One-loop four-graviton string amplitude at finite $\alpha'$

We evaluate the one-loop four-graviton scattering amplitude in type-II superstring theory exactly in \alpha' <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mi>′</mml:mi> </mml:mrow> </mml:math> . This result is achieved by combining physical insights into the i\varepsilon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>ε</mml:mi> </mml:mrow> </mml:math> prescription in string theory with a new technical application of the Rademacher expansion of modular integrals. We provide an implementation of our formula in \texttt{C++} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mtext mathvariant="monospace">𝙲++</mml:mtext> </mml:math> and use it to study the behavior of the amplitude at finite \alpha' <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mi>′</mml:mi> </mml:mrow> </mml:math> and in different kinematic regimes. Our analysis reveals a tension between explicit computations and the saddle point analysis of Gross and Mende in the high-energy limit and suggests the presence of additional saddle points.