String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry

TL;DR

The paper develops a string-inspired framework to construct 1-loop amplitudes for 6D gauge theories with half-maximal supersymmetry, enforcing locality and gauge invariance to obtain unique 3- and 4-point results. It builds a comprehensive alphabet of kinematic blocks (parity-even/odd, BG currents, and invariants) and shows how they assemble into local, gauge-invariant integrands, with a detailed treatment of IR regularization and anomaly structure. The work connects to BCJ duality and double-copy concepts, illustrating a BCJ-consistent 3-point sector and identifying deviations at 4 points, while also matching 4D spinor-helicity expressions and reproducing the field-theory limit of open-string orbifold amplitudes. By reducing to D=4 and comparing with known results, it validates the formalism and clarifies how model-dependent content (vector vs hypermultiplets) shapes the one-loop amplitudes, with implications for string-derived supergravity via double copy. The methods open a path to higher-multiplicity amplitudes and potential extensions to other supersymmetry fractions and to gravity via a refined BCJ program.

Abstract

We compute one-loop amplitudes in six-dimensional Yang-Mills theory with half-maximal supersymmetry from first principles: imposing gauge invariance and locality on an ansatz made from string-theory inspired kinematic building blocks yields unique expressions for the 3- and 4-point amplitudes. We check that the results are reproduced in the field-theory limit $α' \rightarrow 0$ of string amplitudes in K3 orbifolds, using simplifications made in a companion string-theory paper 1603.05262.

Figures (8)

• Figure 1: Cubic-vertex subdiagrams with an off-shell leg $\cdots$ can be represented by local multiparticle polarizations $e^{m}_{12},f^{mn}_{12}$ and $e^{m}_{123},f^{mn}_{123}$, respectively.
• Figure 2: Berends--Giele currents $\mathfrak e^m_{12\ldots p}$ and $\mathfrak f^{mn}_{12\ldots p}$ combine multiparticle polarizations with appropriate propagators so as to reproduce the cubic-vertex subdiagrams in a color-ordered $(p{+}1)$-point tree amplitude with an off-shell leg $\cdots$.
• Figure 3: Cubic diagrams of bubble topology with kinematic numerator $T_{A,B}$ at ${A=12\ldots p}$ and $B=n,n{-}1 \ldots p{+}1$, where $\ell$ denotes the loop momentum.
• Figure 4: The singular propagator $s_{12}^{-1}$ in the one-mass bubble diagram is compensated by the formally vanishing numerator $T_{12,3}= s_{12}(e_1 \cdot e_2)(k_1 \cdot e_3)$. If the diagram on the left is a "snail", then the diagram on the right is a "shy snail". All our snails are shy.
• Figure 5: The kinematic factors $T_{123,4}$ and $T_{321,4}$ of an external bubble compensate the singular propagator $s_{123}^{-1}$ upon minahaning. The leftover poles in $s_{12}$ and $s_{23}$ correspond to tree-level subdiagrams whose local numerators in the ellipsis can be assembled from (\ref{['red49']}).