresponding fermions are localized on the IR brane, the former condition can
be naturally ful lled, while the latter condition may require a coincidence of parameters
as discussed around eq. (4.18). After imposing this, the electroweak scale is determined
Gf , y, y~, mN and mL (plus IR and MPL) as follows from eqs. (3.17) and (4.3). Using
eq. (4.19) and the requirement that Mf . vEW as well as imposing that mL & vEW to satisfy
electroweak precision tests [72], we see that
vEW . mL .
4 vEW
plog( IR=vEW)
yy~ mL
log( IR=mL)
16 2
2
. mN . yy~ vEW :
mL
Using the range for mL in the range for mN , we then nd
yy~ vEW
log( IR=vEW)
16 2
. mN . yy~ vEW :
The fact that the electroweak doublets need to be close to the electroweak scale is the
coincidence problem discussed after eq. (4.20). Note that the condition for the mass of the
singlets can be naturally ful lled if it dominantly arises from the loop process in gure 3
(cf. eq. (4.19)). Demanding that the right electroweak scale is obtained, we then see from
eq. (4.3) that
3
Gf
mL
yy~ vE2W MIP2RL ;
6
where y and y~ need to be chosen such that eqs. (4.21) and (4.22) for
limits discussed below eq. (4.22) are ful lled.
Gf as well as the
In the in ationary sector, the allowed window of Hubble scales and the minimum
number of efolds are given by
2
MPL
IR . HI .
5=3
MIP2RL=3 ;
Ne &
M2P2L :
IR
In table 3, we give numerical values for two benchmark points. For the rst one, we
set the cuto to its maximal allowed value in our model,
IR = 4 103 TeV, and choose
y = 2y~ = 0:2 and mL = 700 GeV. For the second one, we choose the intermediate cuto
IR = 500 TeV as well as y = 2y~ = 0:04 and mL = 450 GeV. For both benchmark points,
we assume that mN is dominantly generated by the loop process in gure 3 in which case
the lower bound in eq. (4.28) is saturated (while our choices for mL satisfy the bound in
eq. (4.27)). This in particular leads to Mf
vEW as used for table 2. Both benchmark
(4.27)
(4.28)
(4.29)
(4.30)
(4.31)
g; g0
GeV
[7 10 6 ; 0:06]
min
IR = 4 103 TeV, y = 2y~ = 0:2 and mL = 700 GeV, while for the second line, IR = 500 TeV,
y = 2y~ = 0:04 and mL = 450 GeV.
points satisfy the constraints in eqs. (4.21) and (4.22) in addition to the relevant constraints
from colliders and big bang nucleosynthesis as can be seen from
gure 10 in ref. [72]. Note
that for cuto s
IR . 500 TeV, constraints from big bang nucleosynthesis can become
problematic. Indeed from eqs. (4.29) and (4.30) and the requirement that mN .
see that lower cuto s necessitate smaller values for yy~. If y
y~, this leads to longer lifetimes
for the lightest N N c bound states which arise from the con nement of Gf (see [72]). For
too long lifetimes, these decay during big bang nucleosynthesis. One way out is to choose
y
1
y~. The large coupling y then allows for relatively fast decays via an o shell
Z [72]. For example for IR = 10 TeV, y = 1; y~ = 10 9; mL = 800 GeV and assuming that
Gf , we
the mass of the lightest N N c bound state is
3 Gf , we nd that its lifetime is of order
1000 s while it can kinematically only decay into electron pairs or lighter states. This then
satis es the corresponding limit on the lifetime of order 104 s [76]. Alternatively, one could
add new decay channels for the bound states which can allow them to decay faster and
su ciently long before big bang nucleosynthesis. We leave a further investigation of this
possibility for future work.
Warping the doublescanner mechanism
A UV completion
5
5.1
potential
As discussed in section 3.1, the Higgsdependent barrier in the relaxion potential needs to
dominate over the Higgsindependent one once the Higgs VEV has reached the electroweak
scale. This requires that Mf . vEW which in turn necessitates to introduce new particles
coupled to the Higgs near the electroweak scale. We now discuss an interesting alternative
presented in ref. [2]. The idea is to have another axionlike scalar
with couplings in the
g
3
+
4
f
1
g~
+ g~
+
H2 !
Mf2
cos
f
(5.1)
and arrange its evolution such that it cancels o the Higgsindependent barrier. Note that
we have also included a term
cos( =f ) in the potential which will be important. The
remaining terms involving the relaxion are as in eq. (1.1). Similar to the relaxion, the
shiftsymmetry breaking couplings g and g~ of the eld
are taken to be very small.
Let us assume that
begins its evolution at some initial value
& (
+ g~ )=g~ so
that the Higgsindependent term in brackets in eq. (5.1) is unsuppressed. Provided that
f4 =f , the barrier term for the relaxion then dominates over its sliding term and the
to slide and it eventually reaches the value
+ g~ )=g~ . This removes the barrier
for the relaxion which can subsequently also slide down the potential. Both
and
then
roll down if they track each other according to the relation
+ g~ )=g~ . The resulting
growth of
after a while causes the Higgs mass parameter to turn tachyonic and H begins
to grow too. Shortly afterwards, the Higgsdependent barrier in eq. (5.1) then becomes so
' (
big that the relaxion stops again. Provided that
can no longer cancel this barrier, the
relaxion remains stuck. This mechanism works for certain ranges of parameters which we
review below. It then allows the backreaction from the Higgs to stop the relaxion once its
VEV has reached the electroweak scale even if Mf
vEW.
We rst present a construction to generate the required terms in the potential (see
also [10, 12]). This construction is, in fact, largely independent of the embedding into
warped space and can thus be used in other UV completions of the relaxion as well. It is
meant to serve as a proof of principle, and does not preclude the existence of simpler or
more complete models. Let us introduce an additional U(1) gauge symmetry in the bulk.
