Scalar amplitudes from fiber bundle geometry

We compute tree-level n-point scattering amplitudes in scalar field theories in terms of geometric invariants on a fiber bundle. All 0- and 2-derivative interactions are incorporated into a metric on this bundle. The on-shell amplitudes can be efficiently pieced together from covariant Feynman rules, and we present a general closed formula for obtaining the n-point amplitude in this way. The covariant Feynman rules themselves can be derived using a generalization of the normal coordinate expansion of the fiber bundle metric. We demonstrate the efficiency of this approach by computing the covariant Feynman rules up to n ¼ 10 points, from which one can obtain the full amplitudes using our general formula. The formalism offers a prototype for obtaining geometric amplitudes in theories with higher-derivative interactions, by passing from the fiber bundle to its jet bundles.