Decays of $W$ bosons: possible mixing of spin $0$ and spin $1$ states

TL;DR

This paper explores the possibility that $W$ bosons can exhibit a mixture of spin 0 and spin 1 states, using Hagen-Hurley equations to analyze decay processes and potential spin mixing effects.

Contribution

It introduces a formalism based on Hagen-Hurley equations to describe $W$ bosons with mixed spin states, providing a new perspective on their decay mechanisms.

Findings

Proposes $W$ boson as a mixture of spin 0 and spin 1 states.

Analyzes beta and top quark decays considering spin mixing.

Suggests that $W$ bosons can be described within the same Dirac representation.

Abstract

We study decays of a spin $1$ boson within the formalism of Hagen-Hurley equations. Such particle can decay into two spin $\frac{1}{2}$ particles, a Weyl neutrino and a massive fermion, whose spins can couple to $S=0$ or $S=1$. Since spin $0$ and spin $1$ bosons can be described by the Dirac equation within the same representation of $\gamma ^{\mu }$ matrices mixing of $S=0$ and $S=1$ states is possible. We argue that the Hagen-Hurley equations describe $W$ boson with spin $S\in 0\oplus 1$ space and analyse mixed beta decays as well as top quark decays from this perspective.

Equations

$${\gamma }_{\mu }{p}^{\mu }\Psi =m\Psi ,$$

$$\Psi =c_1{\Psi }_{\left(1\right)}+c_0{\Psi }_{\left(0\right)}$$

$$Q{\Psi }_{\left(0\right)}={\Psi }_{\left(1\right)}\text{vskip}3.0ptplus1.0ptminus1.0pt$$

$$\displaystyle Q=\left(\begin{array}[]{ll}I_{2\times 2}&0_{2\times 2}\\ 0_{2\times 2}&-\psi^{-1}\eta_{A}^{~{}B}\end{array}\right)$$

$$Q^{2}=\left(\begin{array}[]{ll}I_{2\times 2}&0_{2\times 2}\\ 0_{2\times 2}&\psi^{-2}\left(\eta^{2}-\eta_{11}\eta_{22}\right)I_{2\times 2}\end{array}\right)$$

$$Q{\Psi }_{\left(1\right)}={\Psi }_{\left(0\right)}$$

$$n⟶p+W^{-}⟶p+e+{\bar{\nu }}_e$$

$$n\left(\uparrow\right)\longrightarrow\left\{\begin{array}[]{l}p\left(\downarrow\right)+\left[e\left(\uparrow\right)\bar{\nu}_{e}\left(\uparrow\right)\right]\qquad\text{GT transition (}82\text{\%)\vskip 3.0pt plus 1.0pt minus 1.0pt}\\ p\left(\uparrow\right)+\left[e\left(\uparrow\right)\bar{\nu}_{e}\left(\downarrow\right)\right]\qquad\text{F transition (}18\text{\%)}\end{array}\right.$$

$$t\left(\uparrow \right)⟶b\left(\downarrow \right)+W\left(\Uparrow \right)⟶b\left(\downarrow \right)+\left[l\left(\uparrow \right)+{\nu }_l\left(\uparrow \right)\right],$$

$$t\left(\uparrow \right)⟶b\left(\uparrow \right)+W\left(\text{vbox}\text{scalebox}0.5\bullet \right)⟶b\left(\uparrow \right)+\left[l\left(\uparrow \right)+{\nu }_l\left(\downarrow \right)\right].$$

Full text

Keywords: $W$ boson, top quark, beta decay

1 Introduction

Recently, we have split the $7\times 7$ Hagen-Hurley equations describing spin $1$ particles in the interacting case, into two Dirac equations with nonstandard solutions [1]. More exactly, we have obtained the Weyl and the Dirac equations, describing two spin $\frac{1}{2}$ particles, a Weyl neutrino and a massive fermion, with spins coupling to $S=0$ or $S=1$ [1].

In our recent paper we have demonstrated that spin $\frac{1}{2}$ fermions as well as spin [math] Duffin-Kemmer-Petiau bosons as well as spin $1$ Hagen-Hurley bosons can be described by the Dirac equation within the same representation of $\gamma^{\mu}$ matrices [2]. We show that within this formalism a transition between integer spins (mixing) of these bosons is possible. This finding supports hypothesis that a particle obeying the the $7\times 7$ Hagen-Hurley equations has spin in $0\oplus 1$ space.

