Download Duality Theory of Weak Interaction

DUALITY THEORY OF WEAK INTERACTIONS
TIAN MA AND SHOUHONG WANG
Abstract. The main objectives of this article are 1) to derive a field theory of
weak interactions, 2) to study the duality of weak interactions, and 3) to derive
the long overdue weak potential and force formula. The new field equations for
weak interactions is derived by applying the principle of interaction dynamics (PID) and the principle of representation invariance (PRI) to a standard
SU (2) gauge action coupled with the action for the Dirac spinors. The new
field equations establish a natural duality between weak gauge fields {Wµa },
representing the W ± and Z intermediate vector bosons, and three bosonic
scalar fields φa , representing the neutral Higgs field H 0 as recently discovered
by LHC and two charged Higgs H ± . With the duality, for the first time, we
derive layered weak interaction potentials. These potentials clearly demonstrate many features of weak interactions consistent with observations, such
as the short-range nature of weak interactions. Also, PID provides an entirely
different way of introducing Higgs fields from first principles, and gives rise to
a different spontaneous symmetry breaking mechanism.
Contents
1. Introduction
2. Recapitulation of Two Basic Principles
2.1. Principle of Representation Invariance
2.2. Principle of Interaction Dynamics (PID)
2.3. Gauge symmetry breaking
3. Field Equations for Weak Interactions Based on PID and PRI
4. Weak Interaction Potentials
4.1. Dual equations of weak interaction potentials
4.2. Layered formulas of weak forces
4.3. Physical conclusions for weak forces
5. PID Induced Spontaneous Symmetry Breaking
References
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1. Introduction
The weak interaction is one of the four fundamental interactions in Nature.
The success of the standard model in particle physics demonstrates that the weak
interaction obeys SU (2) gauge invariance; see among many others [5, 2, 14].
Key words and phrases. weak interaction, duality theory, field equations of weak interaction,
weak interaction potential, weak interaction force formula, unified field equations, principle of
interaction dynamics, principle of representation invariance, electroweak theory, Higgs mechanism.
The work was supported in part by the Office of Naval Research, by the US National Science
Foundation, and by the Chinese National Science Foundation.
1
2
MA AND WANG
The main objectives of this article are 1) to derive a field theory of weak interactions, 2) to study the duality of weak interactions, and 3) to derive the long
overdue weak potential and force formula. Hereafter we describe briefly the main
ingredients of this article.
First, the key ingredient of the study is two basic principles: the principle of
interaction dynamics (PID) and principle of representation invariance (PRI), postulated recently by the authors in [9, 7, 8]. Intuitively, PID amounts to taking
the variation of the action functional under energy-momentum conservation constraint. PRI requires that physical laws be independent of representations of the
gauge groups. PRI is a basic logic requirement for any SU (N ) gauge theory, and
is universally valid. There are many strong evidences supporting PID, including in
particular dark energy and dark matter, and the Higgs field. In fact, as we shall
see later in this article, PID offers a new mechanism, based on first principles, for
spontaneous symmetry breaking.
Second, the weak interaction obeys the SU (2) gauge symmetry, which dictates
the Lagrangian action. Namely, the weak interaction fields are the SU (2) gauge
fields
Wµa = (W0a , W1a , W2a , W3a )
for 1 ≤ a ≤ 3,
and their action is
1
W a W µνb ,
(1.1)
LW = − Gw
4 ab µν
coupled with the standard action for the fermions as Dirac spinors ψ. Here
a
Wµν
= ∂µ Wνa − ∂ν Wµa + gw λabc Wµb Wνc
1 d c
1
λ λ .
Gab = Tr(τa τb† ) =
2
4N ac db
for 1 ≤ a ≤ 3,
Third, by PID and PRI, the field equations of the weak interaction are given
by:
µ b
b αβ
c
Gw
Wαν
Wβd − gw ψ̄ w γ ν σa ψ w
(1.2)
ab ∂ Wµν − gw λcd g
i
h
1
= ∂µ + γb1 Wµb − m2w xµ φw
a,
4
(1.3)
(iγ µ Dµ − ml )ψ w = 0.
The right-hand side of (1.2) is due to PID, leading naturally to the introduction
of three scalar dual fields. The left-hand side of (1.2) represents the intermediate
vector bosons W ± and Z, and the dual fields represent two charged Higgs H ± (to
be discovered) and the neutral Higgs H 0 , with the later being discovered by LHC
in 2012.
In a nutshell, the new field equations (1.2)
• offer a natural approach to introduce Higgs fields based solely on fundamental principles, and
• establish a natural duality between weak gauge fields {Wµa }, representing
the W ± and Z intermediate vector bosons, and three bosonic scalar fields
±
0
φw
a , representing both two charged and one neutral Higgs particles H , H .
It is worth mentioning that the right-hand side of (1.2), involving the Higgs fields,
can not be generated by directly adding certain terms in the Lagrangian action, as
in the case for the new gravitational field equations derived in [9].
DUALITY THEORY OF WEAK INTERACTIONS
3
Fourth, as mentioned earlier, PID induces naturally spontaneous symmetry
breaking mechanism. By construction, it is clear that the Lagrangian action LW
in (1.1) obeys the SU (2) gauge symmetry, the PRI and the Lorentz invariance.
Both the Lorentz invariance and PRI are universal principles, and, consequently,
the field equations (1.2) and (1.3) are covariant under these symmetries. The gauge
symmetry is spontaneously breaking in the field equations (1.2), due to the presence
of the terms in the right-hand side, derived by PID.
Spontaneous symmetry breaking is a phenomenon appearing in various physical
fields. In 2008, the Nobel Prize in Physics was awarded to Y. Nambu for the discovery of the mechanism of spontaneous symmetry breaking in subatomic physics.
In 2013, F. Englert and P. Higgs were awarded the Nobel Prize for the theoretical
discovery of a mechanism that contributes to our understanding of the origin of
mass of subatomic particles.