We identify the eld
with the 5th component of the gauge eld after imposing
appropriate boundary conditions. In order to generate the sliding term in eq. (5.1), we add an
anomalous coupling of
to a nonabelian gauge group GF on the UV brane using the
construction in section 2.2. We also introduce two chiral fermions
and c on the UV brane,
with a Dirac mass m
and in respectively the fundamental and antifundamental
representation of GF . These fermions have no explicit coupling to . Such a coupling is then
generated if we perform a chiral rotation of
to GF . If the gauge group con nes at some scale
GF
or c to remove the anomalous coupling of
> m , this gives rise to the potential
f is the decay constant resulting from the anomalous coupling and b = arg(m )
is the phase of the mass term. As we see later, we again have
=
IR. Expanding in
around the linear part of the trigonometric potential gives the sliding term in eq. (5.1) with
g
3
= jm j GF
F
3
IR
up to factors of order one.
Generating the coupling of to the periodic potential for
is somewhat more involved.
Notice that in eq. (5.1), the periodic potential for
appears with the same phase in the
last four terms (which for de niteness we have chosen as cos( =f )). Having the same phase
to a high precision in these a priori independent terms is in fact necessary for the
doublescanner mechanism to work. Let us assume that
instead couples to sin( =f ). Keeping
the phases for the other periodic terms xed, the barrier in eq. (5.1) then reads
(5.2)
(5.3)
4
f
g~
tan
+ g~
+
f
H2 !
Mf2
cos
f
:
(5.4)
Even if
can then initially cancel o the Higgsindependent terms (which depending on
the initial value for
may require
=g~ ), this cancellation is generically irreversibly
spoiled once
starts rolling. The same holds for a phase di erence less than
=2, if the
other periodic terms have di erent phases or if the decay constants in the periodic terms
di er from each other (in all cases down to values which are determined by the small
couplings in the potential).
In order to ensure the required phase and period structure, we extend the gauge
symmetry Gf in the bulk from section 3.2 to the product group Gf1
Gf2
Gf3
Gf4 . In
addition, we impose discrete symmetries Z2 and Z02 that interchange the groups as follows:
HJEP07(218)3
Gf1
x
Z02?y
Gf3
Z2
!
!
Z2
Gf2
?yxZ02
Gf4 :
This in particular imposes that the underlying groups (e.g. SU(N )) are the same for
Gf1 ; Gf2 ; Gf3 and Gf4 . We couple the 5D gauge eld AM that gives rise to
to the gauge eld
strengths of these four groups via ChernSimons terms as in section 2.1. We impose that
in the resulting anomalous couplings,
transforms as
under Z2, while it is even
under Z02 (by choosing the coe cients cB in eq. (2.13) to transform accordingly). This gives
S4D
Z
Tr hGf1 Gf1 i
Tr hGf2 Gf2 i + Tr hGf3 Gf3 i
Tr hGf4 Gf4 i
;
where the decay constant f
IR is equal for all gauge groups by virtue of the symmetries.
We also add anomalous couplings of
construction in section 2.2. We choose
to Gf3 and Gf4 on the UV brane, using the
to be even under Z2. This gives
S4D
Z
d4x
1
16 2 F~
Tr hGf3 Gf3 i + Tr hGf4 Gf4 i
;
where the decay constant F~
Z02 on the UV brane.
We do not add corresponding couplings to Gf1 and Gf2 though. This explicitly breaks the
On the IR brane, we next introduce four pairs of chiral fermions 1; 1c, 2; 2c, 3; 3c
and 4; 4c in the fundamental and antifundamental representation of Gf1 , Gf2 , Gf3 and
Gf4 , respectively. The fermion pairs interchange under Z2 consistent with eq. (5.5) but we
choose Z02 to be explicitly broken on the IR brane too. Including Dirac masses for the pairs
of chiral fermions and higherdimensional couplings to the Higgs, this gives
f is equal for the two gauge groups by virtue of the Z2.
(5.5)
(5.6)
(5.7)
S4D
Z
d4x
m 1 [ 1 1c + 2 2] 1 + c 1 2
c
+ m 3 [ 3 3c + 4 4] 1 + c 3 2
c
H2
+ h:c: ;
(5.8)
where the elds are already canonically normalized and m 1 ; m 3 .
IR. The coe cients
c 1 and c 3 are a priori di erent from each other and could be of order one or be suppressed
by a loop factor. We can now perform the chiral rotations
i
1 ! e f 1
3 ! ei f +i F~
3
2 ! e i f 2
4 ! e i f +i F~
while leaving 1c
, 2c
, 3c and 4c invariant. This moves
and
from eqs. (5.6) and (5.7)
into eq. (5.8). We assume that the gauge groups con ne at energies below the IR scale.
By virtue of the Z2 which is unbroken everywhere, the con nement scales of Gf1 and
Gf2 are identical, as are those of Gf3 and Gf4 . The condensates then are pairwise equal,
3
Gf1 and h 3 3ci = h 4 4ci =
3Gf3 . The resulting potential at low energies
IR
(5.9)
H2
2
IR
(5.10)
;
h 1 1ci = h 2 2ci =
reads
f
cos(b 1 ) + jc 1 j cos(d 1 ) 2
cos
~
F
+ b 3
+ jc 3 j cos
~
F
+ d 3
where b 1 = arg(m 1 ), d 1 = arg(m 1 c 1 ), b 3 = arg(m 3 ) and d 3 = arg(m 3
c 3 ) are given
by the complex phases of the parameters. We have kept track of the phases in order to
show that all terms are proportional to cos( =f ) without relative phase shifts as required.
This is guaranteed by the Z2 under which
!
and the potential is invariant. However,
note that we have tacitly assumed that the fermionic condensates are real. As we have
discussed at the end of section 3.2 and in appendix C, these phases are pionlike elds and
thus dynamical. Doubling the spectrum in order to ensure that the potential for these
pions factorizes from the remaining potential then
xes their phases to the same value for
all four condensates and leads to an additional overall minus sign in eq. (5.10).
