A clockwork WIMP
Revised: June
clockwork WIMP
Thomas Hambye 0 1 2
Daniele Teresi 0 1 2
Michel H.G. Tytgat 0 1 2
0 ories in Higher Dimensions , Neutrino Physics
1 Boulevard du Triomphe , CP225, 1050 Brussels , Belgium
2 Service de Physique Theorique
3 Universite Libre de Bruxelles
We embed a thermal dark matter (DM) candidate within the clockwork framework. This mechanism allows to stabilize the DM particle over cosmological time because it suppresses its decay into Standard Model (SM) particles. At the same time, pair annihilations are unsuppressed, so that the relic density is set by the usual freezeout of the DM particle from the thermal bath. The slow decay of the DM candidate is induced by clockwork" particles that can be quite light (rather than at some GUT or Planck scale) and could be searched for at current or future colliders. According to the scenario considered, the very same particles also mediate the annihilation process, thus providing a connection between DM annihilation and DM decay, and xing the mass scale of the clockwork states, otherwise unconstrained, to be in the TeV range or lighter. We then show how this setup can minimally emerge from the deconstruction of an extra dimension in at spacetime. Finally, we argue that the clockwork mechanism that we consider could induce Majorana neutrino masses, with a seesaw scale of order TeV or less and Yukawa couplings of order unity.
Beyond Standard Model; Cosmology of Theories beyond the SM; Field The

A
1 Introduction
2
3
4
5
6
1
Dark matter phenomenology
Clockwork WIMP from deconstruction
Majorana neutrino masses
Conclusions
Introduction
model, it could be absolutely stable or very long lived (see e.g. [1] for a general discussion
of the stability issue). Absolute stability may be due, for instance, to a gauge symmetry,
as for the electron in the SM. Alternatively, DM could be longlived for accidental reasons,
as the proton, whose stability is associated to a global continuous symmetry, such as the
baryon number. The possibility that DM may be long lived is actually very interesting as
its decay could be probed through indirectdetection searches. Constraints from
uxes of
cosmic rays impose that the lifetime of DM candidates in the WIMP mass range, which
we will consider in the sequel, must be typically larger than
1026 sec. The most
straightforward way to get such a large lifetime is to assume that the decay is induced
by the exchange of very heavy particles, encapsulated at low energies by masssuppressed
e ective operators. Assuming couplings of order one, instability associated to a
dimension5 interaction would require that the heavy degrees of freedom lie way above the Planck
scale. In contrast, a dimension6 operator only requires that these particles are around the
GUT scale. This is appealing but also means that the nature of these very heavy degrees
of freedom would be impossible to test directly.
Alternatively, one could assume that
the particles that trigger the decay of DM are much lighter, but then with much smaller
couplings (something that we are usually reluctant to assume, unless protected from large
quantum corrections by a symmetry). However this cannot be tested either.
In the following we propose that, along a clockwork mechanism [2{7], a DM particle
could be made accidentally stable but have a decay into SM particles induced by order
unity interactions with particles that could be produced at colliders. Such a clockwork
mechanism does not prevent the DM to undergo fast annihilations into SM particles (or
{ 1 {
hiddensector particles) so that its relic density is determined by the standard freezeout
mechanism, i.e. it is a WIMP.
As we will see, we need to introduce a lot of new
elds (and speci c couplings). One
may right away wonder whether this is worth explaining the mere stability of dark matter.
the presence of all these elds can be the 4dimensional consequence of theories with
extradimensions. We will show in particular that a fermion clockwork chain can arise in a simple
way from the deconstruction of a single eld in
at spacetime.
The plan of the article is as follows. In the next section 2 we build the model, based
on the clockwork mechanism and expose the basic properties of the DM candidate. The
basic phenomenology is developed in section 3. In section 4 we discuss how this kind of
setup can be minimally obtained from the deconstruction of an extra dimension, whereas
in section 5 we comment on the possibility of inducing Majorana neutrino masses from a
similar mechanism. We nally draw our conclusions in section 6.
2
Clockwork dark matter
One may  but does not have to  think of the clockwork mechanism as arising from the
deconstruction [8] of an extra spatial dimension. The resulting \theory space" consists of a
series of elds with adjacent interactions. Here we start with N Dirac fermions i
(Li; Ri)
(with i = 1 : : : N ), similarly to [4], together with a single righthanded chiral fermion R0.
