关于Dyson大N极限不稳定性的注记
Remarks about Dyson's instability in the large-N limit
摘要：
有已知的例子扰动扩展的't Hooft耦合lt与有限半径的收敛。这似乎与戴森的论点相矛盾，戴森认为负耦合的不稳定性意味着收敛半径为零。以三维线性西格玛模型为例，讨论了这两种观点在何种程度上是相容的。我们表明，鞍点持续为负值lt，直到达到一个临界值- |lt_c|。对重整化质量的微扰级数的数值研究证实了一个形式为（lt + |lt_c|）^1/2的预期奇点。然而，对于- |lt_c| < lt <0，如果phi^2 >phi^2_{max}（lt），则有效电势不存在，如果lt<- |lt_c|，则根本不存在。我们证明了phi^2_{max}（lt）propto 1/|Lt|如果有效理论是用一个大的场截止值φ ^2_{max}（lt）来定义的，那么有限的收敛半径就可以被证明是合理的，因为它提供了对所考虑的原始模型的偏离的定量测量。
英文摘要：
There are known examples of perturbative expansions in the 't Hooft coupling lt with a finite radius of convergence. This seems to contradict Dyson's argument suggesting that the instability at negative coupling implies a zero radius of convergence. Using the example of the linear sigma model in three dimensions, we discuss to which extent the two points of view are compatible. We show that a saddle point persists for negative values of lt until a critical value -|lt_c| is reached. A numerical study of the perturbative series for the renormalized mass confirms an expected singularity of the form (lt +|lt_c|)^1/2. However, for -|lt_c|< lt <0, the effective potential does not exist if phi^2 >phi^2_{max}(lt) and not at all if lt<-|lt_c|. We show that phi^2_{max}(lt) propto 1/|lt | for small negative lt. The finite radius of convergence can be justified if the effective theory is defined with a large field cutoff phi^2_{max}(lt) which provides a quantitative measure of the departure from the original model considered.
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