Classical black hole scattering from a worldline quantum field theory
Gustav Mogull, Jan Plefka, Jan Steinhoff
Abstract
A bstract A precise link is derived between scalar-graviton S-matrix elements and expectation values of operators in a worldline quantum field theory (WQFT), both used to describe classical scattering of black holes. The link is formally provided by a worldline path integral representation of the graviton-dressed scalar propagator, which may be inserted into a traditional definition of the S-matrix in terms of time-ordered correlators. To calculate expectation values in the WQFT a new set of Feynman rules is introduced which treats the gravitational field h μν ( x ) and position $$ {x}_i^{\mu}\left({\tau}_i\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> <mml:mi>μ</mml:mi> </mml:msubsup> <mml:mfenced> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mfenced> </mml:math> of each black hole on equal footing. Using these both the 3PM three-body gravitational radiation 〈 h μv ( k )〉 and 2PM two-body deflection $$ \Delta {p}_i^{\mu } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> <mml:mi>μ</mml:mi> </mml:msubsup> </mml:math> from classical black hole scattering events are obtained. The latter can also be obtained from the eikonal phase of a 2 → 2 scalar S-matrix, which we show corresponds to the free energy of the WQFT.