V International Symposium on Quantum Theory and Symmetries
Journal of Physics: Conference Series 128 (2008) 012005
IOP Publishing
doi:10.1088/1742-6596/128/1/012005
Higgsless electroweak model due to the spherical
geometry
Nikolay A Gromov
Department of Mathematics, Komi Science Center UrD RAS, Kommunisticheskaya st., 24,
Syktyvkar, 167982, Russia
E-mail: [email protected]
Abstract. A new formulation of the Electroweak Model with 3-dimensional spherical
geometry in the target space is suggested. The free Lagrangian in the spherical field space
along with the standard gauge field Lagrangian form the full Higgsless Lagrangian of the model,
whose second order terms reproduce the same fields with the same masses as the Standard
Electroweak Model. The vector bosons masses are automatically generated, so there is no need
in special mechanism of spontaneous symmetry breaking.
1. Introduction
The Standard Electroweak Model (SEWM) based on gauge group SU (2) × U (1) gives a good
description of electroweak processes. One of the unsolved problems is the origin of electroweak
symmetry breaking. In the standard formulation the scalar field (Higgs boson) performs this
task via Higgs mechanism, which generates a mass terms for vector bosons. However, it is not
yet experimentally verified whether electroweak symmetry is broken by such a Higgs mechanism,
or by something else.
The emergence of large number Higgsless models [1]–[10] was stimulated by difficulties with
Higgs boson. These models are mainly based on extra dimensions of different types or larger
gauge groups. The construction given in [11] is based on an observation: the underlying group
of SEWM can be represented as a semidirect product of U (1) and SU (2).
In the present paper a new formulation of the Higgsless Electroweak Model is suggested.
Firstly we observe that the quadratic form φ† φ = φ∗1 φ1 + φ∗2 φ2 = R2 of the complex matter
field φ ∈ C2 is invariant with respect to gauge group transformations SU (2) × U (1) and we can
restrict fields on the quadratic form without the loss of gauge invariance of the model. This
quadratic form define three dimensional sphere in four dimensional Euclidean space of the real
components of φ, where the noneuclidean spherical geometry is realized.
Secondly we introduce the free matter field Lagrangian in this spherical field space, which
along with the standard gauge field Lagrangian form the full Higgsless Lagrangian of the model.
Its second order terms reproduce the same fields as the Standard Electroweak Model but without
the remaining real dynamical Higgs field. The vector bosons masses are automatically generated
and are given by the same formulas as in the Standard Electroweak Model, so there is no need
in special mechanism of spontaneous symmetry breaking.
The fermion Lagrangian of SEWM are modified by replacing of the fields φ with the restricted
on the quadratic form fields in such a way that its second order terms provide the electron mass
c 2008 IOP Publishing Ltd
1
V International Symposium on Quantum Theory and Symmetries
Journal of Physics: Conference Series 128 (2008) 012005
IOP Publishing
doi:10.1088/1742-6596/128/1/012005
and neutrino remain massless. The preliminary versions are in [12],[13].
2. Standard electroweak model
The bosonic sector of SEWM is SU (2) × U (1) gauge theory in the space Φ2 (C) of fundamental
representation of SU (2). The bosonic Lagrangian is given by the sum
LB = LA + Lφ ,
where
LA =
(1)
1
1
1
1
1 2
2 2
3 2
Tr(Fµν )2 − (Bµν )2 = − [(Fµν
) + (Fµν
) + (Fµν
) ] − (Bµν )2
2
8g
4
4
4
(2)
is the gauge field Lagrangian for SU (2) × U (1) group and
1
Lφ = (Dµ φ)† Dµ φ − V (φ)
2
(3)
is the matter
field
Lagrangian (summation on the repeating Greek indexes is always understood).
φ1
Here φ =
∈ Φ2 (C), Dµ are the covariant derivatives
φ2
Dµ φ = ∂µ φ − ig
3
X
!
Tk Akµ
φ − ig 0 Y Bµ φ,
(4)
k=1
where
1
1
T1 = σ1 =
2
2
0 1
1 0
,
1
1
T2 = σ2 =
2
2
0 −i
i 0
,
1
1
T3 = σ3 =
2
2
1 0
0 −1
,
with σk being Pauli matrices, are linear representation of generators in the complex space Φ2 (C)
of the fundamental representation of SU (2) and Y = 21 1 is generator of U (1) in the same space.
The gauge fields
Aµ (x) = −ig
3
X
Bµ (x) = −ig 0 Bµ (x)
Tk Akµ (x),
k=1
take their values in Lie algebras su(2), u(1) respectively, and the stress tensors are
Fµν (x) = Fµν (x) + [Aµ (x), Aν (x)],
Bµν = ∂µ Bν − ∂ν Bµ .
