Curve integral formula for the Möbius strip
arXiv:2603.03393
The paper extends the curve integral formula for scattering amplitudes of colored scalars to non‑orientable surfaces such as the Möbius strip, using quasi‑cluster algebras and a doubled orientable embedding, and checks the construction via a superstring amplitude limit.
Abstract
The scattering amplitudes for colored scalars can be calculated using the so-called curve integral formula, relying on simple combinatorics. It introduces a set of global Schwinger parameters for all Feynman diagrams that contribute to an amplitude. We extend this construction to non-orientable surfaces by making use of the quasi-cluster algebras defined for non-orientable surfaces. We embed the non-orientable surface in a doubled orientable surface, and project the appropriate features onto the non-orientable surface. The curve integral formula can also be thought of as the high-tension limit of an appropriate string amplitude. As a check of our construction, we take a superstring amplitude with the Möbius strip topology and take its field theory limit to obtain the same Feynman diagrams as in the corresponding curve integral. Our construction can be generalized to arbitrary higher genus non-orientable surfaces. To illustrate this, we list the possible curves and their dual momenta for a two-loop non-orientable surface, and construct the surface Symanzik polynomials using the surface generalization of spanning trees.
37 pages + appendices, 18 figures. Minor clarifications added to version 2