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We perform a high precision measurement of the static $q\bar{q}$ potential inthree-dimensional SU($N$) gauge theory with $N=2,3$ and compare the results tothe potential obtained from the effective string theory. In particular, we showthat 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 andthe non-universal boundary coefficient $\bar{b}_2$, including an extensiveanalysis of all types of systematic uncertainties. We find that the magnitudeof $\bar{b}_2$ decreases with increasing $N$, leading to the possibility of avanishing $\bar{b}_2$ in the large $N$ limit. In the standard form of theeffective string theory possible massive modes and the presence of a rigidityterm are usually not considered, even though they might give a contribution tothe energy levels. To investigate the effect of these terms, we perform asecond analysis, including these contributions. We find that the associatedexpression for the potential also provides a good description of the data. Theresulting continuum values for $\bar{b}_2$ are about a factor of 2 smaller thanin the standard analysis, due to contaminations from an additional $1/R^4$term. However, $\bar{b}_2$ shows a similar decrease in magnitude withincreasing $N$. In the course of this extended analysis we also obtaincontinuum results for the masses appearing in the additional terms and we findthat they are around twice as large as the square root of the string tension inthe continuum and compatible between SU(2) and SU(3) gauge theory. In thefollow up papers we will extend our investigations to the large $N$ limit andexcited states of the open flux tube.

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