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Abstract

In a classical scattering problem, the classical eikonal is defined as the
generator of the canonical transformation that maps in-states to out-states. It
can be regarded as the classical limit of the log of the quantum S-matrix. In a
classical analog of the Born approximation in quantum mechanics, the classical
eikonal admits an expansion in oriented tree graphs, where oriented edges
denote retarded/advanced worldline propagators. The Magnus expansion, which
takes the log of a time-ordered exponential integral, offers an efficient
method to compute the coefficients of the tree graphs to all orders. We exploit
a Hopf algebra structure behind the Magnus expansion to develop a fast
algorithm which can compute the tree coefficients up to the 12th order (over
half a million trees) in less than an hour. In a relativistic setting, our
methods can be applied to the post-Minkowskian (PM) expansion for gravitational
binaries. We demonstrate the methods by computing the 3PM eikonal and find
agreement with previous results based on amplitude methods.