Four Loop Scattering in the Nambu-Goto Theory

TL;DR

This work investigates multiloop scattering on the non‑critical Nambu–Goto worldsheet, starting with a two‑loop analysis and revealing polynomial on‑shell amplitudes despite UV counterterms. By invoking gravitational dressing and near‑integrable models, the authors explain the two‑loop cancellations and extend the framework to three and four loops, where logarithmic dependence first arises. They introduce an undressed (PS) theory and a weight‑based counting to organize counterterms, using dressing/undressing to predict higher‑loop structures and identify universal logarithms. The results hint at a systematic approach to perturbative expansions around the GGRT theory and have potential implications for non‑integrable, asymptotically fragile worldsheet dynamics in confining strings.

Abstract

We initiate the study of multiloop scattering amplitudes in the Nambu-Goto theory on the worldsheet of a non-critical string. We start with a brute force calculation of two loop four particle scattering. Somewhat surprisingly, even though non-trivial UV counterterms are present at this order, on-shell amplitudes remain polynomial in the momenta of colliding particles. We show that this can be understood as a consequence of existence of certain close by (semi)integrable models. Furthermore, these arguments can be extended to obtain the answer for three and four loop scattering, bypassing the brute force calculation. The resulting amplitudes develop non-polynomial (logarithmic) dependence on the momenta starting at three loops.

Figures (4)

• Figure 1: Kinematics considered in the paper.
• Figure 2: Topologies contributing at order $l_s^6$.
• Figure 3: Three loop quartic large D diagram.
• Figure 4: Three loop sextic diagram.