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The leading method to study the running coupling constant of non-abeliangauge theories is based on the Schr\"odinger functional scheme. However, theboundary conditions and $\mathcal{O}(a)$ improvement have not beensystematically generalized for theories with more than three colors. Thesetheories have applications in BSM model building as well as in the large $N$limit. We have studied the boundary conditions and improvement for the pureYang-Mills theory within the Schr\"odinger functional scheme. We havedetermined for all values of $N$ the boundary fields which provide highsignal/noise ratio. Additionally, we have calculated the improvementcoefficient $c_t$ for the pure gauge to one loop order for SU($N$) gaugetheories with $N=2,\ldots,8$ from which $N\geq 4$ are previously unknown.

Schr\"odinger functional boundary conditions and improvement of the SU($N$) pure gauge action for $N>3$