The running of the coupling in SU(N) pure gauge theories

The running of the coupling is studied in SU(4) gauge theory using the Schr\"odinger functional technique. Up to energies of the order of the square root of the string tension $\sigma$, the running is found to agree with the two-loop perturbative formula. Relating the perturbative to the non-perturbative regime of the running and converting to the $\msbar$ scheme allows one to extract the ratio $\Lambda_{\msbar}/\sqrt{\sigma}$. The result is then used in combination with similar calculations present in the literature for SU(2) and SU(3) to extract $\Lambda_{\msbar}/\sqrt{\sigma}$ in the large $N$ limit. Our results for $N=3,4$ agree with a recent study of the same quantity performed using the Parisi mean field scheme as an intermediate scheme, while $\Lambda_{\msbar}/\sqrt{\sigma}$ in SU(2) turns out to differ by 2.5%. Possible explanations of this discrepancy are discussed.