标题： 用于最大同位旋的$Nππ$的有限体积形式 显示英文标题

标题： Finite-volume formalism for $Nππ$ at maximal isospin

摘要： 我们将相对论场论的有限体积形式主义扩展到最大同位旋的$N \pi \pi$散射态，$I=5/2$。 如同之前使用相对论场论方法的工作一样，我们考虑一个通用低能有效理论的所有阶次，并确定将有限体积能量与中间 K 矩阵相关联的量化条件，以及将后者与物理散射振幅联系起来的积分方程。 我们讨论了 K 矩阵的参数化，并详细解释了由于核子自旋与非简并粒子的使用相结合而出现的新特性。 作为一个具体例子，我们提供了一个数值应用示例，包括$\Delta$模式在$N\pi$子通道中的共振。 扩展到$I=3/2$和$1/2$通道更为复杂，这是由于与$N\pi$状态的混合，我们不为这些情况提供完整的形式主义。 我们解释为什么不能将$N\pi$状态包括在内，这是通过将核子视为$p$波$N\pi$散射中的极点来处理的，这种方法在使用三粒子$DD\pi$公式研究$D D^*$散射时已取得成功。 我们还提供了在假设没有两到三混合情况下的所有同位旋结果，从而为后续论文中全面处理所有$N\pi\pi \leftrightarrow N\pi$系统奠定了基础。 最后，我们研究了来自$N\pi\pi\pi$中间态的$N\pi\pi$幅度中的奇点，并发现我们的亚阈值截断函数必须进行修改以避免这些奇点。 显示英文摘要

摘要： We extend the relativistic field theoretic finite-volume formalism to $N \pi \pi$ scattering states at maximal isospin, $I=5/2$. As in previous work using the relativistic field theory approach, we work to all orders in a generic low-energy effective theory, and determine the quantization condition that relates finite-volume energies to intermediate K matrices, and the integral equations connecting the latter to the physical scattering amplitudes. We discuss the parametrization of the K matrices, and explain in detail the new features that arise in implementing the quantization condition due to the spin of the nucleon in combination with the use of non-degenerate particles. As a concrete example, we provide a sample numerical application including the $\Delta$ resonance in the $N\pi$ subchannel. The extension to the $I=3/2$ and $1/2$ channels is more involved, due to mixing with $N\pi$ states, and we do not provide a complete formalism for these cases. We explain why $N\pi$ states cannot be included by treating the nucleon as a pole in $p$-wave $N\pi$ scattering, an approach that has been successful in studying $D D^*$ scattering using the three-particle $DD\pi$ formalism. We additionally provide results for all isospins under the assumption of no two-to-three mixing, thereby laying the groundwork for a follow-up paper in which all $N\pi\pi \leftrightarrow N\pi$ systems are fully treated. Finally, we study the singularities in $N\pi\pi$ amplitudes arising from $N\pi\pi\pi$ intermediate states, and find that our subthreshold cutoff functions must be modified to avoid such singularities.