Removal of the Tachyons from the Fermionic Sector of the Quadratic ZX Action for the Exotic Model (E8)

TL;DR

This work investigates tachyonic instabilities in the fermionic ZX sector of the Exotic Model (XM), a SUSY framework built from the SSM plus an Exotic Invariant. By constraining 12 initial parameters to 7, via explicit rules that enforce equal masses for the four ZX fermions, the author demonstrates a tachyon-free tree-level spectrum in the ZX fermionic sector, with 16 propagators organized into four quartets sharing a common denominator and a single mass value $m^2 = -\frac{k_5 m_1 m_2}{2 k_6}$. The analysis relies on a renormalized BRS framework that distinguishes simple versus non-simple cohomology, and is implemented through a Mathematica notebook (mnbk) that computes the propagators and demonstrates the normal-form structure $r_i/(\Box - m^2)$. While promising, the results hinge on renormalization of the BRS transformations in the non-simple case, and the extension to the ZX bosons remains for future work; the findings offer a concrete step toward a SUSY-splitting mechanism without spontaneous SUSY breaking and illuminate potential experimental predictions of the XM. Overall, the paper provides a concrete, computable path to removing fermionic tachyons in the ZX sector, strengthening the XM as a viable alternative to the conventional SSM, pending full bosonic analysis and renormalization validation.

Abstract

This paper contains an encouraging result for the development of the Exotic Model (the `XM'). The XM is made from the Supersymmetric Standard Model (the `SSM') by coupling it to the `Exotic Invariant'. This rather unique `Exotic Invariant' is suggested by the detailed form of the SSM, combined with the detailed form of the BRS cohomology of SUSY. The motivation for examining the XM is that the usual SSM has some well-known, and serious, problems in its generally accepted, but unsatisfactory, SUSY breaking mechanism. That mechanism is based on the spontaneous breaking of SUSY. The Exotic Model avoids all of these problems, easily, because its SUSY splitting mechanism does not use, or need, any kind of spontaneous breaking of SUSY. The XM contains a mechanism for SUSY mass splitting because the addition of the Exotic Invariant modifies the algebra of SUSY. This paper is E8 in a series of papers denoted En, n= 1,2, ... all devoted to the XM. There is a lot of work still required to elucidate how the XM functions, and to see if the XM is really a viable alternative to the SSM. The first crucial question, raised in E7, is whether it is possible to constrain the initial XM action, so that the result excludes tachyonic excitations at tree level. If the XM is to be a viable theory, it is essential to first remove the tachyonic excitations, in a sensible way, from the fermionic and the bosonic sectors of the quadratic ZX action. This paper E8 explains how to remove the tachyons from the basic fermionic `ZX sector' at tree level. There are four fermion fields in this ZX sector. The removal requires that all of these fields have the same mass. This result is shown using a Mathematica notebook, which is issued with this paper.