# Minimal independent couplings at order $\alpha'^2$

## TL;DR

This paper determines the minimal set of independent couplings at order α'^2 in string theory effective actions, using field redefinitions and Bianchi identities, and presents these couplings in two different schemes.

## Contribution

It identifies the minimal number of independent couplings at order α'^2 and expresses them in two schemes with different derivative and dilaton structures.

## Key findings

- Minimum of 60 independent couplings at order α'^2 identified.
- Two schemes for expressing couplings are developed, with different dilaton inclusion.
- 20 couplings are common in both schemes, simplifying the effective action representation.

## Abstract

Using field redefinitions and Bianchi identities on the general form of the effective action for metric, $B$-field and dilaton, we have found that the minimum number of independent couplings at order $\alpha'^2$ is 60. We write these couplings in two different schemes in the string frame. In the first scheme, each coupling does not include terms with more than two derivatives and it does not include structures $R,\,R_{\mu\nu},\,\nabla_\mu H^{\mu\alpha\beta}$, $ \nabla_\mu\nabla^\mu\Phi$. In this scheme, 20 couplings which are the minimum number of couplings for metric and $B$-field, include dilaton trivially as the overall factor of $e^{-2\Phi}$, and all other couplings include derivatives of dilaton. In the second scheme, the dilaton appears in all 60 coupling only as the overall factor of $e^{-2\Phi}$. In this scheme, 20 of the couplings are exactly the same as those in the previous scheme.