重整化与奇点分解
Renormalization and resolution of singularities
摘要：
自从爱泼斯坦和格拉泽的开创性工作以来，人们已经很好地建立了位置空间中紫外发散的微扰重整化相当于将分布扩展到对角线上。对于一般的费曼图，相关的对角线形成线性子空间的非平凡排列。因此，人们可能会问，如果将这种排列分解为一个正常的交叉因子，重正化是否会变得更简单。本文研究了de Concini和Procesi的奇妙模型上的分布扩张问题，推广了位形空间的Fulton-MacPherson紧化。我们表明，一个典型的扩展到光滑模型符合通常的Epstein-Glaser重整化。为此，我们使用位置空间的解析正则化。与极点系数相关的t Hooft恒等式可以从分层中恢复，Zimmermann森林公式被编码在紧化的几何中。因此，沿着除数的每个不可约分量的一个减法足以得到使用局部反项的有限结果。作为推论，我们确定了位置空间中至多对数Feynman图的Hopf代数，并讨论了高发散度的情况。
英文摘要：
Since the seminal work of Epstein and Glaser it is well established that perturbative renormalization of ultraviolet divergences in position space amounts to extension of distributions onto diagonals. For a general Feynman graph the relevant diagonals form a nontrivial arrangement of linear subspaces. One may therefore ask if renormalization becomes simpler if one resolves this arrangement to a normal crossing divisor. In this paper we study the extension problem of distributions onto the wonderful models of de Concini and Procesi, which generalize the Fulton-MacPherson compactification of configuration spaces. We show that a canonical extension onto the smooth model coincides with the usual Epstein-Glaser renormalization. To this end we use an analytic regularization for position space. The 't Hooft identities relating the pole coefficients may be recovered from the stratification, and Zimmermann's forest formula is encoded in the geometry of the compactification. Consequently one subtraction along each irreducible component of the divisor suffices to get a finite result using local counterterms. As a corollary, we identify the Hopf algebra of at most logarithmic Feynman graphs in position space, and discuss the case of higher degree of divergence.
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