8
1 Introduction
the model and thus possible to relate different limits that are instead independent or
not present under the portal interaction.
1.2.1 Massive Dark Photon: Origin and Size of the Mixing
Parameter
The size of the mixing parameter ε is arbitrary. It is this feature that makes the charge
not quantized. At the same time, it cannot be O(1) because, if so, the massive dark
photon would have already been discovered.
A natural suppression of ε is achieved if the mixing only comes as a correction at
one- or two-loop level in some UV completion. This is achieved in a natural manner
if the tree-level mixing is set to zero. One looks for the renormalization of the model
and introduces the necessary counter-terms, of which the mixing in Eq. (1.1) is one.
If there are states in the UV completion carrying both ordinary and dark charges, the
loop of these states generates the mixing but it comes suppressed by the loop factor
(neglecting logarithmic terms) and therefore of order, say, 1/(16π
2
) times the square
of the coupling constant and therefore approximately O(10
−3
), for a perturbative
value of such a coupling. One can further suppress such a term by assuming that the
states carrying both charges come in doublets of opposite dark charges. In this case,
the first contribution is at the two-loop level, and approximately of order O(10
−5
).
If the mixing originates in the exchange of heavy messenger fields [30] or in a
multi-loop contribution [31, 32], its value can be smaller.
Even smaller values of the parameter ε are expected if the origin of the mixing
is non-perturbative; for example, values between O(10
−12
) and O(10
−6
) have been
discussed—mostly within the broad heading of string compactification [33–38], or in
scenarios of SUSY breaking [39] and hidden valley [40]. These arguments are often
cited to motivate experimental searches in the region of small mixing parameter ε
in the case of the massive dark photon—regardless of the large uncertainties in the
predictions of the corresponding theoretical approaches.
1.2.2 Massless Dark Photon: Higher-Order Operators
The massless dark photon does not interact directly with the currents of the SM
fermions. The higher-order operators through which the interaction with ordinary
matter ψ
i takes place start with the dimension-five operators in the Lagrangian
L =
e D
2Λ 5
ψ
i σ μν
D
i j
M + iγ 5 D
i j
E
ψ
j F
μν
,
(1.14)
where F
μν is the field strength associated to the dark photon field A
μ , and σ μν =
i/2 [γ μ , γ ν ]. The operator proportional to the coefficient D M is the magnetic dipole
1 Introduction
the model and thus possible to relate different limits that are instead independent or
not present under the portal interaction.
1.2.1 Massive Dark Photon: Origin and Size of the Mixing
Parameter
The size of the mixing parameter ε is arbitrary. It is this feature that makes the charge
not quantized. At the same time, it cannot be O(1) because, if so, the massive dark
photon would have already been discovered.
A natural suppression of ε is achieved if the mixing only comes as a correction at
one- or two-loop level in some UV completion. This is achieved in a natural manner
if the tree-level mixing is set to zero. One looks for the renormalization of the model
and introduces the necessary counter-terms, of which the mixing in Eq. (1.1) is one.
If there are states in the UV completion carrying both ordinary and dark charges, the
loop of these states generates the mixing but it comes suppressed by the loop factor
(neglecting logarithmic terms) and therefore of order, say, 1/(16π
2
) times the square
of the coupling constant and therefore approximately O(10
−3
), for a perturbative
value of such a coupling. One can further suppress such a term by assuming that the
states carrying both charges come in doublets of opposite dark charges. In this case,
the first contribution is at the two-loop level, and approximately of order O(10
−5
).
If the mixing originates in the exchange of heavy messenger fields [30] or in a
multi-loop contribution [31, 32], its value can be smaller.
Even smaller values of the parameter ε are expected if the origin of the mixing
is non-perturbative; for example, values between O(10
−12
) and O(10
−6
) have been
discussed—mostly within the broad heading of string compactification [33–38], or in
scenarios of SUSY breaking [39] and hidden valley [40]. These arguments are often
cited to motivate experimental searches in the region of small mixing parameter ε
in the case of the massive dark photon—regardless of the large uncertainties in the
predictions of the corresponding theoretical approaches.
1.2.2 Massless Dark Photon: Higher-Order Operators
The massless dark photon does not interact directly with the currents of the SM
fermions. The higher-order operators through which the interaction with ordinary
matter ψ
i takes place start with the dimension-five operators in the Lagrangian
L =
e D
2Λ 5
ψ
i σ μν
D
i j
M + iγ 5 D
i j
E
ψ
j F
μν
,
(1.14)
where F
μν is the field strength associated to the dark photon field A
μ , and σ μν =
i/2 [γ μ , γ ν ]. The operator proportional to the coefficient D M is the magnetic dipole