1.1 Massless and Massive Dark Photons
5
The case of spontaneously broken symmetry can be distinguished from the Stueckelberg mass terms because the former will give rise to processes in which the dark
photon is produced together with the dark Higgs boson, the vacuum expectation
value of which hides the symmetry.
Whereas the Lagrangian in Eq. (1.9) is the most general, the simplest and most
frequently discussed case consists in giving mass directly to only one of the U (1)
gauge bosons so that, for instance, M b = 0 in Eq. (1.7), the mass states are already
diagonal. Even in this simple case, the mass term removes the freedom of choosing
the angle θ in Eq. (1.3). With this choice, δ = 0 in Eq. (1.9), the ordinary photon
couples only to ordinary matter and the massive dark photon is characterized by a
direct coupling to the electromagnetic current of the the SM particles (in addition to
that to dark-sector matter) and described by the Lagrangian
L ⊃ −
eε
√
1 − ε 2
J μ A
μ
−e ε J μ A
μ
,
(1.10)
as in Eq. (1.5) above. This is the choice defining the massive dark photon. The
coupling of the massive dark photon to SM particles is not quantized—taking the
arbitrary value eε. Because of this direct current-like coupling to ordinary matter,
it is the spontaneously broken or Stueckelberg massive dark photon that is mostly
discussed in the literature and considered in the experimental proposals.
Notice that the massive dark photon has the same couplings as the massless
dark photon after choosing sin θ = 0 (right-side of Fig. 1.1); this case therefore
represents the limit of vanishing mass of the massive dark photon. On the contrary, the
massless dark photon proper—corresponding to the choice tan θ =
ε/
√
1 − ε 2
—
is not related to any limiting case of the massive dark photon.
There are no electromagnetic milli-charged particles in the massive case; they are
present only if both U (1) gauge groups are spontaneously broken (or equivalently
M b = 0 in the Stueckelberg Lagrangian in Eq. (1.7))—which is not the case of our
world where the photon is massless.
1.1.1 Kinetic Mixing: Electric or Hyper-Charge?
There seems to be the choice in the kinetic mixing in Eq. (1.1) between the U (1) e.m.
group of electric charge and the U (1) Y group of the hyper-charge, with mixing
parameter ε defined as in Eq. (1.1). Concerning the massless dark photon, these
two choices give rise to the same physics, since the dark photon remains decoupled
from the SM fields at the tree-level. The only difference is that the photon and Z -
boson are now both coupled to the dark-sector current, with e D ε/
√
1 − ε 2 cos θ W and
e d ε/
√
1 − ε 2 sin θ W strength, respectively.
5
The case of spontaneously broken symmetry can be distinguished from the Stueckelberg mass terms because the former will give rise to processes in which the dark
photon is produced together with the dark Higgs boson, the vacuum expectation
value of which hides the symmetry.
Whereas the Lagrangian in Eq. (1.9) is the most general, the simplest and most
frequently discussed case consists in giving mass directly to only one of the U (1)
gauge bosons so that, for instance, M b = 0 in Eq. (1.7), the mass states are already
diagonal. Even in this simple case, the mass term removes the freedom of choosing
the angle θ in Eq. (1.3). With this choice, δ = 0 in Eq. (1.9), the ordinary photon
couples only to ordinary matter and the massive dark photon is characterized by a
direct coupling to the electromagnetic current of the the SM particles (in addition to
that to dark-sector matter) and described by the Lagrangian
L ⊃ −
eε
√
1 − ε 2
J μ A
μ
−e ε J μ A
μ
,
(1.10)
as in Eq. (1.5) above. This is the choice defining the massive dark photon. The
coupling of the massive dark photon to SM particles is not quantized—taking the
arbitrary value eε. Because of this direct current-like coupling to ordinary matter,
it is the spontaneously broken or Stueckelberg massive dark photon that is mostly
discussed in the literature and considered in the experimental proposals.
Notice that the massive dark photon has the same couplings as the massless
dark photon after choosing sin θ = 0 (right-side of Fig. 1.1); this case therefore
represents the limit of vanishing mass of the massive dark photon. On the contrary, the
massless dark photon proper—corresponding to the choice tan θ =
ε/
√
1 − ε 2
—
is not related to any limiting case of the massive dark photon.
There are no electromagnetic milli-charged particles in the massive case; they are
present only if both U (1) gauge groups are spontaneously broken (or equivalently
M b = 0 in the Stueckelberg Lagrangian in Eq. (1.7))—which is not the case of our
world where the photon is massless.
1.1.1 Kinetic Mixing: Electric or Hyper-Charge?
There seems to be the choice in the kinetic mixing in Eq. (1.1) between the U (1) e.m.
group of electric charge and the U (1) Y group of the hyper-charge, with mixing
parameter ε defined as in Eq. (1.1). Concerning the massless dark photon, these
two choices give rise to the same physics, since the dark photon remains decoupled
from the SM fields at the tree-level. The only difference is that the photon and Z -
boson are now both coupled to the dark-sector current, with e D ε/
√
1 − ε 2 cos θ W and
e d ε/
√
1 − ε 2 sin θ W strength, respectively.