C. Englert et al. / Physics Letters B 802 (2020) 135261 3
The matter fields come into effect via bulk and UV brane actions which have the general form
S =
d
6
x
√
−G
[
L
6D
+δ(y)L
5D
]
, (3)
where
√
−G = 1/(kz
6
). Starting of with the 6D Lagrangian, the gauge sector has the usual form for a Yang-Mills theory, accompanied by
a gauge fixing term and ghost fields
L
gauge
bulk
=−tr
1
4
F
MN
F
MN
+
1
2ξ
(
f
gf
)
2
+L
ghost
.
(4)
The bulk 6D action for the fermions is
L
ferm
bulk
=
4
α=1
α
32
D (c
α
32
)
α
32
+
3
β=1
β
11
D (c
β
11
)
β
11
+
3
β=1
β
11
D (c
β
11
)
β
11
, (5)
with bulk mass parameters c
α
32
, c
β
11
, c
β
11
for the fermions in their respective representation along with the generational index included
in the covariant derivative definition (e.g. [29]).
The
brane-localised scalar in the spinorial representation
32
(x, w) has a Higgs-like scalar potential
L
scalar
brane
=−(D
μ
32
)
†
(D
μ
32
)−(D
w
32
)
†
(D
w
32
) − λ(
†
32
32
−|r|
2
)
2
.
λ, |r| determine the vacuum expectation value (VEV) that
32
(x, w) develops along the SU(5) direction. This is then responsible for the
breaking of the SO(11) gauge symmetry on the UV brane.
On
the same 5D brane, we have the brane symplectic Majorana fermions χ
β
1
(x, w), which facilitate the 6D seesaw mechanism [35]via
L
Maj.
brane
=
1
2
χ
β
1
(γ
μ
∂
μ
+γ
6
∂
w
)χ
β
1
−
1
2
M
ββ
χ
β
1
χ
β
1
, (6)
where M
ββ
is a constant matrix. Finally we have the Lagrangian terms that specify the coupling between the bulk 6D fermions and
the 5D fields on the SO(11) brane which induce effective Dirichlet boundary conditions, and lift the mass degeneracy of the quark and
lepton sector on the IR brane. The brane-localised action contains eight allowed couplings between
32
,
32
,
11
which are consistent
with gauge symmetry, parity assignments and keeping the action dimensionless.
Symmetry
breaking
Symmetry
breaking in this model consists of 3 stages which break SO(11) down to SU(3)
C
× U (1)
EM
on the IR brane:
1) Symmetry
breaking via orbifold parity assignments, which break SO(11) to the Pati-Salam [37]group SO(11) → SU(4)
C
× SU(2)
L
×
SU(2)
R
≡ G
PS
on the IR brane.
2) Symmetry
breaking via 5D brane interactions between the bulk gauge fields and
32
, which break the SO(11) symmetry down to
SU(5) on the UV brane. The zero mode spectrum on the IR brane has a SM symmetry content SU(5) ∩ G
PS
= G
SM
.
3) Hosotani
breaking [38–40], which acts as the electroweak symmetry breaking mechanism on the IR brane, breaking G
SM
to SU(3)
C
×
U (1)
EM
through a non-vanishing expectation value θ
H
of the associated Wilson loop. More specifically this happens through the A
z
component of the gauge field, which is a bi-doublet under the SU(2)
L
× SU(2)
R
and therefore plays the role of the usual SM Higgs
boson [41].
Effective
Higgs potential
The
equations of motion for the relevant towers, and how they relate to SU(3)
C
× U (1)
EM
via the twisted gauge imposed by the
Hosotani mechanism, along with the computation of the effective potential is summarised in Appendix A.
The
free parameter set in charge of controlling the solution space consists of
P =
k, z
L
, c
0
, c
1
, c
2
, c
0
, μ
1
,
˜
μ
2
, μ
11
, μ
11
, M,m
B
.
(7)
k is the AdS
5
curvature, z
L
the warp factor, c
0
, c
1
, c
2
, c
0
are the fermion bulk masses along the warped dimension y; μ
1
,
˜
μ
2
, μ
11
, μ
11
are
couplings localised on the 5D UV brane (at y = 0in Fig. 1) between the 5D scalar
32
and the bulk fermion fields
α
32
,
β
11
,
β
11
, which
have the effect of reducing the PS symmetry down to the SM on the IR brane. Finally M, m
B
are 5D Majorana masses confined to the UV
brane. All remaining parameters (see Sec. 2) are not relevant for the gauge boson and fermion equations of motion and, hence, do not
impact our analysis.
The
parameters determine the dynamical value of order parameter θ
H
for electroweak symmetry breaking following the Hosotani
mechanism. The shape of the effective potential V
eff
(θ
H
) is sculpted by the bosonic and fermionic contributions. Following [29], we focus
on the 3rd generation, and identify c
α
32
= c
0
, c
β
11
= c
1
, c
β
11
= c
2
, c
4
32
= c
0
. We have also set m
B
, M to the sample values stated by the
authors in the original paper, M =−10
7
GeV, m
B
= 1.145 · 10
12
GeV, which is done to simplify the analysis and ensure the correct order
of magnitude for neutrino masses (i.e. < 0.1eV).
The
effective Higgs potential consists of the fermionic and bosonic contributions V
eff
(θ
H
) = V
Bosons
eff
+V
Fermions
eff
, arising from the relevant
KK towers. For the explicit form of the contributions we refer the reader to the effective potential section in [29]. The mass of the Higgs
boson is given by the second derivative of the effective potential