2.1 Limits on the Dark Dipole Scale d M /Λ 2
25
where the factor of 2 in front takes into account the two polarizations of the dark
photon (with respect to the axion), v h = 174 GeV and m e is the electron’s mass.
To satisfy the limit in Eq. (2.7), the dark photon parameters in Eq. (2.8) must
satisfy
Λ
2
√
α D d
e
M
∼ > 4.5 × 10
6 TeV
2
,
(2.9)
after having included the numerical values of m e and v h . The limit in Eq. (2.9) updates
the one found in [4].
• Supernovae. An additional limit is found from the neutrino signal of supernova
1987A, for which the length of the burst constrains anomalous energy losses in the
explosion.
As before, a bound can be derived from that for the coupling between axions and
nucleons. The corresponding averaged square amplitude is given in [10, 11] as
spin
|M|
2
=
16(4π)
3
α
2
π α
aN
3 m
2
N
k
2
k 2 + m 2
π
2
+
l
2
l 2 + m 2
π
2
+
k
2 l
2
− 3(k · l)
2
(k 2 + m 2
π )(l 2 + m 2
π )
(2.10)
where α π = (2m N f /m π )
2
/4π 15 is the pion-nucleon coupling and k = p 2 − p 4
and l = p 2 − p 3 and p i the momenta of the nucleons. The coefficient α
aN is the
coupling constant of the axion to the nucleons.
In the thermal medium k
2
3m N T and we can neglect the pion mass to obtain
spin
|M|
2
=
32(4π)
3
α
2
π α
aN
m
2
N
(2.11)
and the energy-loss rate per unit mass in the degenerate case is [10, 12]
Q/ρ α
aN 1.74 × 10
33 ρ
10 15
T
MeV
6
,
(2.12)
in units of erg g
−1 s
−1 , which should not exceed the neutrino luminosity. This limit
yields, taking the most conservative estimate in [13, 14],
α
aN ≤ 1.3 × 10
−18
.
(2.13)
The combination that controls energy transfer to dark photons in this process from
ordinary matter (the quarks in the nucleons) is
α
aN = 2
1
4π
2 e D d
q
M
v h m N
Λ 2
2 ,
(2.14)
25
where the factor of 2 in front takes into account the two polarizations of the dark
photon (with respect to the axion), v h = 174 GeV and m e is the electron’s mass.
To satisfy the limit in Eq. (2.7), the dark photon parameters in Eq. (2.8) must
satisfy
Λ
2
√
α D d
e
M
∼ > 4.5 × 10
6 TeV
2
,
(2.9)
after having included the numerical values of m e and v h . The limit in Eq. (2.9) updates
the one found in [4].
• Supernovae. An additional limit is found from the neutrino signal of supernova
1987A, for which the length of the burst constrains anomalous energy losses in the
explosion.
As before, a bound can be derived from that for the coupling between axions and
nucleons. The corresponding averaged square amplitude is given in [10, 11] as
spin
|M|
2
=
16(4π)
3
α
2
π α
aN
3 m
2
N
k
2
k 2 + m 2
π
2
+
l
2
l 2 + m 2
π
2
+
k
2 l
2
− 3(k · l)
2
(k 2 + m 2
π )(l 2 + m 2
π )
(2.10)
where α π = (2m N f /m π )
2
/4π 15 is the pion-nucleon coupling and k = p 2 − p 4
and l = p 2 − p 3 and p i the momenta of the nucleons. The coefficient α
aN is the
coupling constant of the axion to the nucleons.
In the thermal medium k
2
3m N T and we can neglect the pion mass to obtain
spin
|M|
2
=
32(4π)
3
α
2
π α
aN
m
2
N
(2.11)
and the energy-loss rate per unit mass in the degenerate case is [10, 12]
Q/ρ α
aN 1.74 × 10
33 ρ
10 15
T
MeV
6
,
(2.12)
in units of erg g
−1 s
−1 , which should not exceed the neutrino luminosity. This limit
yields, taking the most conservative estimate in [13, 14],
α
aN ≤ 1.3 × 10
−18
.
(2.13)
The combination that controls energy transfer to dark photons in this process from
ordinary matter (the quarks in the nucleons) is
α
aN = 2
1
4π
2 e D d
q
M
v h m N
Λ 2
2 ,
(2.14)