  We formulate scalar field theories in a curved braided $L_\infty$-algebra
formalism and analyse their correlation functions using Batalin-Vilkovisky
quantization. We perform detailed calculations in cubic braided scalar field
theory up to two-loop order and three-point multiplicity. The divergent tadpole
contributions are eliminated by a suitable choice of central curvature for the
$L_\infty$-structure, and we confirm the absence of UV/IR mixing. The
calculations of higher loop and higher multiplicity correlators in homological
perturbation theory are facilitated by the introduction of a novel diagrammatic
calculus. We derive an algebraic version of the Schwinger-Dyson equations based
on the homological perturbation lemma, and use them to prove the braided Wick
theorem.