On the other hand, the decay constants that appear in cos( =f ) between the rst and
second line of eq. (5.10) are the same due to the Z02 in the bulk. However, note that this
symmetry is broken on the UV brane by the couplings for
in eq. (5.7). Nevertheless we
expect that this does not a ect the decay constants for
in eq. (5.10) by virtue of the
nonrenomalization properties of anomalous couplings (see e.g. ref. [77]). Also any such
e ect would be strongly suppressed since F~
f . We leave a detailed study of this for
future work. Furthermore, we have allowed for the masses m 1 and m 3 being di erent
which breaks the Z02 also on the IR brane. This generically leads to a di erent running
of the gauge couplings of Gf1 and Gf2 compared to those of Gf3 and Gf4 and accordingly
di erent con nement scales
Gf1 and
Gf3 . However, it does not a ect the decay constants
for
in eq. (5.10) either as these are de ned not involving the gauge couplings of the
underlying gauge groups (cf. eqs. (2.15) and (2.16)). As follows from eqs. (3.7) to (3.9),
it is precisely the decay constants de ned in this way which determine the period of the
periodic potentials. These periods are thus not a ected by the di ering running of the gauge
couplings. Note also that the resulting di erence between the con nement scales can be
made arbitrarily small for example by increasing the number of colours of the gauge groups.
We can match with the potential in eq. (5.1) after expanding both eqs. (5.2) and (5.10)
in
around regions where the corresponding trigonometric potentials are linear. Both
trigonometric potentials can be in the linear part simultaneously for example for F
and b
b 3
. This also ensures that the right signs in the potential are obtained. In
addition to eq. (5.3), we can then identify
f4 = jm 1 j 3Gf1 ;
Mf =
IR
pjc 1 j
;
g~
3
jm 3 j Gf3 IR
jm 1 j 3Gf1
~
F
up to factors of order one. Notice that eq. (5.10) contains a term cos( =f ) cos( =F~ )H2
which is not included in eq. (5.1).
However,
provided
that for example
3
jm 3 j Gf3
3
jm 1 j Gf1 and jc 3 j is somewhat suppressed compared to jc 1 j, this only gives a
small correction to the Higgsdependent barrier and therefore does not a ect the dynamics.
Note that this would not be possible if the Z02 would be unbroken on the IR brane.
As in section 3.2, we next introduce fermions
and
c in the fundamental and
antifundamental representation of a nonabelian gauge symmetry GF to generate the sliding
term for the relaxion and its coupling to the Higgs. These fermions also allow us to generate
the term
cos( =f ) in eq. (5.1). To this end, we consider the higherdimensional operator
S4D
Z
d4x
c 1
m m 1
4
which we expect to be present since the relevant fermions live on the IR brane. The elds
are already canonically normalized and m ; m 1 .
IR. The coe cient c 1 is again of
order one or suppressed by a loop factor. Performing the chiral rotations in eqs. (3.7)
and (5.9), we nd below the con nement scales
GF
Gf1
Gf2
Gf3
Gf4
F
(5.11)
(5.12)
(5.13)
(5.14)
HJEP07(218)3
S4D
Z
d4x 4jc 1 j
3 3
jm j GF jm 1 j Gf1 cos
4
IR
F
+ b 1
cos
f
where b 1 = arg(c 1 m m 1 ). Expanding the trigonometric function of =F around its
linear part, we can identify
up to factors of order one. Note that the coupling in eq. (5.12) with 1 1c; 2 2c replaced by
3 3c; 4 4c gives an additional term cos( =F + =F~ ) cos( =f ) in the potential. We expect
g~ = jc 1 j
3
jm j GF
I3RF
3
that for example for jm 3 j Gf3
somewhat suppressed compared to c 1 , this does not signi cantly a ect the dynamics.
and the corresponding coe cient c 3 being
A summary of the matter content on the IR brane is given in table 4.
We have now generated all terms in the potential of eq. (5.1) as well as the sliding term
and coupling to the Higgs of the relaxion. In order to see if the potential parameters in
eqs. (5.3), (5.11) and (5.14) (plus eqs. (3.3) and (3.10) for g and g0) can take on values which
allow the doublescanner mechanism to work, we next discuss various constraints. We again
need to ensure that the conditions discussed in section 4.1 are ful lled. In particular, the
Higgs VEV once the relaxion stops is as before given by eq. (4.3). One di erence between
the potential parameters for the electroweakscale barrier and the double scanner is that
Mf
vEW in the former and Mf
IR in the latter. But in both scenarios, by construction
the Higgsindependent barrier plays no role and therefore only the combination
f2 =Mf is
relevant for the dynamics of the relaxion and Higgs. Using eq. (4.3) to x the Higgs VEV,
we can express this combination in terms of the decay constants and
F . Constraints
on these parameters therefore apply for both the electroweakscale barrier and the double
scanner. We can therefore conclude that the allowed range for the IR scale is again given
by table 2. Note that
f and Mf are di erent from those given in the table but the
combination
f2 =Mf and the other parameters in the table agree for both scenarios. In
particular, we again
nd that
On the other hand, the constraint on
IR and that
F
GF
m
IR is required.
Gf in eq. (4.17) can always be ful lled as follows
from eq. (4.4). Similarly, we see using eqs. (4.3), (4.6) and (4.15) that the constraints in
eqs. (4.22) and eq. (4.25) are automatically ful lled.
There are new conditions that are speci c to the doublescanner mechanism: the elds
and
track each other according to the relation
+ g~ )=g~ once the barrier is
' (
su ciently small provided that [2]
where g is given by eqs. (3.3) and (3.10). On the other hand,
can no longer cancel the
barrier that the Higgs generates once it obtains a VEV if [2]
g g~ & g g~ ;
g
2
g g~
. g g~
(5.15)
(5.16)
with
being the Higgs quartic coupling. We have F
F
F~ since these decay constants
all arise from anomalous couplings on the UV brane. Comparing eqs. (3.3) and (5.14), we
also see that g~
j
c 1 jg. On the other hand, the couplings g and g~ can a priori be quite
di erent. The gauge group GF that gives rise to the sliding term for
can in principle be
localized on the UV brane. Nevertheless we should still demand that its con nement scale
is below the IR scale to ensure that the e ective description for
is valid at the energy scale
where the potential is generated. In addition, we need to require that jm j .