Although the phenomenon was discovered in superconductivity by GinzburgLandau in 1951, the mechanism of spontaneous symmetry breaking in particle
physics was first proposed by Y. Nambu in 1960; see [11, 12, 13]. The Higgs mechanism, discovered in [4, 1, 3], is a special case of the Nambo-Jona-Lasinio spontaneous
symmetry breaking, leading to the mass generation of sub-atomic particles.
PID provides a new mechanism for gauge symmetry breaking and mass generation. The difference between both the PID and the Higgs mechanisms is that
the first one is a natural sequence of the first principle, and the second is to add
artificially a Higgs field in the Lagrangian action. Also, the PID mechanism obeys
PRI, and the Higgs mechanism violates PRI.
Fifth, another key point of the study is that the field equations must satisfy
PRI, which induces an important SU (2) constant vector {ωa }. The components
of this vector represent the portions distributed to the gauge potentials Wµa by the
weak charge gw . Hence the weak interaction potential is given by the following PRI
representation invariant
(1.4)
Wµ = ωa Wµa = (W0 , W1 , W2 , W3 ),
and the weak charge potential and weak force are as
(1.5)
Φw = W 0
the time component of Wµ ,
Fw = −gw (ρ)∇Φw ,
where gw (ρ) is the weak charge of a particle with radius ρ.
Sixth, with the above physical meaning of the gauge potentials and the associated forces, for the first time, we deduce the following layered formulas for the
weak interaction potential:
B
−kr
−kr 1
− (1 + 2kr)e
,
Φw = gw (ρ)e
r
ρ
(1.6)
3
ρw
gw (ρ) = N
gw ,
ρ
where Φw is the weak force potential of a particle with radius ρ and carrying N
weak charges gw , taken as the unit of weak charge gs for each weakton [10], ρw is the
weakton radius, B is a parameter depending on the particles, and 1/k = 10−16 cm
represents the force-range of weak interactions.
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MA AND WANG
The layered weak interaction potential formula (1.6) shows clearly that the weak
interaction is short-ranged. Also, it is clear that the weak interaction is repulsive,
asymptotically free, and attractive when the distance of two particles increases.
The paper is organized as follows. The two basic principles PID and PRI are
briefly introduced in Section 2. Section 3 derives the new field model and duality of
weak interactions. Section 4 derives the weak potential and force formulas. Section
5 addresses the spontaneous symmetry breaking mechanism based on PID.
2. Recapitulation of Two Basic Principles
We recall in this section two basic principles, principle of interaction dynamics
(PID) and principle of representation invariance (PRI), postulated recently by the
authors [7, 8].
2.1. Principle of Representation Invariance. The Lagrangian action of an
SU (N ) gauge theory is given by
Z
(2.1)
L = Ldx,
with the action density
(2.2)
1
a
F bµν + Ψ̄(iγ µ Dµ − m)Ψ,
L = − Gab Fµν
4
where
a
Fµν
= ∂µ Aaν − ∂ν Aaµ + gλabc Abµ Acν ,
(2.3)
Dµ Ψ = (∂µ + igAaµ τa )Ψ,
1
1 d c
Gab = Tr(τa τb† ) =
λ λ ,
2
4N ac db
and τa (1 ≤ a ≤ N 2 − 1) are the generators of SU (N ), λcab are the structure
constants with respect to τa , {Gab } is a Riemannian metric on SU (N ) (see [8]),
Aaµ = (Aa0 , Aa1 , Aa2 , Aa3 )T (1 ≤ a ≤ N 2 − 1) are the SU (N ) gauge fields, Ψ =
(Ψ1 , · · · , ΨN )T are the N Dirac spinors.
In a neighborhood U ⊂ SU (N ) of the unit matrix, a matrix Ω ∈ U can be
written as
a
Ω = eiθ τa ,
where
(2.4)
τa = {τ1 , · · · , τK } ⊂ Te SU (N ),
K = N 2 − 1,
is a basis of generators of the tangent space Te SU (N ). An SU (N ) representation
transformation is a linear transformation of the basis in (2.4) as
(2.5)
τea = xba τb ,
where X = (xba ) is a nondegenerate complex matrix.
Mathematical logic dictates that a physically sound gauge theory should be
invariant under the SU (N ) representation transformation (2.5). Consequently, the
following principle of representation invariance (PRI) must be universally valid and
was first postulated in [8].
DUALITY THEORY OF WEAK INTERACTIONS
5
Principle 2.1 (Principle of Representation Invariance). All SU (N ) gauge theories
are invariant under the transformation (2.5). Namely, the actions of the gauge
fields are invariant and the corresponding gauge field equations as given by (2.6)(2.7) are covariant under the transformation (2.5).
Direct consequences of PRI include the following; see also [8] for details:
• The physical quantities such as θa , Aaµ , and λcab are SU (N )-tensors under
the generator transformation (2.5).
• The tensor Gab in (2.3) is a symmetric positive definite 2nd-order covariant
SU (N )-tensor, which can be regarded as a Riemannian metric on SU (N ).
• The Lagrangian action L in (2.1) and (2.2) is representation invariant under
the generator transformation (2.5), and the representation invariant gauge
field equations are
b
c
Gab ∂ ν Fνµ
(2.6)
− gλbcd g αβ Fαµ
Adβ − g Ψ̄γµ τa Ψ = (∂µ + αb Abµ )φa ,
(2.7)
(iγ µ Dµ − m)Ψ = 0.
Here the terms on the right hand side are derived using the principle of interaction dynamics (PID), to be stated in the next subsection. As we indicated in [8],
the field models based on PID appear to be the only model which obeys PRI. In
fact, based on PRI, for the gauge fields Aaµ and Gkµ , corresponding to two different
gauge groups SU (N1 ) and SU (N2 ), the following combinations
(2.8)
αAaµ + βGkµ
are prohibited. The reason is that Aaµ is an SU (N1 )-tensor with tensor index a,
and Gkµ is an SU (N2 )-tensor with tensor index k. The above combination violates
PRI. This point of view clearly shows that the classical electroweak theory violates
PRI. Hence the classical electroweak theory is only an approximate model.
