华东师范大学学报（自然科学版） >

2023 , Vol. 2023 >Issue 4: 137 - 150

DOI: https://doi.org/10.3969/j.issn.1000-5641.2023.04.015

物理学与电子学

Faddeev方程与三玻色子系统低能短程有效场论研究

  • 王凯 ,
  • 杨继锋
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  • 华东师范大学 物理与电子科学学院, 上海　200241

收稿日期: 2022-05-18

  网络出版日期: 2023-07-25

基金资助

国家自然科学基金 (11435005)

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Faddeev equation for three-boson system in low-energy short-distance effective field theory

  • Kai WANG ,
  • Jifeng YANG
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  • School of Physics and Electronic Science, East China Normal University, Shanghai　200241, China

Received date: 2022-05-18

  Online published: 2023-07-25

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摘要

在低能短程 (接触型势能) 有效场论中二体 $\rm{t }$ 矩阵闭合形式解的基础上, 用Faddeev方程计算了二体接触型势能前提下的零自旋三玻色子系统三体T矩阵的闭合形式近似解. 在动量表示里接触型势能是多项式, 其二体t矩阵满足的Lippmann-Schwinger方程可用因子化方法约化为闭合的代数方程, 从而可得到该方程的解析解, 并方便地对其进行非微扰重整化. 但是在Faddeev方程中难以直接沿用上述因子化方法. 为此, 对三体T矩阵元的外动量依赖采用“分流”处理, 从而仍可利用因子化方法将Faddeev方程转化为代数方程. 在此基础上求得了领头阶二体势能下的三体T矩阵的闭合形式近似解, 并推广到次领头阶势能下的情形, 进而完成了自洽性验证. 与二体问题一样, 由于因子化和一般参数化, 这样的三体T矩阵非微扰解析解同样可以允许方便地进行非微扰重整化.

关键词： 有效场论; Faddeev方程; 非微扰近似; 一般参数化

本文引用格式

王凯 , 杨继锋 . Faddeev方程与三玻色子系统低能短程有效场论研究[J]. 华东师范大学学报（自然科学版）, 2023 , 2023(4) : 137 -150 . DOI: 10.3969/j.issn.1000-5641.2023.04.015

Abstract

Based on the closed-form t matrix of a two-body system in low-energy short-distance effective field theory, the approximate closed-form three-body T matrix for a zero-spin three-boson system is obtained using the Faddeev equation under two-body contact interactions. In momentum representation, the contact potentials are polynomials, and the Lippmann-Schwinger equation can be simplified to algebraic equations using a factorization trick, facilitating nonperturbative renormalization. However, it is impossible to apply such a factorization trick directly to the Faddeev equation. Therefore, the momenta dependence of the T matrix is “split” such that the factorization trick can still be applied. The closed-form T matrices are then obtained as nonperturbative approximate solutions of the Faddeev equation under the leading and next-to-leading order contact potentials with verified consistency. As in a two-body case, such a closed-form T matrix also facilitates the convenient implementation of the nonperturbative renormalization.

Key words： effective field theory; Faddeev equations; nonperturbative approximation; general parametrization

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