Abstract

We provide estimates of both the cross section and rate coefficient for the radiative attachment of a second positron to create the H̅⁺ ion, H̅(1s)+e⁺→H̅⁺(1s^{2 1}S^e)+ℏω, for which the polarization of the initial state H̅(1s) is taken into account. We show how to analytically integrate the resulting six-dimensional, three-body integrals for wave functions composed of explicitly correlated exponentials, a result that may be extended to Hylleraas wave functions. We extend Bhatia’s polarization results for the equivalent matter problem down to the low temperatures required for the Gravitational Behaviour of Antihydrogen at Rest (GBAR) experiment at CERN. The two-electron polarization cross-term is of intrinsic interest because it has every appearance of being singular at the origin, but non-singular when integrated numerically. We show that conventional approaches lead to a final integral with two singular terms that may be made to cancel in lowest order. However, higher-order terms in such approaches defy analytical integration. We use an integro-differential transform based on Gaussian transforms to bypass this blockage to yield a fully analytic result. Even in this method, one avoids the singular form only by integrating out the radial integrals before solving the second Gaussian integral.

Graphical abstract

中文翻译：

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产生氢离子时发生极化

摘要

我们提供第二个正电子的辐射附着的横截面和速率系数的估计，以产生H ^{+ +}离子H ⁺（1 s）+ e ⁺ →H ^{+ +}（1 s ^{2 1} S ^e）+ ℏω，为此初始状态H̅（1 s）。我们展示了如何分析地合成波函数的六维三体积分，这些波函数由显式相关的指数组成，该结果可能扩展到Hylleraas波函数。我们将等效问题的巴蒂亚极化结果扩展到欧洲核子研究组织（CERN）静止氢反重力实验（GBAR）所需的低温。两电子极化的交叉项具有内在的意义，因为它的所有出现在原点都是奇异的，但在数值积分时却不是奇异的。我们表明，常规方法导致具有两个奇异项的最终积分，可以使它们按最低顺序抵消。但是，这种方法中的高阶术语不利于分析集成。我们使用基于高斯变换的积分微分变换来绕过此障碍，以产生完整的分析结果。即使在这种方法中，也只能通过在求解第二高斯积分之前对径向积分进行积分来避免奇异形式。

图形概要