Cuts and Contours

TL;DR

Cuts and Contours develops a positive, surfaceology-based parametrization of string amplitudes to reveal the full singularity structure that the traditional worldsheet moduli hide. It introduces global integration contours in $y$-space that render tree-level integrals finite across all kinematics, via two approaches: a Feynman $i\varepsilon$-like deformation and a generalized Pochhammer contour, with explicit treatment at $n=4$ and $n=5$, and generalization to arbitrary $n$. At loop level, the paper formulates integrand-level unitarity constraints by analyzing locus-based threshold expansions and leading singularities, showing how these constrain UV regularizations such as $\hat{D}_n$ and other surface integrals, and it develops a loop contour consistent with Schwinger-type parametrizations and threshold decomposition. The framework enables direct extraction of discontinuities from the integrand, reduces loop problems to a finite set of Feynman-like integrals, and offers practical contours for numerical evaluation, while outlining challenges posed by cancellations in extreme kinematics and potential extensions to higher loops and different string theories. Overall, the work provides a cohesive contour- and residue-based toolkit to study string amplitudes' singularity structure and unitarity within a positive, blow-up parametrization, with implications for UV regularization and analytic structure.

Abstract

The traditional formulation of string amplitudes via worldsheet integrals provides a parametrization of the moduli space that fails to expose the complete singularity structure of the amplitudes. This problem is solved by the positive parametrization of string amplitudes given by surfaceology. In this work, we use this formalism to study a number of properties of string amplitudes at tree-level and one-loop. We introduce several global prescriptions for an integration contour for which the integrals are finite everywhere in kinematic space. At tree-level, this is done in two ways: one directly implements the Feynman $i\varepsilon$ to analytically continue from Euclidean to Lorentzian worldsheets; the other is a generalization of the closed Pochhammer contour to arbitrary number of points. At loop-level, we present a systematic way of extracting cuts directly from the worldsheet integrand. This provides a powerful set of unitarity constraints, which we use to test the consistency of different "stringy" UV regularizations of field theory amplitudes. In addition, we identify the massive threshold expansion of the integrand, which allows us to reduce the problem to a finite set of Feynman integrals in Schwinger parametrization and provide a straightforward contour prescription reminiscent of its field-theory version.

Figures (19)

• Figure 1: The original integration region in worldsheet coordinates (left) is "blown up" via local changes of variables into a geometry which exposes the complete set of singularities of the amplitude (right).
• Figure 2: From curves on surfaces to the $g$-vector fan. In 1, we show a choice of the triangulation of the $5$-point disk containing chords $\{(1,3),(1,4)\}$ and the respective dual fatgraph. In 2, we show how the curves $(2,4)$ (red), $(2,5)$ (blue), and $(3,5)$ (green) can be drawn as paths on the fatgraph, as well as the respective words we associate with each path. In 3, we present the $g$-vector fan which divides the plane into $5$-cones, each of which corresponding to one of the five cubic planar diagrams entering the $5$-point amplitude.
• Figure 3: (Left) Integration contour at $4$-points.$\ \Gamma_0$ (in blue) is the region where we integrate along the original contour, and $\Gamma_{(1,3)} \cup \Gamma_{(2,4)}$ (in red) are the deformations where $t_{1,3}$ becomes imaginary. (Right) Illustration of the cancellation between the real part of the contributions from $\Gamma_0$ and $\Gamma_{(1,3)}\cup \Gamma_{(2,4)}$, for $X_{1,3} = x+0.4i$ and $X_{2,4} = -8 + 0.4i$. In solid blue and red we show the result from $\Gamma_0$ and $\Gamma_{(1,3)} \cup \Gamma_{(2,4)}$, respectively, for $R_\star =0.8$. In dashed we represent the analogous plots but for $R_\star =0.2$. In gray we show the real part of the $4$-point amplitude, given by the sum of the red and blue contributions.
• Figure 4: Left: division of the $\{(1,3),(1,4)\}$ cone into region with deformed/undeformed variables as in \ref{['eq:regions 13,14']}. Right: Division of the whole Feynman fan. The shaded octagon corresponds to the region where all variables are undeformed.
• Figure 5: Conditions on the minimum values for the region boundaries $R^{ij}_\star$ in each cone, which are determined by the $\mathcal{F}$-polynomials appearing in the corresponding representation of the integrand.