We study the space of all kinematically allowed four photon and four graviton S-matrices, polynomial in scattering momenta. We demonstrate that this space is the permutation invariant sector of a module over the ring of polynomials of the Mandelstam invariants s , t and u . We construct these modules for every value of the spacetime dimension D , and so explicitly count and parameterize the most general four photon and four graviton S-matrix at any given derivative order. We also explicitly list the local Lagrangians that give rise to these S-matrices. We then conjecture that the Regge growth of S-matrices in all physically acceptable classical theories is bounded by s 2 at fixed t . A four parameter subset of the polynomial photon S-matrices constructed above satisfies this Regge criterion. For gravitons, on the other hand, no polynomial addition to the Einstein S-matrix obeys this bound for D ≤ 6. For D ≥ 7 there is a single six derivative polynomial Lagrangian consistent with our conjectured Regge growth bound. Our conjecture thus implies that the Einstein four graviton S-matrix does not admit any physically acceptable polynomial modifications for D ≤ 6. A preliminary analysis also suggests that every finite sum of pole exchange contributions to four graviton scattering also violates our conjectured Regge growth bound, at least when D ≤ 6, even when the exchanged particles have low spin.

中文翻译：

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局部四光子和四引力子 S 矩阵的分类和约束

我们研究了所有运动学上允许的四个光子和四个引力子 S 矩阵的空间，散射动量多项式。我们证明了这个空间是一个模块在 Mandelstam 不变量 s 、 t 和 u 的多项式环上的置换不变扇区。我们为时空维度 D 的每个值构建这些模块，因此在任何给定的导数阶数上明确地计算和参数化最通用的四光子和四引力子 S 矩阵。我们还明确列出了产生这些 S 矩阵的局部拉格朗日函数。然后我们推测，在所有物理上可接受的经典理论中，S 矩阵的 Regge 增长在固定 t 时以 s 2 为界。上面构建的多项式光子 S 矩阵的四参数子集满足这个 Regge 标准。另一方面，对于引力子，当 D ≤ 6 时，爱因斯坦 S 矩阵的多项式加法不服从这个界限。对于 D ≥ 7，有一个单一的 6 导数多项式拉格朗日与我们推测的 Regge 增长界限一致。因此，我们的猜想意味着爱因斯坦四引力子 S 矩阵不允许对 D ≤ 6 进行任何物理上可接受的多项式修改。初步分析还表明，对四引力子散射的极点交换贡献的每个有限和也违反了我们猜想的 Regge 增长界限，至少当 D ≤ 6 时，即使交换粒子具有低自旋。