The renormalization of the string operator in QCD

It is shown that the string operator U(x1,x2,C) = P exp[ig∫sub(x1)sup(x2) dxsub(μ)Asup(μ) (x] with an open path (smooth and non-intersecting) can be renormalized path-independently in any order in the form Z2 Perx[ig z2/z1 √Zsub(3)sup(A) ∫sub(x1)sup(x2) dxsub(μ) Asub(R)sup(μ)(x)]. To demonstrate this, the renormalization is carried out up to order g4. Next it is argued that the renormalization preserves the algebraic identity U(x1,x2;C) U(x2,x3;C tilde) = U(x1,x3;C U C tilde) when the paths C and C tilde are connected smoothly 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 Z2 which multiplicavely renormalizes the string operator in the case of the open path, is cancelled in any order of g by the divergence appearing in the coincidence of the end points. That is, the Wilson loop operator can be renormalized by the coupling renormalization gsubR=Z2/Z1√Zsub(3)sup(A)g alone