The One-Loop H^2R^3 and H^2(DH)^2R Terms in the Effective Action

TL;DR

This work computes one-loop amplitudes involving the NS-NS 2-form $B$ and gravitons in type II string theory and expands them to low energy in $\alpha'$. By subtracting diagrams generated by known quartic terms and covariantizing the remainder, it identifies new higher-derivative couplings $H^2R^3$ and $H^2(\nabla H)^2R$, each multiplied by the universal tensor factor $(t_8t_8 \pm \tfrac{1}{8}\epsilon_{10}\epsilon_{10})$, with signs depending on IIB/IIA. The results extend the familiar $R^4$ corrections to include B-field contributions and clarify the precise tensor structure of loop-induced terms, constraining supersymmetric completions and duality-related consistency checks. Together, they provide a coherent framework for how NS-NS fluxes enter higher-derivative corrections in the string effective action at one loop.

Abstract

We consider the one-loop B^2h^3 and B^4h amplitudes in type II string theory, where B is the NS-NS two-form and h the graviton, and expand to lowest order in alpha'. After subtracting diagrams due to quartic terms in the effective action, we determine the presence and structure of both an H^2R^3 and H^2(DH)^2R term. We show that both terms are multiplied by the usual (t_8t_8\pm{1/8}ε_{10}ε_{10}) factor.

Figures (3)

• Figure 1: Field theory vertices relevant for the $B^2h^3$ amplitude. From left to right: a three-vertex from $R$, two three-vertices from $e^{-\phi}H^2$, and three four-vertices and a five-vertex from $\bar{R}^4$.
• Figure 2: Field theory diagrams contributing to the $B^2h^3$ amplitude.
• Figure 3: Field theory diagrams contributing to the $B^4h$ amplitude: (a)-(c) pole diagrams, and (d) a contact diagram.