We thus have the following set of (2N + 1) chiral elds
R0; L1; R1; : : : ; LN ; RN :
We refer to this set as the clockwork chain. Each chiral fermion is charged under a U(
1
)
chiral symmetry. The total symmetry group is thus
U(
1
)R0
U(
1
)L1
U(
1
)R1
: : :
U(
1
)LN
U(
1
)RN ;
and we identify the last abelian factor with the leptonnumber symmetry of the SM:
U(
1
)RN
U(
1
)LSM :
The latter choice makes natural a Yukawa coupling y of RN with the SM lefthanded
leptons, here denoted by LSM . We also introduce, as in ref. [4], two sets of N spurions Si
and Ci with Si
(1; 1) under U(
1
)Li+1
U(
1
)Ri and Ci
(1; 1) under U(
1
)Li
U(
1
)Ri ,
which break the symmetry (2.2) to U(
1
). With these ingredients, the Lagrangian takes on
the form
L = LSM + Lkinetic
N
X
i=1
ySSiLiRi 1
yC CiLiRi + h.c.
(2.1)
(2.2)
(2.3)
{ 2 {
1
2
y LSM He RN + h.c.
mN R0c R0 + h.c. :
(2.4)
For simplicity we take all spurions to be universal
ySSi = m ;
yC Ci = M
q m
We have also introduced a soft breaking of U(
1
)R0 which gives a Majorana mass mN to
R0 (thus breaking the residual U(
1
) symmetry). The fermionic
For the phenomenological analysis in the next section we will focus on the regime q
1,
for simplicity. Physically, we thus have a chain of N massive Dirac fermions, with relatively
weaker nearestneighbour couplings (that allow the fermions to hop from one site to the
other). The Majorana particle R0 lies at one extremity of the chain, while the SM fermions
live at the other extremity. The lightest state is (essentially) the Majorana particle
R0,
but it communicates to the SM through a chain of (relatively heavy, with respect to the
hopping scale m) particles, which, as in ref. [4], we dub \clockwork gears" in the sequel.
Going to the mass eigenbasis we get a band of N Dirac fermions ( Li Ri ), i = 1 : : : N ,
with masses
m i = p
i m
(q
1)m
(q + 1)m ;
(the exact form of i can be found in [4]) and a light Majorana eld N which will be our
DM candidate. The mixing matrices are
L = U L
L ;
R = U R
R ;
with (in the limit N
1, q
1, for the general formulas see [4])
L
m
LiRi 1
q LiRi
2
mN R0c R0 + h.c.
(2.6)
f =N
qg ;
(N !
h; Z; lW )
mN y
8
2
q2N :
{ 3 {
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
By requiring that the lifetime is larger than the typical indirectdetection bound
we get instead
As an example, for mN
100 GeV and y
1, this can be satis ed by
For mN < mW the decay processes are at 1loop level, and the bound on q and N below is
slightly weaker. By requiring that the decays (2.11) are slower than the age of the Universe
AU ' 4
1017s, we get the bound:
giving rise to the usual clockwork suppression by a factor qN .
So far the Lagrangian of eq. (2.4) leads to a long lived DM candidate but does not
provide any DM annihilation process, apart from N N ! LL and N N ! HHy, which are
doubly suppressed by the clockwork chain and thus far too slow to account for the DM
relic density. Therefore a clockwork WIMP DM scenario requires extra interaction(s) that
are not suppressed by the clockwork chain. In the following we will consider two simple
possibilities.
Scenario 1.
A rst one consists in simply promoting the spurions into dynamical scalar
elds, acquiring a vev so that (again, taking them universal for simplicity) the m and M
parameters above are now de ned as
yShSii = m ;
yC hCii = M
q m
S, the sums in eq. (2.19) are dominated by the
i = 1 term for q
For simplicity we consider here that all the quartic couplings are of the same order (and
take them to have the same value
C;S;CS along the clockwork chain). Depending on the
mass spectrum of the model, the quartic couplings may or may not be necessary to have
an e cient annihilation channel. For instance, as will be discussed below, if mS1 < mN the
dominant annihilation is DM DM
! S1S1, which proceeds even if the quartic couplings
vanish. In the opposite situation the quartic couplings are necessary in order to induce Sih
and/or Cih mixing, leading to annihilation into a pair of h or into h+Si or h+Ci. Similarly,
such quartic couplings are interesting because they lead to possibilities of direct detection.
