![](https://csdnimg.cn/release/download_crawler_static/12345712/bg4.jpg)
For simplicity we take all spurions to be universal
y
S
S
i
= m , y
C
C
i
= M ≡ q m with q > 1 . (2.5)
We have also introduced a soft breaking of U(1)
R
0
which gives a Majorana mass m
N
to
the light eigenstate N ∼ R
0
(thus breaking the residual U(1) symmetry). The fermionic
mass Lagrangian thus reads
L ⊃ −m
N
X
i=1
¯
L
i
R
i−1
− q
¯
L
i
R
i
−
m
N
2
R
c
0
R
0
+ h.c. (2.6)
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 nearest-neighbour couplings (that allow the fermions to hop from one site to the
other). The Majorana particle R
0
lies at one extremity of the chain, while the SM fermions
live at the other extremity. The lightest state is (essentially) the Majorana particle ∼ R
0
,
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 (ψ
L
i
ψ
R
i
), i = 1 . . . N,
with masses
m
ψ
i
=
p
λ
i
m ≈ (q − 1)m ÷ (q + 1)m , (2.7)
(the exact form of λ
i
can be found in [4]) and a light Majorana field N which will be our
DM candidate. The mixing matrices are
L = U
L
ψ
L
, R = U
R
ψ
R
, (2.8)
with (in the limit N 1, q 1, for the general formulas see [4])
U
R
i0
≈
1
q
i
, |U
R
ik
| ≈
r
2
Nq
2
× {π/N ÷ q} ,
U
L
ik
≈
r
2
N
× {π/N ÷ 1}. (2.9)
In particular, the coupling of the light state with the SM is
L ⊃ −
y
q
N
¯
L
SM
e
HP
R
N + h.c. (2.10)
which is the first key result: through the clockwork mechanism, the effective Yukawa
coupling between the DM particle and SM fields is suppressed by a factor of q
N
≫ 1.
Ultimately the value of this parameter is to be set by the requirement of stability on
cosmological scales. Assuming that m
N
is significantly larger than the relevant thresholds,
the Goldstone-boson equivalence theorem gives the decay width
Γ(N → νh, νZ, lW ) ≈
m
N
8π
y
2
q
2N
. (2.11)
– 3 –