Advances in High Energy Physics

Volume 2016, Article ID 2689742, 4 pages

http://dx.doi.org/10.1155/2016/2689742

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

Chair of Mathematics and Physics, Politechnika Świȩtokrzyska, Al. 1000-lecia PP 7, 25-314 Kielce, Poland

Received 11 May 2016; Accepted 19 July 2016

Academic Editor: Smarajit Triambak

Copyright © 2016 Andrzej Okniński. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study the 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 boson in beta decay.

#### 1. Introduction

Recently, we have shown that in the free case covariant solutions of the and 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 case the generalized solutions describe decay of a pion [2]. The aim of this work is to interpret spin solutions, possibly in the context of weakly decaying particles.

There are several relativistic equations describing spin particles; see [3, 4] for the reviews. The most common approach to study properties of spin bosons is based on the DKP equations (the DKP particles are bosons [5]). Several classes of potentials were used in DKP equations to investigate interactions of spin particles [6–15]. However, we will apply the 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 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 the and DKP equations are generalized solutions of the Dirac equation [1]. In our derivation we have split the DKP equations for into two 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 formulation; see (18) and (19) in [1] and Subsection 6(ii) in [19]. These equations violate parity [19], where , . Since parity is violated in weak interactions [21], these equations seem to describe weakly interacting particles.

We write one of these equations (Eq. (19) of [1]), in the interacting case, in the form where (1b) is the spin constraint [19]. In (1a) we have , , where are the Pauli matrices and is the unit matrix. Equations (1a) and (1b), which were first proposed by Dirac [20], can be written in the Hagen-Hurley form:with matrices given in [3], with , where and stands for transposition of a matrix.

Equations (1a) in explicit form read where condition (1b), , is not imposed. We thus get two Dirac equations or, alternatively, a single Dirac equation with generalized solution : 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 , . To interpret (3a) and (3b) we put where is the Weyl spinor, describing massless neutrinos, while and 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 now and, accordingly, the spin is not determined: more exactly, the spin is in the 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 case [2]). After the substitution of (5a) and (5b) into (3a) and (3b) we obtain two equations:where and, after substituting solution of the Weyl equation,, , we get a single Dirac equation for spinors and :where , since components and cancel out.

Equations (7) and (8) describe two spin particles, whose spins can couple to or : that is, .

#### 4. Conclusions

Results obtained in Sections 2 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 particle, to (7) and (8), via substitution of (5a) and (5b), which means that now , 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 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 virtual boson, decaying into a lepton and antineutrino. This is most conveniently explained in the case of a mixed beta decay [28]:where products of the boson decay (see [29]) are shown in square brackets and denotes spin : 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 boson belongs to the space, their spin can be or . 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 (%) undergo the Fermi transition [28]. This mixed mechanism is explained by decoupled spins of the just born products: indeed, the condition for the spinor , due to substitution (5a), does not hold and spin of the products is in the space. More about properties of bosons can be found in [21].

It is now obvious that another set of equations, involving spinor rather than (see (18) of [1]), describes a decay with intermediate 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.

#### Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### References

- A. Okniński, “Synthesis of relativistic wave equations: the noninteracting case,”
*Advances in Mathematical Physics*, vol. 2015, Article ID 528484, 5 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Okniński, “Neutrino-assisted fermion-boson transitions,”
*Acta Physica Polonica B*, vol. 46, no. 2, pp. 221–229, 2015. View at Publisher · View at Google Scholar · View at Scopus - J. Beckers, N. Debergh, and A. G. Nikitin, “On parasupersymmetries and relativistic descriptions for spin one particles: I. The free context,”
*Fortschritte der Physik*, vol. 43, no. 1, pp. 67–80, 1995. View at Publisher · View at Google Scholar · View at MathSciNet - J. Beckers, N. Debergh, and A. G. Nikitin, “On parasupersymmetries and relativistic descriptions for spin one particles. {II}. The interacting context with (electro)magnetic fields,”
*Fortschritte der Physik*, vol. 43, no. 1, pp. 81–96, 1995. View at Publisher · View at Google Scholar · View at MathSciNet - A. F. Bennett, “Duffin-Kemmer-Petiau particles are bosons,”
*Foundations of Physics*, 2016. View at Publisher · View at Google Scholar - R. E. Kozack, B. C. Clark, S. Hama, V. K. Mishra, R. L. Mercer, and L. Ray, “Spin-one Kemmer-Duffin-Petiau equations and intermediate-energy deuteron-nucleus scattering,”
*Physical Review C*, vol. 40, no. 5, pp. 2181–2194, 1989. View at Publisher · View at Google Scholar · View at Scopus - V. K. Mishra, S. Hama, B. C. Clark, R. E. Kozack, R. L. Mercer, and L. Ray, “Implications of various spin-one relativistic wave equations for intermediate-energy deuteron-nucleus scattering,”
*Physical Review C*, vol. 43, no. 2, pp. 801–811, 1991. View at Publisher · View at Google Scholar · View at Scopus - Y. Nedjadi and R. C. Barrett, “Solution of the central field problem for a Duffin-Kemmer-Petiau vector boson,”
*Journal of Mathematical Physics*, vol. 35, no. 9, pp. 4517–4533, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - T. R. Cardoso, L. B. Castro, and A. S. de Castro, “On the nonminimal vector coupling in the Duffin-Kemmer-Petiau theory and the confinement of massive bosons by a linear potential,”
*Journal of Physics A: Mathematical and Theoretical*, vol. 43, no. 5, Article ID 055306, 18 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. Molaee, M. Ghominejad, H. Hassanabadi, and S. Zarrinkamar, “S-wave solutions of spin-one DKP equation for a deformed Hulthén potential in (1 + 3) dimensions,”
*The European Physical Journal Plus*, vol. 127, no. 9, pp. 1–8, 2012. View at Google Scholar - H. Hassanabadi, Z. Molaee, M. Ghominejad, and S. Zarrinkamar, “Spin-one DKP equation in the presence of Coulomb and harmonic oscillator interactions in $\left(1+3\right)$-dimension,”
*Advances in High Energy Physics*, vol. 2012, Article ID 489641, 10 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - H. Hassanabadi and M. Kamali, “The spin-one Duffin-Kemmer-Petiau equation in the presence of pseudo-harmonic oscillatory ring-shaped potential,”
*Chinese Physics B*, vol. 22, no. 10, Article ID 100304, 5 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus - H. Hassanabadi, M. Kamali, and B. H. Yazarloo, “Spin-one Duffin-Kemmer-Petiau equation in the presence of Manning-Rosen potential plus a ring-shaped-like potential,”
*Canadian Journal of Physics*, vol. 92, no. 6, pp. 465–471, 2014. View at Publisher · View at Google Scholar · View at Scopus - L. B. Castro and L. P. de Oliveira, “Remarks on the spin-one Duffin-Kemmer-Petiau equation in the presence of nonminimal vector interactions in $(3+1)$ dimensions,”
*Advances in High Energy Physics*, vol. 2014, Article ID 784072, 8 pages, 2014. View at Publisher · View at Google Scholar · View at Scopus - H. Hassanabadi, Z. Molaee, M. Ghominejad, and S. Zarrinkamar, “On remarks on the spin-one Duffin-Kemmer-Petiau equation in the presence of nonminimal vector interactions in (3+1) dimensions,” https://arxiv.org/abs/1404.2223.
- C. R. Hagen and W. J. Hurley, “Magnetic moment of a particle with arbitrary spin,”
*Physical Review Letters*, vol. 24, no. 24, pp. 1381–1384, 1970. View at Publisher · View at Google Scholar · View at Scopus - W. J. Hurley, “Relativistic wave equations for particles with arbitrary spin,”
*Physical Review D*, vol. 4, no. 12, pp. 3605–3616, 1971. View at Publisher · View at Google Scholar · View at Scopus - W. J. Hurley, “Invariant bilinear forms and the discrete symmetries for relativistic arbitrary-spin fields,”
*Physical Review D*, vol. 10, no. 4, pp. 1185–1200, 1974. View at Publisher · View at Google Scholar · View at Scopus - J. T. Lopuszański, “The representations of the Poincaré group in the framework of free quantum fields,”
*Fortschritte der Physik*, vol. 26, no. 4, pp. 261–288, 1978. View at Publisher · View at Google Scholar · View at MathSciNet - P. A. Dirac, “Relativistic wave equations,”
*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 155, no. 886, pp. 447–459, 1936. View at Publisher · View at Google Scholar - J. F. Donoghue, E. Golowich, and B. R. Holstein,
*Dynamics of the Standard Model*, Cambridge University Press, Cambridge, UK, 2014. - E. Marsch and Y. Narita, “Fermion unification model based on the intrinsic SU(8) symmetry of a generalized Dirac equation,”
*Frontiers in Physics*, vol. 3, article 82, 8 pages, 2015. View at Publisher · View at Google Scholar - A. Okniński, “Duffin-Kemmer-Petiau and Dirac equations—a supersymmetric connection,”
*Symmetry*, vol. 4, no. 3, pp. 427–440, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Okniński, “Splitting the Kemmer-Duffin-Petiau equations,”
*Proceedings of Institute of Mathematics of NAS of Ukraine*, vol. 50, part 2, pp. 902–908, 2004. View at Google Scholar - M. Thomson,
*Modern Particle Physics*, Cambridge University Press, New York, NY, USA, 2013. - L. de Broglie,
*Théorie Générale des Corpuscules á Spin*, Gauthier-Villars, Paris, France, 1943. - E. M. Corson,
*Introduction to Tensors, Spinors, and Relativistic Wave Equations*, Blackie and Son, London, UK, 1953. - K. S. Krane,
*Introductory Nuclear Physics*, John & Wiley Sons, New York, NY, USA, 1988. - K. A. Olive, “Review of particle physics,”
*Chinese Physics C*, vol. 38, no. 9, Article ID 090001, 2014. View at Publisher · View at Google Scholar