特殊时空缺陷上散射的奇异透镜效应
Singular lensing from the scattering on special space-time defects
摘要：
众所周知，某些特殊类别的自引力点状缺陷，如全球（非规范）单极子，会产生非渐近平坦的时空，其特征是立体角赤字，其大小取决于底层微观模型的细节。电中性粒子在这样的时空上的散射是由当散射角和亏损角重合时表现出共振行为的振幅来描述的。这又导致环状结构，其中截面形式上是发散的（“奇异透镜效应”）。在这项工作中，我们重新审视这一特殊现象，有两个目的，一是将其置于当代和更普遍的背景下，鉴于对这种缺陷的理论和一般现象学的新兴趣，更重要的是，解决导致上述效果的特定计算中出现的某些微妙之处。特别是，通过采用一个特定的正规化过程中遇到的形式无限勒让德系列，我们设法确保恢复的闵可夫斯基时空，从而消失的透镜现象，在无缺陷的限制，和有效性的光学定理的弹性总截面。此外，该现象的奇异性还通过另一种计算得到证实，与原来的方法不同，该计算没有使用勒让德多项式的母函数，而是利用了菲涅耳积分的渐近性质。
英文摘要：
It is well known that certain special classes of self-gravitating point-like defects, such as global (non gauged) monopoles, give rise to non-asymptotically flat space-times characterized by solid angle deficits, whose size depends on the details of the underlying microscopic models. The scattering of electrically neutral particles on such space-times is described by amplitudes that exhibit resonant behaviour when the scattering and deficit angles coincide. This, in turn, leads to ring-like structures where the cross sections are formally divergent ("singular lensing"). In this work, we revisit this particular phenomenon, with the twofold purpose of placing it in a contemporary and more general context, in view of renewed interest in the theory and general phenomenology of such defects, and, more importantly, of addressing certain subtleties that appear in the particular computation that leads to the aforementioned effect. In particular, by adopting a specific regularization procedure for the formally infinite Legendre series encountered, we manage to ensure the recovery of the Minkowski space-time, and thus the disappearance of the lensing phenomenon, in the no-defect limit, and the validity of the optical theorem for the elastic total cross section. In addition, the singular nature of the phenomenon is confirmed by means of an alternative calculation, which, unlike the original approach, makes no use of the generating function of the Legendre polynomials, but rather exploits the asymptotic properties of the Fresnel integrals.
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