arXiv 1908.023242019-08quant-phhep-ph

$D=3$和$D=3-2ε$维度中的库仑期望值

Coulomb expectation values in $D=3$ and $D=3-2ε$ dimensions

Gregory S. Adkins, Md Faisal Alam, Conor Larison, Ruosi Sun

备注：: 41 pages, 2 figures; v2 includes a sample calculation
发表信息：: Phys. Rev. A 101, 042511 (2020)
DOI：: 10.1103/PhysRevA.101.042511

摘要：

我们探讨了量子库仑问题的两体束缚态，在$D=3$和$D=3-2ε$尺寸，详细，并给出了广泛的列表中出现的期望值的评估QED修正束缚态能量。我们描述了用于获得这些期望值的技术，并给出了涉及相关拉盖尔多项式的积分的评价的一般公式。此外，我们给出了公式的积分，涉及减相关的拉盖尔多项式-那些与低功率的变量减去关闭-出现时，评估不同的期望值。我们目前的微扰结果（在参数$ε$），显示如何在$D=3-2ε$维度的束缚态能量和波函数不同，从他们的$D=3$维度的同行，并使用这些公式来寻找正则表达式的发散期望值，如$\big \langle \bar V^3 \big \rangle$和$\big \langle (\bar V')^2 \big \rangle$，其中$\bar V$是$D$维库仑势。我们评估了一些有限的$D$ -维期望值，如$\big \langle r^{-2+4ε} \partial_r^2 \big \rangle$和$\big \langle r^{4ε} p^4 \big \rangle$，它们的$ε\rightarrow 0$限值不同于它们的三维对应物$\big \langle r^{-2} \partial_r^2 \big \rangle$和$\big \langle p^4 \big \rangle$。我们探讨使用递归关系，费曼-赫尔曼定理，和动量空间括号结合$D$维傅立叶变换的$D$维期望值的评价。本文的结果对于用量纲正则化方法计算库仑束缚系统的性质是有用的。

英文摘要：

We explore the quantum Coulomb problem for two-body bound states, in $D=3$ and $D=3-2ε$ dimensions, in detail, and give an extensive list of expectation values that arise in the evaluation of QED corrections to bound state energies. We describe the techniques used to obtain these expectation values and give general formulas for the evaluation of integrals involving associated Laguerre polynomials. In addition, we give formulas for the evaluation of integrals involving subtracted associated Laguerre polynomials--those with low powers of the variable subtracted off--that arise when evaluating divergent expectation values. We present perturbative results (in the parameter $ε$) that show how bound state energies and wave functions in $D=3-2ε$ dimensions differ from their $D=3$ dimensional counterparts and use these formulas to find regularized expressions for divergent expectation values such as $\big \langle \bar V^3 \big \rangle$ and $\big \langle (\bar V')^2 \big \rangle$ where $\bar V$ is the $D$-dimensional Coulomb potential. We evaluate a number of finite $D$-dimensional expectation values such as $\big \langle r^{-2+4ε} \partial_r^2 \big \rangle$ and $\big \langle r^{4ε} p^4 \big \rangle$ that have $ε\rightarrow 0$ limits that differ from their three-dimensional counterparts $\big \langle r^{-2} \partial_r^2 \big \rangle$ and $\big \langle p^4 \big \rangle$. We explore the use of recursion relations, the Feynman-Hellmann theorem, and momentum space brackets combined with $D$-dimensional Fourier transformation for the evaluation of $D$-dimensional expectation values. The results of this paper are useful when using dimensional regularization in the calculation of properties of Coulomb bound systems.

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