The process of renormalization in nonperturbative Hamiltonian effective field theory (HEFT) is examined in the $\mathrm{\Delta }$-resonance scattering channel. As an extension of effective field theory incorporating the Lüscher formalism, HEFT provides a bridge between the infinite-volume scattering data of experiment and the finite-volume spectrum of energy eigenstates in lattice QCD. HEFT also provides phenomenological insight into the basis-state composition of the finite-volume eigenstates via the state eigenvectors. The Hamiltonian matrix is made finite through the introduction of finite-range regularization. The extent to which the established features of this regularization scheme survive in HEFT is examined. In a single-channel $\pi N$ analysis, fits to experimental phase shifts withstand large variations in the regularization parameter $\mathrm{\Lambda }$, providing an opportunity to explore the sensitivity of the finite-volume spectrum and state composition on the regulator. While the Lüscher formalism ensures the eigenvalues are insensitive to $\mathrm{\Lambda }$ variation in the single-channel case, the eigenstate composition varies with $\mathrm{\Lambda }$; the admission of short-distance interactions diminishes single-particle contributions to the states. In the two-channel $\pi N$, $\pi \mathrm{\Delta }$ analysis, $\mathrm{\Lambda }$ is restricted to a small range by the experimental data. Here the inelasticity is particularly sensitive to variations in $\mathrm{\Lambda }$ and its associated parameter set. This sensitivity is also manifest in the finite-volume spectrum for states near the opening of the $\pi \mathrm{\Delta }$ scattering channel. Future high-quality lattice QCD results will be able to discriminate $\mathrm{\Lambda }$, describe the inelasticity, and constrain a description of the basis-state composition of the energy eigenstates. Finally, HEFT has the unique ability to describe the quark-mass dependence of the finite-volume eigenstates. The robust nature of this capability is presented and used to confront current state-of-the-art lattice QCD calculations.

中文翻译：

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有效场论非微扰扩展中的正则化

非微扰哈密顿有效场论 (HEFT) 中的重整化过程在$\mathrm{\Delta }$-共振散射通道。作为结合 Lüscher 形式主义的有效场论的扩展，HEFT 在实验的无限体积散射数据和晶格 QCD 中能量本征态的有限体积谱之间架起了一座桥梁。HEFT 还通过状态特征向量提供了对有限体积本征态的基态组成的现象学洞察。通过引入有限范围正则化使哈密顿矩阵变得有限。检验了该正则化方案的既定特征在 HEFT 中存在的程度。在单通道$\pi ñ$分析，适合实验相移承受正则化参数的大变化$\mathrm{\Lambda }$，提供了一个机会来探索有限体积谱的灵敏度和调节器上的状态组成。而 Lüscher 形式主义确保特征值对$\mathrm{\Lambda }$在单通道情况下，本征态成分随$\mathrm{\Lambda }$; 短距离相互作用的承认减少了单粒子对状态的贡献。在双通道$\pi ñ$,$\pi \mathrm{\Delta }$分析，$\mathrm{\Lambda }$被实验数据限制在一个很小的范围内。这里的非弹性对变化特别敏感$\mathrm{\Lambda }$及其相关的参数集。这种敏感性也体现在靠近开口处的有限体积谱中。$\pi \mathrm{\Delta }$散射通道。未来高质量的晶格QCD结果将能够判别$\mathrm{\Lambda }$，描述非弹性，并约束对能量本征态的基态组成的描述。最后，HEFT 具有描述有限体积本征态的夸克质量依赖性的独特能力。这种能力的稳健性被提出并用于应对当前最先进的晶格 QCD 计算。