Nonabelian gauge theories and range of the corresponding force

Question

Do all nonabelian gauge theories produce short range force?

This post imported from StackExchange Physics at 2014-03-22 17:07 (UCT), posted by SE-user Revo

Comments

QCD forces only look "short range" because they're so powerful at long distances that they're confining. That's why only color-neutral objects may exist in isolation and the residual forces acting on such color-neutral objects (like nucleons) decreases quickly with the distance.

This post imported from StackExchange Physics at 2014-03-22 17:07 (UCT), posted by SE-user Luboš Motl

Answer 1

No, pure (unbroken) SU(2) Yang Mills for example is massless, i.e. long range

This post imported from StackExchange Physics at 2014-03-22 17:07 (UCT), posted by SE-user twistor59

Answer 2

One can use the running of the coupling constant as a proxy for confinement. The $\beta$-function for $SU(N)$ to first order gives (see http://en.wikipedia.org/wiki/Asymptotic_freedom#Calculating_asymptotic_freedom): $$\beta_1(\alpha) = { \alpha^2 \over \pi} \left( -{11N \over 6} + {n_f \over 3} \right).$$ So for $N=3$ confinement (i.e. short range force) occurs with fewer than 16 species of fermions, but for $N=2$ it has to be fewer than 11.

This post imported from StackExchange Physics at 2014-03-22 17:07 (UCT), posted by SE-user genneth