The One-Loop Five-Graviton Amplitude and the Effective Action

TL;DR

This work computes the one-loop five-graviton amplitude in type II string theory within the light-cone GS framework and expands it to reveal the low-energy effective action up to $D^6R^4$ terms. By subtracting contributions from known $D^{2n}R^4$ terms, it shows the absence of $R^5$ and $D^2R^5$ corrections and identifies the precise $D^4R^5$ structure required at this order, with coefficients matching the $D^6R^4$ sector. The analysis also uncovers the role of $\epsilon_{10}$ tensors, showing that $\epsilon_8\epsilon_8$ pieces combine with $t_8t_8$ in a $(t_8t_8\pm\tfrac18\epsilon_{10}\epsilon_{10})$ pattern, and discusses modular-function dependencies (e.g., $Z_{3/2}$ in IIB) and possible universal relations across orders and backgrounds. The results inform the construction of higher-derivative corrections in the string effective action and motivate conjectures about all-orders structures and tree-loop correspondences for five-graviton amplitudes in type II string theory.

Abstract

We consider the one-loop five-graviton amplitude in type II string theory calculated in the light-cone gauge. Although it is not possible to explicitly evaluate the integrals over the positions of the vertex operators, a low-energy expansion can be obtained, which can then be used to infer terms in the low-energy effective action. After subtracting diagrams due to known D^{2n}R^4 terms, we show the absence of one-loop R^5 and D^2R^5 terms and determine the exact structure of the one-loop D^4R^5 terms where, interestingly, the coefficient in front of the D^4R^5 terms is identical to the coefficient in front of the D^6R^4 term. Finally, we show that, up to D^6R^4 ~ D^4R^5, the ε_{10} terms package together with the t_8 terms in the usual combination (t_8t_8\pm{1/8}ε_{10}ε_{10}).

Figures (3)

• Figure 1: One of the fundamental domains of $SL(2,\mathbb{Z})$.
• Figure 2: The origin of the massless poles in the five-graviton amplitude.
• Figure 3: The two field theory diagrams contributing to the five-graviton amplitude: (a) a pole diagram and (b) a contact diagram.