We have suggested that these solutions describe decay of a virtual $W$ boson in beta decay. Although the spin is conserved in the process of beta decay, spin conservation of the $W$ boson as a virtual particle may be disputable. It is argued that in pion decay as well as in the Fermi mechanism the $W$ boson is in $S_{z}=0$ state (is longitudinally polarized) and behaves as spin [math] particle. On the other hand, assuming that the virtual $W$ boson obeys the Hagen-Hurley equations, we propose that spin of the $W$ boson belongs to $0\oplus 1$ space. Therefore, in this work we are looking for a mechanism permitting transition between spins [math] and $1$ explaining existence of Gamow-Teller and Fermi competing mechanisms of beta decay.

New information may be provided by the top quark decay. Since the mass of the top quark is greater than mass of the $W$ boson the $W$ boson is a real particle in $t\longrightarrow bW$ decay. Therefore, it can be expected that the spin of the $W$ boson is $S=1$ and conserved. Alternatively, also the real $W$ boson is described by the Hagen-Hurley equations and, due to spin [math] and $1$ mixing, has only partly determined spin, i.e. $S\in 0\oplus 1$ space.

In the next Section we describe shortly generalized solutions of the Dirac equation using the same conventions and notation as in [2]. Transformation, describing transition between integer spins and making spin mixing possible, is also constructed. Then, in the next two Sections the Gamow-Teller and Fermi mechanisms as well as top quark decay are analysed in the light of possibility of spin [math] and $1$ mixing. We analyse nature of the $W$ boson, virtual as well as real, in the last Section.

2 Generalized solutions of the Dirac equation and mixing of the $S=0$

and $S=1$ spin states

We have shown that in the non-interacting case full covariant solutions of the $S=0$ Duffin-Kemmer-Petiau and $S=1$ Hagen-Hurley equations are generalized (matrix) solutions of the same Dirac equation:

[TABLE]

see [2] for representation of $\gamma^{\mu}$ matrices. More exactly, the wave functions $\Psi_{\left(0\right)}=\left(\psi_{A\dot{B}},\ \psi I_{2\times 2}\right)^{T}$, $\Psi_{\left(1\right)}=\left(\zeta_{A\dot{B}},\ -\eta_{C}^{\ D}\right)^{T}$ $\left(\eta_{CD}=\eta_{DC}\right)$, are solutions of Dirac equation (1) and correspond to $S=0$,$\ S=1$ cases, respectively [2].

It follows that a function

[TABLE]

fulfills Eq. (1) and thus describes a mixture of spin [math] and spin $1$ bosonic states. Moreover, transformation of spin states, $S=0\rightleftarrows S=1$, can be constructed explicitly. An operator $Q$ transforming function $\Psi_{\left(0\right)}$ into $\Psi_{\left(1\right)}$ is:

[TABLE]

where I_{2\times 2}=\left(\begin{array}[]{cc}1&0\\ 0&1\end{array}\right), 0_{2\times 2}=\left(\begin{array}[]{cc}0&0\\ 0&0\end{array}\right), \eta_{A}^{~{}B}=\left(\begin{array}[]{cc}\eta&\eta_{22}\\ -\eta_{11}&-\eta\end{array}\right) and $\eta\equiv\eta_{12}=\eta_{21}$. Obviously, we have to demand also $\zeta_{A\dot{B}}=\psi_{A\dot{B}}$.

The inverse transformation can be also computed. Indeed, we if demand $Q\Psi_{\left(1\right)}=Q^{2}\Psi_{\left(0\right)}$ then we get:

[TABLE]

and thus we also have:

[TABLE]

provided that $\det\left(\eta_{A}^{~{}B}\right)=-\eta^{2}+\eta_{11}\eta_{22}=-\psi^{2}$.

3 $W$ boson in beta decay

Interestingly, there are two mechanisms of beta decay, Gamow-Teller (GT) and Fermi (F). In some cases, known as mixed beta decay, both mechanisms are present. For example, decay of a neutron is a mixed decay [3]:

[TABLE]

More exactly, there are two possible alignments of the electron and neutrino spins:

[TABLE]

where products of the $W^{-}$ boson decay (see [4]) are shown in square brackets and $\left(\uparrow\right)$ denotes spin $\frac{1}{2}$.