2.2. Principle of Interaction Dynamics (PID). PID was first postulated by
the authors in [7], and motivated by the presence of dark matter and dark energy
[9]. We now briefly recall PID and the dark matter and dark energy motivation.
The law of gravity is then represented by the gravitational field equations solving
for the gravitational potentials {gµν }, the Riemannina metric of the space-time
manifold M . The principle of general relativity is a fundamental symmetry of
Nature, which dictates the Lagrangian action for gravity as the Einstein-Hilbert
functional:
Z √
8πG
(2.9)
LEH ({gµν }) =
R+ 4 S
−gdx.
c
M
Here R stands for the scalar curvature of M , and S is the energy-momentum density
of matter field in the universe. Due to the presence of dark energy and dark
matter, the energy-momentum tensor Tµν of normal matter is in general no longer
conserved: ∇µ (Tµν ) 6= 0.
Consequently, we have shown in [9] that the Euler-Lagrangian variation of the
Einstein-Hilbert functional LEH must be taken under energy-momentum conservation constraints; see [9] for details:
d (2.10)
LEH (gµν + λXµν ) =(δLEH (gµν ), X)
dλ λ=0
=0 ∀X = {Xµν } with ∇µ Xµν = 0.
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MA AND WANG
The resulting gravitational field equations are then given by
(2.11)
1
8πG
Rµν − gµν R = − 4 Tµν + ∇µ Φν ,
2
c 8πG
∇µ
Tµν − ∇µ Φν = 0.
c4
which give rise to a unified theory for dark matter and dark energy [9].
Note that the term ∇µ Φν does not correspond to any Lagrangian action density,
and is the direct consequence of energy-momentum conservation constraint of the
variation element X in (2.10). Also this new term, derived using energy-momentum
conservation constraint variation (2.10), plays a similar role as the Higgs field in the
standard model. It is natural then for us to postulate PID as a general principle
for all four interactions.
To state PID, let (M, gµν ) be a Riemannian manifold of space-time with Minkowski
signature, and A = (A1 , · · · , An ) be a vector field. We define two differential operators acting on an (r, s)-tensor u as
(2.12)
∇A u = ∇u + u ⊗ A,
(2.13)
divA u = divu − A · u.
For a functional F (u), a tensor u0 is an extremum point of F under the divA -free
constraint, if u0 satisfies
d
= (δF (u0 ), X) = 0
for all X with divA X = 0.
F (u0 + λX)
dλ
λ=0
In [8], we have shown that if u0 is an extremum point of F (u) with the divA -free
constraint, then there exists an (r − 1, s) tensor (or (r, s − 1) tensor) φ such that
u0 is a solution of the following equation
δF (u) = DA φ.
We remark that the divA -free constraint is equivalent to energy-momentum conservation. The original motivation of divA -free constraints was to take into consideration the presence of dark energy and dark matter in the modified gravitational
field equations [9], and then it is natural for us to postulate the following principle
of interaction dynamics (PID) [7]:
Principle of Interaction Dynamics (PID). For all physical interactions there
are Lagrangian actions
Z
√
(2.14)
L(g, A, ψ) =
L(gij , A, ψ) −gdx,
M
where g = {gij } is the Riemann metric representing the gravitational potential, A is
a set of vector fields representing gauge potentials, and ψ are the wave functions of
particles. The action (2.14) satisfy the invariance of general relativity (or Lorentz
invariance), the gauge invariance, and PID. Moreover, the states (g, A, ψ) are the
extremum points of (2.14) with the divA -free constraint.
DUALITY THEORY OF WEAK INTERACTIONS
7
2.3. Gauge symmetry breaking. In physics, symmetries are displayed in two
levels in the laws of Nature:
(2.15)
the invariance of Lagrangian actions L,
(2.16)
the covariance of variation equations of L.
The implication of the following three symmetries:
Einstein General Relativity,
(2.17)
Lorentz Invariance,
Gauge Representation Invariance (PRI),
is the universality of physical laws, i.e. the validity of laws of Nature is independent
of the coordinate systems expressing them. Consequently, the symmetries in (2.17)
cannot be broken at both levels of (2.15) and (2.16).
However, the physical implication of the gauge symmetry is different at the two
levels (2.15) and (2.16):
(1) The gauge invariance of the Lagrangian action, (2.15), amounts to saying
that the energy contributions of particles in a physical system are indistinguishable.
(2) The gauge invariance of the variation equations, (2.16), means that the
particles involved in the interaction are indistinguishable.
It is clear that the first aspect (1) above is universally true, while the second
aspect (2) is not universally true. In other words, the Lagrange actions obey the
gauge invariance, but the corresponding variation equations break the gauge symmetry. This suggests us to postulate the following principle of gauge symmetry
breaking for interactions described by a gauge theory.
Principle 2.2 (Gauge Symmetry Breaking). The gauge symmetry holds true only
for the Lagrangian actions for the electromagnetic, week and strong interactions,
and it will be violated in the field equations of these interactions.
The principle of gauge symmetry breaking can be regarded as part of the spontaneous symmetry breaking, which is a phenomenon appearing in various physical
fields. In 2008, the Nobel Prize in Physics was awarded to Y. Nambu for the discovery of the mechanism of spontaneous symmetry breaking in subatomic physics.
In 2013, F. Englert and P. Higgs were awarded the Nobel Prize for the theoretical
discovery of a mechanism that contributes to our understanding of the origin of
mass of subatomic particles.
Although the phenomenon was discovered in superconductivity by GinzburgLandau in 1951, the mechanism of spontaneous symmetry breaking in particle
physics was first proposed by Y. Nambu in 1960; see [11, 12, 13]. The Higgs mechanism, discovered in [4, 1, 3], is a special case of the Nambo-Jona-Lasinio spontaneous
symmetry breaking, leading to the mass generation of sub-atomic particles.
