二体有限体积矩阵元的一致性检验：II。扰动系统
Consistency checks for two-body finite-volume matrix elements: II. Perturbative systems
摘要：
使用在参考文献中提出的一般形式主义。[1，2]中，我们研究了耦合到具有微扰相互作用的全同标量两粒子态的外电流的$\mathbf{2}+\mathcal{J}\to\mathbf{2}$矩阵元的有限体积效应.在具有周期性$L$的有限立方体积中，我们通过$\mathcal O(1/L^5)$推导出了矩阵元的$1/L$展开式，发现它受两个普适的电流依赖性参数，标量电荷和阈值两粒子形状因子的控制.我们确认的结果，通过数值研究的一般形式主义，并通过一个独立的微扰计算。我们进一步证明了与Feynman-Hellmann定理的一致性，该定理可用于联系基态能量和矩阵元的$1/L$展开式。后者给出了一个简单的洞察，为什么领先的体积校正矩阵元素具有相同的缩放的能量，$1/L^3$，在矛盾的早期工作，发现一个$1/L^2$的矩阵元素的贡献。我们在这里表明，这样一个长期出现在中间阶段的微扰计算，但取消在最后的结果。
英文摘要：
Using the general formalism presented in Refs. [1,2], we study the finite-volume effects for the $\mathbf{2}+\mathcal{J}\to\mathbf{2}$ matrix element of an external current coupled to a two-particle state of identical scalars with perturbative interactions. Working in a finite cubic volume with periodicity $L$, we derive a $1/L$ expansion of the matrix element through $\mathcal O(1/L^5)$ and find that it is governed by two universal current-dependent parameters, the scalar charge and the threshold two-particle form factor. We confirm the result through a numerical study of the general formalism and additionally through an independent perturbative calculation. We further demonstrate a consistency with the Feynman-Hellmann theorem, which can be used to relate the $1/L$ expansions of the ground-state energy and matrix element. The latter gives a simple insight into why the leading volume corrections to the matrix element have the same scaling as those in the energy, $1/L^3$, in contradiction to earlier work, which found a $1/L^2$ contribution to the matrix element. We show here that such a term arises at intermediate stages in the perturbative calculation, but cancels in the final result.
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