Thus, important ingredients along this scenario 1 are the e ective Yukawa couplings of
N with the Higgs boson h and the clockwork gears. In the 4component notation for the
spinors they are given by
where
Thus
h
2
X UjLiyUiR 1; 0 = N0
X UjLiyUiR0 = N0
N
i=1
j
2
r
2
N + 1
N
X sin
i=1
r
r
2
2
sin
N + 1
r
2
N + 1
N + 1 q
1
sin
j
N + 1
N
X sin
i=1
j
N + 1
:
h
2
ij
N + 1
1
qi 1
ij
N + 1
1
qi
r
2
N + 1
sin
j
N + 1
whereas the UiR0 overlap between NR and the Ri eld is approximately as UiR0 in (2.9). This
leads to the couplings
and, in particular, to the e ective coupling to the Higgs boson, which is relevant for direct
detection:
L
L
mmqN2 N cN
ySS1
1
q yC C1
+ h.c.
mmqN2 N cN p
h
2
p
mN
2mq2
N cN h + h.c.
For the same reason, the clockwork gears are short lived and do not constitute a thermal
relic, or spoil BigBang Nucleosynthesis predictions.
Although these formulas are obtained from the diagonalization of the clockwork mass
Lagrangian in the limit mN ! 0, numerically we
nd that they are rather accurate as
long as mN . 0:7{0:8 m . An exact diagonalization gives that the Dirac spinors
i split
into pseudoDirac Majorana spinors, and the clockwork mechanism is active as long as
mN < (q
1) m
m . The exact diagonalization also gives that the light Majorana state
N Rc acquires an overlap UiL0 with the LH eld Li
L
Ui0
mN
m qi+1 ;
Note that, whereas the exact amount of clockwork suppression and the precise masses
of the states in the band depend on the details of the couplings along the whole clockwork
chain (which, as said above, we assumed to be universal for simplicity), for q
1 the
couplings relevant for DM freezeout and direct detection essentially depend only on the
coupling of N with the rst clockwork chain state, L1, therefore being quite insensitive to
the exact pro le of the clockwork.
Scenario 2.
scalar eld
The second possibility for annihilations that we will consider is from a single
N to couple directly with R0 (with all Si and Ci being nondynamical)
L0 =
yN
2
N R0cR0 + h.c.
N yN N HyH ;
(2.28)
whose vev induces both the R0 soft mass of eq. (2.4), mN = yN h N i, and mixing of N with
the Higgs boson (through the
N interaction). For what concerns the DM annihilation,
this scenario is similar to a standard Majorana DM Higgsportal scenario. The annihilation
may proceed through N N !
N N or into SM
elds through h N mixing.
3
Dark matter phenomenology
In this section, we discuss the basic phenomenology of the above DM model, in particular its
abundance set by thermal freezeout and the constraints from direct detection through the
Higgs coupling. Some comments on collider limits are discussed at the end of this section.
{ 6 {
(2.25)
and the scalars h and Si (we assume in the sequel that mh < mSi ). Here and in what
follows, with a slight abuse of notation, Si denote the (canonically normalized) real part
of the respective complex elds.
Scenario 1A: mS1 < mN . In this regime, the dominant channel that drives the relic
abundance is annihilation into the scalar S1
with t and uchannel exchange of the i, the same ones involved in the DM decay. Even
if kinematically allowed, all other channels (into other S or C states) are suppressed by
HJEP07(21)4
extra powers of 1=q. The relevant coupling of S1 is
L
p
S
2
j S1 j PR N + h.c. ;
is found to be (using the Higgsnucleon coupling of [9])
with
as given in eq. (2.24), fN = 0:30, mnuc = 0:946 GeV and the directdetection nucleon
crosssection is
GSI = p
2mN
2mq2 v m2
h
fN mnuc ;
SI =
1
m2N m2nuc
(mN + mnuc)
2 G2SI :
with
and the crosssection is
In gure 1 we plot the coupling yS (solid black lines) required for the relic density, for
mS = 150 GeV and q
1 (so that one is in the regime of validity of (3.4) and (3.6)). In the
region mN & 0:8 m , as discussed above, the relations given in the previous section become
less accurate, whereas the clockwork mechanism stops working altogether for mN
m
(hence the green/shaded region is excluded). Moreover, the plot should be considered valid
as long as the phasespace is su ciently open so that processes other than N N ! S1S1
may be neglected for determining the relic abundance of DM. Direct detection is excluding
{ 7 {
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
3000
0
.0
0
S
θ
/q=
8
1. We also show directdetection limits (blue, continuous) and future
prospects (red, dashed), from LUX 2016 and XENON1T, respectively. For a xed value of S=q,
the excluded (probed) region is below the blue (red) lines.
regions below the blue lines, from LUX 2016 [10]. Future prospects for XENON1T [11]
scenario is compatible with the perturbativity bound yS < p
4
are also given, this probing the regions below the red dashed lines. One observes that this
3:5 for mN ; m
< 2 TeV.
