Unitarity cuts of the worldsheet

Abstract

We compute the imaginary parts of genus-one string scattering amplitudes. Following Witten’s i\varepsiloniε prescription for the integration contour on the moduli space of worldsheets, we give a general algorithm for computing unitarity cuts of the annulus, Möbius strip, and torus topologies exactly in \alpha'α′. With the help of tropical analysis, we show how the intricate pattern of thresholds (normal and anomalous) opening up arises from the worldsheet computation. The result is a manifestly-convergent representation of the imaginary parts of amplitudes, which has the analytic form expected from Cutkosky rules in field theory, but bypasses the need for performing laborious sums over the intermediate states. We use this representation to study various physical aspects of string amplitudes, including their behavior in the (s,t)(s,t) plane, exponential suppression, decay widths of massive strings, total cross section, and low-energy expansions. We find that planar annulus amplitudes exhibit a version of low-spin dominance: at any finite energy, only a finite number of low partial-wave spins give an appreciable contribution to the imaginary part.

Natural boundaries for scattering amplitudes

“…Let us first consider the case of the forward limit, t = 0 (or backward, u = 0). Since it is the same as the case of the fixed angle θ = 0 (or θ = π), we immediately conclude that a natural boundary develops for (45) in those cases. Notice that the forward/backward limit is even more singular, because positions of singularities depend solely on the ratio l n in (43).…”

“…Let us notice that for any physical (real) θ , all s * k,l,n are real and hence all the singularities lie on the half-line (45). As shown above, the natural boundary still exists after complexifying the scattering angle, but the singularities are more spread out around it and approach the real axis as n → ∞, see Fig.…”

“…4]. However, the assumption that the integrand is well-behaved around the threshold does not hold up to scrutiny in practical one-loop examples once spinning particles are included, leading to different threshold behavior, see, e.g., [45,Sec. 6.6].…”

Tropological sigma models

“…It should be intriguing to analyze how this construction withstands the process of tropicalization. 7 Tropical geometry has already made its appearance in the study of equilibrium string amplitudes, first in the α ′ → 0 limit [65] and more recently at general α ′ [66,67]. 8 See remarks in section 3.1 of [69].…”

Evaluating one-loop string amplitudes

“…Instead, γ ∪ Γ should be treated as a new proposal for the integration contour in the τ-space. A more precise description on the whole complex moduli space M 1,n ( ) will be given in [13]. As a matter of fact, the essential singularities are another way of determining that the contour on the left of Figure 1 could not have been the correct one: it simply gives a divergence close…”