Abstract
We applied a recently published modified Fock-Schwinger (MFS) method to find the exact solution of the propagator equation for a charged vector boson in the presence of a constant magnetic field directly in the momentum space as a sum over Landau levels in arbitrary $\xi$-gauge. In contrast to the standard approaches for finding propagators, MFS method demonstrated several improvements in terms of computational complexity reduction and revealed simple internal structures in intermediate and final expressions, thus allowing to obtain new useful representations of the propagator.
Highlights
Analysis of elementary particle loop processes in extreme conditions, such as strong magnetic fields, requires a knowledge of particle propagators where the field effects are taken into account exactly.There exist at least two naturally arising scales of strong magnetic fields
Each of the exponential operators accounted for the contribution to the final expression due to the (i) propagation of charged particle, (ii) its spin interaction with magnetic field, and (iii) the choice of ξ-gauge
A particular form of δ-function decomposition reduced the action of these exponentials either to the eigenvalue substitution or to the use of the quantum harmonic oscillator (QHO) ladder operators
Summary
Analysis of elementary particle loop processes in extreme conditions, such as strong magnetic fields, requires a knowledge of particle propagators where the field effects are taken into account exactly. The solution of the propagator equation already contains all the spin parts summed over, and the δ-function in the right-hand side ensures the correct normalization The latter approach seems to require less computational effort to obtain a propagator. We applied a recently published modified Fock-Schwinger (MFS) approach [13] to the solution of the massive vector boson propagator equation in the presence of a constant magnetic field in arbitrary ξ-gauge. We briefly mention known results and approaches for finding quantum field propagators in external electromagnetic fields, and provide a known expression for the case of W-boson, obtained using the East-Coast metric convention. We derive the propagator equation in the West-Coast metric convention (commonly used in modern particle physics) and provide the corresponding solution
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