The contents of renormalization group invariance and equations under tight constraints are explored and demonstrated with closed-form on-shell T matrices of pionless effective field theory for nuclear forces right within the effective field theory philosophy. The ‘running’ couplings under such tight constraints are presented in 1S 0 and uncoupled P channels up to truncation order 𝒪(Q4). Some linear relations are exposed and in turn employed in the pursuit of nonperturbative ‘running’ solutions which serves as an alternative choice without resorting to special prescription and additional operations or treatments. The utility of such ‘running’ behaviors inherent in the closed-form T-matrices of pionless effective field theory is remarked and a number of important issues related to effective field theory constructed with various truncations are interpreted or discussed from the underlying theory perspective.

中文翻译：

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在无子有效场论的严格约束下运行

紧约束下重整化群不变性和方程的内容用壳上封闭形式进行了探索和论证吨有效场论哲学中核力的无粒子有效场论矩阵。在这种严格约束下的“运行”联轴器呈现在1小号 0和解耦磷截断顺序的通道𝒪(问4). 一些线性关系被暴露出来，反过来又被用于追求非微扰的“运行”解决方案，这是一种替代选择，无需求助于特殊的处方和额外的操作或治疗。封闭形式中固有的这种“运行”行为的效用吨评论了无子有效场论的矩阵，并从基本理论的角度解释或讨论了与用各种截断构造的有效场论相关的许多重要问题。