For the benchmark model parameters of
gure 1, XENON1T will test completely the
allowed region, for S=q > 0:0085. Note also that in the presence of quartic couplings, the
S and C states decay into SM elds via mixing with h and thus do not provide any relic on top of the DM.
Scenario 1B: mN < mSi and 2mN < mSi + mh.
When mN > mh, we are far from
the Higgs resonance and the relic density is mainly determined by the process
N N ! hh ;
with t and uchannel exchange of i. The relevant couplings are given in (2.18) and the
crosssection can be obtained from (3.4) with the replacements
mS1 ! mh ;
yS !
= yS S :
This is the main di erence with respect to scenario 1A: the e ective coupling
is now
suppressed by mixing with the SM Higgs. The directdetection cross section is through the
Higgs boson, as in scenario 1A, and so is as in eq. (3.6).
In
gure 2 we show in the mN
m
plane the e ective coupling
(black solid lines)
required to have the observed relic density, as well as the directdetection exclusion limit
from LUX 2016. Again, in the green region mN & 0:8 m
our approximate expressions
become less accurate, while the clockwork mechanism is active up to mN
m .
Directdetection exclusion limits require a large value of q. As gure 2 reveals, for q & 50 or
so the parameter space above the line mN
0:8 m
is a priori acceptable. Now, relic
{ 8 {
(3.7)
(3.8)
1.8
1.7
1B. Directdetection limits from LUX 2016 are shown in blue.
abundance requires that the e ective coupling is
& 0:8. For 130 GeV < mS < 300 GeV
the phenomenological bound on the mixing of a single scalar singlet with the Higgs boson
is [12{14]
. 0:3{0:4. By assuming universal mixing of all the Si with h, this implies
S . p
0:4
N
;
p
4
with N & 16 to account for the lifetime of the DM particle, eq. (2.13). Thus
> 0:8
corresponds to nonperturbative values of yS.
Therefore, for mN > mh, scenario 1B
does not work, unless one gives up the assumption of universality and the mixing
is bigger than the average of S;i. For instance, by taking S;1 = 0:3 (0:4), perturbativity
requires
. 0:3 (0:4)
' 1:1(1:4), which is compatible with all the constraints, for
mN . 225 (350) GeV, m
. 275 (425) GeV. We see that this scenario has a smaller
parameter space available, as compared to scenario 1A, due to the extra Higgs boson mixing
suppression, which in scenario 1A only a ects direct detection.
Alternatively, this scenario works near the Higgs resonance mN
mh=2, also for
universal mixing. To see this, notice that in this regime the relevant coupling for both
the relic density and direct detection is (2.27), i.e. essentially the one of a Higgsportal
Majorana DM. By adapting the results of [15], by means of the replacement
the results of [15] give 55 GeV < mN < 62:5 GeV and
h
!
q
4mN
p2vqm
;
where cres varies between 1 (for mN ' 55 GeV) and 0.2 (for mN ' 62 GeV).
Finally, this scenario also works near the S1 resonances, i.e. for 2mN
in this case the phenomenology depends strongly on the unknown values of mSi , this option
mSi but, since
will not be discussed further here.
{ 9 {
(3.9)
S;1
(3.10)
(3.11)
5.5
5
this scenario interpolates between the previous two, away from the resonances.
For universal mixing of all the Si with h, we
nd that only mN < 240 GeV (with
mS1 < 300 GeV) is compatible with perturbativity, although in a narrow region of the
parameter space. Relaxing the universality assumption, the parameter space opens up.
For instance, in
gure 3 we show the results for S = 0:1, mS1 = 200 GeV. We see that
in this case directdetection constraints require q ' 13, which is a signi cantly milder
constraint, with respect to Scenario 1B.
Scenario 2: nondynamical spurions. If the spurions m and qm in (2.6) do not have
dynamical origin (or the
elds Si and Ci are decoupled), one can still have a successful
channel for the annihilation of the DM if, for instance, mN has a dynamical origin, being
generated by the vev of a scalar
N , as given in eq. (2.28). For heavy
N , this scenario
e ectively reduces to a Higgsportal model, which works only near the Higgs resonance [13].
