AI 中文总结
本研究构建了带非局域形状因子的四费米子理论，推导了平均场能隙方程，通过红外/紫外匹配分析两种形状因子对能隙和临界耦合的不同影响，并给出有限温密推广形式。
AI 中文摘要
在本工作中，我们构建并分析了一种非局域四费米子理论，其中常规的狄拉克算符被一个整函数形式的非局域形状因子所形变。在引入辅助标量场后，我们推导了平均场有效作用量以及对应的动力学质量能隙方程。该分析的一个核心技术要点是：非局域形状因子是狄拉克算符的矩阵函数，因此逆传播子必须被视为由恒等算符和$\not{\text{\\!p}}$算符生成的闭代数中的元素，而非纯标量。我们针对两种代表性的形状因子选择——$f_{I}(\not{\text{\\!\partial}})=e^{-\not{\partial}/\Lambda}$和$f_{II}(\not{\text{\\!\partial}})=e^{-i\not{\partial}/\Lambda}$——得到了能隙核的显式表达式。沿用近期类狄拉克非局域旋量理论[1]中使用的红外/紫外匹配方法，我们将动量积分在中间标度$M\ll \Omega\ll \Lambda$处拆分，分别在红外和紫外区域进行解析展开，并与常规的局域NJL/Gross-Neveu结果进行对比。我们发现，双曲型形状因子会增强能隙积分并降低临界耦合，而振荡型形状因子则会抑制能隙积分并提高临界耦合。我们通过松原求和和修正的围道表示构建了有限温度和有限密度的推广形式，且在$\Lambda\to\infty$的极限下可恢复局域热能隙方程。
英文摘要
In this work, we formulate and analyze a nonlocal four-fermion theory in which the usual Dirac operator is deformed by an entire nonlocal form factor. After introducing an auxiliary scalar field, we derive the mean-field effective action and the corresponding gap equation for the dynamical mass. A central technical point of the analysis is that the nonlocal form factor is a matrix function of the Dirac operator, so the inverse propagator must be treated as an element of the closed algebra generated by the identity and $\not{\!p}$ operators, rather than as a purely scalar quantity. We obtain explicit expressions for the gap kernel for two representative choices of a form factor, $f_{I}(\not{\!\partial})=e^{-\not{\partial}/Λ}$ and $f_{II}(\not{\!\partial})=e^{-i\not{\partial}/Λ}$. Following the IR/UV matching method used in the recent Dirac-like nonlocal spinor theory [1], the momentum integral is split at an intermediate scale $M\ll Ω\ll Λ$, expanded analytically in the infrared and ultraviolet regions, and compared with the usual local NJL/Gross-Neveu result. We show that the hyperbolic form factor enhances the gap integral and lowers the critical coupling, whereas the oscillatory form factor suppresses it and raises the critical coupling. The finite-temperature and finite-density extension is formulated through Matsubara sums and a corrected contour representation, with the local thermal gap equation recovered in the limit $Λ\to\infty$.
Comments31 pages
