  Tachyon condensation in quantum field theory (QFT) plays a central role in
models of fundamental interactions and cosmology. Inspired by tower truncation
in string field theory, ultraviolet completions were proposed with
infinite-derivative form factors that preclude the appearance of pathological
ghosts in the particle spectrum, contrary to other local higher-derivative
QFT's. However, if the infinite-derivative QFT exhibits other vacua, each of
them has its own spectrum, which is generally not ghost-free: an infinite tower
of ghost-like resonances pops up above the nonlocal scale at tree-level, whose
consistency is unclear. In this article, a new weakly nonlocal deformation of a
generic local QFT is introduced via a Lorentz and gauge covariant star-product
of fields, which is commutative but nonassociative in general. This framework
realizes tachyon condensation without ghosts at the perturbative level, with
applications for spontaneous symmetry breaking.