For light
N , this scenario can work, for instance, when the annihilation into two
N is
kinematically open. However, we will not discuss this scenario in detail, since the relevant
phenomenology is decoupled from the clockwork mechanism. In particular, in scenario 2
the clockwork chains
do not need to be at reach of current or nearfuture experiments,
even if this is a perfectly viable possibility.
Other existing limits and future prospects. In the Scenario 1 discussed above, the
clockwork gears i need to be at TeV or lighter. In particular, in Scenario 1B and 1C, the
combination of perturbativity considerations and the existing limits on Higgssinglet mixing
requires m
to be at most at a few hundreds GeV, whereas in Scenario 1A the clockwork
gears can be as heavy as
TeV, if one allows large values of yS at the boundary of the
perturbativity region. On the other hand, in the philosophy of the clockwork mechanism
the coupling to the SM doublet y should not be too small either. Therefore, a natural
consequence of the framework discussed in this note is the presence of observable e ects of
the states i
. The coupling of the mass eigenstates i to LSM proceeds via their (large)
overlap with the clockwork state RN . Thus, as long as q
1, we may reason in terms of a
single state with mass m
and Yukawa coupling y to the SM. If the relevant experiments
y=p
are instead able to resolve the di erent eigenstates
i, one should think in terms of a
collection of N states with couplings
N . The intermediate regime, in which the
e ective description of the band of gears in terms of a single state is not valid, but at the
same time the gears are not well separated, is much more involved and a detailed analysis
goes beyond the purposes of this work.
Phenomenologically, the states
i are essentially a collection of pseudoDirac heavy
neutrinos. Their collider signatures are studied in detail [16]. The most stringent existing
constraint come from electroweak precision tests (EWPT) and gives
jBl j2
v2 jylj2
avour, and slightly weaker for the electron
avour. The most sensitive
channel for their direct production at LHC is the trilepton plus missing energy [16]. In
s = 14 TeV data with 300 fb 1 integrated luminosity will probe couplings
allowed by EWPT up to m
200 GeV, which is the region required in Scenarios 1B
A stronger indirect bound is provided by LFV in processes like
conversion in nuclei. The nal MEG limit [17] on the branching ratio for
! e and
! e
! e is given by
BR(
! e )
8
10 4 jBe j2 jB
j2 < 4:2
Although this limit is stronger than the one from EWPT for jBe j ' j
B
j, it crucially
depends on both the muon and electron couplings being sizeable.
This result will be
improved in the near future by 4 orders of magnitudes by the
! e conversion experiments
Mu2e [18] and COMET [19]. Thus, under the reasonable assumption of not having a large
hierarchy between the muon and electron couplings, LFV provides the best prospects to
probe the framework in the near future.
If mN is not much smaller than m , the pseudoDirac states split into Majorana pairs
with di erent masses and couplings to the SM leptons. In this case, the most sensitive
additional channel is provided by samesign dilepton plus jets with no missing energy. The
300 fb 1 dataset will allow to probe couplings allowed by EWPT up to m
300 GeV [20].
Finally, future electroweak precision tests at lepton colliders would provide a strong indirect
probe of the mechanism.
As for the scalar sector, Scenario 1B and 1C require a large mixing between the scalar
S1 and the Higgs boson, whereas in Scenario 1A (2) the mixing of h with S1 ( N ) can
be either large or small. A detailed analysis of the relevant constraints is given in [12{
14]. For mS1 < 500 GeV the most stringent constraint comes from the direct searches of
Clockwork WIMP from deconstruction
We now want to investigate whether the scenario we advocate can be related to the
deconstruction of an extra dimension [4] in a simple way. In this section we show how this
can be achieved in a
at spacetime for a fermionic clockwork chain, di erently from the
construction in [4], which makes use of a curved metric.
We start from the Lagrangian of a free massive fermion in 5D
!
M
i
M
;
= (L R)T and we get
where Z is the coordinate in the extra dimension, compacti ed with length
R, Z
L5
L5
1
2
i
M LR + h.c. :
Higgslike resonances, giving S < 0:3{0:4. For higher masses the most stringent limit
is again given by EWPT: S / 0:3. Therefore, the scalar sector of our framework can
give observable signatures at Higgs searches, for mS1 of few hundreds GeV, or at future
electroweak precision measurements, for larger mS1 .