PID provides a new mechanism for gauge symmetry breaking and mass generation. The difference between both the PID and the Higgs mechanisms is that
the first one is a natural sequence of the first principle, and the second is to add
artificially a Higgs field in the Lagrangian action. Also, the PID mechanism obeys
PRI, and the Higgs mechanism violates PRI.
8
MA AND WANG
3. Field Equations for Weak Interactions Based on PID and PRI
In this section, we derive the general field equations for the weak interaction
based on PID and PRI. We know that the weak interaction obeys the SU (2) gauge
symmetry, which dictates the Lagrangian action. Namely, the weak interaction
fields are the SU (2) gauge fields
Wµa = (W0a , W1a , W2a , W3a )
for 1 ≤ a ≤ 3,
and their action is
1
LW = − Gw
W a W µνb ,
4 ab µν
which also stands for the scalar curvature of spinor bundle: M ⊗p (C4 )2 . Here
(3.1)
a
Wµν
= ∂µ Wνa − ∂ν Wµa + gw λabc Wµb Wνc
for 1 ≤ a ≤ 3.
By PID and PRI, the field equations of the weak interaction are given by (see
also [7]):
µ b
b αβ
c
(3.2)
Gw
Wαν
Wβd − gw ψ̄ w γ ν σa ψ w
ab ∂ Wµν − gw λcd g
i
h
1
= ∂µ + γb1 Wµb − m2w xµ φw
a,
4
(3.3)
(iγ µ Dµ − ml )ψ w = 0.
Using the Pauli matrices as the SU (2) representation matrices, and taking the
physical units into consideration, the above field equations can be rewritten as
gw a αβ b
a
(3.4)
∂ ν Wνµ
−
ε g Wαµ Wβc − gw Jµa
~c bc
1 m H c 2
gw w b a
γb Wµ φw ,
= ∂µ −
xµ +
4
~
~c
h
gw a i
mc
µ
(3.5)
ψ = 0,
iγ ∂µ + i Wµ σa ψ −
~c
~
where mH represents the mass of the Higgs particle, σa = σ a (1 ≤ a ≤ 3) are the
Pauli matrices, and
gw a b c
a
Wµν
= ∂µ Wνa − ∂ν Wµa +
ε W W ,
~c bc µ ν
(3.6)
a
µ a
Jµ = ψ̄γ σ ψ.
Taking divergence on both sides of (3.4) we get
m c 2
gw w µ
1 mH c 2
H
(3.7)
∂ µ ∂µ φaw −
φaw +
γb ∂ (Wµb φaw ) − (
) xµ ∂ µ φaw
~
~c
4 ~
gw
b
= − εabc g αβ ∂ µ (Wαµ
Wβc ) − gw ∂ µ Jµa .
~c
Also, we need to supplement (3.4)-(3.5) with three additional 3 gauge equations
to compensate the induced dual fields φaw :
(3.8)
Fwa (Wµ , φw , ψ) = 0
for 1 ≤ a ≤ 3.
Duality between W ± , Z Bosons and Higgs Bosons H ± , H 0
The three massive vector bosons, denoted by W ± , Z 0 , has been discovered experimentally. The field equations (3.4) give rise to a natural duality:
(3.9)
Z 0 ↔ H 0,
W ± ↔ H ±,
DUALITY THEORY OF WEAK INTERACTIONS
9
where H 0 , H ± are three dual scalar bosons, called the Higgs particles. The neutral
Higgs H 0 has been discovered by LHC in 2012, and the charged Higgs H ± have yet
to found experimentally.
In Section 4, we shall introduce the dual bosons (3.9) by applying the field
equations (3.4)-(3.7).
Weak force
If we consider the weak interaction force, we have to use the equations (3.4),
(3.5) and (3.7), which are recast as follows:
1 2
gw
gw a αβ b
c
a
ν
a
(3.10)
ε g Wαµ Wβ − gw Jµ = ∂µ − kw xµ +
γWµ φaw ,
∂ Wνµ −
~c bc
4
~c
gw µ
1
(3.11)
∂ µ ∂µ φaw − k 2 φ2w +
γ∂ (Wµ φaw ) − k 2 xµ ∂ µ φaw
~c
4
gw a αβ µ
b
µ a
ε g ∂ (Wαµ Wβc ),
= −gw ∂ Jµ −
~c bc
gw
mc
(3.12)
iγ µ (∂µ + i Wµa σa )ψ −
ψ = 0,
~c
~
where γ, kw are constants, Wµ is as in (1.4) and γωa = γaw .
4. Weak Interaction Potentials
4.1. Dual equations of weak interaction potentials. According to the standard model, the weak interaction is described by the SU (2) gauge theory. Also we
know that the weak interaction potential is given by the following PRI representation invariant
(4.1)
Wµ = ωa Wµa = (W0 , W1 , W2 , W3 ),
where {ωa | 1 ≤ a ≤ 3} is the SU (2) tensor as defined at the end of last section.
Also, the weak charge potential and weak force are as
(4.2)
Φw = W 0
the time component of Wµ ,
Fw = −gw (ρ)∇Φw ,
where gw (ρ) is the weak charge of a particle with radius ρ.
In this subsection, we are devoted to establish the field equations for the dual
potential Φw and φw of the weak interaction from the field equations (3.10)-(3.12).
Taking inner products of (3.10) and (3.11) with ωa respectively we derive the
field equations of the two dual weak interaction potentials Φw and φw as follows
gw
1
gw
a
∂ ν Wνµ −
κab g αβ Wαµ
Wβb − gw Jµw = (∂µ − k 2 xµ +
γWµ )φw ,
~c
4
~c
2
1 ∂
(4.4)
− ∆ φw + k02 φw − gw ∂ µ Jµw
c2 ∂t2
1
gw µ a
=
∂ κab g αβ Wαµ
Wβb − γWµ φw − k02 xµ ∂ µ φw ,
~c
4
c
a
where κab = εab ωc , Wµ is as in (4.1), φw = ωa φw , and
(4.3)
Jµw = ωa ψ̄γµ σa ψ,
(4.5)
Wµν = ∂µ Wν − ∂ν Wµ +
gw
κab Wµa Wνb .
