Generalized solutions of the Dirac equation, W bosons, and beta decay

We study the 7x7 Hagen-Hurley equations describing spin 1 particles. We split these equations, in the interacting case, into two Dirac equations with non-standard solutions. It is argued that these solutions describe decay of a virtual W boson in beta decay.


Introduction
Recently, we have shown that in the free case covariant solutions of the s = 0 and s = 1 Duffin-Kemmer-Petiau (DKP) equations are generalized solutions of the Dirac equation [1]. These wavefunctions are non-standard 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.
In the next Section we transform the Hagen-Hurley equations, in the interacting case, into two Dirac equations with non-standard 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 [21]. In Section 3 we describe transition from non-standard 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 Ref. [22].

Generalized solutions of the Dirac equation in the interacting case
We have shown recently that, in the non-interacting case, solutions of the s = 0 and s = 1 DKP equations are generalized solutions of the Dirac equation [1]. In our derivation we have splitted the 10 × 10 DKP equations for s = 1 into two 7 × 7 Hagen-Hurley equations [16][17][18]. Let us note here that in the case of interaction with external fields such splitting is not possible since the identities (27) of Ref. [23], enabling the splitting, are not valid in the interacting case. Therefore, we shall base our theory on the 7 × 7 formulation, see Eqs. (18), (19) in [1] and Subsection 6 ii) in [19]. These equations violate parity P , where P : , and thus one should expect a link with weak interactions. We write one of these 7×7 equations (Eq. (19) of Ref. [1]), in the interacting case, in form: and it is assumed that χḂḊ = χḊḂ (2) what is the s = 1 constraint. In Eqs.

Conclusions
Results obtained in Sections 2, 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 equation (1), describing a spin s = 1 particle, to equations (7), (8), via substitution (5) -which means that now s ∈ 0 ⊕ 1, corresponds to decay of this particle into a Weyl antineutrino, cf. Eq. (7), and a Dirac lepton, cf. Eq. (8). Indeed, it should be a weak decay since Eq. (1) violates 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 antineutrino, should have total spin 0 or 1 and there should be a third particle to secure spin conservation. The above descritption fits a (three-body) beta decay with formation of a virtual W − boson, decaying into a lepton and antineutrino. This is most conveniently explained in the case of a mixed beta decay [26]: Fermi transition (9) where products of the W − boson decay (see [27]) are shown in square brackets and (↑) denotes spin 1 2 -this seems to correspond well to the proposed transition from Eq. (1) to Eqs. (7), (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 spinflip in the decaying nucleon. Let us add here, that in the reaction (9) some neutrons (82%) decay according to the Gamow-Teller mechanism while some (18%) undergo the Fermi transition [26]. This mixed mechanism is explained by decoupled spins of the just born products -indeed, the condition χ12 = χ21 for the spinor χȦḂ, due to the substitution (5a), does not hold and spin of the products is in the 0 ⊕ 1 space.
It is now obvious that another set of 7 × 7 equations, involving spinor η AB rather than χȦḂ, see Eq. (18) of Ref. [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.