6
1 Introduction
Let now consider the massive dark-photon coupling to hyper-charge. In this case
it is convenient to parametrize the coupling of the dark photon to the hyper-charge as
˜
L = −
ε
2 cos θ W
˜
F
μν B
μν
.
(1.11)
The usual diagonalization of the gauge bosons W
3
μ and B μ now includes also the
dark photon ˜
A
μ (in the non-diagonal basis) so that the physical gauge bosons Z μ
and A μ also contain a dark-photon component A
μ in the mass eigenstate basis. In
particular, at the O(ε) in the expansion, we have
⎛
⎝
W
3
μ
B μ
˜
A
μ
⎞
⎠ =
⎛
⎝
c W s W −s W ε
−s W c W −c W ε
t W ε 0
1
⎞
⎠
⎛
⎝
Z μ
A μ
A
μ
⎞
⎠ ,
(1.12)
where c W , s W and t W are the usual cosine, sine, and tangent of the Weinberg angle
θ W , respectively. New couplings of the massive dark photon to the SM fermions
appear for the photon and the Z gauge boson up to O(ε
2
):
L ⊃ −e ε J
μ A
μ + e
ε t W J
μ Z μ + e
J
μ A
μ ,
(1.13)
where J μ is the EM current, while J
μ and e
are the matter current and coupling of
the dark-photon in the dark sector, respectively. After integrating out the Z boson,
we see that the coupling of the massive dark photon to the SM fermions is recovered
as −eε.
Which coupling is used depends then only on the energy of the processes considered, with the direct coupling to the photon for all processes below the electroweak
scale breaking, and the hyper-charge above it. Since all limits are to be considered
approximately within the order of magnitude, the presence of the factor c W in the
definition in Eq. (1.11) does not matter. The Lagrangian in Eq. (1.13) shows that,
if the mixing is between the dark photon and the hyper-charge, the Z gauge boson
acquires a milli-charged coupling strength e
t W ε to the dark sector current.
For completeness, let us also recall two other possibilities that have been discussed
in the literature:
– There is no kinetic mixing as in Eq. (1.1) but the mass term between the dark
photon and the Z -boson is taken non-diagonal and therefore giving a mixing
between these two states [20–24]. The dark photon is named the dark Z and there
are characteristic experimental signatures in parity violating processes and the
coupling to neutrinos;
– The B − L global symmetry (or other conserved flavor symmetries) are gauged
and taken to be the U (1) group of the dark photon, which mixes with the hypercharge [25–27]. There is direct coupling to the SM fermions in this case and the
dark photon is no longer dark.
1 Introduction
Let now consider the massive dark-photon coupling to hyper-charge. In this case
it is convenient to parametrize the coupling of the dark photon to the hyper-charge as
˜
L = −
ε
2 cos θ W
˜
F
μν B
μν
.
(1.11)
The usual diagonalization of the gauge bosons W
3
μ and B μ now includes also the
dark photon ˜
A
μ (in the non-diagonal basis) so that the physical gauge bosons Z μ
and A μ also contain a dark-photon component A
μ in the mass eigenstate basis. In
particular, at the O(ε) in the expansion, we have
⎛
⎝
W
3
μ
B μ
˜
A
μ
⎞
⎠ =
⎛
⎝
c W s W −s W ε
−s W c W −c W ε
t W ε 0
1
⎞
⎠
⎛
⎝
Z μ
A μ
A
μ
⎞
⎠ ,
(1.12)
where c W , s W and t W are the usual cosine, sine, and tangent of the Weinberg angle
θ W , respectively. New couplings of the massive dark photon to the SM fermions
appear for the photon and the Z gauge boson up to O(ε
2
):
L ⊃ −e ε J
μ A
μ + e
ε t W J
μ Z μ + e
J
μ A
μ ,
(1.13)
where J μ is the EM current, while J
μ and e
are the matter current and coupling of
the dark-photon in the dark sector, respectively. After integrating out the Z boson,
we see that the coupling of the massive dark photon to the SM fermions is recovered
as −eε.
Which coupling is used depends then only on the energy of the processes considered, with the direct coupling to the photon for all processes below the electroweak
scale breaking, and the hyper-charge above it. Since all limits are to be considered
approximately within the order of magnitude, the presence of the factor c W in the
definition in Eq. (1.11) does not matter. The Lagrangian in Eq. (1.13) shows that,
if the mixing is between the dark photon and the hyper-charge, the Z gauge boson
acquires a milli-charged coupling strength e
t W ε to the dark sector current.
For completeness, let us also recall two other possibilities that have been discussed
in the literature:
– There is no kinetic mixing as in Eq. (1.1) but the mass term between the dark
photon and the Z -boson is taken non-diagonal and therefore giving a mixing
between these two states [20–24]. The dark photon is named the dark Z and there
are characteristic experimental signatures in parity violating processes and the
coupling to neutrinos;
– The B − L global symmetry (or other conserved flavor symmetries) are gauged
and taken to be the U (1) group of the dark photon, which mixes with the hypercharge [25–27]. There is direct coupling to the SM fermions in this case and the
dark photon is no longer dark.