GF . In order
3
to study one concrete example, let us assume that jm 1 j Gf1
to Z02 being only weakly broken on the IR brane). This gives g~
3
jm 3 j Gf3 (corresponding
g and g & g . The
conditions in eqs. (5.15) and (5.16) then simplify to
1
2
g jc 1 j & g ;
g j
c 1 j
. g :
(5.17)
This can be ful lled for a wide range of g if jc 1 j . 1=(2 ). This example shows that the
conditions for the doublescanner mechanism to work can be easily satis ed.
Finally, let us consider loop corrections to the potential. The doublescanner
mechanism cannot remove barriers from terms like cos2( =f ) [2]. Therefore these must be smaller
than the Higgsdependent barrier when the Higgs reaches the electroweak scale. For loop
corrections from the Higgs, this translates to the condition
f2 . 4 Mf vEW and in turn
to eq. (4.12) which is less stringent than the already imposed eq. (4.4). This also means
that eq. (4.11) can always be ful lled. Furthermore, in addition to eq. (5.12) we expect
higherdimensional operators like
S4D
Z
d4x c
4 (
IR
c)2 + c 1 1 4
and similar terms involving 3
; 3c; 4; 4c since the relevant fermions are all localized on the
IR brane. The coe cients are again of order one or suppressed by a loop factor and are
partly determined by the Z2. Assuming all parameters to be real for simplicity, below the
con nement scales this gives
The rst term gives a correction to the sliding term for the relaxion which is negligible for
c
. 1. The second term, on the other hand, gives another type of barrier that cannot be
cancelled by the doublescanner mechanism. It is su ciently suppressed compared to the
Higgsdependent barrier provided that
condition which for example for c 1 1
c 1
ful lled for the entire range of IR scales in table 2.
f2 . vEW I2R=(Mf pc 1 1 ). This in turn leads to a
1 is the same as eq. (4.4) and which is then
(5.18)
(5.19)
6
Conclusions
We have implemented the cosmological relaxation mechanism in a warped extra dimension.
The relaxion potential trades the hierarchy between the Planck and electroweak scale for
a technically natural hierarchy of decay constants. Warped extra dimensions are then a
natural choice for its UV completion as they can generate a large hierarchy of scales purely
from geometry. In our construction, the relaxion is identi ed with the scalar component of
an abelian gauge eld in the bulk, whose pro le automatically has a small overlap with the
UV brane. The warping generates the hierarchy from the Planck scale down to the scale
of the IR brane, which is then identi ed with the cuto
of the relaxion potential. From
there onwards, the Higgs mass is relaxed down to its physical value.
In section 2, we have presented a modelbuilding toolkit for generating anomalous
couplings of the relaxion to new, strong sectors. Depending on the localization of the
anomalous terms in the warped interval, hierarchically di erent decay constants for these
couplings may be obtained, including decay constants which are superPlanckian. A
benchmark model coupling the relaxion to the Higgs was constructed in section 3. The sliding
term and its coupling to the Higgs is generated through the condensation of a Dirac pair of
SM singlet fermions that live on the IR brane. The barrier term, on the other hand, is
generated close to the electroweak scale by the condensation of vectorlike fermions with the
same quantum numbers as one generation of SM leptons. These are also localized at the IR
brane, and have masses near or below the weak scale, but are a priori unrelated to it, leading
to the wellknown coincidence problem. In order to avoid this and achieve a larger scale for
the barrier term, a more elaborate construction is required. In section 5, we have presented
a warped UV completion for one such scenario, the doublescanner mechanism of ref. [2].
The constraints for the model, both in general and those speci c to the construction
of section 3, were discussed thoroughly in section 4, as well as the stability of the
potential under radiative corrections. The requirement of obtaining the correct Higgs VEV
may be used to
x the scale where the barrier term is generated in terms of the other
parameters. Then, we have found that the scale where the sliding and scanning terms are
generated needs to be of order the IR scale. Since the SM
elds live in the bulk, standard
avor constraints of RandallSundrum models push the minimum value of the IR scale to
& 10 TeV. The maximum cuto that we can achieve while ensuring that all theoretical
and phenomenological constraints are ful lled is
4 106 GeV.
In this work, we have focused on in ation to provide a friction term for the
slowroll of the relaxion, but interesting alternatives such as the particleproduction mechanism
of ref. [14] exist. It would be interesting to explore how such constructions may be
implemented in warped space. The framework that we have described naturally allows for
hierarchical decay constants for axionlike elds to be generated. As such it presents many
further opportunities for model building, not limited to relaxion models, such as
applications to in ation or dark matter. Another interesting possibility for generating this
hierarchy is to consider a more general geometry with more than one AdS5 throat [
78
].
Acknowledgments
LdL thanks DESY for hospitality during his stay, where part of this work was completed
and acknowledges support by the S~ao Paulo Research Foundation (FAPESP) under grants
2012/214369 and 2015/253930. BvH thanks Fermilab for hospitality while part of this
work was completed. This visit has received funding/support from the European Union's
Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant
agreement No 690575. BvH also thanks the Fine Theoretical Physics Institute at the
University of Minnesota for hospitality and partial support. The work of CSM was supported
by the Alexander von Humboldt Foundation, in the framework of the Sofja Kovalevskaja
Award 2016, endowed by the German Federal Ministry of Education and Research. The
authors would like to thank Aqeel Ahmed, Enrico Bertuzzo, Zackaria Chacko, Giovanni Grilli
di Cortona, Adam Falkowski, Gero von Gersdor , Tony Gherghetta, Christophe Grojean,
Roni Harnik, Arthur Hebecker, Ricardo D. Matheus, Enrico Morgante, Eduardo Ponton,
Pedro Schwaller, Marco Serone, Geraldine Servant, Alexander Westphal and Alexei Yung
for useful discussions and comments.
