UV Divergence and Tensor Reduction

Abstract: We present an efficient algorithm to decompose the ultraviolet (UV) divergences of Feynman integrals to local divergences and various types of sub-divergences. With some reasonable assumptions the local divergences of Feynman integrals can be uniquely defined in dimensional regularization scheme. By an asymptotic expansion in the hard momenta, the computation of local and sub-divergences is reduced to the computation of local divergences of massless vacuum integrals. In theories with spin $\le\frac{1}{2}$, the beta functions and anomalous dimensions can be extracted directly from the local divergence of integrals. We also propose two methods to reduce the tensor structures which can be used in the computation of local divergence. The first method is based on dimensional shift and is extremely powerful for integrals with loop number $L\le3$. The second method is based on a PV reduction in a $d_{\infty}$ dimension subspace, and it is more suited in four and more loops.