Abstract

Here, rather detailed numerical comparisons of energies and momenta for$\ $\ positive $m_{+}=m\succ 0$ \ and negative \ $m_{-}=-m\prec 0$ \ \particle masses with $m_{+}^{2}=m_{-}^{2}=$ $m^{2}$ , within the bicubic equation limiting particle velocity formalism with three particle limiting velocities $c_{1},$ $c_{2}$\ and $c_{3}$, are done. Already these limiting velocities, on a global scale, can differentiate positive, $m_{+}$ and negative \ $m_{-}$ particle masses. While $c_{1}(m_{+})$, $c_{2}(m_{+})$ and $c_{3}(m_{+})$ are real, imaginary and real, corresponding, respectively, to primary, obscure and normal particles; $c_{1}(m_{-})$, $c_{2}(m_{-})$ and $% c_{3}(m_{-})$ are respectively imaginary, real and real, now representing respectively, obscure, primary and normal particles. In fact, from limiting velocity solutions, one identifies: $c_{1}^{2}(m_{+})=$ $% c_{2}^{2}(m_{-}),c_{1}^{2}(m_{-})=c_{2}^{2}(m_{+})$, $c_{3}^{2}(m_{-})=$ $% c_{3}^{2}(m_{+})$.\ \ The unified particle mass-shell like forms with particle energies and momenta are readily expressible for $m_{+}=m\succ 0$ and $m_{-}=-m\prec 0$ masses with respective limiting velocities, separating $c_{3}(m_{+})$ from $c_{3}(m_{-})$ as well as $c_{1}(m_{+})$ from $% c_{1}(m_{-})$ and $c_{2}(m_{+})$ from $c_{2}(m_{-}).$ We assume that flavor neutrinos, which, while in process do not change flavor, belong to normal limiting velocity $c_{3}$ class. Then the muon neutrino from OPERA velocity measurement should maintain the same velocity squares $v^{2}$ and $c_{3}^{2}$ when one changes the positive neutrino mass $% m_{+\nu }\left( \mu \right) \succ 0$ into the negative neutrino mass $% m_{-\nu }\left( \mu \right) \prec 0$ , since theoretically $% c_{3}^{2}(m_{+\nu }(\mu ))=c_{3}^{2}(m_{-\nu }(\mu ))$. For OPERA measurements this is verified perturbatively by simultaneously evaluating squares of normal limiting velocities with $m_{+\nu }(\mu )$ and $m_{-\nu }(\mu )$ masses, yielding the same result $c_{3}^{2}(m_{+\nu }(\mu )=c_{3}^{2}(m_{-\nu }(\mu )\simeq v_{\nu }^{2}\left( \mu \right) \simeq c^{2} $.

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