~c
10
MA AND WANG
Experiments showed that the SU (2) gauge fields Wµa for weak interaction field
particles possess masses. In addition, the dual Higgs fields φaw of Wµa also have
masses. In (4.3), there is no massive term of Wµ . However, we see that on the
right-hand side of (4.3) there is a term
gw
(4.6)
γφw Wµ ,
~c
which is spontaneously generated by PID, and breaks the gauge-symmetry. Namely,
(4.6) will vary under the following SU (2) gauge transformation
Wµa → Wµa − εabc θb Wµc −
1
∂µ θa
gw
We shall show that it is the spontaneous symmetry breaking term (4.6) that generates mass from the ground state of φw .
It is clear that the following state
(Wµa , φaw , ψ) = (0, φa0 , 0)
with φa0 being constants,
is a solution of (4.3) and (4.4), which is a ground state of φaw . Let a0 = φa0 ωa , which
is a constant. Take the transformation
φw → φw + a0 ,
Wµa → Wµa ,
ψ → ψ.
Then the equations (4.3) and (4.4) are rewritten as
gw
a
(4.7)
κab g αβ Wαµ
Wβb − gw Jµw
∂ ν Wνµ − k12 Wµ −
~c
1 2
gw
1
= ∂µ − k0 xµ +
γWµ φµ − a0 k02 xµ ,
4
~c
4
h1 ∂
i
(4.8)
− ∆ φw + k02 φw − gw ∂ µ Jµw + k02 a0
c2 ∂t2
1
gw µ a
∂ κab g αβ Wαµ
Wβb − γWµ (φµ + a0 ) − k02 xµ ∂ µ φw ,
=
~c
4
p
where k1 = gw γa0 /~c represents mass.
Thus, (4.7) and (4.8) have masses as
(4.9)
k0 = mH c/~,
k1 = mW c/~,
where mH and mW are the masses of Higgs and W ± bosons. Physical experiments
measured the values of mH and mW as
(4.10)
mH w 160 GeV/c2 ,
mW w 80 GeV/c2 .
By (3.8), equations (4.7) and (4.8) need to add three gauge fixing equations.
Based on the superposition property of the weak charge forces, the dual potentials
W0 and φw should satisfy linear equations, i.e. the time-component µ = 0 equation
of (4.7) and the equation (4.8) should be linear. Therefore, we have to take the
three gauge fixing equations in the following forms
(4.11)
~c a0 k02
a
x0 = 0,
κab g αβ Wα0
Wβb − ∂ µ (Wµa W0b ) + γW0 φw −
gw 4
k2
gw µ a
∂ κab g αβ Wαµ
Wβb − γWµ φw − 0 xµ ∂ µ φw − k02 a0 = 0,
~c
4
∂ µ Wµ = 0,
DUALITY THEORY OF WEAK INTERACTIONS
11
and with the static conditions
(4.12)
∂
Φw = 0,
∂t
∂
φw = 0,
∂t
(Φw = W0 ).
With the equations (4.11) and the static conditions (4.12), the time-component
µ = 0 equation of (4.7) and its dual equation (4.8) become
(4.13)
(4.14)
1
− ∆Φw + k12 Φw = gw Qw − k02 cτ φw ,
4
− ∆φw + k02 φw = gw ∂ µ Jµw ,
where cτ is the wave length of φw , Qw = −J0w , and Jµw is as in (4.5). The two
dual equations (4.13) and (4.14) for the weak interaction potentials Φw and φw are
coupled with the Dirac equations (3.12), written as
mc
gw
ψ = 0.
(4.15)
iγ µ ∂µ + i Wµa σa ψ −
~c
~
In addition, by (4.9) and (4.10) we can determine values of the parameters k0 and
k1 as follows
(4.16)
k0 = 2k1 ,
k1 = 1016 cm−1 .
The parameters 1/k0 and 1/k1 represent the attracting and repulsive radii of weak
interaction forces.
In the next subsection, we shall apply the equations (4.13)-(4.15) to derive the
layered formulas of weak interaction potentials.
4.2. Layered formulas of weak forces. Now we deduce from (4.13)-(4.16) the
following layered formulas for the weak interaction potential:
1 B
Φw = gw (ρ)e−kr
− (1 + 2kr)e−kr ,
r
ρ
(4.17)
3
ρw
gw (ρ) = N
gw ,
ρ
where Φw is the weak force potential of a particle with radius ρ and carrying N
weak charges gw (gw is the unit of weak charge gs for each weakton, an elementary
particle), ρw is the weakton radius, B is a parameter depending on the particles,
and
1
(4.18)
= 10−16 cm,
k
represents the force-range of weak interactions.
To derive the layered formulas (4.17), first we shall deduce the following weak
interaction potential for a weakton
B0
0
−kr 1
−kr
−
(1 + 2kr)e
.
(4.19)
Φw = gs e
r
ρw
To derive the solution φw of (4.14), we need to compute the right-hand term of
(4.14). By (4.5) we have
∂ µ Jµw = ωa ∂µ ψ̄γ µ σa + ωa ψ̄γ µ σa ∂µ ψ.
12
MA AND WANG
Due to the dirac equation (4.15),
gw b µ
mc
Wµ ψ̄γ σb σa ψ + i
ψ̄σa ψ,
~c
~
mc
gw
ψ̄σa ψ.
ψ̄γ µ σa ∂µ ψ = i Wµb ψ̄γ µ σa σb ψ − i
~c
~
∂µ ψ̄γ µ σa ψ = −i
Thus we obtain
gw
gw
ωa Wµb ψ̄γ µ [σa , σb ] ψ = −2 εcab ωa Wµb Jcµ .
