Spectrum of the open QCD flux tube and its effective string description I: 3d static potential in SU(N=2,3)

We perform a high precision measurement of the static $q\bar{q}$ potential in three-dimensional SU($N$) gauge theory with $N=2,3$ and compare the results to the potential obtained from the effective string theory. In particular, we show that the exponent of the leading order correction in $1/R$ is 4, as predicted, and obtain accurate results for the continuum limits of the string tension and the non-universal boundary coefficient $\bar{b}_2$, including an extensive analysis of all types of systematic uncertainties. We find that the magnitude of $\bar{b}_2$ decreases with increasing $N$, leading to the possibility of a vanishing $\bar{b}_2$ in the large $N$ limit. In the standard form of the effective string theory possible massive modes and the presence of a rigidity term are usually not considered, even though they might give a contribution to the energy levels. To investigate the effect of these terms, we perform a second analysis, including these contributions. We find that the associated expression for the potential also provides a good description of the data. The resulting continuum values for $\bar{b}_2$ are about a factor of 2 smaller than in the standard analysis, due to contaminations from an additional $1/R^4$ term. However, $\bar{b}_2$ shows a similar decrease in magnitude with increasing $N$. In the course of this extended analysis we also obtain continuum results for the masses appearing in the additional terms and we find that they are around twice as large as the square root of the string tension in the continuum and compatible between SU(2) and SU(3) gauge theory. In the follow up papers we will extend our investigations to the large $N$ limit and excited states of the open flux tube.