2.3 A Minimal Model of the Dark Sector
39
m
2
φ /m χ μ
√
α L α R |ρ μμ | 2 G M (x μ , η φ )
∼ > 6.3 × 10
3 TeV ,
(2.65)
where x μ = m
2
χ μ /m
2
φ , with m χ μ the dark-fermion mass associated to the muon. Again,
for a quick estimate of the bounds above and those that follow, the loop function
G M (x, y) can be considered a coefficient of order O(10
−1
) as long as η φ is not too
small. For instance, for x 1 and y 0.5, the loop function G M 0.05.
The various Yukawa couplings and messenger and fermion masses are probed in
a selective manner in flavor physics where we must distinguish among the various
couplings and states. Mixing (proportional to a coefficient ρ i j in the equations below)
between different flavor states must be included.
The strongest bound comes from the limit on the BR (μ → eγ) < 4.2 × 10
−13
(CL 90%) [58] of the MEG experiment. From this result, we find that
m
2
φ /m χ μ
√
α L α R |ρ μμ ρ
μe | G M (x μ , η φ )
∼ > 4.9 × 10
8 TeV .
(2.66)
A weaker bound can be extracted, in the hadronic sector, from the difference
between the experimental limit on the BR (B → X s γ) < (3.21 ± 0.33) × 10
−4 [59]
of the BaBar collaboration and its SM estimate [60]. It yields
m
2
S /m Q b
√
α L α R |ρ bb ρ
bs | G M (x b , η S )
∼ > 1.3 × 10
4 TeV ,
(2.67)
where x b = m
2
Q b /m
2
S , with m Q b the mass of dark fermion associated to the b-quark,
The limits in Eqs. (2.66) and (2.67) apply specifically to the off-diagonal terms in
the Yukawa couplings ρ i j of the muon-electron and b-s quark mixing respectively,
and to the corresponding mass of messenger states.
The mass mixing in the Kaon system [53, 61] gives a further limit
m
2
S
α
2
L + α
2
R
ρ ss ρ
∗
sd
2 ∼ > 3 × 10
5 TeV
2
,
(2.68)
which is not related to the dark photon and its coupling α D because it comes from
the box-diagram insertion of the dark scalars and fermions.
The limit in Eq. (2.68) is obtained by requiring that the messenger contribution
to the box diagram for the K
0 - ¯
K
0 mixing does not exceed the experimental value of
the mixing parameter Δm K = 3.48 × 10
−12 MeV [62]. Due to chirality arguments,
the leading contribution to the box diagram in Eq. (2.68) does not depend on the
dark fermion mass, which is assumed to be much smaller than the corresponding
messenger mass in the down sector and therefore very weakly on the loop function.
The limit in Eq. (2.68) applies specifically to the off-diagonal term in the Yukawa
coupling of d-s quark mixing and the corresponding messenger state. A similar but
weaker bound can be found from B-meson mixing.
39
m
2
φ /m χ μ
√
α L α R |ρ μμ | 2 G M (x μ , η φ )
∼ > 6.3 × 10
3 TeV ,
(2.65)
where x μ = m
2
χ μ /m
2
φ , with m χ μ the dark-fermion mass associated to the muon. Again,
for a quick estimate of the bounds above and those that follow, the loop function
G M (x, y) can be considered a coefficient of order O(10
−1
) as long as η φ is not too
small. For instance, for x 1 and y 0.5, the loop function G M 0.05.
The various Yukawa couplings and messenger and fermion masses are probed in
a selective manner in flavor physics where we must distinguish among the various
couplings and states. Mixing (proportional to a coefficient ρ i j in the equations below)
between different flavor states must be included.
The strongest bound comes from the limit on the BR (μ → eγ) < 4.2 × 10
−13
(CL 90%) [58] of the MEG experiment. From this result, we find that
m
2
φ /m χ μ
√
α L α R |ρ μμ ρ
μe | G M (x μ , η φ )
∼ > 4.9 × 10
8 TeV .
(2.66)
A weaker bound can be extracted, in the hadronic sector, from the difference
between the experimental limit on the BR (B → X s γ) < (3.21 ± 0.33) × 10
−4 [59]
of the BaBar collaboration and its SM estimate [60]. It yields
m
2
S /m Q b
√
α L α R |ρ bb ρ
bs | G M (x b , η S )
∼ > 1.3 × 10
4 TeV ,
(2.67)
where x b = m
2
Q b /m
2
S , with m Q b the mass of dark fermion associated to the b-quark,
The limits in Eqs. (2.66) and (2.67) apply specifically to the off-diagonal terms in
the Yukawa couplings ρ i j of the muon-electron and b-s quark mixing respectively,
and to the corresponding mass of messenger states.
The mass mixing in the Kaon system [53, 61] gives a further limit
m
2
S
α
2
L + α
2
R
ρ ss ρ
∗
sd
2 ∼ > 3 × 10
5 TeV
2
,
(2.68)
which is not related to the dark photon and its coupling α D because it comes from
the box-diagram insertion of the dark scalars and fermions.
The limit in Eq. (2.68) is obtained by requiring that the messenger contribution
to the box diagram for the K
0 - ¯
K
0 mixing does not exceed the experimental value of
the mixing parameter Δm K = 3.48 × 10
−12 MeV [62]. Due to chirality arguments,
the leading contribution to the box diagram in Eq. (2.68) does not depend on the
dark fermion mass, which is assumed to be much smaller than the corresponding
messenger mass in the down sector and therefore very weakly on the loop function.
The limit in Eq. (2.68) applies specifically to the off-diagonal term in the Yukawa
coupling of d-s quark mixing and the corresponding messenger state. A similar but
weaker bound can be found from B-meson mixing.