~c
~c
c
µ
µ
Here we used [σa , σb ] = i2εab σc and Jc = ψ̄γ σc ψ. Note that
~i
~k ~j
~ µ,
~ ×W
εcab ωa Wµb = ω1
ω2
ω3 = ω
W 1 W 2 W 3 µ
µ
µ
(4.20)
∂ µ Jµw = i
~ µ = (Wµ1 , Wµ2 , Wµ3 ). Hence, (4.20) can be rewritten as
where ω
~ = (ω1 , ω2 , ω3 ), W
(4.21)
∂ µ Jµw = −
2gw
~ µ ) · J~µ ,
(~
ω×W
~c
and J~µ = (J1µ , J2µ , J3µ ). The weak current density Jaµ is as
Jaµ = θaµ δ(r),
θaµ the constant tensor,
~ µ in (4.21) is replaced by the average value
and W
Z
1
~ µ dx,
W
(4.22)
|Bρw | Bρw
where Bρw ⊂ Rn is the ball with radius ρw . As in [6] for the strong interaction, the
average value (4.22) is
~ µ = ζ~µ /ρw ,
W
ζ~µ = (ζµ1 , ζµ2 , ζµ3 ).
Thus, (4.21) can be expressed as
(4.23)
∂ µ Jµw = −κδ(r)/ρw ,
and κ is a parameter, written as
2gw ~µ
θ · (~
ω × ζ~µ ).
~c
Putting (4.23) in (4.14) we deduce that
gw κ
−∆φw + k02 φw = −
δ(r),
ρw
whose solution is given by
gw κ 1 −k0 r
(4.25)
φw = −
e
.
ρw r
(4.24)
κ=
Therefore we obtain the solution of (4.14) in the form (4.25).
Inserting (4.25) into (4.13) we get
(4.26)
− ∆Φw + k12 Φw = gw Qw +
gw B 1 −k0 r
e
,
ρw r
where B is a parameter with dimension 1/L, given by
1
B = κk02 cτ, κ is as in (4.24).
4
DUALITY THEORY OF WEAK INTERACTIONS
13
Since gw Qw = −gw J0w is the weak charge density, we have
Qw = βδ(r),
and β is a scaling factor. We can take proper unit for gs such that β = 1. Thus,
putting Qw = δ(r) in (4.26) we derive that
− ∆Φw + k12 Φw = gw δ(r) +
(4.27)
gw B 1 −k0 r
e
.
ρw r
Let the solution of (4.27) be radially symmetric, then the equation (4.27) can be
equivalently written as
1 d
gw B 1 −k0 r
2 d
(4.28)
− 2
r
Φw + k12 Φw = gw δ(r) +
e
.
r dr
dr
ρw r
The solution of (4.28) can be expressed as
(4.29)
Φw =
gw −k1 r gw B
e
−
ϕ(r)e−k0 r ,
r
ρw
where ϕ(r) satisfies the equation
1
2k0
1
(4.30)
ϕ00 + 2
− k0 ϕ0 −
+ k12 − k02 ϕ = .
r
r
r
Let ϕ be in the form
ϕ=
∞
X
βn r n
( the dimension of ϕ is L).
n=0
Inserting ϕ into (4.30), comparing coefficients of rn , we get
1
β1 = k0 β0 + ,
2
1
1 2
β2 = (k0 − k12 )β0 + k0 ,
2
3
..
.
2k0
βn−1 − (k02 − k12 )βn−2 , n ≥ 2.
n+1
Note that the dimensions of B and β0 in ϕ(r) are 1/L and L. The parameter B =
Bβ0 is dimensionless. Physically, we can only measure the value of B. Therefore
we take ϕ in its second-order approximation as follows
βn =
ϕ = β0 (1 + k0 r).
In addition, by (4.16) we take
k1 = k,
k0 = 2k,
k = 1016 cm−1 .
Thus, the formula (4.29) is written as
B
1
−
(1 + 2kr)e−kr .
Φs = gw e−kr
r
ρw
This is the weak interaction potential of a weakton, which is as given in (4.19).
In the same fashion as used in the layered formula for strong interactions in [6],
for a particle with radius ρ and N weak charges gw , we can deduce the layered
formula of weak interaction potentials as in the form given by (4.17).
14
MA AND WANG
4.3. Physical conclusions for weak forces. As mentioned earlier, the layered
weak interaction potential formula (4.17) plays the same fundamental role as the
Newtonian potential for gravity and the Coulomb potential for electromagnetism.
Hereafter we explore a few direct physical consequences of the weak interaction
potentials.
Short-range nature of weak interactions
By (4.17) it is easy to see that for all particles, their weak interaction force-range
is as
1
r = = 10−16 cm,
k
which is consistent with experimental data.
Weak force formula
For two particles with radii ρ1 , ρ2 , and with N1 , N2 weak charges gw . Their weak
charges are given by
3
ρw
gw
for j = 1, 2.
(4.31)
gw (ρj ) = Nj
ρj
The weak potential energy for the two particles is
1 B̄
− (1 + 2kr)e−kr ,
(4.32)
V = gw (ρ1 )gw (ρ2 )e−kr
r
ρ̄
where gw (ρ1 ) and gw (ρ2 ) are as in (4.31), ρ̄ is a radius depending on ρ1 and ρ2 , and
B̄ is a constant depending on the types of these two particles.
The weak force between the two particles is given by
d
1
k 4B̄ 2 −kr
−kr
(4.33)
F = − V = gw (ρ1 )gw (ρ2 )e
+ −
k re
.
dr
r2
r
ρ̄
Repulsive condition
For the two particles as above, if their weak interaction constant B̄ satisfies the
inequality
1
k
4B̄ 2 −kr
1
+ ≥
k re
, ∀0 < r ≤ ,
r2
r
ρ̄
k
or equivalently B̄ satisfies
e
(4.34)
B̄ ≤ ρ̄k (e = 2.718, k = 1016 cm−1 ),
2
then the weak force between these two particles is always repulsive.