The potential V (φ) in (3) is introduced by hand in a special form (sombrero)
V (φ) =
2
λ †
φ φ − v2 ,
4
(5)
where λ, v are constants.
The Lagrangian LB (1) describe massless fields. To generate mass terms for the vector bosons
without breaking the gauge invariance one uses the Higgs mechanism. One of LB ground states
φvac =
0
v
Akµ = Bµ = 0
,
is taken as a vacuum state of the model, and small field excitations
φ1 (x),
φ2 (x) = v + χ(x),
2
Aaµ (x),
Bµ (x)
V International Symposium on Quantum Theory and Symmetries
Journal of Physics: Conference Series 128 (2008) 012005
IOP Publishing
doi:10.1088/1742-6596/128/1/012005
1 0
with respect to the vacuum are regarded. The matrix Q = Y
=
, which annihilates
0 0
the ground state Qφvac = 0, is the generator of the electromagnetic subgroup U (1)em . The new
fields
+T 3
1 Wµ± = √ A1µ ∓ iA2µ ,
2
1
3
0
,
gA
−
g
B
µ
µ
g 2 + g 02
Zµ = p
1
0 3
Aµ = p 2
g
A
+
gB
µ
µ
g + g 02
are introduced, where Wµ± are complex (Wµ− )∗ = Wµ+ and Zµ , Aµ are real.
For small fields the Lagrangian LB (1) can be rewritten in the form
LB = L0 + Lint ,
(2)
where L0 = LB is the usual second order Lagrangian for free vector and scalar bosons, and
higher order terms Lint are regarded as field interactions. The second order terms of the
Lagrangian (1) are as follows
1 + −
1
1
1
1
1
(2)
LB = − Wµν
Wµν + m2W Wµ+ Wµ− − Fµν Fµν − Zµν Zµν + m2Z Zµ Zµ + (∂µ χ)2 − m2χ χ2 ,
2
4
4
2
2
2
(6)
where Zµν = ∂µ Zν − ∂ν Zµ , Fµν = ∂µ Aν − ∂ν Aµ , W ± µν = ∂µ Wν± − ∂ν Wµ± and describe massive
vector fields Wµ± with identical mass mW = 12 gv (W -bosons), massless vector field Aµ , mA = 0
p
(photon), massive vector field Zµ , with the mass mZ = v2 g 2 + g 02 (Z-boson) and massive scalar
√
field χ, mχ = 2λv (Higgs boson).
W - and Z-bosons have been observed and have the masses mW = 80GeV, mZ = 91GeV.
Higgs boson has not been experimentally verified up to now.
3. Higgsless electroweak model with 3D spherical matter space
The complex 2D space Φ2 can be regarded as 4D real Euclidean space R4 . Let us introduce the
real fields r, ψ̄ = (ψ1 , ψ2 , ψ3 ) by
φ1 = r(ψ2 + iψ1 ),
φ2 = r(1 + iψ3 ).
(7)
It is easy to see that the quadratic form
φ† φ = φ∗1 φ1 + φ∗2 φ2 = R2
(8)
is invariant with respect to group transformations SU (2) × U (1) and we can restrict fields on
this quadratic form without the loss of gauge invariance of the model. Similar restriction was
appeared in the unified conformal model for fundamental interactions [14] in the case when the
length scale is defined in such a way that elementary particle masses are the same for all times
and in all places.
For the real fields the form (8) is written as r2 (1 + ψ̄ 2 ) = R2 , where ψ̄ 2 = ψ12 + ψ22 + ψ32 ,
therefore
R
r=q
.
(9)
1 + ψ̄ 2
Hence there are three independent real fields ψ̄. These fields belong to the space Ψ3 with
noneuclidean spherical geometry which is realized on the 3D sphere in 4D Euclidean space
R4 . The fields ψ̄ are intrinsic Beltrami coordinates on Ψ3 .
3
V International Symposium on Quantum Theory and Symmetries
Journal of Physics: Conference Series 128 (2008) 012005
IOP Publishing
doi:10.1088/1742-6596/128/1/012005
2
The potential (5) is the constant V (φ) = λ R2 − v 2 /4 on the sphere and what is more
V (φ) = 0 for R = v. Therefore, let us define the free gauge invariant matter field Lagrangian
Lψ with the help of the metric tensor [12] of the spherical space Ψ3
gkk (ψ̄) =
1 + ψ̄ 2 − ψk2
,
(1 + ψ̄ 2 )2
gkl (ψ̄) =
−ψk ψl
(1 + ψ̄ 2 )2
in the form
3
R2 X
R2 (1 + ψ̄ 2 )(Dµ ψ̄)2 − (ψ̄, Dµ ψ̄)2
Lψ =
gkl Dµ ψk Dµ ψl =
.