We conclude this section with a brief comment regarding indirect detection. In the
scenarios discussed here, the annihilation of DM is pwave suppressed, and thus it is relevant
only at the time of thermal freezeout. However, through the clockwork gears, the DM
particle can decay on a time scale that may be relevant for indirect searches,
As the decay is through the coupling with the SM Higgs and leptons, the most notable
potential spectral feature is the presence of monoenergetic neutrinos from the DM decay
into h + , a signature that has been studied in the literature (see [21] and references
therein).
a
2
N 1
X 1
i=1
a
(4.1)
i 5,
(4.2)
(4.3)
(4.4)
In a theory with fermions, it is well known that a naive discretization gives rise to the
famous doubling problem on the lattice (see e.g. [22]), which may be removed using a
Wilson term (see the discussion in [4]):
L
a
2
where a =
R=N
! 0 is the lattice spacing in the extra dimension. This term, which
vanishes in the continuum limit, guarantees that the discrete theory tends to the continuum
dimension and rede ning conveniently the elds1 we nd
one (4.1) in the appropriate limit N ! 1. By discretizing (4.2) and (4.3) along the extra
L
N
X i i
i=1
N
X
i=1
i +
LiRi+1 + h.c.
+ M
LiRi + h.c. :
1To obtain a form that matches exactly (2.6), we have added boundary counterterms, following [4].
However, this is not necessary for the clockwork mechanism to work, but only for convenience of the
presentation.
The last ingredient to obtain the clockwork chain as in (2.6) is the addition of a chiral
eld R0 localized at Z = 0, with a Majorana mass mN , interacting with the bulk fermion
, whereas the SM is localized at Z =
R. Notice that the precise value of the coupling
of the interaction term LR0 (Z) does not a ect signi cantly the clockwork mechanism
since, although the overall clockwork suppression is proportional to this coupling, the
exponential part of the suppression does not depend on it. Equivalently, one can instead
impose the Dirichlet boundary condition L(0) = 0 which e ectively removes L0 and leaves
one unpaired rightchiral mode R0 in the discretized theory. We thus end up with the
clockwork Lagrangian in (2.6) with the identi cations
HJEP07(21)4
qm
+ y h i = m + y h i
:
In the N ! 1 we obtain a nite clockwork suppression:
m
1
a
;
1
a
qm
+ M :
q
N =
! e RM :
with :
L5
y
;
The spectrum of the clockwork states in the continuum limit consists of the DM, with
a mass
mN and a tower of KaluzaKlein modes (the clockwork gears) starting at the
mass M . Since, for the e ectiveness of the clockwork, M
1=R, the KaluzaKlein modes
will have mass di erences
M
1=R
M , i.e. they will not be largely separated one
from each other.
The clockwork setup that we envision super cially resembles that of domain wall
fermions (see e.g. [23]), with a chiral mode localized on one wall, and the (chiral) SM
degrees of freedom localized on another wall. The correspondance is only approximate
because we assume only one chiral mode, strictly located at the Z = 0 wall, which furthermore
must have a Majorana mass.
As for realizing the dynamical scalar elds needed for DM annihilation, we distinguish
between the two scenarios discussed in the previous sections.
Scenario 1.
One possibility is to add a scalar
in the bulk, with a Yukawa interaction
which can also generate the Dirac bulk mass M , if it acquires a vev. After discretization,
this will give rise, e ectively, to the interaction with the
elds Ci discussed in section 2.
Thus the Ci are dynamical elds, unlike the Si which all come from the discretization of
the fth dimension, Si = 1=a. This framework predicts the universality of the clockwork
chain. In this case obviously the annihilation cannot proceed into Si states but it can into
Ci states, just in the same way as considered in sections 2 and 3, replacing S1 by C1 and yS
by yC =q in eqs. (3.2){(3.4). The scenario is perfectly viable provided that q is not too large.
Note that in the limit N ! 1, i.e. q ' 1, these Ci interactions are not suppressed with
respect to the ones of Si in eq. (3.2) and in this case all the DM DM
! CiCi processes
are relevant. If the bulk mass term M is generated by the vev of , eq. (4.5) gives
(4.5)
(4.6)
(4.7)
(4.8)
con ned to this brane.2
rise to couplings of the form
As for the quartic couplings of the Ci and the Higgs boson in (2.17), these can be sizeable
if the eld
has a sizeable overlap with the brane at Z =
R, or if the Higgs eld is not Another possibility is to gauge the fermionic eld with an abelian group, which gives
LiUiRi+1
with
Ui = exp(iAi) ;
(4.9)
where Ai is the component of the gauge eld (technically, averaged along the link between
sites i and i + 1 times the lattice spacing a). Under a gauge transformation, i ! ei i i,
i). This is of course well known. The relevant point is that here the Ci
are spurions generated by the M mass term and the link variables Ui play the role of the
elds Si in section 2. The relevant Ui degrees of freedom are nevertheless not real elds,
like instead the Si considered in eqs. (3.2){(3.4), but phases, which from the 4D perspective
are pseudoscalar elds. In this case one expects that their mixing with the Higgs boson
is at least suppressed, and therefore only scenario IA would work (in the same way as in
section 3).