This is puzzling because the $W$ boson has spin $1$. However, the $W$ boson is a virtual particle in beta decay and thus spin conservation is open to discussion. It may be thus assumed that in the case of Fermi decay the $W$ boson is in $S_{z}=0$ state (longitudinal boson) and behaves as a spin [math] particle. Similar explanation applies to the pion decay [5].

We have proposed alternative explanation, based on the analysis of Hagen-Hurley equations, suggesting that spin of a virtual $W$ boson belongs to the $0\oplus 1$ space [1].

4 $W$ boson in top quark decay

It can be expected that decay of the top quark may cast new light on nature of the $W$ boson. The top quark, being the heaviest known fundamental particle, decays with formation of a real spin $1$ $W$ boson. Moreover, it decays before hadronization process can occur and thus spin information of the top quark is not destroyed [6, 7]. It can be shown that in the decay $t\longrightarrow bW\longrightarrow bl\nu_{l}$ the $b\nu_{l}$ subsystem is in a $J=0$ state and thus the entire spin of the top quark is transferred to the lepton [8]. Therefore, the following process should occur exclusively:

[TABLE]

where the $W$ boson has spin $1$, denoted as $\left(\Uparrow\right)$, see Section 15.1.1 in [6] or Section 2.2 in [7].

Suppose now that the $W$ boson is described by the Hagen-Hurley equations. We have shown that decay of the Hagen-Hurley boson requires that its spin belongs to $0\oplus 1$ space [1]. It follows from results obtained in Section 2 that spin [math] and spin $1$ states can mix. Therefore, it is possible that the decaying $W$ boson has not fully determined spin $S$, i.e. $S\in 0\oplus 1$, and can decay also in the $S=0$ state analogously to the Fermi transition, where \left(\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{\displaystyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{\textstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{\scriptstyle\bullet}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{\scriptscriptstyle\bullet}}}}}\right) below denotes spin [math]:

[TABLE]

5 Discussion

Our considerations are based on assumption that the $W$ boson obeys the (spin $1$) Hagen-Hurley equations. This is possible since these equations do not conserve parity and can thus describe a weakly interacting particle. We have demonstrated that such particle can decay into a lepton and neutrino but its spin is only partly determined, belonging to the $0\oplus 1$ space. Furthermore, it follows from Section 2 that spin $1$ and spin [math] states can mix, see Eq. (2).

There are two cases of the $W$ boson decay.

When the $W$ boson is virtual (as in the pion decay or in beta decay) its spin is not conserved. Therefore, there are two mechanisms of mixed beta decay, cf. Eq. (7), moreover the pion decay $\pi^{+}\longrightarrow W^{+}\longrightarrow\mu^{+}+\nu_{\mu}$ is allowed – this is because longitudinally polarized $W$ boson behaves as a spin [math] particle in the pion decay as well as in the Fermi mechanism of beta decay. We offer alternative explanation. The virtual $W$ boson is described by the Hagen-Hurley equations so when it decays its spin is in $0\oplus 1$ space and hence can decay in spin [math] state. Ratio of Gamow-Teller to Fermi decay is $\left|c_{1}\right|^{2}/\left|c_{0}\right|^{2}$ with $c_{1}$, $c_{0}$ defining the linear combination $\Psi$, see (2). 2. 2.

When the $W$ boson is real (as in the top quark decay) the spin is conserved and, according to the Standard Model, the decay should proceed via channel (8) only. More exactly, in about $30$% of the decays the $W$ boson has negative helicity, in $70$ % of the events it has longitudinal helicity, while positive helicity is negligible [6]. We propose alternative possibility. The real $W$ is described by the Hagen-Hurley equations and due to mechanism of decay (which requires that its spin belongs to $0\oplus 1$ space [1]) and mixing of spin [math] and spin $1$ states, cf. Section 2, decays as a linear combination $\Psi$, cf. (2). In this case both channels (8), (9) should be observed.

We assume that decay of the $W$ boson (both real and virtual) is described by the Hagen-Hurley equations. If confirmed, this would cast new light on mixed beta decay and on the pion decay, as well as on the top quark decay. Measurement of spins of the products in reaction $t\longrightarrow bW\longrightarrow bl\nu_{l}$, especially spin correlation of $t$ and $b$ quarks, see Eqs. (8), (9), may confirm or rule out this hypothesis.