径向对称非阿贝尔背景下重正化费米子有效作用量的计算
Calculation of renormalized fermion effective actions in radially symmetric non-Abelian backgrounds
摘要：
本文将我们最近提出的计算重整化泛函行列式的分波截断方法推广到计算某些类型的径向对称、非阿贝尔、背景规范场（包括类瞬子和类瞬子-反瞬子组态）中任意质量的四维费米子单圈有效作用量。本文对矩阵值径向微分算子的泛函行列式进行了详细的研究，阐明了我们对高阶分波贡献的解析处理和广义Gel'fand-Yaglom公式在判定低阶分波贡献中的应用。一般来说，低分波部分需要一些数值计算。然而，在无质量极限下，我们的分波径向微分算子的可因式分解性质可以用来半解析地计算甚至低分波部分，因此我们在一类非阿贝尔背景规范场中明确地计算了完整的费米子有效作用量。对于非零质量，我们还进行了必要的数值分析，关于低分波的贡献，产生大量的有效作用的数值精确的结果。将这些结果与大质量膨胀的结果进行比较，讨论了大质量膨胀的有效范围。还研究了有效的瞬子-反瞬子相互作用对费米子质量的依赖性。
英文摘要：
Our recent method to calculate renormalized functional determinants, the partial wave cutoff method, is extended for the evaluation of 4-D fermion one-loop effective action with arbitrary mass in certain types of radially symmetric, non-Abelian, background gauge fields (including instanton-like and instanton-antiinstanton-like configurations). A detailed study on functional determinants for matrix-valued radial differential operators is presented, explicating both our analytic treatment on the high partial wave contribution and the application of the generalized Gel'fand-Yaglom formula to determine the low partial wave contribution. In general, some numerical work is needed for the low partial wave part. In the massless limit, however, the factorizable nature of our partial-wave radial differential operators can be exploited to evaluate semi-analytically even the low partial wave part, and we thus have the full fermion effective action calculated explicitly in a class of non-Abelian background gauge fields. With nonzero mass, we also perform necessary numerical analysis as regards the low partial wave contribution to produce numerically exact results for the massive effective action. Comparing these against the results of the large mass expansion, the validity range of the large mass expansion is addressed. Also studied is the fermion mass dependence of the effective instanton-antiinstanton interaction.
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