An important element of the $S$-matrix bootstrap program is the relationship between the modulus of an $S$-matrix element and its phase. Unitarity relates them by an integral equation. Even in the simplest case of elastic scattering, this integral equation cannot be solved analytically and numerical approaches are required. We apply modern machine learning techniques to studying the unitarity constraint. We find that for a given modulus, when a phase exists it can generally be reconstructed to good accuracy with machine learning. Moreover, the loss of the reconstruction algorithm provides a good proxy for whether a given modulus can be consistent with unitarity at all. In addition, we study the question of whether multiple phases can be consistent with a single modulus, finding novel phase-ambiguous solutions. In particular, we find a new phase-ambiguous solution which pushes the known limit on such solutions significantly beyond the previous bound.

翻译：$S$矩阵自举程序的一个重要组成部分是$S$矩阵元的模与其相位之间的关系。幺正性通过一个积分方程将两者联系起来。即使在弹性散射的最简单情况下，该积分方程也无法解析求解，因此需要数值方法。我们应用现代机器学习技术来研究幺正性约束。研究发现，对于给定的模，若存在相位，通常可以用机器学习以较高精度重构该相位。此外，重构算法的损失函数为判断给定模是否可能满足幺正性提供了良好指标。同时，我们探究了多个相位是否可能与单个模一致的问题，并发现了新的相位歧义解。特别地，我们找到了一种新的相位歧义解，该解将此类型解的已知界限显著推进到超出此前范围。