A
An anomalous coupling on the UV brane from two throats
The interaction in eq. (2.17) should be understood as an e ective coupling that can for
example arise from a ChernSimons term in a second throat as we now brie y discuss.
More details will be presented in [
78
]. To this end, we consider a setup with two warped
spaces which are glued together at a common UV brane but each slice still has its own
IR brane. For simplicity, we assume that both slices have the same AdS scale k. Let us
denote the coordinates along the extra dimension in the two throats as z1 and z2, with
metric in each throat again given by eq. (2.1). The coordinates match at the common UV
brane at zUV1 = zUV2 = 1=k, while the IR branes are at zIR1 = ekL1=k and zIR2 = ekL2=k.
We then introduce an abelian gauge boson which propagates in both throats (see e.g. [44]).
We break the gauge symmetry on the two IR branes by imposing the boundary conditions
in eq. (2.6) but leave it unbroken on the UV brane. This allows for one massless mode
from A5 which lives in both throats with wavefunction A5 = N a(zi) 1 in a given throat
(the wavefunction is continuous at the UV brane). We will be interested in the case where
one throat is signi cantly longer than the other. The normalization constant N is then
dominated by the longer throat. Choosing L1 > L2 without loss of generality, we have
zIR2, which gives N
' g4p2kL1e kL1 with g4 de ned as before. Let us next
introduce a ChernSimons coupling of AM to a nonabelian gauge group, where we choose
the coupling to be localized in the second throat:
S5D
Z
d x
4 Z zIR2
cb2 MNP QRAM Tr [GNP GP Q] :
Notice that the coupling to A5 from this resembles eq. (2.17) with the function replaced
by the integral over A5 in the second throat. In the limit of a very short second throat
with zIR2
O(few) zUV, we can think of this integral as a smearedout function.
Correspondingly we expect the decay constant of
in this limit to agree with eq. (2.19). Let us
again restrict ourselves to the zeromode of the nonabelian gauge eld. Integrating over
the extra dimension, we in particular nd
with decay constant given by
S4D
Z
d4x
1
Tr [G
fB2 ' cb2 g4p2kL1
k ekL1 2kL2
(A.1)
(A.2)
(A.3)
or fB2
I2R2= IR1. For a very short second throat with L1
2L2, this indeed agrees with
eq. (2.19). On the other hand, the twothroat construction allows for more general choices
for the decay constant, with a continuum between MP2L= IR1 and
IR1 (as
IR1 <
IR2 by
assumption). The resulting phenomenology and the details of the construction will be
presented in [
78
].
In this appendix, we brie y review how charged bulk fermions can give rise to
ChernSimons terms. We consider a bulk fermion
which couples to both the nonabelian gauge
group and the U(1) from section 2.1. The action reads
S5D
Z
d4xdz pg
iD=
+ m
;
iAM with GM being the nonabelian
gauge eld (and AM the U(1) gauge eld). In order to see that this gives the same anomaly
as a ChernSimons term, we can perform a eld rede nition [
79, 80
]
(B.1)
(B.2)
zIR) :
(B.3)
(B.4)
Z z
z0
! exp i
dz~A5(x; z~)
;
where the constant z0 can be chosen according to convenience. However, the eld rede
nition is anomalous on the branes18 and transforms the action into (see [81{84])
S5D ! S5D +
Z
d4xdz
Z z
z0
dz~A5(x; z~)
48 2 Tr [G
G ]
UV (z
zUV ) + IR (z
The coe cients UV and IR depend on the boundary conditions on the two branes for the
lefthanded component
L of the bulk fermion (which in turn xes the boundary conditions
of the righthanded component
R). If
L is even (odd) on a given brane,
= 1( 1).
Let us rst assume
UV =
IR in which case
does not have a massless mode. From
eq. (B.3), we then get the anomalous coupling of in eq. (2.15) with
cB =
IR :
4
Notice that this is independent of z0. In the opposite case
UV =
IR, on the other hand,
cB depends on z0. But then
has a massless mode which contributes to the anomaly
and which cancels the dependence on z0. If the ChernSimons term arises from such a
bulk fermion, any perturbative contribution to the potential for A5 can be su ciently
suppressed by making the bulk mass of the fermion somewhat larger than the AdS scale
(see e.g. [34, 47]).
C
Pionlike elds in the relaxion potential
In this appendix, we include the pionlike elds which arise from the condensing fermions on
the IR brane and which contribute to the potential. Let us focus on ; c for de niteness.
As usual, we can parametrize the pseudoNambuGoldstone boson corresponding to the
breaking of the chiral symmetry of ; c by the model eld U = exp(i
=f ) with a
18We note that, e.g. for SU(N), there is an additional SU(N)3 anomaly. It can be canceled by adding
another bulk fermion, uncharged under U(1), with opposite boundary conditions from .
decay constant of order f
this gives
GF . After con nement then h
3 U . From eq. (3.8),
m
where for simplicity we again ignore phases and prefactors. Since F
f , generically
settles into its minimum
min = f
f
rst after which the potential becomes
independent of . This problem is remedied for example by introducing another pair of
chiral fermions ~ ~c with the same quantum numbers. Instead of eq. (3.6) we then have
;
S4D
Z
1 +
H2
2
IR
[m
c + m ~ ~ ~c] + h.c. :
(C.2)
Similar to the up and down quark in the Standard Model, the fermions transform under
an approximate SU(2)L
SU(2)R symmetry which is spontaneously broken to a
diagonal SU(2)V by the condensates and explicitly but weakly broken by their masses. The
corresponding pseudoNambuGoldstone bosons are parametrized as
U = ei =f
with
0
p
2 +!
0
:
We next perform the chiral rotation
! ei 2F ;
~ ! ei 2F ~
with
c and ~c left invariant to remove the coupling of
to Tr GF GF
in eq. (3.4).
