Dirac fermions under rainbow gravity effects in the Bonnor-Melvin-Lambda spacetime

In this paper, we study the relativistic energy spectrum for Dirac fermions under rainbow gravity effects in the $(3+1)$-dimensional Bonnor-Melvin-Lambda spacetime, where we work with the curved Dirac equation in cylindrical coordinates. Using the tetrads formalism of General Relativity and considering a first-order approximation for the trigonometric functions, we obtain a Bessel equation. To solve this differential equation, we also consider a region where a hard-wall confining potential is present (i.e., some finite distance where the radial wave function is null). In other words, we define a second boundary condition (Dirichlet boundary condition) to achieve the quantization of the energy. Consequently, we obtain the spectrum for a fermion/antifermion, which is quantized in terms of quantum numbers $n$, $m_j$ and $m_s$, where $n$ is the radial quantum number, $m_j$ is the total magnetic quantum number, $m_s$ is the spin magnetic quantum number, and explicitly depends on the rainbow functions $F(\xi)$ and $G(\xi)$, curvature parameter $\alpha$, cosmological constant $\Lambda$, fixed radius $r_0$, and on the rest energy $m_0$, and $z$-momentum $p_z$. So, analyzing this spectrum according to the values of $m_j$ and $m_s$, we see that for $m_j>0$ with $m_s=-1/2$ (positive angular momentum and spin down), and for $m_j<0$ with $m_s=+1/2$ (negative angular momentum and spin up), the spectrum is the same. Besides, we graphically analyze in detail the behavior of the spectrum for the three scenarios of rainbow gravity as a function of $\Lambda$, $r_0$, and $\alpha$ for three different values of $n$ (ground state and the first two excited states).