[论文解读] Mass hierarchy and mass gap on thick branes with Poincare symmetry

Nandinii Barbosa-Cendejas, Alfredo Herrera–Aguilar+3 more|arXiv (Cornell University)|Dec 19, 2007

Black Holes and Theoretical Physics参考文献 24被引用 12

一句话总结

本文提出了一种具有厚膜配置的5维分支宇宙模型，通过规范场理论中的Kaluza-Klein（KK）谱的能隙自然地解决了层次问题，避免了轻量级KK模态。通过求解描述横向迹消失涨落的Schrödinger型方程，该研究在两种情形下导出了质量态的精确谱：一种具有能隙及两个束缚态（包括4维引力子），另一种为平滑的RS型模型，无能隙但具有由合流合流超几何方程控制的连续模态，两种模型均无奇点，并对牛顿定律产生修正。

ABSTRACT

We consider a scalar thick brane configuration arising in a 5D theory of gravity coupled to a self-interacting scalar field in a Riemannian manifold. We start from known classical solutions of the corresponding field equations and elaborate on the physics of the transverse traceless modes of linear fluctuations of the classical background, which obey a Schroedinger-like equation. We further consider two special cases in which this equation can be solved analytically for any massive mode with m^2>0, in contrast with numerical approaches, allowing us to study in closed form the massive spectrum of Kaluza-Klein (KK) excitations and to compute the corrections to Newton's law in the thin brane limit. In the first case we consider a solution with a mass gap in the spectrum of KK fluctuations with two bound states - the massless 4D graviton free of tachyonic instabilities and a massive KK excitation - as well as a tower of continuous massive KK modes which obey a Legendre equation. The mass gap is defined by the inverse of the brane thickness, allowing us to get rid of the potentially dangerous multiplicity of arbitrarily light KK modes. It is shown that due to this lucky circumstance, the solution of the mass hierarchy problem is much simpler and transparent than in the (thin) Randall-Sundrum (RS) two-brane configuration. In the second case we present a smooth version of the RS model with a single massless bound state, which accounts for the 4D graviton, and a sector of continuous fluctuation modes with no mass gap, which obey a confluent Heun equation in the Ince limit. (The latter seems to have physical applications for the first time within braneworld models). For this solution the mass hierarchy problem is solved as in the Lykken-Randall model and the model is completely free of naked singularities.

研究动机与目标

• 通过构建具有自然能隙的厚膜配置，解决分支宇宙模型中的层次问题。
• 消除薄膜模型（如Randall-Sundrum模型）中普遍存在的快子不稳定性及任意轻量级KK模态的多重性。
• 为质量态涨落的KK谱提供解析解，避免数值近似。
• 探索特殊函数——勒让德函数与合流合流超几何方程——在分支宇宙动力学中的物理实现。
• 确保模型在保持无裸奇点的同时，重现4维引力子并修正牛顿引力定律。

提出的方法

• 推导线性化涨落在经典标量厚膜背景下的横向迹消失模态的Schrödinger型方程。
• 采用5维引力-标量场系统在黎曼流形中的已知经典解作为背景几何结构。
• 对 m^2 > 0 的情形解析求解所得Schrödinger方程，实现KK谱的精确计算。
• 识别两种特殊情况：一种具有能隙且连续模态由勒让德型方程描述；另一种在Ince极限下由合流合流超几何方程描述。
• 通过薄膜极限计算两种模型中牛顿定律的修正。
• 通过构建具有Poincaré对称性的光滑、正则膜轮廓，确保无裸奇点。

实验结果

研究问题

• RQ1 具有KK谱能隙的厚膜模型是否能比薄膜Randall-Sundrum模型更清晰地解决层次问题？
• RQ2 与数值方法相比，Schrödinger型方程在质量态涨落中的解析解如何提升对KK谱的理解？
• RQ3 特殊函数——特别是勒让德函数与合流合流超几何方程——在描述分支宇宙模型中连续KK模态区段中起什么作用？
• RQ4 是否存在一种光滑、无奇点的分支宇宙模型，具有单一无质量4维引力子及连续质量态？其与Lykken-Randall模型相比有何异同？
• RQ5 对于这些可解析求解的厚膜配置，在薄膜极限下牛顿定律的修正量是多少？

主要发现

• 具有能隙的模型表现出两个束缚态：无质量4维引力子与一个质量态KK激发态，其余模态形成由勒让德方程控制的连续谱。
• 能隙由膜厚度的倒数定义，有效消除了任意轻量级KK模态，从而简化了层次问题的解决方案。
• 第二种模型提供了RS模型的平滑实现，具有单一无质量束缚态及由Ince极限下合流合流超几何方程描述的连续质量态。
• 解析解使得KK谱及薄膜极限下牛顿定律的修正量可精确计算，无需依赖数值积分。
• 两种模型均无裸奇点，确保了物理上的可行性与稳定性。
• 在涨落方程中使用特殊函数，首次在分支宇宙情景中实现了合流合流超几何方程的物理应用。