For this choice of chiral rotation, no kinetic mixing between the relaxion and the pions is
induced (see ref. [85]). Choosing m
= m ~ for simplicity, from eq. (C.2) we get below the
con nement scale
V ( ; H)
m
3
GF
1 +
H2
2
IR
cos
2F
cos
f
;
where
q
( 0 )2 + 2 +
. The potential for the pions and relaxion thus factorizes and
no longer vanishes once the pions settle into their minimum. This is similar to what happens
for the axion and the pion of the Standard Model, see ref. [59]. For the generalization of
the potential to the case m 6
= m ~, see also ref. [59]. The potential after minimization
with respect to the pion then still leads to a nonvanishing potential for the relaxion but
the latter is no longer a simple cosine.
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[1] P.W. Graham, D.E. Kaplan and S. Rajendran, Cosmological relaxation of the electroweak
scale, Phys. Rev. Lett. 115 (2015) 221801 [arXiv:1504.07551] [INSPIRE].
(C.1)
(C.3)
(C.4)
(C.5)
[2] J.R. Espinosa, C. Grojean, G. Panico, A. Pomarol, O. Pujolas and G. Servant, Cosmological
Higgsaxion interplay for a naturally small electroweak scale, Phys. Rev. Lett. 115 (2015)
251803 [arXiv:1506.09217] [INSPIRE].
JHEP 02 (2016) 077 [arXiv:1507.08649] [INSPIRE].
[3] S.P. Patil and P. Schwaller, Relaxing the electroweak scale: the role of broken dS symmetry,
[4] J. Jaeckel, V.M. Mehta and L.T. Witkowski, Musings on cosmological relaxation and the
hierarchy problem, Phys. Rev. D 93 (2016) 063522 [arXiv:1508.03321] [INSPIRE].
(2016) 166 [arXiv:1509.00047] [INSPIRE].
162 [arXiv:1509.00834] [INSPIRE].
063 [arXiv:1509.03583] [INSPIRE].
[arXiv:1510.00710] [INSPIRE].
[INSPIRE].
[7] O. Matsedonskyi, Mirror cosmological relaxation of the electroweak scale, JHEP 01 (2016)
two eld relaxion mechanism, JHEP 09 (2016) 150 [arXiv:1602.04812] [INSPIRE].
[11] T. Kobayashi, O. Seto, T. Shimomura and Y. Urakawa, Relaxion window, Mod. Phys. Lett.
A 32 (2017) 1750142 [arXiv:1605.06908] [INSPIRE].
[12] J.L. Evans, T. Gherghetta, N. Nagata and M. Peloso, Lowscale Dterm in ation and the
relaxion mechanism, Phys. Rev. D 95 (2017) 115027 [arXiv:1704.03695] [INSPIRE].
[13] N. ArkaniHamed, T. Cohen, R.T. D'Agnolo, A. Hook, H.D. Kim and D. Pinner, Solving the
hierarchy problem at reheating with a large number of degrees of freedom, Phys. Rev. Lett.
117 (2016) 251801 [arXiv:1607.06821] [INSPIRE].
[14] A. Hook and G. MarquesTavares, Relaxation from particle production, JHEP 12 (2016) 101
[arXiv:1607.01786] [INSPIRE].
[arXiv:1507.07525] [INSPIRE].
[15] E. Hardy, Electroweak relaxation from nite temperature, JHEP 11 (2015) 077
[hepph/0409138] [INSPIRE].
Lett. B 738 (2014) 301 [arXiv:1404.3511] [INSPIRE].
[16] K. Choi and S.H. Im, Realizing the relaxion from multiple axions and its UV completion with
high scale supersymmetry, JHEP 01 (2016) 149 [arXiv:1511.00132] [INSPIRE].
[17] D.E. Kaplan and R. Rattazzi, Large eld excursions and approximate discrete symmetries
from a clockwork axion, Phys. Rev. D 93 (2016) 085007 [arXiv:1511.01827] [INSPIRE].
[18] N. Fonseca, L. de Lima, C.S. Machado and R.D. Matheus, Large eld excursions from a few
site relaxion model, Phys. Rev. D 94 (2016) 015010 [arXiv:1601.07183] [INSPIRE].
[19] J.E. Kim, H.P. Nilles and M. Peloso, Completing natural in ation, JCAP 01 (2005) 005
[20] K. Harigaya and M. Ibe, Simple realization of in aton potential on a Riemann surface, Phys.
07 (2014) 074 [arXiv:1404.6923] [INSPIRE].
JHEP 11 (2014) 147 [arXiv:1407.4893] [INSPIRE].
[21] K. Choi, H. Kim and S. Yun, Natural in ation with multiple subPlanckian axions, Phys.
[22] T. Higaki and F. Takahashi, Natural and multinatural in ation in axion landscape, JHEP
[23] K. Harigaya and M. Ibe, Phase locked in ation. E ectively transPlanckian natural in ation,
[24] M. Peloso and C. Unal, Trajectories with suppressed tensortoscalar ratio in aligned natural
in ation, JCAP 06 (2015) 040 [arXiv:1504.02784] [INSPIRE].
monodromy, JHEP 02 (2018) 124 [arXiv:1610.05320] [INSPIRE].
[25] L. McAllister, P. Schwaller, G. Servant, J. Stout and A. Westphal, Runaway relaxion
JHEP 10 (2017) 018 [arXiv:1704.07831] [INSPIRE].
arXiv:1705.10162 [INSPIRE].
[29] G.F. Giudice and M. McCullough, Comment on \disassembling the clockwork mechanism",
[30] K. Choi, S.H. Im and C.S. Shin, General continuum clockwork, arXiv:1711.06228 [INSPIRE].
[31] C.T. Hill, S. Pokorski and J. Wang, Gauge invariant e ective Lagrangian for KaluzaKlein
modes, Phys. Rev. D 64 (2001) 105005 [hepth/0104035] [INSPIRE].
[32] N. ArkaniHamed, A.G. Cohen and H. Georgi, (De)constructing dimensions, Phys. Rev. Lett.
86 (2001) 4757 [hepth/0104005] [INSPIRE].
Rev. Lett. 83 (1999) 3370 [hepph/9905221] [INSPIRE].
[33] L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension, Phys.
