# Generalized Solutions of the Dirac Equation,W Bosons, and Beta Decay

Authors: Andrzej Okniński
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

We study the7
×
7 Hagen-Hurley equations describing spin 1 particles. We split these equations, in the interacting case, into two Dirac equations with nonstandard solutions. It is argued that these solutions describe decay of a virtual W boson in beta decay.


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## Body

## 1. Introduction

Recently, we have shown that in the free case covariant solutions of thes
=
0 and s
=
1 Duffin-Kemmer-Petiau (DKP) equations are generalized solutions of the Dirac equation [1]. These wavefunctions are nonstandard since they involve higher-order spinors. We have demonstrated recently that in the s
=
0 case the generalized solutions describe decay of a pion [2]. The aim of this work is to interpret spin 1 solutions, possibly in the context of weakly decaying particles.There are several relativistic equations describing spin1 particles; see [3, 4] for the reviews. The most common approach to study properties of spin 1 bosons is based on the 10
×
10 DKP equations (the DKP particles are bosons [5]). Several classes of potentials were used in DKP equations to investigate interactions of spin 1 particles [6–15]. However, we will apply the 7
×
7 Hagen-Hurley equations [16–18] in spinor form [1, 19, 20]. Our motivation stems from the observation that these equations violate parity and thus should describe weakly interacting particles (see [21] for weak interactions and violation of parity).In the next section we transform the Hagen-Hurley equations, in the interacting case, into two Dirac equations with nonstandard solutions involving higher-order spinors, extending our earlier results described in [1]. These generalized solutions bear some analogy to generalized solutions of the Dirac equation argued to describe a lepton and three quarks [22]. In Section 3 we describe transition from nonstandard solutions of two Dirac equations to the Dirac equation for a lepton and the Weyl equation for a neutrino. In the last section we show that the transition is consistent with decay of a virtual W boson in beta decay. In what follows we are using definitions and conventions of [23].


