The renormalization of the string operator in QCD

We shall observe that the renormalization of the string operator U(x₁, x₂; C) = Pexp{ig∫_x1^x2dx_μA^μ(x)} with an open path C (smooth and non-intersecting) is path-independently performed in any order of perturbation. To demonstrate this, the renormalization constants will be calculated up to order g⁴. Next the renormalization effect on the algebraic identity $\text{U(x}{}_1\text{, x}{}_2\text{;}\text{C}\text{)U(x}{}_2\text{, x}{}_3\text{;}\text{C}\text{) = U(x}{}_1\text{, x}{}_3\text{;}\text{C}\text{∪}\text{C}\text{)}$ will be discussed and it will be proved that the renormalization preserves the algebraic identity in any order of perturbation if the paths C and $\text{C}$ are smoothly connected at x₂. Finally, the string operator renormalization is extended to the case when the path C is smoothly closed (the Wilson loop operator). It is then shown that the renormalization factor which multiplicatively renormalizes the string operator in the case of the open path, is cancelled in any order of perturbation by the divergence appearing in the coincidence of the end points. As a results, the Wilson loop operator can be renormalized by the coupling constant renormalization alone.