[34] R. Contino, Y. Nomura and A. Pomarol, Higgs as a holographic pseudoGoldstone boson,
Nucl. Phys. B 671 (2003) 148 [hepph/0306259] [INSPIRE].
[35] Y. Hosotani and M. Mabe, Higgs boson mass and electroweakgravity hierarchy from
dynamical gaugeHiggs uni cation in the warped spacetime, Phys. Lett. B 615 (2005) 257
[hepph/0503020] [INSPIRE].
[hepph/0610336] [INSPIRE].
[36] A. Falkowski, About the holographic pseudoGoldstone boson, Phys. Rev. D 75 (2007) 025017
[37] N. ArkaniHamed, H.C. Cheng, P. Creminelli and L. Randall, Extra natural in ation, Phys.
Rev. Lett. 90 (2003) 221302 [hepth/0301218] [INSPIRE].
[38] I.R. Klebanov and M.J. Strassler, Supergravity and a con ning gauge theory: duality cascades
and SB resolution of naked singularities, JHEP 08 (2000) 052 [hepth/0007191] [INSPIRE].
[39] S.B. Giddings, S. Kachru and J. Polchinski, Hierarchies from
uxes in string
compacti cations, Phys. Rev. D 66 (2002) 106006 [hepth/0105097] [INSPIRE].
[40] T. Gherghetta, B. von Harling and N. Setzer, A natural little hierarchy for RS from
accidental SUSY, JHEP 07 (2011) 011 [arXiv:1104.3171] [INSPIRE].
[41] T. Gherghetta, A holographic view of beyond the Standard Model physics, in Physics of the
large and the small, TASI 09, proceedings of the Theoretical Advanced Study Institute in
Elementary Particle Physics, Boulder, CO, U.S.A., 1{26 June 2009, World Scienti c,
Singapore, (2011), pg. 165 [arXiv:1008.2570] [INSPIRE].
[42] W.D. Goldberger and M.B. Wise, Modulus stabilization with bulk elds, Phys. Rev. Lett. 83
(1999) 4922 [hepph/9907447] [INSPIRE].
061 [hepph/0611278] [INSPIRE].
their own, Phys. Rev. D 72 (2005) 095018 [hepph/0505001] [INSPIRE].
[45] C. Csaki, J. Hubisz and P. Meade, TASI lectures on electroweak symmetry breaking from
extra dimensions, in Physics in D
4. Proceedings, Theoretical Advanced Study Institute in
elementary particle physics, TASI 2004, Boulder, CO, U.S.A., 6 June{2 July 2004, World
Scienti c, Singapore, (2005), pg. 703 [hepph/0510275] [INSPIRE].
[46] K.W. Choi, A QCD axion from higher dimensional gauge eld, Phys. Rev. Lett. 92 (2004)
101602 [hepph/0308024] [INSPIRE].
[hepph/0302087] [INSPIRE].
[48] B. Grzadkowski and J. Wudka, Note on the strong CP problem from a 5dimensional
perspective, Phys. Rev. D 77 (2008) 096004 [arXiv:0705.4307] [INSPIRE].
[49] N. ArkaniHamed, L. Motl, A. Nicolis and C. Vafa, The string landscape, black holes and
gravity as the weakest force, JHEP 06 (2007) 060 [hepth/0601001] [INSPIRE].
[50] A. Hebecker, F. Rompineve and A. Westphal, Axion monodromy and the weak gravity
conjecture, JHEP 04 (2016) 157 [arXiv:1512.03768] [INSPIRE].
[51] L.E. Iban~ez, M. Montero, A. Uranga and I. Valenzuela, Relaxion monodromy and the weak
gravity conjecture, JHEP 04 (2016) 020 [arXiv:1512.00025] [INSPIRE].
[52] B. Heidenreich, M. Reece and T. Rudelius, Weak gravity strongly constrains large eld axion
in ation, JHEP 12 (2015) 108 [arXiv:1506.03447] [INSPIRE].
[53] B. Heidenreich, M. Reece and T. Rudelius, Sharpening the weak gravity conjecture with
dimensional reduction, JHEP 02 (2016) 140 [arXiv:1509.06374] [INSPIRE].
[54] T. Rudelius, Constraints on axion in ation from the weak gravity conjecture, JCAP 09
(2015) 020 [arXiv:1503.00795] [INSPIRE].
[55] J. Brown, W. Cottrell, G. Shiu and P. Soler, Fencing in the swampland: quantum gravity
constraints on large eld in ation, JHEP 10 (2015) 023 [arXiv:1503.04783] [INSPIRE].
[56] A. Hebecker, P. Mangat, F. Rompineve and L.T. Witkowski, Winding out of the swamp:
evading the weak gravity conjecture with Fterm winding in ation?, Phys. Lett. B 748 (2015)
455 [arXiv:1503.07912] [INSPIRE].
[57] P. Saraswat, Weak gravity conjecture and e ective eld theory, Phys. Rev. D 95 (2017)
025013 [arXiv:1608.06951] [INSPIRE].
[58] L.E. Iban~ez and M. Montero, A note on the WGC, e ective eld theory and clockwork within
string theory, JHEP 02 (2018) 057 [arXiv:1709.02392] [INSPIRE].
[59] G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G. Villadoro, The QCD axion, precisely,
JHEP 01 (2016) 034 [arXiv:1511.02867] [INSPIRE].
96 (2017) 113007 [arXiv:1708.00010] [INSPIRE].
arXiv:1711.00858 [INSPIRE].
HJEP07(218)3
B 586 (2000) 141 [hepph/0003129] [INSPIRE].
[64] T. Flacke, C. Frugiuele, E. Fuchs, R.S. Gupta and G. Perez, Phenomenology of
relaxionHiggs mixing, JHEP 06 (2017) 050 [arXiv:1610.02025] [INSPIRE].
[65] K. Choi and S.H. Im, Constraints on relaxion windows, JHEP 12 (2016) 093
[arXiv:1610.00680] [INSPIRE].
[INSPIRE].
[66] M. Sher, Electroweak Higgs potentials and vacuum stability, Phys. Rept. 179 (1989) 273
Rev. D 71 (2005) 016002 [hepph/0408134] [INSPIRE].
