New look at the modified Coulomb potential in a strong magnetic field

The static Coulomb potential of quantum electrodynamics (QED) is calculated in the presence of a strong magnetic field in the lowest Landau level approximation using two different methods. First, the vacuum expectation value of the corresponding Wilson loop is calculated perturbatively in two different regimes of dynamical mass $m_{\mathrm{dyn}}$, i.e., $|{\mathbf{q}}_{\parallel }^2|\ll m_{\mathrm{dyn}}^2\ll |eB|$ and $m_{\mathrm{dyn}}^2\ll |{\mathbf{q}}_{\parallel }^2|\ll |eB|$, where ${\mathbf{q}}_{\parallel }$ is the longitudinal component of the momentum relative to the external magnetic field $B$. The result is then compared with the static potential arising from Born approximation. Both results coincide. Although the arising potentials show different behavior in the aforementioned regimes, a novel dependence on the angle $\theta$ between the particle-antiparticle’s axis and the direction of the magnetic field is observed. In the regime $|{\mathbf{q}}_{\parallel }^2|\ll m_{\mathrm{dyn}}^2\ll |eB|$, for strong enough magnetic field and depending on the angle $\theta$, a qualitative change occurs in the Coulomb-like potential; whereas for $\theta =0$, $\pi$ the potential is repulsive, it exhibits a minimum for angles $\theta \in ]0,\pi [$.

Figure 1

Potential $V(\mathbf{x})$ from (2.19) of a particle-antiparticle pair for different magnetic fields. According to this result, whereas for $B\le {10}^9$ at each given distance the absolute value of the potential increases with increasing the magnetic field, the shape of the potential does not change for $B\ge {10}^9$.

Figure 2

The ratio $\frac{M_{\mathrm{eff}}}{M_{\gamma }}=g(\theta )$ for $\theta \in [0,\pi ]$ in the regime $m_{\mathrm{dyn}}^2\ll |{\mathbf{q}}_{\parallel }^2|\ll |eB|$ of LLL dominance.

Figure 3

Potential $V_1(R,\theta )$ for different $B$ and $\theta$. No qualitative changes occur by varying the angle $\theta$.

Figure 4

Coefficients ${\mathcal{A}}_1$, $\gamma {\mathcal{A}}_2$, and ${\gamma }^2{\mathcal{A}}_3$ for different magnetic fields. As it turns out, ${\mathcal{A}}_3$ decreases rapidly with increasing magnetic field.

Figure 5

Potential $V_3(R,\theta )$ for different $B$ and $\theta$. Bound states can be formed for $\theta \in ]0,\pi [$ and for strong magnetic fields, $B\ge {10}^5$, in the regime $R\le 0.005\mathrm{fm}$. The depth of the potential at $R_{\min }$ increases with the magnetic field.

Figure 6

Behavior of the minimum of the potential $V_3$ as a function of ${10}^7\le B\le {10}^{16}$ for different $0<\theta \le \pi /2$. For $\pi /2\le \theta <\pi$, $R_{\min }$ shows a symmetry in changing $\theta \to \theta +\pi /2$ (see the three-dimensional figure on the right-hand side).

Figure 7

Potential $V_2(R,\theta )$ for different $B$ and $\theta$. No qualitative changes occur by varying the angle $\theta$. This potential is comparable with the potential (1.1) from 7 in the scaling regime.