It follows from this conclusion that for the neutrinos:
(ν1 , ν2 , ν3 ) = (νe , νµ , ντ ),
(ν̄1 , ν̄2 , ν̄3 ) = (ν̄e , ν̄µ , ν̄τ ),
the weak interaction constants
B̄ij for νi and νj
B̄ij for νi and anti-neutrinos ν̄j
satisfy the exclusion condition (4.34).
Value of weak charge gw
∀1 ≤ i, j ≤ 3,
∀i 6= j,
DUALITY THEORY OF WEAK INTERACTIONS
15
Based on the Standard Model, the coupling constant Gw of the β-decay of nucleons and the Fermi constant Gf have the following relation
8 mW c 2
(4.35)
G2w = √
Gf ,
~
2
and Gf is given by
m c 2
p
,
(4.36)
Gf = 10−5 ~c/
~
where mW and mp are masses of W ± bosons and protons. By the gauge theory,
Gw is also the coupling constant of SU (2) gauge fields. Therefore we can regard
Gw as the weak charge of nucleons, i.e.
Gw = gw (ρn ),
ρn the nucleon radius.
In addition, it is known that
gw (ρn ) = 9
ρw
ρn
3
gs .
Hence, we deduce from (4.35) and (4.36) that
6
2
mw
ρw
8
2
81
gw
=√
× 10−5 ~c.
ρn
2 mp
Then we derive the relation
(4.37)
2
gw
8
= √
81 2
mw
mp
2
× 10
−5
×
ρn
ρw
6
~c.
Recall the strong charge derived in [6], we find that the strong charge gs and the
weak charge gw have the same order of magnitude. Direct computation shows that
2
gw
' 0.35
gs2
equivalently
gw
' 0.6.
gs
5. PID Induced Spontaneous Symmetry Breaking
In the derivation of equations (4.7) and (4.8) we have used the PID mechanism
of spontaneous symmetry breaking. In this subsection we shall discuss the intermediate vector bosons W ± , Z and their dual scalar bosons H ± , H 0 , called the Higgs
particles by using the PID mechanism of spontaneous symmetry breaking.
We know that the interaction field equations are oriented toward two directions:
i) to derive interaction forces, and ii) to describe the field particles and derive
the particle transition probability. The PID-PRI weak interaction field equations
describing field particles are given by (3.4)-(3.7). Here, for convenience, we write
them again as follows:
gw a αβ b
a
(5.1)
∂ ν Wνµ
−
ε g Wαµ Wβc − gw Jµa
~c bc
1 mH c 2
gw
= (∂µ −
xµ +
γb Wµb )φa ,
4
~
~c
m c 2
gw
H
− ∂ µ ∂µ φa +
φa −
γb ∂ µ (Wµb φa )
(5.2)
~
~c
gw a αβ µ
1 mH c 2
b
=
ε g ∂ (Wαµ
Wβc ) + gw ∂ µ Jµa − (
) x µ ∂ µ φa ,
~c bc
4 ~
16
MA AND WANG
where mH is the Higgs particle mass, and Wµa (1 ≤ a ≤ 3) describe the intermediate
vector bosons as follows
(5.3)
W ± : Wµ1 ± iWµ2 ,
Z:
Wµ3 ,
and φa describe the dual Higgs bosons as
(5.4)
H ± : φ1 ± iφ2 ,
H0 :
φ3 .
By (3.8), equations (5.1)-(5.2) need to be supplemented with three gauge fixing
equations. According to physical requirement, we take these equations as
(5.5)
∂ µ Wµa = 0
for 1 ≤ a ≤ 3.
To match (5.3) and (5.4), we take the SU (2) generator transformation as follows

 
1
σ
e1
1
σ
e2  = √ 1
2 0
σ
e3
i
−i
0
 
0
σ1
  σ2  .
0
√
σ3
2
Under this transformation, Wµa and φa are changed into
f1, W
f2, W
f 3 ) = (W ± , Zµ ) = √1 (W 1 ± iW 2 ), W 3 ,
(W
µ
µ
µ
µ
µ
µ
µ
2
1
(φe1 , φe2 , φe3 ) = (φ± , φ0 ) = √ (φ1 ± iφ2 ), φ3 .
2
The equations (5.1) and (5.2) obey PRI, and under the above transformation they
becomes
(5.6)
(5.7)
(5.8)
(5.9)
igw αβ
±
∂ ν Wνµ
±
g (W ± )αµZβ − Zαµ Wβ± ) − gw Jµ±
~c
1 mH c 2
2
±
2
x µ φ± ,
= ∂µ + kW Wµ + kZ Zµ −
4
~
igw αβ
+
−
∂ ν Zνµ −
g (Wαµ
Wβ− − Wαµ
Wβ+ ) − gw Jµ0
~c
1 mH c 2
2
±
2
= ∂µ + kW Wµ + kZ Zµ −
x µ φ0 ,
4
~
m c
gw µ f b ±
H
H ± +
H± −
∂ (e
γb Wµ H )
~
~c
gw αβ µ ± f b f c
1 mH c 2
=
g ∂ (e
εbc Wαµ Wβ ) + gw ∂ µ Jµ± −
xµ ∂ µ H ± ,
~c
4
~
m c 2
gw µ f b 0
H
H 0 +
H0 −
∂ (e
γb Wµ H )
~
~c
gw αβ µ 0 f b f c
1 m H c 2
=
g ∂ (e
εbc Wαµ Wβ ) + gw ∂ µ Jµ0 −
xµ ∂ µ H 0 ,
~c
4
~
DUALITY THEORY OF WEAK INTERACTIONS
where H ± =
√1 (φ1
2
± iφ2 ), H 0 = φ3 in (5.8) and (5.9), and
1
Jµ± = √ (Jµ1 ± iJµ2 )
2
Jµ0 = Jµ3
(5.10)
17
the charged current,
the neutral current,
±
Wνµ
= ∂ν Wµ± − ∂µ Wν± ±
Zνµ = ∂ν Zµ − ∂µ Zν +
igw
(Zµ Wν± − Zν Wµ± ),
~c
igw
(Wµ+ Wν− − Wν+ Wµ− ),
~c
and
(5.11)
gw γ1
2
,
kW
=√
2~c
2
kZ
=
gw γ3
,
~c
which are derived by the following transformation
 2 

 
1 i
0
kW
γ1
g
w 
2 
kW
1 −i √0  γ2  ,
=√
2~c 0 0
2
kZ
γ3
2
with γ2 = 0 in the Pauli matrix representation.