## 2. Generalized Solutions of the Dirac Equation in the Interacting Case

We have shown recently that, in the noninteracting case, solutions of thes
=
0 and s
=
1 DKP equations are generalized solutions of the Dirac equation [1]. In our derivation we have split the 10
×
10 DKP equations for s
=
1 into two 7
×
7 Hagen-Hurley equations [16–18]. Let us note here that in the case of interaction with external fields such splitting is not possible since the identities (27) of [24], enabling the splitting, are not valid in the interacting case. Therefore, we will base our theory on the 7
×
7 formulation; see (18) and (19) in [1] and Subsection  6(ii) in [19]. These equations violate parity P [19], where P
:
x
0
→
x
0, x
i
→
-
x
i
i
=
1,2
,
3. Since parity is violated in weak interactions [21], these equations seem to describe weakly interacting particles.We write one of these7
×
7 equations (Eq. (19) of [1]), in the interacting case, in the form (1a)
π
B
˙
A
ζ
A
D
˙
=
m
χ
B
˙
D
˙
,
π
A
D
˙
χ
B
˙
D
˙
=
-
m
ζ
A
B
˙
,
(1b)
χ
B
˙
D
˙
=
χ
D
˙
B
˙
,where (1b) is the spin 1 constraint [19]. In (1a) we have π
A
B
˙
=
σ
0
π
0
+
σ
→
·
π
→
A
B
˙, π
μ
=
p
μ
-
q
A
μ, where σ
k
k
=
1,2
,
3 are the Pauli matrices and σ
0 is the 2
×
2 unit matrix. Equations (1a) and (1b), which were first proposed by Dirac [20], can be written in the 7
×
7 Hagen-Hurley form:(2)
β
μ
p
μ
Ψ
=
m
Ψ
,with β
μ matrices given in [3], with Ψ
=
χ
1
˙
1
˙
,
χ
,
χ
2
˙
2
˙
,
ζ
1
1
˙
,
ζ
1
2
˙
,
ζ
2
1
˙
,
ζ
2
2
˙
T, where χ
=
d
f
χ
1
˙
2
˙
=
χ
2
˙
1
˙ and T stands for transposition of a matrix.Equations (1a) in explicit form read (3a)
-
π
1
+
i
π
2
χ
1
˙
1
˙
-
π
0
-
π
3
χ
2
˙
1
˙
=
-
m
ζ
1
1
˙
,
π
0
+
π
3
χ
1
˙
1
˙
+
π
1
-
i
π
2
χ
2
˙
1
˙
=
-
m
ζ
2
1
˙
,
-
π
1
-
i
π
2
ζ
1
1
˙
-
π
0
-
π
3
ζ
2
1
˙
=
m
χ
1
˙
1
˙
,
π
0
+
π
3
ζ
1
1
˙
+
π
1
+
i
π
2
ζ
2
1
˙
=
m
χ
2
˙
1
˙
,
(3b)
-
π
1
+
i
π
2
χ
1
˙
2
˙
-
π
0
-
π
3
χ
2
˙
2
˙
=
-
m
ζ
1
2
˙
,
π
0
+
π
3
χ
1
˙
2
˙
+
π
1
-
i
π
2
χ
2
˙
2
˙
=
-
m
ζ
2
2
˙
,
-
π
1
-
i
π
2
ζ
1
2
˙
-
π
0
-
π
3
ζ
2
2
˙
=
m
χ
1
˙
2
˙
,
π
0
+
π
3
ζ
1
2
˙
+
π
1
+
i
π
2
ζ
2
2
˙
=
m
χ
2
˙
2
˙
,where condition (1b), χ
B
˙
D
˙
=
χ
D
˙
B
˙, is not imposed. We thus get two Dirac equations or, alternatively, a single Dirac equation with generalized solution B
=
ζ
1
1
˙
ζ
1
2
˙
ζ
2
1
˙
ζ
2
2
˙
χ
1
˙
1
˙
χ
1
˙
2
˙
χ
2
˙
1
˙
χ
2
˙
2
˙: (4)
π
0
γ
0
-
π
1
γ
1
-
π
2
γ
2
-
π
3
γ
3
B
=
m
B
,generalizing (24) of [1].


## 3. Decay of Spin 1 Bosons

We note that solutions of two Dirac equations (3a) and (3b) are nonstandard since they involve higher-order spinors rather than spinors ξ
A, η
B
˙. To interpret (3a) and (3b) we put (5a)
χ
B
˙
D
˙
x
=
η
B
˙
x
α
D
˙
x
,
(5b)
ζ
A
B
˙
x
=
ξ
A
x
α
B
˙
x
,where α
A
˙
x is the Weyl spinor, describing massless neutrinos, while η
B
˙
x and ξ
A
x are the Dirac spinors. Our analysis below will not be exact since neutrinos are massive [21]. On the other hand, their masses are very small so this approximation should not lead to significant errors.Note that nowχ
1
˙
2
˙
≠
χ
2
˙
1
˙ and, accordingly, the spin is not determined: more exactly, the spin is in the 0
⊕
1 space. It means that we consider not real but virtual (off-shell) bosons [25]. This substitution is in the spirit of the method of fusion of de Broglie [26, 27] (similar ansatz was used in the s
=
0 case [2]). After the substitution of (5a) and (5b) into (3a) and (3b) we obtain two equations:(6)
-
π
1
+
i
π
2
η
1
˙
α
A
˙
-
π
0
-
π
3
η
2
˙
α
A
˙
=
-
m
ξ
1
α
A
˙
,
π
0
+
π
3
η
1
˙
α
A
˙
+
π
1
-
i
π
2
η
2
˙
α
A
˙
=
-
m
ξ
2
α
A
˙
,
-
π
1
-
i
π
2
ξ
1
α
A
˙
-
π
0
-
π
3
ξ
2
α
A
˙
=
m
η
1
˙
α
A
˙
,
π
0
+
π
3
ξ
1
α
A
˙
+
π
1
+
i
π
2
ξ
2
α
A
˙
=
m
η
2
˙
α
A
˙
,where A
˙
=
1
˙
,
2
˙ and, after substituting solution of the Weyl equation,(7)
p
A
B
˙
α
B
˙
=
0
,
α
A
˙
x
=
α
^
A
˙
e
i
k
·
x, k
μ
k
μ
=
0, we get a single Dirac equation for spinors ξ
A
x and η
B
˙
x:(8)
-
π
~
1
+
i
π
~
2
η
1
˙
-
π
~
0
-
π
~
3
η
2
˙
=
-
m
ξ
1
,
π
~
0
+
π
~
3
η
1
˙
+
π
~
1
-
i
π
~
2
η
2
˙
=
-
m
ξ
2
,
-
π
~
1
-
i
π
~
2
ξ
1
-
π
~
0
-
π
~
3
ξ
2
=
m
η
1
˙
,
π
~
0
+
π
~
3
ξ
1
+
π
~
1
+
i
π
~
2
ξ
2
=
m
η
2
˙
,where π
~
μ
≡
π
μ
+
k
μ, since components α
1
˙
x and α
2
˙
x cancel out.Equations (7) and (8) describe two spin 1
/
2 particles, whose spins can couple to s
=
0 or s
=
1: that is, 1
/
2
⊗
1
/
2
=
0
⊕
1.


