Gravi-Scalarballs for Dark Matter

    We study bound states of massless particles, in different theories. The semi-classical method, namely the interaction is described by the potential energy extracted from the scattering amplitude, is used. We reduce the particles system to a one-dimensional problem of a particle moving in an “effective potential” of the potential energy. By this way, we find the necessary and sufficient conditions for bound states. Numerically solving the equations of motion and plotting the trajectories of the particles, we find bound states are indeed allowed. Specially, we show that in string theory, local and nonlocal higher derivative theories, as well as general asymptotically-free or finite theories, gravitationally interacting bound states are allowed when the energy is larger than the Planck energy. While in higher derivative or nonlocal theories with interaction which is governed by a dimensionless or a dimensionful coupling constant, bound states are allowed for the energy less than the Planck energy. These bound states can form because the scattering amplitudes is soft in the ultraviolet region. Indeed, in such theories, the divergency of the potential energy at the singularity is removed while the force is zero or constant at this point. Besides, we show that gravi-scalarballs and stringballs likely form in the early universe, while electroballs and scalarballs possibly form after inflation. Since the energy of bound states which form in the early universe may ranges from the Planck energy to any arbitrarily large or small value, these bound states can be regarded as the candidates of dark matter and/or as the seeds of the structure formation at large scale in the universe. 

    As for how to detect these bound states, we propose to study the gravitational deflection. The deflection of a massless scalar particle by a gravi-scalarball is discussed. We calculate the deflection angle and find that it relates to the internal motions of the gravi-scalarball. In a special case, it is similar to one and a half of the deflection angle of light by the sun. Finally, we study a gravitationally induced quantum interference experiment in a spacetime with a potential energy related to gravi-scalarballs. The corresponding phase is called gravitational phase. We find a way to calculate such phase in a general spacetime, then apply it to this example. The interference of two beams along a parallelogram is studied. We calculate the fringe shift of a process when the parallelogram is flipped.

      在不同的理论框架中，我们研究了由无质量粒子构成的束缚态。这个工作采用了半经典的计算，即粒子体系的相互作用由一个取决于散射振幅的势能来描述。我们把粒子体系的运动约化成了一个一维的问题，然后通过势能重新定义了一个“有效势”，从而把复杂的粒子体系简化为一个粒子在有效势场中的运动。借助“有效势”的方法，我们发现了形成一个束缚态的充分和必要条件。通过数值求解粒子的相对论运动方程，我们得到了粒子的运动轨迹，结果表明束缚态确实可以存在。我们的分析显示：在弦理论、局域和非局域高阶导数理论、一般的渐进自由或者有限理论中，当粒子的能量大于普朗克能量时，由引力相互作用导致的束缚态可以存在；而对于其它种类的相互作用，在高阶导数或非局域理论中，如果耦合常数没有量纲或者具有正的质量量纲，则只有当能量小于普朗克能量时，才允许存在束缚态。这些束缚态可以存在的原因是，散射振幅在紫外区域具有柔和的性质，在这些理论中，相应的势能在原先的奇点处不再发散，相应的力在此处为零或者常数。另外，我们讨论了早期宇宙中这些束缚态的产生。我们发现，引力-标量球和弦球很可能出现在早期宇宙阶段，而电球和标量球则可能产生于暴涨之后。由于这些产生于早期宇宙的束缚态其能量具有从普朗克能量到各种能量的广泛范围，它们可以作为暗物质的候选者或者宇宙大尺度结构形成过程的种子。

      在观测方面，我们建议通过粒子的引力偏折效应来检验这些束缚态的存在。考虑一个无质量标量粒子被一个引力-标量球散射的过程，我们计算出了相应的偏折角。计算结果表明，粒子的偏折角和引力-标量球内部的运动有关。在特殊情况下，偏折角的表达式比较简单，类似于光被太阳引力偏折的角度乘上二分之三。最后，根据引力-标量球中的势能，我们研究了一个由引力导致的量子干涉实验，对应的相位我们称之为引力相位。我们首先找到了在一般时空中计算引力相位的方法，然后把它应用到这个例子中。具体地，在引力场中的两束粒子沿着一个平行四边形的两侧前进和汇合，在汇合处发生干涉，我们计算了这两条路径的相位差，并推导出了由翻转四边形导致的干涉条纹移动的表达式。

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