On string momentum in effective string theories

TL;DR

This paper calculates the center of mass momentum in effective string theories of the Polchinski-Strominger type using Noether's theorem, showing that corrections to free theory vanish at certain orders due to an improvement property.

Contribution

It provides a detailed calculation of string momentum corrections in two conformally invariant effective string theories, demonstrating the vanishing of these corrections at specific orders.

Findings

Corrections to free bosonic string momentum are of the improvement type.

At order R^{-2} and R^{-3}, corrections vanish upon integration.

The analysis applies to Polyakov-Liouville and extended Polchinski-Strominger theories.

Abstract

Centre of mass momentum for strings is an important ingredient in the calculation of the spectrum of string theories. It is calculated by the N\"other prescription for two types of conformally invariant effective string theories of the Polchinski-Strominger type. One is the so called Polyakov-Liouville theory \cite{liouville} and the other an extension of the original Polchinski-Strominger action analysed upto order $R^{-3}$ where $2\pi R$ is the length of the string. In the first case analysis is carried out to order $R^{-2}$ and in the second case to order $R^{-3}$. In both cases the correction to the free bosonic theory result is shown to be of the {\em improvement} type so that the correction to the total string momentum, obtained by integrating over the spatial coordinate of the string world sheet, vanishes.