Renormalization of loop functions for all loops

It is shown that the vacuum expectation values W(C₁,...,C_n) of products of the traces of the path-ordered phase factors P exp[igC_iA_μ(x)dx^μ] are multiplicatively renormalizable in all orders of perturbation theory. Here A_μ(x) are the vector gauge field matrices in the non-Abelian gauge theory with gauge group U(N) or SU(N), and C_i are loops (closed paths). When the loops are smooth (i.e., differentiable) and simple (i.e., non-self-intersecting), it has been shown that the generally divergent loop functions W become finite functions W~ when expressed in terms of the renormalized coupling constant and multiplied by the factors e^-KL(C_i), where K is linearly divergent and L(C_i) is the length of C_i. It is proved here that the loop functions remain multiplicatively renormalizable even if the curves have any finite number of cusps (points of nondifferentiability) or cross points (points of self-intersection). If C_γ is a loop which is smooth and simple except for a single cusp of angle γ, then W_R(C_γ)=Z(γ)W~(C_γ) is finite for a suitable renormalization factor Z(γ) which depends on γ but on no other characteristic of C_γ. This statement is made precise by introducing a regularization, or via a loop-integrand subtraction scheme specified by a normalization condition W_R(C¯_γ)=1 for an arbitrary but fixed loop C¯_γ. Next, if C_β is a loop which is smooth and simple except for a cross point of angles β, then W~(C_β) must be renormalized together with the loop functions of associated sets Sⁱ_β={Cⁱ₁,...,Cⁱ_pi} (i=2,...,I) of loops Cⁱ_q which coincide with certain parts of C_β≡C₁¹. Then W_R(Sⁱ_β)=Z^ij(β)W~(S^j_β) is finite for a suitable matrix Z^ij(β). Finally, for a loop with r cross points of angles β₁,...,β_r and s cusps of angles γ₁,...,γ_s, the corresponding renormalization matrices factorize locally as Z^i₁j₁(β₁)...Z^i_rj_r (β_r)Z(γ₁)...Z(γ_s).