Finally, notice that in the case of a curved spacetime along the extra dimension, as
considered in [4], with a metric of the form ds2 = e 43 kjZj(dx2 + dZ2) originating from a
model with a dilaton S, a similar scenario could also be realized, provided that the UV
completion of the model ful lls certain conditions. For instance, in order to obtain a Yukawa
interaction with a bulk scalar
not exponentiallysuppressed for the DM light mode, one
has to choose conveniently the spurion charge under dilatations of the (dimensionful) bulk
Yukawa coupling. In particular, one can show that the unique successful choice is to take
y
! e y , under the global Weyl transformation [4] gMN ! e 2 gMN , S ! S + 3 .
With this choice, the warping coming from the metric and the interaction with the dilaton
combine to give an unsuppressed interaction with the DM mode, localized at Z '
R in
the construction of [4].
Scenario 2. In this case one only needs a scalar eld localized on the brane Z = 0, with
a \Majorana" interaction with R0:
L5
N R0cR0 (Z) :
(4.10)
To generate the quartic scalar coupling N in (2.28), which need to be nonzero (to provide
decay channels of
N into the SM) but not necessarily sizeable, one may assume that
another (free) scalar eld is present in the bulk, with a signi cant overlap with both the
branes. In this case the structure consists of two branes where interactions occur, connected
by free elds in the bulk. In alternative, one may take the Higgs eld as living in the bulk
and no extra bulk scalar.
Finally, notice that this scenario 2 would also work in the extradimensional
construction of [4] because the interaction is essentially 4D, and therefore not exponentially
suppressed by the metric, provided that a decay channel of N into the SM is present.
2In the latter case, the dimension5 scalar doublet leads to N copies of the Higgs doublet at the di erent
sites in theory space. At this stage, we may simply assume that the SM Higgs is the lightest mass eigenstate.
As the SM lefthanded neutrinos have large Yukawa couplings with the TeVscale clockwork
gears
i, one could worry that these couplings induce far too large SM neutrino masses.
However, this is not the case because, if there were no R0 at the other end of the clockwork
chain, the lefthanded neutrinos would have no chiral partner to get a mass from. Since
the interactions of R0 with the SM fermions have to proceed along the whole clockwork
chain, the lightneutrino masses receive a suppression from the clockwork mechanism
m
m2D
' q2N mN
;
(5.1)
where mD = yv, and too large neutrino masses are not generated. On the contrary,
since the suppression needed for the DM lifetime is much larger than for the neutrino
masses (i.e. typically, in the ordinary language of e ective operators, a dimension6 versus
dimension5 GUT scale suppression) the mass induced in the above framework is far below
the masses needed: m
10 40 eV for mN
However, it is worth to point out that a fermion clockwork chain can induce a Majorana
neutrino mass in agreement with data through eq. (5.1) if this chain involves a smaller q2N
factor. For instance for q = 10, m
10 1 eV requires 2N
15
log10(mN =GeV).
Alternatively if a R0 has no Majorana mass, a Dirac mass is induced as in [4], which for
q = 10 requires N
12. As one could have expected from the start, the Dirac case needs
more clockwork suppression than the Majorana case.
Experimentally we know that there are at least two nonzero neutrino masses, and to
generate at least two of them there are essentially two simple options, as we discuss now.
Option 1. If one assumes that only the last site (i.e. RN ) couples to the SM leptons,
in both the Majorana and Dirac cases a single clockwork chain is not enough, because it
can only induce one neutrino mass, given that the chain couples to a single combination of
Le; ; . In order to induce 2 (3) neutrino masses one therefore needs 2 (3) clockwork chains,
including a di erent
chiral partner for each chain.
Notice that the N
R0 eigenstates of the two neutrino mass chains could be used to
generate the observed baryon asymmetry of the Universe via resonant leptogenesis [24]. If
one assumes that the Majorana breaking is universal for two neutrino mass chains, the mass
splitting between the two unstable N
R0, necessary for resonant leptogenesis, can be
generated by the interaction with the clockwork chains, since the exact mass eigenvalues
di er slightly from mN and are function of the q and N of both chains, which are not
necessarily the same. By literally repeating the standard argument for leptogenesis, one
can
gure out that the washout induced by the Yukawa couplings with the SM neutrinos
is not large, and therefore leptogenesis is in principle possible.
