Abstract
It is shown that the vacuum expectation values of products of the traces of the path-ordered phase factors are multiplicatively renormalizable in all orders of perturbation theory. Here are the vector gauge field matrices in the non-Abelian gauge theory with gauge group or , and 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 become finite functions when expressed in terms of the renormalized coupling constant and multiplied by the factors , where is linearly divergent and is the length of . 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 is a loop which is smooth and simple except for a single cusp of angle , then is finite for a suitable renormalization factor which depends on but on no other characteristic of . This statement is made precise by introducing a regularization, or via a loop-integrand subtraction scheme specified by a normalization condition for an arbitrary but fixed loop . Next, if is a loop which is smooth and simple except for a cross point of angles , then must be renormalized together with the loop functions of associated sets () of loops which coincide with certain parts of . Then is finite for a suitable matrix . Finally, for a loop with cross points of angles and cusps of angles , the corresponding renormalization matrices factorize locally as .
- Received 30 March 1981
DOI:https://doi.org/10.1103/PhysRevD.24.879
©1981 American Physical Society