Abstract

The complex particle energy, appearing in this article, with the suggestive choices of physical parameters,is transformed simply into the real particle energy. Then with the bicubic equation limiting particle velocity formalism, one evaluates the three particle limiting velocities, $c_{1},$ $c_{2}$\ and $% c_{3},$ (primary, obscure and normal) in terms of the ordinary particle velocity, $v$, and derived positive $m_{+}=m\succ 0$ \ and negative \ $% m_{-}=-m\prec 0$ \ \ particle masses with $m_{+}^{2}=m_{-}^{2}=$ $m^{2}$.

In general, the important quantity in solving this bicubic equation is the real square value $\ z^{2}(m)$ of the congruent parameter, $z(m)$, that connects real or complex value of particle energy, $E,$ and the real or complex value of particle velocity squared, $v^{2}$, $2Ez(m)=3\sqrt{3}mv^{2}$% . With real $z^{2}(m)$ one determines the real value of discriminant, $D,$ of the bicubic equation, and they together influence the connection between $% E$ and $v^{2}.$ Hence, when $z^{2}\prec 1$ and \ $D\prec 0$ one has simply that $E\gg mv^{2}$. However,with $D\succeq 0$ and $z^{2}\succeq 1$ , both $E$ and $v^{2}$ may become complex simultaneously through connecting relation $% E=3\sqrt{3}mv^{2}/2z(m)$, with their real values satisfying \

Re $E\succcurlyeq m\left( \func{Re}v^{2}\right) $, keeping, however $z^{2}$ the same and real.

In this article, this new situation with $D\succeq 0$ is discussed in detail.by looking as how to adjust the particle\ parameters to have $\func{Im% }E=0$ with implication that automatically also Im$v^{2}=0.$.In fact, after having adjusted the particle\ parameters successfully this way, one simply writes Re$E=E$ and Re$v^{2}=v^{2}$. \ \ This way one arrives at that the limiting velocities satisfy $c_{1}=c_{2}$\ $\#$ $c_{3}$, which shows the degeneracy of $c_{1}$ and $c_{2}$ as the same numerical limiting velocity for two particles. This degeneracy $c_{1}$ =$c_{2}$ is simply due to the absence of $\func{Im}E$. It would start disappearing with just an infinitesimal $\func{Im}E$. Now,while $c_{1}=c_{2}$ is real, $c_{3}$ is imaginary and all of them associated with the same particle energy, $E$. With these velocity values the congruent parameter becomes quantized as $% z(m_{\pm })=3\sqrt{3}m_{\pm }v^{2}/2E=\pm 1$ which, with the bicubic discriminant $D=0$ value, implies the quantization also of the particle mass, $m,$ into $m_{\pm }=\pm m$ values . The numerically equal energies,from $E=\func{Re}E$ can be expressed as $\ \ \ \ \ \ \ \ \ \ \ $$E(c_{1,2}($ $m_{\pm }))=E(c_{3}(m_{\pm }))$ either directly in terms of $% c_{1}(m_{\pm })=c_{2}(m_{\pm })$ and $c_{3}(m_{\pm })$ or also indirectly in terms of particle velocity, $v$, as well as in the Lorentzian fixed forms with $v^{2}\#$ $c_{1}^{2},$ $c_{2}^{2}$\ or $c_{3}^{2}$ assuring different from zero mass, $m$ $\#$ $0$.

At the end, with here developed formalism, one calculates for a light sterile neutrino dark matter particle, the energies associated with $m_{\pm} $ masses and $c_{1,2}$and $c_{3}$ limiting velocities.

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