1.1 Massless and Massive Dark Photons
3
– the massless kind, which, as we are about to show, does not couple directly to
any of the SM currents and interacts instead with ordinary matter only through
operators of dimension higher than four;
– the massive kind, which couples to ordinary matter through a current (with arbitrary charge), that is, a renormalizable operator of dimension four. The massless
limit of this case does not correspond to the massless case above.
Because of their different coupling to SM particles, the two kinds are best discussed
separately.
Let us first consider the massless case.
As first discussed in [1] in this case the classical Lagrangian can be diagonalized.
What happens at the quantum level and how the mixing manifests itself has been
analyzed in detail in [16] for the unbroken gauge theory as well as the spontaneously
broken case (see, also, the appendix of [17] which we mostly follow).
The kinetic terms in Eq. (1.1) can be diagonalized by rotating the gauge fields as
A
μ
a
A
μ
b
=
⎛
⎜
⎝
1
√
1 − ε 2
0
−
ε
√
1 − ε 2
1
⎞
⎟
⎠
cos θ − sin θ
sin θ cos θ
A
μ
A μ
,
(1.3)
where now we can identify A
μ with the ordinary photon and A
μ with the dark photon.
The additional orthogonal rotation in Eq. (1.3) is always possible and introduces an
angle θ which is arbitrary as long as the gauge bosons are massless.
After the rotation in Eq. (1.3), the interaction Lagrangian in Eq. (1.2) becomes
L
=
e
cos θ
√
1 − ε 2
J
μ + e
sin θ −
ε cos θ
√
1 − ε 2
J μ
A
μ
+
−
e
sin θ
√
1 − ε 2
J
μ + e
cos θ +
ε sin θ
√
1 − ε 2
J μ
A
μ
.
(1.4)
By choosing sin θ = 0 (cos θ = 1) (see right-side of Fig. 1.1) we can have the ordinary photon A μ coupled only to the ordinary current J μ while the dark photon couples
to both the ordinary and the dark current J
μ , the former with strength εe/
√
1 − ε 2
proportional to the mixing parameter ε. The Lagrangian is therefore:
L
=
e
√
1 − ε 2
J
μ −
eε
√
1 − ε 2
J μ
A
μ
+ e J μ A
μ
.
(1.5)
Vice versa, with the choice sin θ = ε and cos θ =
√
1 − ε 2 (see left-side of
Fig. 1.1), we have the opposite situation with the dark photon only coupled to the
dark current and the ordinary photon to both currents, with strength εe/
√
1 − ε 2 to
the dark one. This latter coupling between the dark-sector matter to the ordinary
photon is called a milli-charge. Its value is experimentally known to be small [18].
3
– the massless kind, which, as we are about to show, does not couple directly to
any of the SM currents and interacts instead with ordinary matter only through
operators of dimension higher than four;
– the massive kind, which couples to ordinary matter through a current (with arbitrary charge), that is, a renormalizable operator of dimension four. The massless
limit of this case does not correspond to the massless case above.
Because of their different coupling to SM particles, the two kinds are best discussed
separately.
Let us first consider the massless case.
As first discussed in [1] in this case the classical Lagrangian can be diagonalized.
What happens at the quantum level and how the mixing manifests itself has been
analyzed in detail in [16] for the unbroken gauge theory as well as the spontaneously
broken case (see, also, the appendix of [17] which we mostly follow).
The kinetic terms in Eq. (1.1) can be diagonalized by rotating the gauge fields as
A
μ
a
A
μ
b
=
⎛
⎜
⎝
1
√
1 − ε 2
0
−
ε
√
1 − ε 2
1
⎞
⎟
⎠
cos θ − sin θ
sin θ cos θ
A
μ
A μ
,
(1.3)
where now we can identify A
μ with the ordinary photon and A
μ with the dark photon.
The additional orthogonal rotation in Eq. (1.3) is always possible and introduces an
angle θ which is arbitrary as long as the gauge bosons are massless.
After the rotation in Eq. (1.3), the interaction Lagrangian in Eq. (1.2) becomes
L
=
e
cos θ
√
1 − ε 2
J
μ + e
sin θ −
ε cos θ
√
1 − ε 2
J μ
A
μ
+
−
e
sin θ
√
1 − ε 2
J
μ + e
cos θ +
ε sin θ
√
1 − ε 2
J μ
A
μ
.
(1.4)
By choosing sin θ = 0 (cos θ = 1) (see right-side of Fig. 1.1) we can have the ordinary photon A μ coupled only to the ordinary current J μ while the dark photon couples
to both the ordinary and the dark current J
μ , the former with strength εe/
√
1 − ε 2
proportional to the mixing parameter ε. The Lagrangian is therefore:
L
=
e
√
1 − ε 2
J
μ −
eε
√
1 − ε 2
J μ
A
μ
+ e J μ A
μ
.
(1.5)
Vice versa, with the choice sin θ = ε and cos θ =
√
1 − ε 2 (see left-side of
Fig. 1.1), we have the opposite situation with the dark photon only coupled to the
dark current and the ordinary photon to both currents, with strength εe/
√
1 − ε 2 to
the dark one. This latter coupling between the dark-sector matter to the ordinary
photon is called a milli-charge. Its value is experimentally known to be small [18].