Abstract
Massive QED, in contrast with its massless counterpart, possesses two conserved charges; one is a screened (vanishing) Maxwell charge which is directly associated with the massive vector mesons through the identically conserved Maxwell current. A somewhat peculiar situation arises for couplings of Hermitian matter fields to massive vector potentials; in that case the only current is the screened Maxwell current and the coupling disappears in the massless limit. In case of selfinteracting massive vector mesons the situation becomes even more peculiar in that the usually renormalizability guaranteeing validity of the first order power-counting criterion breaks down in second order and requires the compensatory presence of additional Hermitian H-fields. Some aspect of these observation have already been noticed in the BRST gauge theoretic formulation, but here we use a new setting based on string-local vector mesons which is required by Hilbert space positivity. The coupling to H-fields induces Mexican hat like selfinteractions; they are not imposed and bear no relation with spontaneous symmetry breaking; they are rather consequences of the foundational causal localization properties realized in a Hilbert space setting. In case of selfinteracting massive vectormesons their presence is required in order to maintain the first order power-counting restriction of renormalizability also in second order. The presentation of the new Hilbert space setting for vector mesons which replaces gauge theory and extends on-shell unitarity to its off-shell counterpart is the main motivation for this work. The new Hilbert space setting also shows that the second order Lie-algebra structure of selfinteracting vector mesons is a consequence of the principles of QFT and promises a deeper understanding of the origin of confinement.
Highlights
The theoretical interest in massive vector mesons can be traced back to Schwinger’s conjecture [1], which states that “massive QED” leads to “charge screening”
Massive QED, in contrast with its massless counterpart, possesses two conserved charges; one is a screened Maxwell charge which is directly associated with the massive vector mesons through the identically conserved Maxwell current, while the presence of a particle-antiparticle counting charge depends on the matter
A somewhat peculiar situation arises for couplings of Hermitian matter fields to massive vector potentials; in that case the only current is the screened Maxwell current and the coupling disappears in the massless limit
Summary
The theoretical interest in massive vector mesons can be traced back to Schwinger’s conjecture [1], which states that “massive QED” leads to “charge screening”. This leads to a rather clear distinction between quasiclassical pictures of symmetry breaking and gauge-induced H -self-interactions This CGI setting permits a direct perturbative construction of a unitary gauge-invariant S-matrix for a g A · AH coupling of a massive vector meson with a Hermitian field; in this way it highlights the second order induction mechanism which leads to the Mexican hat H self-interaction. The presentation of the general formalism in the presence of string-crossings will be contained a forthcoming paper by Jens Mund; further details as regards applications to interacting string-local fields will be addressed in joint work by Mund and the present author It had been known for a long time [21,22] that within the setting of algebraic QFT the field-particle relation in the presence of a mass gap can always be described in terms of operators localized in arbitrary narrow space-like cones (whose cores are semi-infinite space-like strings). The concluding remarks present an outlook about what can be expected from this new setting concerning unsolved problems of infrared divergencies such as “infraparticles” in QED and confinement in QCD
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