2 k,l=1
2(1 + ψ̄ 2 )2
(10)
Let us note that the Lagrangian on 3D sphere in 4D Euclidean space was considered in [15],
where the invariant perturbative theory was developed which does not depend on some particular
parametrization.
The covariant derivatives (4) are obtained using the representations of generators for the
algebras su(2), u(1) in the space Ψ3 [12]
−(1 + ψ12 )
i
ψ3 − ψ1 ψ2  ,
T1 ψ̄ =
2
−(ψ2 + ψ1 ψ3 )

−(ψ3 + ψ1 ψ2 )
i
T2 ψ̄ =  −(1 + ψ22 )  ,
2
ψ1 − ψ2 ψ3


−ψ2 + ψ1 ψ3
i
T3 ψ̄ =  ψ1 + ψ2 ψ3  ,
2
1 + ψ32


−(ψ2 + ψ1 ψ3 )
i
Y ψ̄ =  ψ1 − ψ2 ψ3 
2
−(1 + ψ32 )



and are as follows:
Dµ ψ1 = ∂µ ψ1 +
i g0
gh
−(1 + ψ12 )A1µ − (ψ3 + ψ1 ψ2 )A2µ − (ψ2 − ψ1 ψ3 )A3µ − (ψ2 + ψ1 ψ3 )Bµ ,
2
2
i g0
gh
(ψ3 − ψ1 ψ2 )A1µ − (1 + ψ22 )A2µ + (ψ1 + ψ2 ψ3 )A3µ + (ψ1 − ψ2 ψ3 )Bµ ,
2
2
h
i
g
g0
Dµ ψ3 = ∂µ ψ3 +
−(ψ2 + ψ1 ψ3 )A1µ + (ψ1 − ψ2 ψ3 )A2µ + (1 + ψ32 )A3µ − (1 + ψ32 )Bµ . (11)
2
2
The gauge fields Lagrangian (2) does not depend on the fields φ and therefore remains unchanged.
For small fields, the second order Lagrangian (10) is written as
Dµ ψ2 = ∂µ ψ2 +
(2)
Lψ =
3 h
i2
i2
R2 h
R2 X
(Dµ ψ̄)(1) =
(Dµ ψk )(1) ,
2
2 k=1
where linear terms in covariant derivates (11) have the form
g
g
2
g
(Dµ ψ1 )(1) = ∂µ ψ1 − A1µ = −
A1µ − ∂µ ψ1 = − Â1µ ,
2
2
g
2
g
g
2
g
(Dµ ψ2 )(1) = ∂µ ψ2 − A2µ = −
A2µ − ∂µ ψ2 = − Â2µ ,
2
2
g
2
(1)
(Dµ ψ3 )
g 3 g0
1q 2
= ∂µ ψ3 + Aµ − Bµ =
g + g 02 Zµ .
2
2
2
4
(12)
V International Symposium on Quantum Theory and Symmetries
Journal of Physics: Conference Series 128 (2008) 012005
IOP Publishing
doi:10.1088/1742-6596/128/1/012005
For the new fields
1 Wµ± = √ Â1µ ∓ iÂ2µ ,
2
gA3µ − g 0 Bµ + 2∂µ ψ3
Zµ =
p
g 2 + g 02
g 0 A3µ + gBµ
Aµ = p 2
g + g 02
,
Lagrangian (12) is rewritten as follows
(2)
Lψ = +
R2 g 2 + − R2 (g 2 + g 02 )
Wµ Wµ +
(Zµ )2 .
4
8
The quadratic part of the full Lagrangian
1 + −
1
1
m2
(2)
(2)
Wµν + m2W Wµ+ Wµ− − (Fµν )2 − (Zµν )2 + Z (Zµ )2 ,
L0 = LA + Lψ = − Wµν
2
4
4
2
where
Rq 2
Rg
, mZ =
g + g 02 ,
(13)
2
2
and Zµν = ∂µ Zν − ∂ν Zµ , Fµν = ∂µ Aν − ∂ν Aµ , W ± µν = ∂µ Wν± − ∂ν Wµ± , describes all the
experimentally verified parts of SEWM but does not include the scalar Higgs field. For R = v
the masses (13) are identical to those of the SEWM (6).
