The Feynman $i ε$ in String Theory

TL;DR

The paper provides a covariant formulation of the Feynman $i\varepsilon$ in string perturbation theory by identifying a deformation of the integration cycle in the complexified moduli space that encodes Lorentzian propagation near on-shell degenerations. It articulates how this cycle can be implemented for open and closed strings, using both Euclidean-to-Lorentzian continuations and contour/Schwinger-parameter techniques to reproduce the correct $i\varepsilon$-shift of propagator poles. By analyzing tree-level amplitudes (Veneziano and Virasoro–Shapiro) and generalizing to generic degenerations, it lays out a framework that extends to loops via long-strip/long-tube regions and discusses the role of complexification, Pochhammer-like structures, and superstring degenerations. The approach has implications for unitarity, CPT symmetry, and gauge invariance in string theory, and provides a starting point for understanding the analytic structure and high-energy behavior of string amplitudes within a consistent, cycle-based formalism.

Abstract

The Feynman $i\varepsilon$ is an important ingredient in defining perturbative scattering amplitudes in field theory. Here we describe its analog in string theory. Roughly one takes the string worldsheet to have Lorentz signature when a string is going on-shell although it has Euclidean signature generically.

Figures (13)

• Figure 1: A disc with marked points $0,x,1,\infty$ on its boundary is conformally equivalent, for $x\to 0$, to the Riemann surface with boundary depicted here, which describes propagation of an open string through a proper time of order $|\log x|$.
• Figure 2: The Pochhammer contour is a closed contour that winds back and forth around the branch points at 0 and 1 in such a way that the form $\omega={\mathrm d} x\,x^{-\alpha' s -2}(1-x)^{-\alpha' t-2}$ is single-valued.
• Figure 3: A string world sheet in closed-string field theory, with three propagators -- represented by simple tubes -- joining trivalent vertices that in general are represented by rather complicated worldsheets. (The vertices here are depicted as smooth genus 1 worldsheets, though in closed-string field theory the vertices carry Kahler metrics that often are not smooth.)
• Figure 4: This contour represents integration over Euclidean proper time $\tt$ up to $\tt=\tt_0$ for some large $\tt_0$, and then continuing with $\tt=\tt_0+i\tau$, $0\leq\tau<\infty$.
• Figure 5: An integration contour for the Veneziano amplitude that incorporates the Feynman $i\varepsilon$. The part of the contour near $x=0$ and 1 is drawn as a tight spiral to make it legible, but actually (to ensure convergence whenever $s,t,$ and $u$ are real) one wants to repeat the same little circle around 0 or 1 infinitely many times, with the help of an $\exp(-\varepsilon\Phi)$ convergence factor.