Abstract
The low-energy light-by-light cross section as determined by the nonlinear Euler-Heisenberg QED Lagrangian is evaluated in the presence of constant magnetic fields in the center-of-mass system of the colliding photons. This cross section has a complicated dependence on directions and polarizations. The overall magnitude decreases as the magnetic field is increased from zero, but this trend is reversed for ultrastrong magnetic fields $B\gtrsim B_c$, where the cross section eventually grows quadratically with the magnetic field strength perpendicular to the collision axis. This effect is due to interactions involving the lowest Landau level of virtual Dirac particles; it is absent in scalar QED. An even more dramatic effect is found for virtual charged vector mesons where the one-loop cross section diverges at the critical field strength due to an instability of the lowest Landau level and the possibility of the formation of a superconducting vacuum state. We also discuss (the absence of) implications for the recent observation of light-by-light scattering in heavy-ion collisions.
Highlights
Scattering of light by light is a prediction of quantum electrodynamics (QED) that has been first calculated in 1935, prior to the full development of QED, in the low-energy limit by Euler and Kockel [1,2], and in the ultrarelativistic limit shortly thereafter by Akhiezer, Landau, and Pomeranchuk [3,4]
The former calculations were extended by Heisenberg and Euler [5] who obtained an effective low-energy Lagrangian which includes background electromagnetic fields to all orders in field strength
Working out the effects of magnetic background fields on virtual scalars, we find that magnetic fields lead to a monotonic decrease of the light-by-light scattering cross section in scalar QED, whereas the lowest Landau level of the Dirac spinors contributes a counteracting effect that dominates at large magnetic fields where it leads to a growing cross section
Summary
Scattering of light by light is a prediction of quantum electrodynamics (QED) that has been first calculated in 1935, prior to the full development of QED, in the low-energy limit by Euler and Kockel [1,2], and in the ultrarelativistic limit shortly thereafter by Akhiezer, Landau, and Pomeranchuk [3,4]. Γ þ γ scattering in the presence of weak and strong (constant) magnetic fields in the center-of-mass system of the colliding photons, from B=Bc ≪ 1 to B=Bc ≫ 1 (but, parametrically, B=Bc ≪ α−1=2 so that higher-loop corrections as well as the effects from dispersion and refraction of light in the magnetic field [24] remain negligible). In the following this process will be studied in detail in the low-energy approximation provided by the. At least sufficiently below the mass threshold, where the cross section steeply rises with energy, the EulerHeisenberg Lagrangian permits reliable calculations of the effects of magnetic fields on light-by-light scattering
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