Renormalization Group and Strong Interactions

The renormalization-group method of Gell-Mann and Low is applied to field theories of strong interactions. It is assumed that renormalization-group equations exist for strong interactions which involve one or several momentum-dependent coupling constants. The further assumption that these coupling constants approach fixed values as the momentum goes to infinity is discussed in detail. However, an alternative is suggested, namely, that these coupling constants approach a limit cycle in the limit of large momenta. Some results of this paper are: (1) The $e^{+}-e^{-}$ annihilation experiments above 1-GeV energy may distinguish a fixed point from a limit cycle or other asymptotic behavior. (2) If electrodynamics or weak interactions become strong above some large momentum $\Lambda$, then the renormalization group can be used (in principle) to determine the renormalized coupling constants of strong interactions, except for $U\left(3\right)\mathrm{}\times U\left(3\right)\mathrm{}$ symmetry-breaking parameters. (3) Mass terms in the Lagrangian of strong, weak, and electromagnetic interactions must break a symmetry of the combined interactions with zero mass. (4) The $\Delta I=\frac{1}{2}$ rule in nonleptonic weak interactions can be understood assuming only that a renormalization group exists for strong interactions.