34
2 Phenomenology of the Massless Dark Photon
In general, we can have as many dark fermions as there are in the SM; they can
be classified conveniently according to whether they couple (via the corresponding
messengers) to quarks (q L , u R , d R ) or leptons (l L , e R ): We denote the first (hadronlike) Q and the latter (lepton-like) χ.
The Yukawa-like interaction Lagrangian for flavor-diagonal interactions can be
written as [51, 52]:
L ⊃ −g L
φ
†
L ¯
χ R l L + S
U †
L
¯
Q
U
R q L + S
D†
L
¯
Q
D
R q L
− g R
φ
†
R ¯
χ L e R + S
U †
R
¯
Q
D
L u R + S
U †
R
¯
Q
D
L d R
+ h.c.
(2.46)
where q L (q R ) and e L (e R ) are SU (2) L doublets (singlets) for quarks and leptons
respectively. Sum over flavor and color indices, that we omitted for simplicity, is
understood. The L-type scalars are doublets under SU(2) L , while the R-type scalars
are singlets under SU(2) L . The S L ,R messengers carry color indices (unmarked in
(2.46)), while the messengers φ L ,R are color singlets. The Yukawa coupling strengths
are parameterized by α L ,R ≡ g
2
L ,R /(4π); they can be different for different fermions
and as many as the SM fermions. For simplicity, we take them to be equal and, in
addition, α L = α R .
In order to generate chirality-changing processes, we must have the mixing terms
L ⊃ −λ s S 0
H
†
φ
†
R φ L + ˜
H
† S
U †
R S
U
L + H
† S
D†
R S
D
L
+ h.c. ,
(2.47)
where H is the SM Higgs boson, ˜
H = iσ 2 H
, and S 0 a scalar singlet of the dark
sector. After both S 0 and H take a vacuum expectation value (μ S and v h —the electroweak vacuum expectation value—respectively), the Lagrangian in Eq. (2.47) gives
rise to the mixing between right- and left-handed states.
Dark sector and messenger states are both charged under an unbroken U (1) D
gauge symmetry which is the same of the corresponding massless dark photon, with
coupling strength α D . We assign different dark U (1) D charges to the various dark
sector fermions to ensures, by charge conservation, their stability. Since SM fields
are neutral under U (1) D interactions, messengers and associated dark-fermions field
in Eq. (2.46) must carry the same U (1) D quantum charge.
When the dark sector scalar S 0 and the Higgs boson acquire their vacuum
expectation values, the scalar messengers must be rotated to identify the physical states. Before this rotation, φ Lν , S
U
Ld ,and S
D
Lu are degenerate mass eigenstates
with mass m S . After the rotation, the mass eigenstates (labeled by ±) are given by
φ ± ≡
1
√
2
(φ L ± φ R ) , S
U,D
±
≡
1
√
2
S
U,D
L
± S
U,D
R
, corresponding to masses
m ± = m φ,S
1 ± η s
(2.48)
where we defined the mixing parameters for the S and φ messengers
2 Phenomenology of the Massless Dark Photon
In general, we can have as many dark fermions as there are in the SM; they can
be classified conveniently according to whether they couple (via the corresponding
messengers) to quarks (q L , u R , d R ) or leptons (l L , e R ): We denote the first (hadronlike) Q and the latter (lepton-like) χ.
The Yukawa-like interaction Lagrangian for flavor-diagonal interactions can be
written as [51, 52]:
L ⊃ −g L
φ
†
L ¯
χ R l L + S
U †
L
¯
Q
U
R q L + S
D†
L
¯
Q
D
R q L
− g R
φ
†
R ¯
χ L e R + S
U †
R
¯
Q
D
L u R + S
U †
R
¯
Q
D
L d R
+ h.c.
(2.46)
where q L (q R ) and e L (e R ) are SU (2) L doublets (singlets) for quarks and leptons
respectively. Sum over flavor and color indices, that we omitted for simplicity, is
understood. The L-type scalars are doublets under SU(2) L , while the R-type scalars
are singlets under SU(2) L . The S L ,R messengers carry color indices (unmarked in
(2.46)), while the messengers φ L ,R are color singlets. The Yukawa coupling strengths
are parameterized by α L ,R ≡ g
2
L ,R /(4π); they can be different for different fermions
and as many as the SM fermions. For simplicity, we take them to be equal and, in
addition, α L = α R .
In order to generate chirality-changing processes, we must have the mixing terms
L ⊃ −λ s S 0
H
†
φ
†
R φ L + ˜
H
† S
U †
R S
U
L + H
† S
D†
R S
D
L
+ h.c. ,
(2.47)
where H is the SM Higgs boson, ˜
H = iσ 2 H
, and S 0 a scalar singlet of the dark
sector. After both S 0 and H take a vacuum expectation value (μ S and v h —the electroweak vacuum expectation value—respectively), the Lagrangian in Eq. (2.47) gives
rise to the mixing between right- and left-handed states.
Dark sector and messenger states are both charged under an unbroken U (1) D
gauge symmetry which is the same of the corresponding massless dark photon, with
coupling strength α D . We assign different dark U (1) D charges to the various dark
sector fermions to ensures, by charge conservation, their stability. Since SM fields
are neutral under U (1) D interactions, messengers and associated dark-fermions field
in Eq. (2.46) must carry the same U (1) D quantum charge.
When the dark sector scalar S 0 and the Higgs boson acquire their vacuum
expectation values, the scalar messengers must be rotated to identify the physical states. Before this rotation, φ Lν , S
U
Ld ,and S
D
Lu are degenerate mass eigenstates
with mass m S . After the rotation, the mass eigenstates (labeled by ±) are given by
φ ± ≡
1
√
2
(φ L ± φ R ) , S
U,D
±
≡
1
√
2
S
U,D
L
± S
U,D
R
, corresponding to masses
m ± = m φ,S
1 ± η s
(2.48)
where we defined the mixing parameters for the S and φ messengers