Minimal independent couplings at order α′2

Mohammad R Garousi,Hamid Razaghian

doi:10.1103/physrevd.100.106007

Mohammad R Garousi, Hamid Razaghian

Open Access

https://doi.org/10.1103/physrevd.100.106007

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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.

Highlights

String theory is a quantum theory of gravity with a finite number of massless fields and a tower of infinite number of massive fields reflecting the stringy nature of the gravity
The minimum number of independent couplings is found in the schemes where all the arbitrary parameters are set to zero. We find that this latter method is more convenient to finding the independent couplings systematically, using the Mathematica packages like “xAct” [41]
Writing them in the local inertial frame, we use the arbitrary parameters in the total derivative terms and in the field redefinitions to reduce the 41 couplings to 8 independent couplings that are known in the literature

Summary

INTRODUCTION

String theory is a quantum theory of gravity with a finite number of massless fields and a tower of infinite number of massive fields reflecting the stringy nature of the gravity. The effective action has double expansions: the genus expansion which includes the classical and a tower of quantum corrections, and the stringy expansion which is an expansion in powers of the Regge slope parameter α0. In using the above techniques for finding the effective actions at the higher-derivative orders in the string theory, one needs the most general gauge invariant and minimal independent couplings at each order of α0. To find such couplings, one needs to impose various Bianchi identities and use field redefinitions freedom [33,34,35]. This method has been used to find the 8 independent couplings for gravity, B-field, and dilaton at order α0 in [13], the 7 independent couplings for gravity and dilaton at order α02 in [36,37,38,39], and

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