The interaction part of the full Lagrangian in the first degree of approximation is given by
mW =
(1)
(3)
(3)
Lint = LA + Lψ ,
where the third order terms of the gauge field Lagrangian (2) are
(3)
LA
g
= −p 2
g + g 02
√
(
i
−
Wµν
Wµ+
−
+
Wµν
Wµ−
h
i
0
g Aν + gZν −
2 0
g Aµ + gZµ ×
g
2ig
−
+
+
−
W
W
−
W
W
µν µ
µν µ ∂ν ψ3 −
g 2 + g 02
+
−
× Wµν
(∂ν ψ2 − i∂ν ψ1 ) + Wµν
(∂ν ψ2 + i∂ν ψ1 ) − p
√
i
2 2 h +
−
−p 2
W
(∂
ψ
−
i∂
ψ
)
+
W
(∂
ψ
+
i∂
ψ
)
∂ν ψ3 + g 0 Fµν + gZµν ×
ν
2
ν
1
ν
2
ν
1
µν
µν
02
g +g
))
( √ h
i
4
2
i + 2 − 2
+ 2 ∂µ ψ1 ∂ν ψ2 +
− Wµ
×
Wµ
Wµ+ (∂ν ψ2 − i∂ν ψ1 ) + Wµ− (∂ν ψ2 + i∂ν ψ1 )
4
g
g
and those of the matter field Lagrangian (10) are
(3)
Lψ
R2 g
= √
2 2
(
Wµ+
"
+Wµ−
"
#
g 2 − g 02
g 0 (gAµ − g 0 Zµ )
p
ψ3 (∂µ ψ2 − i∂µ ψ1 ) − 2
(ψ
−
iψ
)∂
ψ
+
(ψ2 − iψ1 ) +
2
1
µ
3
g + g 02
g 2 + g 02
#
g 2 − g 02
g 0 (gAµ − g 0 Zµ )
p
ψ3 (∂µ ψ2 + i∂µ ψ1 ) − 2
(ψ
+
iψ
)∂
ψ
+
(ψ2 + iψ1 ) +
2
1
µ
3
g + g 02
g 2 + g 02
1q 2
+
g + g 02 Zµ (ψ1 ∂µ ψ2 − ψ2 ∂µ ψ1 ) .
g
The fermion Lagrangian of SEWM is taken in the form [16]
LF = L†l iσ̃µ Dµ Ll + e†r iσµ Dµ er − he [e†r (φ† Ll ) + (L†l φ)er ],
5
(14)
V International Symposium on Quantum Theory and Symmetries
Journal of Physics: Conference Series 128 (2008) 012005
IOP Publishing
doi:10.1088/1742-6596/128/1/012005
νe,l
where Ll =
is the SU (2)-doublet, e−
l the SU (2)-singlet, he is constant and er , el , νe are
e−
l
two component Lorentzian spinors. Here σµ are Pauli matricies, σ0 = σ˜0 = 1, σ˜k = −σk . The
covariant derivatives Dµ Ll are given by (4) with Ll instead of φ and Dµ er = (∂µ + ig 0 Bµ )er .
The convolution on the inner indices of SU (2)-doublet is denoted by (φ† Ll ).
The matter fields φ appear in Lagrangian (14) only in mass terms. With the use of (7) and
(9), these mass terms are rewritten in the form
he R
he [e†r (φ† Ll ) + (L†l φ)er ] = q
1+
h
ψ̄ 2
−†
e†r e−
l + el er +
†
†
† −
er + e†r νe,l
er − e†r νe,l + iψ2 νe,l
+ iψ1 νe,l
+iψ3 e−†
l er − er el
i
.
−†
Its second order terms he R e†r e−
l + el er provide the electron mass me = he R, and neutrino
remain massless.
4. Conclusion
The suggested formulation of the Electroweak Model with the gauge group SU (2) × U (1) based
on the 3-dimensional spherical geometry in the target space describes all experimentally observed
fields, and does not include the (up to now unobserved) scalar Higgs field. The free Lagrangian
in the spherical matter field space is used instead of Lagrangian (3) with the potential (5)
of the special form (sombrero). The gauge field Lagrangian is the standard one. There is
no need in Higgs mechanism since the vector field masses are generated automatically. The
renormalizability of the model is the open question at present and will be a subject of further
investigations.
The motion group of the spherical space Ψ3 is isomorphic to SO(4). It is known [17], that
the bosonic sector of SEWM has the custodial SU (2)-symmetry, and potential (5) is globally
invariant with respect of SO(4) which is locally isomorphic to the group SU (2)L × SU (2)R ,
where SU (2)L is the gauge group of SEWM. The appearance of the extra SU (2)R -symmetry
is interpreted as transformation of SU (2)-doublet by pseudoreal representation, that is the
complex conjugate doublet is equivalent to the initial one. The introduction of the real fields ψ̄
is connected in a certain sense with SO(4)-symmetry.
Acknowledgments
The author is grateful to A.V. Zhubr, V.V. Kuratov and I.B. Khriplovich for helpful discussions.
I am indebted to V.N. Pervushin and Yu.M. Zinoviev for pointing out the papers [14], [15],
correspondingly.
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V International Symposium on Quantum Theory and Symmetries
Journal of Physics: Conference Series 128 (2008) 012005
IOP Publishing
doi:10.1088/1742-6596/128/1/012005
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