The equation (5.6)-(5.9) are the model to govern the behaviors of the weak
interaction field particles (5.3) and (5.4). Two observations are now in order.
First, we note that these equations are nonlinear, and consequently, no free weak
interaction field particles appear.
Second, there are two important solutions of (5.6)-(5.7), dictating two different
weak interaction procedures.
The first solution sets
Wµ± = 0,
(5.12)
φ0 = 1.
Then Z satisfies the equation
(5.13)
Zµ + kz2 Zµ = −gw Jµ0 +
1 mH c 2
xµ .
4
~
The second solution takes
(5.14)
Zµ = 0,
φ± = 1.
Then Wµ± satisfy the equations
1 mH c 2
xµ .
4
~
We are now ready to obtain the following physical conclusions for the weak
interaction field particles.
(5.15)
2
Wµ± + kw
Wµ± = −gw Jµ± +
(1) Duality of Field Particles. The field equations (5.6)-(5.9) provide a natural
duality between the field particles:
W ± ↔ H ±,
Z ↔ H 0.
(2) Non-Freedom of Field Particles. Due to the nonlinearity of (5.6)-(5.9), the
weak interaction field particles W ± , Z, H ± , H 0 are non-free bosons.
18
MA AND WANG
(3) PID Mechanism of Spontaneous Symmetry Breaking. In the field equations
(5.13) and (5.15), it is natural that the masses mW and mZ of W ± and Z
bosons appear at the ground states (5.12) and (5.14), with
~
~
kW , mZ = kZ ,
c
c
and kW and kZ are as in (5.11).
(4) Basic Properties of Field Particles. From (5.6)-(5.9) we can obtain some
basic information for the bosons as follows:
mW =
W± :
spin J = 1,
electric charge = ±e,
mass mW ,
Z:
spin J = 1,
electric charge = 0,
mass mZ ,
:
spin J = 0,
electric charge = ±e,
mass m+
H,
H0 :
spin J = 0,
electric charge = 0,
mass m0H .
H
±
Remark 5.1. Under the translation
1 m H c 2 1
2 xµ ,
4
~
kZ
1 mH c 2 1
→ Wµ± + (
) 2 xµ ,
4 ~
kW
Zµ → Zµ +
Wµ±
the equations (5.13) and (5.15) become
(5.16)
2
Zµ + kZ
Zµ = −gw Jµ0 ,
2
Wµ± + kW
Wµ± = −gw Jµ± .
Similarly, if we take the gauge fixing equations
x µ ∂ µ φa = 0
for 1 ≤ a ≤ 3,
instead of (5.5), then under the conditions
Wµa = 0
for 1 ≤ a ≤ 3,
the equations (5.9) and (5.8) are in the form
m c 2
H
H 0 = gw ∂ µ Jµ0 ,
H 0 +
~
(5.17)
m c 2
H
H ± +
H ± = gw ∂ µ Jµ± .
~
The equations (5.16) and (5.17) are the standard Klein-Gordon models describing
the W ± , Z, H ± , H 0 bosons, which are derived only by the PID-PRI unified field
model.
References
[1] F. Englert and R. Brout, Broken symmetry and the mass of gauge vector mesons, Physical
Review Letters, 13 (9) (1964), p. 32123.
[2] D. Griffiths, Introduction to elementary particles, Wiley-Vch, 2008.
[3] G. Guralnik, C. R. Hagen, and T. W. B. Kibble, Global conservation laws and massless
particles, Physical Review Letters, 13 (20) (1964), p. 585587.
[4] P. W. Higgs, Broken symmetries and the masses of gauge bosons, Physical Review Letters,
13 (1964), p. 508509.
[5] M. Kaku, Quantum Field Theory, A Modern Introduction, Oxford University Press, 1993.
DUALITY THEORY OF WEAK INTERACTIONS
19
[6] T. Ma and S. Wang, Duality theory of strong interaction, Indiana University Institute
for Scientific Computing and Applied Mathematics Preprint Series, #1301: http://www.
indiana.edu/~iscam/preprint/1301.pdf, (2012).
[7]
, Unified field equations coupling four forces and principle of interaction dynamics,
arXiv:1210.0448, (2012).
[8]
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69:3, pp 359-392, (2012).
[9]
, Gravitational field equations and theory of dark matter and dark energy, Discrete and
Continuous Dynamical Systems, Ser. A, 34:2 (2014), pp. 335–366; see also arXiv:1206.5078v2.
[10]
, Weakton model of elementary particles and decay mechanisms, Indiana University
Institute for Scientific Computing and Applied Mathematics Preprint Series, #1304: http:
//www.indiana.edu/~iscam/preprint/1304.pdf, (May 30, 2013).
[11] Y. Nambu, Quasi-particles and gauge invariance in the theory of superconductivity, Phys.
Rev., 117 (1960), pp. 648–663.
[12] Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an
analogy with superconductivity. I, Phys. Rev., 122 (1961), pp. 345–358.
[13]
, Dynamical model of elementary particles based on an analogy with superconductivity.
II, Phys. Rev., 124 (1961), pp. 246–254.
[14] C. Quigg, Gauge theories of the strong, weak, and electromagnetic interactions, 2nd edition,
Princeton Unversity Press, 2013.
(TM) Department of Mathematics, Sichuan University, Chengdu, P. R. China
(SW) Department of Mathematics, Indiana University, Bloomington, IN 47405
E-mail address: [email protected], http://www.indiana.edu/ fluid