On string momentum in effective string theories

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.