Renormalization of loop functions for all loops

It is shown that the vacuum expectation values $W(C_1,\cdots ,C_n)$ of products of the traces of the path-ordered phase factors $P\exp \left[ig\oint \int {C_i}^{}{\mathit{A}}_{\mu }\mathrm{}\left(x\right)d{x}^{\mu }\right]$ are multiplicatively renormalizable in all orders of perturbation theory. Here ${\mathit{A}}_{\mu }\mathrm{}\left(x\right)\mathrm{}$ are the vector gauge field matrices in the non-Abelian gauge theory with gauge group $\mathrm{U}\mathrm{}\left(N\right)\mathrm{}$ or $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$, 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 $\tilde{W}$ when expressed in terms of the renormalized coupling constant and multiplied by the factors $e^{-KL\left(C_i\right)}$, where $K$ is linearly divergent and $L\left(C_i\right)$ 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 $C_{\gamma }$ is a loop which is smooth and simple except for a single cusp of angle $\gamma$, then $W_R\left(C_{\gamma }\right)=Z\left(\gamma \right)\tilde{W}\left(C_{\gamma }\right)$ is finite for a suitable renormalization factor $Z\left(\gamma \right)$ which depends on $\gamma$ but on no other characteristic of $C_{\gamma }$. This statement is made precise by introducing a regularization, or via a loop-integrand subtraction scheme specified by a normalization condition $W_R\left({\overline{C}}_{\gamma }\right)=1\mathrm{}$ for an arbitrary but fixed loop ${\overline{C}}_{\gamma }$. Next, if $C_{\beta }$ is a loop which is smooth and simple except for a cross point of angles $\beta$, then $\tilde{W}\left(C_{\beta }\right)$ must be renormalized together with the loop functions of associated sets ${S^i}_{\beta }=\{{C^i}_1,\cdots ,{C^i}_{\mathrm{pi}}\}$ ($i=2,\cdots ,I$) of loops ${C^i}_q$ which coincide with certain parts of $C_{\beta }\equiv {C_1}^1$. Then $W_R\left({S^i}_{\beta }\right)=Z^{\mathrm{ij}}\left(\beta \right)\tilde{W}\left({S^j}_{\beta }\right)$ is finite for a suitable matrix $Z^{\mathrm{ij}}\left(\beta \right)$. Finally, for a loop with $r$ cross points of angles ${\beta }_1,\cdots ,{\beta }_r$ and $s$ cusps of angles ${\gamma }_1,\cdots ,{\gamma }_s$, the corresponding renormalization matrices factorize locally as $Z^{i_1{j}_1}\left({\beta }_1\right)\cdots Z^{i_r{j}_r}\left({\beta }_r\right)Z\left({\gamma }_1\right)\cdots Z\left({\gamma }_s\right)$.