One-loop Amplitudes: String Methods, Infrared Regularization, and Automation

We calculate field theory loop amplitudes by string methods, applied to half-maximal 4-point one-loop graviton amplitudes. Infrared divergences are regulated similarly to soft-collinear effective field theory (SCET): new mass scales are introduced, here by higher-point kinematics. We use an analytically continued single-valued polylogarithm as generating function. The Feynman integrals for the new tensor structures are infrared finite. We provide code as a step towards automation.

💡 Research Summary

This paper presents a comprehensive framework for computing one‑loop field‑theory amplitudes using string‑theoretic methods, with a focus on the half‑maximal supersymmetric four‑point graviton amplitude. The authors introduce an infrared (IR) regularization scheme that mirrors the philosophy of soft‑collinear effective field theory (SCET): instead of introducing explicit mass regulators, they generate new mass scales through higher‑point kinematics. Concretely, a non‑conserved momentum κ is introduced via the relation k₁ + k₂ + k₃ + k₄ = −κ in the four‑point function. This “minahaning” parameter acts as an IR regulator; keeping κ finite renders the would‑be IR divergences into κ‑dependent terms that vanish smoothly as κ → 0, reproducing the expected cancellation without mixing orders of perturbation theory.

The technical core relies on a single‑valued polylogarithm employed as a generating function for all‑mass Feynman integrals. By treating all external legs as massive (the masses are purely regulators, not physical states), the authors obtain box, triangle, and bubble integrals with generic mass dependence. In this all‑mass setting, hidden symmetries such as dual conformal invariance become manifest, simplifying the algebraic structure of the integrands. The generating function approach collapses the polylogarithmic dependence to simple polynomials when the field‑theory limit (α′ → 0) is taken, allowing a clean separation of ultraviolet (UV) and IR singularities.

On the string side, the paper reviews the world‑sheet construction of the amplitude. World‑sheet Wick contractions are automated via a recursive algorithm that handles fermionic contractions, spin‑structure sums, and the Szegö kernel expressed in terms of Jacobi theta functions. For half‑maximal supersymmetry, the orbifold twist vector γ only appears when at least four fermions are inserted, leading to compact expressions for the spin‑sum functions G_{m+2}(γ,−γ,…). These are expanded in a series of functions f^{(n)}(z), of which only f^{(1)} and f^{(2)} are needed for the present calculation.

The field‑theory limit is treated with great care. The authors define a precise ordering of limits: first expand the integrand in powers of α′, then analyse vertex collisions. Three collision regimes are identified—two‑vertex (box), three‑vertex (triangle), and four‑vertex (bubble) collisions—each mapping to a distinct world‑line region. Position integration is performed in a SCET‑inspired manner, dividing the integration domain into “box”, “triangle”, and “bubble” regions. Within each region, the Bern‑Dixon‑Kosower (BDK) differentiation method is employed: vector box integrals serve as generators, and tensor numerators are reduced to scalar master integrals by systematic differentiation with respect to the auxiliary masses.

The authors provide a publicly available code repository (referenced in Section 2) that implements the entire pipeline: world‑sheet Wick contraction, spin‑sum evaluation, field‑theory limit extraction, world‑line integration, and BDK‑based reduction. Users can input arbitrary tensor structures and regulator masses, and the code automatically produces the full set of all‑mass box, triangle, and bubble integrals, isolates UV poles (1/ε terms), and returns the finite remainder.

Results are summarized in Section 7. Infrared divergences are completely absent after taking the κ → 0 limit; the remaining UV divergences appear as standard 1/ε poles. The finite part is presented in two forms: (i) a field‑theory kinematic expression involving ζ(2), ζ(3), and single‑valued polylogarithms, and (ii) a string‑kinematic expression that retains dependence on the modular parameter τ and theta‑function structures. Table 1 and the flowcharts in Figures 1 and 2 give a clear visual map of contributions from boxes, triangles, and bubbles, indicating which terms are IR‑divergent, UV‑divergent, or finite.

The appendices contain detailed derivations of Berends‑Giele currents at one loop, the “skinny” limit (a particular scaling of masses), explicit BDK differentiation identities, a comparison with Mellin‑space representations, and a list of quadratic relations among the master integrals. The authors emphasize that the method does not rely on supersymmetry; the half‑maximal example serves as a demonstrator, and the same machinery can be applied to non‑supersymmetric amplitudes or to other theories such as Yang‑Mills.

In conclusion, the paper delivers a novel, systematic, and automatable approach to compute one‑loop amplitudes that bridges string theory techniques with modern effective‑field‑theory regularization. By integrating SCET‑style IR regulation, single‑valued polylogarithmic generating functions, and the BDK differentiation method, the authors provide a powerful toolkit that can be extended to higher‑point functions, different supersymmetry levels, and possibly to phenomenologically relevant gauge‑theory processes. This work is poised to impact both formal amplitude research and practical high‑energy‑physics computations.