Chapter 5
Appendix
5.1 Dark Sector Portals
In this appendix we give the Lagrangian for the four portals mentioned in the introduction as well as a minimal bibliography to provide their context within the search
of phenomenological models beyond the Standard Model.
The dark sector is assumed to interact with the visible, SM sector through relevant operators of dimension four and five (and possibly sub-leading higher-order
operators).
These portals are classified according to the spin of the mediator field. We can
have
• dark photon (spin 1): The portal operator arises from the kinetic is mixing between
the SM photon field strength F μν and a dark photon F
μν :
ε
2
F μν F
μν
,
it is an operator of dimension four. It is assumed that the dark photon is the main
carrier of the interaction among the dark sector states.
The existence of an independent U (1) group symmetry was originally proposed,
in the context of supersymmetric theories, in [1, 2] and, more in general, in [3, 4];
• axion (spin 0): The operator comes from the interaction between a pseudo-scalar,
the axion a, and the SM photon and fermions ψ:
a
f a
F μν ˜
F
μν
+
1
f a
∂ μ a ψ γ
μ
γ 5 ψ ,
with operators of dimension five; the physics of this portal is based on that of
the axion and related to the strong CP problem as well as axion dark matter. In
many cases, the portal is generalized to an axion-like particle (ALP) with similar
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Fabbrichesi et al., The Physics of the Dark Photon,
SpringerBriefs in Physics, https://doi.org/10.1007/978-3-030-62519-1_5
71
Appendix
5.1 Dark Sector Portals
In this appendix we give the Lagrangian for the four portals mentioned in the introduction as well as a minimal bibliography to provide their context within the search
of phenomenological models beyond the Standard Model.
The dark sector is assumed to interact with the visible, SM sector through relevant operators of dimension four and five (and possibly sub-leading higher-order
operators).
These portals are classified according to the spin of the mediator field. We can
have
• dark photon (spin 1): The portal operator arises from the kinetic is mixing between
the SM photon field strength F μν and a dark photon F
μν :
ε
2
F μν F
μν
,
it is an operator of dimension four. It is assumed that the dark photon is the main
carrier of the interaction among the dark sector states.
The existence of an independent U (1) group symmetry was originally proposed,
in the context of supersymmetric theories, in [1, 2] and, more in general, in [3, 4];
• axion (spin 0): The operator comes from the interaction between a pseudo-scalar,
the axion a, and the SM photon and fermions ψ:
a
f a
F μν ˜
F
μν
+
1
f a
∂ μ a ψ γ
μ
γ 5 ψ ,
with operators of dimension five; the physics of this portal is based on that of
the axion and related to the strong CP problem as well as axion dark matter. In
many cases, the portal is generalized to an axion-like particle (ALP) with similar
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Fabbrichesi et al., The Physics of the Dark Photon,
SpringerBriefs in Physics, https://doi.org/10.1007/978-3-030-62519-1_5
71