## 4. Conclusions

Results obtained in Sections2 and 3 cast new light on the Hagen-Hurley equations as well as on weak decays of spin 1 bosons. We have shown that transition from (1a) and (1b), describing a spin s
=
1 particle, to (7) and (8), via substitution of (5a) and (5b), which means that now s
∈
0
⊕
1, corresponds to decay of this particle into a Weyl antineutrino, cf. (7), and a Dirac lepton, cf. (8). Indeed, it should be a weak decay since (1a) and (1b) violate parity. The spin of this particle becomes undetermined in the process of decay; more exactly it belongs to the 0
⊕
1 space: this suggests that this is a virtual particle. Therefore, the products, a lepton and a Weyl antineutrino (this suggests maximal parity violation [25]), should have total spin 0 or 1 and there should be a third particle to secure spin conservation.The above description fits a (three-body) beta decay with formation of a virtualW
- boson, decaying into a lepton and antineutrino. This is most conveniently explained in the case of a mixed beta decay [28]:(9)
n
↑
⟶
p
↓
+
e
↑
ν
¯
e
↑
,
Gamow-Teller  transition
,
p
↑
+
e
↑
ν
¯
e
↓
,
Fermi  transition
,where products of the W
- boson decay (see [29]) are shown in square brackets and ↑ denotes spin 1
/
2: this seems to correspond well to the proposed transition from (1a) and (1b) to (7) and (8). Since spin of the products of decay of the virtual W
- boson belongs to the 0
⊕
1 space, their spin can be s
=
0 or s
=
1. Moreover, in the case of the Gamow-Teller transition there must be a spin-flip in the decaying nucleon. Let us add here the fact that in reaction (9) some neutrons (82%) decay according to the Gamow-Teller mechanism while some (18%) undergo the Fermi transition [28]. This mixed mechanism is explained by decoupled spins of the just born products: indeed, the condition χ
1
˙
2
˙
=
χ
2
˙
1
˙ for the spinor χ
A
˙
B
˙, due to substitution (5a), does not hold and spin of the products is in the 0
⊕
1 space. More about properties of W
± bosons can be found in [21].It is now obvious that another set of7
×
7 equations, involving spinor η
A
B rather than χ
A
˙
B
˙ (see (18) of [1]), describes a β
+ decay with intermediate W
+ boson. Let us note finally that kinematics of the neutrino appears in the Dirac equation for the electron with arbitrary neutrino four-momentum, suggesting a continuous distribution of neutrino energy.



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*Source: 2689742-2016-08-10.xml*