Color confinement

We argue that the effective charge ḡ(λ) in a color SU(N) gauge theory becomes infinite in the infrared limit λ→0 in the Landau gauge, provided the number F of massless quark flavors satisfies 13N/4<F<11N/2. To deduce this, we must assume that (i) the R transformation ${{{→}}{A}}_{{μ}}(x){→}{{{→}}{A}}_{{μ}}(x)+{{{→}}{r}}_{{μ}}$ ${{{→}}{A}}_{{μ}}=gaugefields$, ${{{→}}{r}}_{{μ}}=constantvectors$, commutes with the gauge-invariant cutoff removal in the renormalized field equations and current definitions, (ii) functional methods can be used to derive the consequences of invariance under the $R$ transformation, and (iii) if $ḡ({λ}){→}{g}_{{∞}} < {∞}$ as ${λ}{→}0$, the vertex functions are well defined and nonvanishing at the value ${g}_{{∞}}$ of the renormalized coupling constant. With these assumptions, generalizations of one-particle irreducible vertex functions involving both gauge fields and color currents are shown to have an infrared behavior which is inconsistent with the existence of the infrared fixed point at g_∞. Therefore, ḡ(0)=∞, suggesting that color is confined in these models. If we assume that the infrared decoupling theorem is valid for the exact theory, our result can be extended to both massive and massless quarks, for all F<11N/2.