We argue that, as long as relativistic quantum particles are in point, the variable $y=E/E_p$ of the rainbow functions pair $g_{ _{ 0}} (y)$ and $g_{ _{ 1}} (y)$ should be fine tuned into $y=|E|/E_p$, 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., $\mathbf{ B}=\mathbf{ \nabla }\times \mathbf{ A}=\frac{ 3}{ 2}B_{ \circ }r\,\hat{ z}$ ). Then we consider KG-particles in cosmic string rainbow gravity spacetime in a uniform magnetic field (i.e., $\mathbf{ B}=\mathbf{ \nabla }\times \mathbf{ A}=\frac{ 1}{ 2}B_{ \circ }\,\hat{ z}$ ). 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) $% g_{ _{ 0}}\left( y\right) =1$, $g_{ _{ 1}}\left( y\right) =\sqrt{ 1-\epsilon y^{ 2}% }$, (ii) $g_{ _{ 0}}\left( y\right) =1$, $g_{ _{ 1}}\left( y\right) =\sqrt{ % 1-\epsilon y}$, (iii) $g_{ _{ 0}}\left( y\right) =g_{ _{ 1}}\left( y\right) =\left( 1-\epsilon y\right) ^{ -1}$, and (iv) $g_{ _{ 0}}\left( y\right) =\left( e^{ \epsilon y}-1\right) /\epsilon y$, $g_{ _{ 1}}\left( y\right) =1$, where $y=|E|/E_p$ 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 $\gamma=\epsilon m/E_p=1$.
Categories: gr-qc astro-ph.HE hep-ph