Propagators Beyond The Standard Model

Abstract

In this paper, we explore the field propagator with a structure that is general enough to encompas both the case of newly-defined mass-dimension 1 fermions and spin-1/2 bosons. The method we employ is to define a map between spinors of different Lounesto classes, and then write the propagator in terms of the corresponding dual structures.

Appendix A: Some Results Based on New Dual Definitions

Quite recently, some new definitions of dual structure opened windows to a new interpretation of how spin-half bosons emerge (from Dirac spinors) and also how they evade the spin-statistic theorem [5]. As claimed, the new dual structure furnishes a local and Lorentz-invariant theory and also provide a positive-definite Hamiltonian. Such aforementioned features are carried by a new dual structure, which reads

$$\begin{aligned} {\mathop {\lambda }\limits ^{\lnot }} = +{\bar{\lambda }}^{\dag }\gamma _0, \quad \text{ and }\quad {\mathop {\chi }\limits ^{\lnot }} = -s{\bar{\chi }}^{\dag }\gamma _0, \end{aligned}$$

(A1)

in which $s=1$ stands for fermionic field and $s=-1$ stands for bosonic field. Bearing in mind the dual definition in (A1), the spin sums (8) and (9) are now replaced by the following set

$$\begin{aligned} \sum _{particle}\lambda _{i}(\varvec{p})\overset{\,{}^{{}^{\varvec{\lnot }}}}{\smash [t]{\lambda }}_{i}(\varvec{p})= & {} \gamma _{\mu }p^{\mu }+ m\mathbbm {I},\end{aligned}$$

(A2)

$$\begin{aligned} \sum _{{anti}{\text {-}}{particle}}\!\!\!\!\!\!\chi _{i}(\varvec{p})\overset{\,{}^{{}^{\varvec{\lnot }}}}{\smash [t]{\chi }}_{i}(\varvec{p})= & {} -(\gamma _{\mu }p^{\mu }- m\mathbbm {I}). \end{aligned}$$

(A3)

With these new results at hands, Eq.(34) is written, accordingly [5, page3], as

$$\begin{aligned}{} & {} {\mathcal {A}}(x-x^{\prime })= -i\xi \mathop {\textrm{lim}} \limits _{\epsilon \rightarrow 0^+}\Bigg \lbrace \Bigg [{\textbf{M}}\int \frac{d^4 p}{(2\pi )^4}\frac{1}{2E({\varvec{p}})\upsilon }\nonumber \\{} & {} \qquad \Bigg (\frac{\sum _{h}\lambda _{h}({\varvec{p}}){\mathop {\lambda }\limits ^{\lnot }}_{h}({\varvec{p}})(p_0+\sqrt{p_{j}p^{j}+m^2})}{p_{\mu }p^{\mu }-m^2+i\epsilon } \Bigg )e^{-ip_{\mu }(x^{\mu }-x^{\prime \mu })}\varvec{\Delta }\Bigg ]_{_{particle}}\nonumber \\{} & {} \qquad + \Bigg [{\textbf{M}}\int \frac{d^4 p}{(2\pi )^4}\frac{1}{2E({\varvec{p}})\upsilon }\Bigg (s\frac{\sum _{h}\chi _{h}(-{\varvec{p}}){\mathop {\chi }\limits ^{\lnot }}_{h}(-{\varvec{p}})(p_0-\sqrt{p_{j}p^{j}+m^2})}{p_{\mu }p^{\mu }-m^2+i\epsilon } \Bigg )\nonumber \\{} & {} \qquad e^{-ip_{\mu }(x^{\mu }-x^{\prime \mu })}\varvec{\Delta }\Bigg ]_{_{{anti}{\text {-}}{particle}}}\Bigg \rbrace , \end{aligned}$$

(A4)

According to the type of field, the statistic will be dictated by the s parameter. If the field is to be fermionic, $s=1$, forcing the anticommutative relations among the creator and annihilator operators. If the field is bosonic, $s=-1$, and commutative relations must be taken into account. For more details, we refer to [5].

Now, following the very same steps as in Sect.(3.1), and bearing in mind the correct relation among the creator/annihilator operators as well as the value of s, the above amplitude of propagation yields

$$\begin{aligned} {\mathcal {S}}_{\text {FD}}(x-x^\prime )= \int \frac{\text {d}^4 p}{(2 \pi )^4}\, e^{-i p_\mu (x^{\prime \mu }-x^\mu )} \frac{\gamma _\mu p^\mu + m\mathbbm {I}}{p_\mu p^\mu -m^2 + i\epsilon }. \end{aligned}$$

(A5)

as expected for Dirac fermions and also for spin-half Therefore, we show the similarity of both results presented in [3] and [5].