arXiv 1906.044272019-06hep-th

Sphaleron的费米子数1/2与谱镜对称性

Fermion Number 1/2 of Sphalerons and Spectral Mirror Symmetry

M. Mehraeen, S. S. Gousheh

备注：: 12 pages. no figures
DOI：: 10.1140/epjc/s10052-020-08451-4

摘要：

本文基于谱镜对称性对标准电弱理论中的$SU(2)_L$ S介子的重子数和轻子数$\frac{1}{2}$进行了重新推导。我们探讨了沿着一个不可收缩的环路场配置，通过sphaleron和其端点是真空的离散变换下的费米子哈密顿量的性质。众所周知，CP变换不是系统在环上任何地方的对称性，除了端点。通过用手征变换增广CP，我们观察到狄拉克哈密顿量在新的变换下恰好在sphaleron处是奇数，这确保了光谱的镜像对称性，包括连续谱。作为一致性检验，我们证明了在新的变换下，由Ringwald提出的费米零模在sphaleron背景下是不变的。我们在这里建立的谱镜对称性，以及零模的存在，是两个必要条件，由此可以用Jackiw和Rebbi提出的推理，或者等价地，用狄拉克海的谱亏$\frac{1}{2}$，来推断出sphaleron的费米子数$\frac{1}{2}$。这种分析的相关性，以其他解决方案进行了讨论。

英文摘要：

We present a rederivation of the baryon and lepton numbers $\frac{1}{2}$ of the $SU(2)_L$ S sphaleron of the standard electroweak theory based on spectral mirror symmetry. We explore the properties of a fermionic Hamiltonian under discrete transformations along a noncontractible loop of field configurations that passes through the sphaleron and whose endpoints are the vacuum. As is well known, CP transformation is not a symmetry of the system anywhere on the loop, except at the endpoints. By augmenting CP with a chirality transformation, we observe that the Dirac Hamiltonian is odd under the new transformation precisely at the sphaleron, and this ensures the mirror symmetry of the spectrum, including the continua. As a consistency check, we show that the fermionic zero mode presented by Ringwald in the sphaleron background is invariant under the new transformation. The spectral mirror symmetry which we establish here, together with the presence of the zero mode, are the two necessary conditions whence the fermion number $\frac{1}{2}$ of the sphaleron can be inferred using the reasoning presented by Jackiw and Rebbi or, equivalently, using the spectral deficiency $\frac{1}{2}$ of the Dirac sea. The relevance of this analysis to other solutions is also discussed.

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