JHEP 09 (2008) 008 [arXiv:0804.1954] [INSPIRE].
[67] K. Agashe, G. Perez and A. Soni, Flavor structure of warped extra dimension models, Phys.
[68] C. Csaki, A. Falkowski and A. Weiler, The avor of the composite pseudoGoldstone Higgs,
[69] S. Casagrande, F. Goertz, U. Haisch, M. Neubert and T. Pfoh, Flavor physics in the
RandallSundrum model: I. Theoretical setup and electroweak precision tests, JHEP 10
(2008) 094 [arXiv:0807.4937] [INSPIRE].
[70] M. Bauer, S. Casagrande, U. Haisch and M. Neubert, Flavor physics in the RandallSundrum
model: II. Treelevel weakinteraction processes, JHEP 09 (2010) 017 [arXiv:0912.1625]
[INSPIRE].
[INSPIRE].
[71] C. Csaki, J. Hubisz and S.J. Lee, Radion phenomenology in realistic warped space models,
Phys. Rev. D 76 (2007) 125015 [arXiv:0705.3844] [INSPIRE].
[72] H. Beauchesne, E. Bertuzzo and G. Grilli di Cortona, Constraints on the relaxion mechanism
with strongly interacting vectorfermions, JHEP 08 (2017) 093 [arXiv:1705.06325]
[73] G.F. Giudice, E.W. Kolb, J. Lesgourgues and A. Riotto, Transdimensional physics and
in ation, Phys. Rev. D 66 (2002) 083512 [hepph/0207145] [INSPIRE].
[74] S.H. Im, H.P. Nilles and A. Trautner, Exploring extra dimensions through in ationary tensor
modes, JHEP 03 (2018) 004 [arXiv:1707.03830] [INSPIRE].
Phys. Rev. D 95 (2017) 075008 [arXiv:1611.08569] [INSPIRE].
[75] K. Choi, H. Kim and T. Sekiguchi, Dynamics of the cosmological relaxation after reheating,
[76] M. Kawasaki, K. Kohri, T. Moroi and Y. Takaesu, Revisiting bigbang nucleosynthesis
constraints on longlived decaying particles, Phys. Rev. D 97 (2018) 023502
[arXiv:1709.01211] [INSPIRE].
[77] A. Eichhorn, H. Gies and D. Roscher, Renormalization ow of axion electrodynamics, Phys.
Rev. D 86 (2012) 125014 [arXiv:1208.0014] [INSPIRE].
125009 [arXiv:1002.3160] [INSPIRE].
large hierarchies, JHEP 01 (2004) 032 [hepth/0311233] [INSPIRE].
HJEP07(218)3
(2001) 395 [hepth/0103135] [INSPIRE].
Phys. Lett. B 525 (2002) 169 [hepth/0110073] [INSPIRE].
Phys. Lett. B 169 (1986) 73 [INSPIRE].
[5] R.S. Gupta , Z. Komargodski , G. Perez and L. Ubaldi , Is the relaxion an axion? , JHEP 02 [6] B. Batell , G.F. Giudice and M. McCullough , Natural heavy supersymmetry , JHEP 12 ( 2015 ) [8] L. Marzola and M. Raidal , Natural relaxation, Mod. Phys. Lett. A 31 ( 2016 ) 1650215 [9] S. Di Chiara , K. Kannike , L. Marzola , A. Racioppi , M. Raidal and C. Spethmann , Relaxion cosmology and the price of netuning , Phys. Rev. D 93 ( 2016 ) 103527 [arXiv: 1511 .02858] [10] J.L. Evans , T. Gherghetta , N. Nagata and Z. Thomas , Naturalizing supersymmetry with a Rev. D 90 ( 2014 ) 023545 [arXiv: 1404 .6209] [INSPIRE].
[26] G.F. Giudice and M. McCullough , A clockwork theory , JHEP 02 ( 2017 ) 036 [27] A. Ahmed and B.M. Dillon , Clockwork Goldstone bosons, Phys. Rev. D 96 ( 2017 ) 115031 [28] N. Craig , I. Garcia Garcia and D. Sutherland , Disassembling the clockwork mechanism, [43] T. Flacke , B. Gripaios , J. MarchRussell and D. Maybury , Warped axions, JHEP 01 ( 2007 ) [44] G. Cacciapaglia , C. Csaki , C. Grojean , M. Reece and J. Terning , Top and bottom: a brane of [47] L. Pilo , D.A.J. Rayner and A. Riotto , Gauge quintessence, Phys. Rev. D 68 ( 2003 ) 043503 [60] A. Nelson and C. PrescodWeinstein , Relaxion: a landscape without anthropics , Phys. Rev . D [61] K.S. Jeong and C.S. Shin , PecceiQuinn relaxion , JHEP 01 ( 2018 ) 121 [arXiv: 1709 .10025] [62] O. Davidi , R.S. Gupta , G. Perez , D. Redigolo and A. Shalit , The NelsonBarr relaxion , [63] T. Gherghetta and A. Pomarol , Bulk elds and supersymmetry in a slice of AdS, Nucl . Phys.
[78] N. Fonseca , B. von Harling , L. de Lima and C.S. Machado , in preparation.
[79] C. Csaki , J. Heinonen , J. Hubisz and Y. Shirman , Odd decays from even anomalies: gauge mediation signatures without SUSY , Phys. Rev. D 79 ( 2009 ) 105016 [arXiv: 0901 .2933] [80] D. Bunk and J. Hubisz , Revealing RandallSundrum hidden valleys , Phys. Rev. D 81 ( 2010 ) [82] N. ArkaniHamed , A.G. Cohen and H. Georgi , Anomalies on orbifolds, Phys. Lett. B 516 [83] C.A. Scrucca , M. Serone , L. Silvestrini and F. Zwirner , Anomalies in orbifold eld theories, [84] B. Gripaios and S.M. West , Anomaly holography, Nucl. Phys. B 789 ( 2008 ) 362