One-loop Amplitudes: String Methods, Infrared Regularization, and Automation

Abstract

We calculate field theory loop amplitudes by string methods, applied to half-maximal 4-point one-loop graviton amplitudes. Infrared divergences are regulated similarly to soft-collinear effective field theory (SCET): new mass scales are introduced, here by higher-point kinematics. We use an analytically continued single-valued polylogarithm as generating function. The Feynman integrals for the new tensor structures are infrared finite. We provide code as a step towards automation.

Figures (10)

• Figure 1: Flow chart that summarizes the method developed in this work to calculate the field-theory limit integrals of amplitude \ref{['eq:our_amplitude']}. The horizontal sections correspond to sections \ref{['sec:FT_limit']}, \ref{['sec:WL_int']}, \ref{['sec:position_int']} and \ref{['sec:results']}, respectively. The circle in the center of the position integration section contains the machinery by which the integrals are calculated. For its expansion, see fig.\ref{['fig:flow_chart_dot']}.
• Figure 2: Flow chart corresponding to the circle in the center of the position integration section of fig.\ref{['fig:flow_chart']}. Here summarizing the method of calculating the field-theory embedded integrals, explained in detail in section \ref{['SCET']}, specifying to the most complicated box integrals.
• Figure 3: Minahaning a 4-point function means including a 5th massless momentum $\kappa$ and taking $\kappa\rightarrow 0$. Having $\kappa \neq 0$ allows 3-index Mandelstam variables $s_{ijk}$ mimicking 5-point kinematics.
• Figure 4: Three-vertex collision in the four-point one-loop low-energy limit, adapted from figure 3 in OchirovTourkine. From left to right: (A) the complex plane, (B) the worldsheet and (C) the worldline graph.
• Figure 5: SCET-inspired field-theory box diagrams. On the left-hand side: string theory kinematics with massless gravitons and minahaning; on the right-hand side: four-mass field-theory diagram.