Effects of the Lorentz symmetry violation on relativistic neutral scalar bosons: Scattering and bound states

TL;DR

This study analyzes Lorentz-symmetry violation in the CPT-even photon sector of the SME by coupling a neutral scalar to a background of a constant magnetic field and a cylindrical electric field, described through a modified Klein–Gordon equation. Using a cylindrical partial-wave method, it derives the radial equation with an effective $V_{ ext{eff}}(r)=rac{ ext{δ}}{r}+rac{ ilde{ ext γ}_l^2-rac{1}{4}}{r^2}$ and analyzes both scattering and bound states via the Whittaker equation and $S$-matrix poles. The work classifies eight potential profiles and shows bound states exist only in a restricted parameter range, dictated by $ ext{κ}_2$, $ ext{κ}_3$, $B$, and the fields, correcting prior claims and highlighting the interplay between LV parameters and relativistic quantum observables. Overall, it provides a comprehensive framework for LV-induced scattering phenomena and bound-state structures in a cylindrically symmetric electromagnetic background, with implications for LV phenomenology in quantum systems.

Abstract

In this letter, we investigate the effects of Lorentz symmetry violation on a relativistic neutral scalar boson within the framework of the Klein-Gordon formalism. We consider a tensor $(K_F)_{μναβ}$ out of the Standard Model Extension, which describes the field configuration consisting of a constant magnetic field $\vec{B}=B\hat{z}$ and a cylindrical electric field $\vec{E}=\fracλ{r}\hat{r}$. We analyze and discuss the effects of Lorentz symmetry violation on the equation of motion and show that the presence of a specific Lorentz symmetry violation parameter is essential to obtain analytical solutions for scattering and bound states. Employing the partial wave approach in cylindrical coordinates, we calculate the relativistic phase shift, scattering amplitude and $S$-matrix, and discuss the effects of Lorentz symmetry violation on the phase shift. Furthermore, we obtain bound-state solutions by examining the poles of the $S$-matrix and compare our findings with those reported in the literature. Our results reveal that bound-state solutions are only feasible for a restrict range of Lorentz symmetry violation parameters, which contradict previous studies.

Figures (4)

• Figure 1: Profiles of effective potential: Classes 1, 3 and 7 (green line), class 2 (red line), class 4 (black line), and classes 5, 6 and 8 (blue line).
• Figure 2: Possible regions of bound and scattering states for $\kappa_{2}g>0$.
• Figure 3: Possible regions of bound and scattering states for $\kappa_{2}g<0$.
• Figure 4: Plots of the energy as function of $\lambda$ for $|l|=1$ and different values of $n$ and $B$. For $n=0$ [solid line], $n=1$ [dotted line] and $n=2$ [dashed line]. For $B=0.7$ [black], $B=0.9$ [blue] and $B=1.1$ [red].