可分离势的HAL QCD方法中的导数展开
Derivative expansion in the HAL QCD method for a separable potential
摘要：
我们研究了HAL QCD方法中的导数展开如何使用量子力学中的可分势来提取物理观测量，该势在坐标系中是可解的但高度非局域的。我们考虑了三种情况的输入来确定HAL QCD势的导数展开：（1）能量本征函数;（2）含时波函数作为含时薛定谔方程在某些边界条件下的解;（3）含时波函数由有限个低能本征函数的线性组合来模拟有限体积效应。我们已经发现，对于所有三种情况下，潜在的提供合理的散射相移，即使在领导阶的导数展开，他们给更准确的结果作为扩展的顺序增加。通过将上述结果与可分势的形式导数展开的结果进行比较，我们得出结论：导数展开不是一种求势的方法，而是一种提取物理观测量如相移和结合能的方法，由HAL QCD方法的导数展开得到的散射相移比由形式导数展开得到的散射相移更快地收敛到精确相移可分离的势能。
英文摘要：
We investigate how the derivative expansion in the HAL QCD method works to extract physical observables, using a separable potential in quantum mechanics, which is solvable but highly non-local in the coordinate system. We consider three cases for inputs to determine the HAL QCD potential in the derivative expansion, (1) energy eigenfunctions (2) time-dependent wave functions as solutions to the time dependent Schrödinger equation with some boundary conditions (3) time-dependent wave function made by a linear combination of finite number of eigenfunctions at low energy to mimic the finite volume effect. We have found that, for all three cases, the potentials provide reasonable scattering phase shifts even at the leading order of the derivative expansion, and they give more accurate results as the order of the expansion increases. By comparing the above results with those from the formal derivative expansion for the separable potential, we conclude that the derivative expansion is not a way to obtain the potential but a method to extract physical observables such as phase shifts and binding energies, and that the scattering phase shifts from the derivative expansion in the HAL QCD method converge to the exact ones much faster than those from the formal derivative expansion of the separable potential.
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