From the continuum 5D point of view, in the
at spacetime construction given in
the previous section, the clockwork suppression is essentially given by the mass M . By
assuming 3 chains as above, i.e. 3 fermions
in the bulk and 3 chiral partners (i.e. 3 neutrino
chiral partners R0 on the other end of the chain with respect to the SM
elds), one could
induce 2 nonnegligible neutrino masses and account for DM. One of the R0 is essentially
the DM particle. The 2 others are necessary to induce 2 nonnegligible neutrino masses.
Such a 2brane setup requires 2 of the bulk fermions to have similar masses (involved in
the neutrino mass generation), whereas the third one (related to DM) with a larger mass.
Instead, in the curved spacetime construction of [4], since the clockwork suppression is
universally determined by the metric, such a setup appears not to be feasible (unless one
introduces an extra mass term for the bulk eld as above). If one allows the chiral partners
to be not all located on the rst brane, but placed on di erent sites (for instance one on
the rst site for DM and 2 much closer to the other end of the chain for neutrino masses)
more possibilities show up. In this case, in both the at and curved cases, 2 bulk fermions
coupling to di erent lepton
avour combinations might be enough to generate 2 neutrino
masses and account for DM, provided that there are still 3 chiral partners. A more detailed
discussion of this may be given elsewhere.
Option 2. If one assumes that the SM
elds interact in di erent avour combinations
with several sites along the chain, a single bulk fermion can be enough to generate two or
three nonzero neutrino masses and account for DM, provided that again there are 3 chiral
partners (with the one responsible for DM placed far from the 2 others as well as from
the SM
elds, so that the clockwork suppression for the neutrino masses gets e ectively
reduced compared to the DM one). This holds for the at case considered above, as well
as for the curved case considered in [4].
6
Conclusions
In this work we have proposed an implementation of a WIMP dark matter candidate within
the clockwork framework. The candidate is a Majorana singlet particle that is stable over
cosmological time because its e ective Yukawa couplings to the SM Higgs and leptons are
suppressed through a clockwork mechanism. For appropriate choices of the parameters, it
can be made very long lived, e.g.
& 1026 s. The very same particles that suppress this
coupling (the clockwork gears) have unsuppressed couplings to the DM candidate, so that
its relic abundance may be set by the standard freezeout mechanism.
Concretely, as proof of existence, we have considered a simple clockwork chain of
fermions, with a single chiral mode at one end of the chain, and the SM leptons at the other
end. In section 3 we have studied the basic DM phenomenology (i.e. relic abundance, direct
detection constraints, collider and lowenergy limits and prospects) for various scenarios.
These constraints, in particular the relic density one, require the clockwork gears to lie at
most at the
TeV scale, i.e. accessible at colliders. Future LFV experiments will also be
able to probe e ciently their e ects. Depending on the assumption one makes about the
nature and relative mass hierarchy of the scalarsector, many viable candidates may be
found. Interestingly, a similar (albeit parametrically distinct) clockwork construction may
be relevant for SM neutrino phenomenology and in particular it can explain the smallness
of the SM neutrinos, see section 5. We pointed out that, in order to have several
nonvanishing neutrino masses, one needs several clockwork chains, if only the last site of the
chains couples to the SM
elds.
Intuitively, the clockwork mechanism imposes that the degrees of freedom at the two
edges of the chain have little overlap, a picture that is supported by an embedding of
the clockwork chain in a 5D spacetime. Interestingly, our implementation departs from
the 5D continuum construction of [4], where the bulk
eld is massless and the mass gap
between the massless mode and the clockwork gears is generated by a curved metric. In the
construction of section 4, instead, the metric is at and the bulk eld is massive. The gap
between the massless mode, introduced on one of the edges (\branes") and the clockwork
gears is, in our case, generated by the mass term of the bulk
eld. Although we have
explicitly shown how the clockwork chain can originate from a at spacetime setup in the
fermionic case, a very similar analysis can be carried out for a scalar clockwork: again, one
needs a massive bulk scalar and extra brane terms.
Acknowledgments
This work is supported by the FNRSFRS, the FRIA, the IISN, a ULBARC, the Belgian
Science Policy (IAP VI11) and a ULB Postdoctoral Fellowship.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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