Symmetry in neutrino oscillation serves for a better understanding of the physical properties of the phenomenon. We present a systematic way of finding symmetry in neutrino oscillation, which we call symmetry finder (SF). By extending the known framework in vacuum into a matter environment, we derive the SF equation, a powerful machinery for identifying symmetry in the system. After learning lessons on symmetry in the Zaglauer-Schwarzer system with matter equivalent to the vacuum symmetry, we apply the SF method to the [P. B. Denton et al., Compact perturbative expressions for neutrino oscillations in matter, J. High Energy Phys. 06 (2016) 051.] (DMP) perturbation theory to first order. We show that the method is so powerful that we uncover the eight reparametrization symmetries with the $1\leftrightarrow 2$ state exchange in DMP, denoted as IA, IB, $\dots$, IVB, all new except for IA. The transformations consist of both fundamental and dynamical variables, indicating their equal importance. It is also shown that all the symmetries discussed in this paper can be understood as the Hamiltonian symmetries, which ensures their all-order validity and applicability to varying density matter.

中文翻译：

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Symmetry finder：一种在中微子振荡中寻找对称性的方法

中微子振荡的对称性有助于更好地理解该现象的物理特性。我们提出了一种在中微子振荡中寻找对称性的系统方法，我们将其称为对称性查找器(SF)。通过将真空中的已知框架扩展到物质环境中，我们推导出 SF 方程，这是一种用于识别系统对称性的强大机制。在学习了 Zaglauer-Schwarzer 系统中具有等效于真空对称性的物质的对称性课程后，我们将 SF 方法应用于 [P. B. 丹顿等人。，物质中中微子振荡的紧凑微扰表达式，J. High Energy Phys。06 ( 2016 年) 051.] (DMP) 一阶微扰理论。我们表明该方法是如此强大，以至于我们发现了八个重新参数化对称性$1\leftrightarrow 2$ DMP 中的状态交换，表示为 IA、IB、 $\dots$，IVB，除IA外都是新的。转换由基本变量和动态变量组成，表明它们同等重要。还表明，本文讨论的所有对称性都可以理解为哈密顿对称性，这确保了它们对变密度物质的全阶有效性和适用性。