（超）域的特征标与群不变多项式：“拉格朗日”之路
Characters and Group Invariant Polynomials of (Super)fields: Road to "Lagrangian"
摘要：
由量子场表示的亚原子基本粒子的动力学及其相互作用由指定的变换性质唯一确定，即，与模型的基本对称性相关的量子数。这些领域构成了一个有限数量的群不变算子，这些算子被组装成一个多项式，称为拉格朗日量。多项式的阶数由质量维数决定。在本文中，我们已经介绍了一个Mathematica软件包，GrIP，计算完整的一组运营商，形成一个基础，在每一个这样的顺序为一个模型，包含任何数量的字段转换下连接紧群。时空对称性仅限于洛伦兹群。本文的第一部分是致力于制定的GrIP算法。在此背景下，详细和明确的结构的字符对应的不同表示的连通紧群和各自的Haar措施已讨论了其各自的极大环面的坐标。在第二部分中，我们记录了GrIP的用户手册，该手册捕获了准备输入文件的通用功能和指南。这个程序可以非常有效地找出与有效场论有关的高质量（非超对称）和正则（超对称）维算子。我们已经用两个例子证明了工作原理：SM和MSSM。我们进一步强调了GrIP的重要特征，例如，识别有效的运营商导致特定的罕见的过程与违反重子和轻子数，使用几个BSM方案。我们还列出了一套完整的6维运营商为每个这样的模型。一些运营商具有丰富的风味结构，详细讨论。这项工作为BSM-EFT铺平了道路。
英文摘要：
The dynamics of the subatomic fundamental particles, represented by quantum fields, and their interactions are determined uniquely by the assigned transformation properties, i.e., the quantum numbers associated with the underlying symmetry of the model. These fields constitute a finite number of group invariant operators which are assembled to build a polynomial, known as the Lagrangian. The order of the polynomial is determined by the mass dimension. In this paper, we have introduced a Mathematica package, GrIP, that computes the complete set of operators that form a basis at each such order for a model containing any number of fields transforming under connected compact groups. The spacetime symmetry is restricted to the Lorentz group. The first part of the paper is dedicated to formulating the algorithm of GrIP. In this context, the detailed and explicit construction of the characters of different representations corresponding to connected compact groups and respective Haar measures have been discussed in terms of the coordinates of their respective maximal torus. In the second part, we have documented the user manual of GrIP that captures the generic features and guides to prepare the input file. This program works very efficiently to find out the higher mass (non-supersymmetric) and canonical (supersymmetric) dimensional operators relevant to the Effective Field Theory. We have demonstrated the working principles with two examples:- the SM and the MSSM. We have further highlighted important features of GrIP, e.g., identification of effective operators leading to specific rare processes linked with the violation of baryon and lepton numbers, using several BSM scenarios. We have also tabulated a complete set of dimension-6 operators for each such model. Some of the operators possess rich flavour structures which are discussed in detail. This work paves the way towards BSM-EFT.
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