Fine tuning of rainbow gravity functions and Klein-Gordon particles in cosmic string rainbow gravity spacetime

We argue that, as long as relativistic quantum particles are in point, the variable $y=E\mathrm{/}{E}_p$ of the rainbow functions pair $g_{{}_0}(y)$ and $g_{{}_1}(y)$ should be fine tuned into $y=\mathrm{\mid }E\mathrm{\mid }\mathrm{/}{E}_p$, where $E_p$ is the Planck’s energy scale. Otherwise, the rainbow functions will be only successful to describe the rainbow gravity effect on relativistic quantum particles and the anti-particles will be left unfortunate. Under such fine tuning, we consider Klein-Gordon (KG) particles in cosmic string rainbow gravity spacetime in a non-uniform magnetic field (i.e., $\mathbf{B}=\mathbf{\nabla }\times \mathbf{A}=\frac{3}{2}{B}_{\circ }r\widehat{z}$ ). Then we consider KG-particles in cosmic string rainbow gravity spacetime in a uniform magnetic field (i.e., $\mathbf{B}=\mathbf{\nabla }\times \mathbf{A}=\frac{1}{2}{B}_{\circ }\widehat{z}$ ). Whilst the former effectively yields KG-oscillators, the later effectively yields KG-Coulombic particles. We report on the effects of rainbow gravity on both KG-oscillators and Coulombic particles using four pairs of rainbow functions: (i) $$, g_{_{1}}\left( y\right) =\sqrt{1-εy^{2}% }, (ii) $g_{{}_0}\left(y\right)=1$, g_{_{1}}\left( y\right) =\sqrt{% 1-εy}, (iii) $g_{{}_0}\left(y\right)=g_{{}_1}\left(y\right)={(1-\epsilon y)}^{-1}$, and (iv) $g_{{}_0}\left(y\right)=(e^{\epsilon y}-1)\mathrm{/}\epsilon y$, $g_{{}_1}\left(y\right)=1$, where $y=\mathrm{\mid }E\mathrm{\mid }\mathrm{/}{E}_p$ and $\epsilon$ is the rainbow parameter. It is interesting to report that, all KG particles’ and anti-particles’ energies are symmetric about $E=0$ value (a natural relativistic quantum mechanical tendency), and a phenomenon of energy states to fly away and disappear from the spectrum is observed for the rainbow functions pair (iii) at $\gamma =\epsilon m\mathrm{/}{E}_p=1$.