Dispersion relations and knot theory

Aninda Sinha

Published 2022-04-29Version 1

I show that the crossing symmetric dispersion relation (CSDR) for 2-2 scattering leads to a fascinating connection with knot theory. In particular, the dispersive kernel can be identified naturally in terms of the generating function for the Alexander polynomials corresponding to the torus knot $(2,2n+1)$ arising in knot theory. In the low energy expansion, the difference between the $(n+1)$-th and $n$-th derivatives of the scattering amplitude with respect to the crossing symmetric variable can be bounded in terms of the torus $(2,2n+1)$-knot invariants and the resulting bounding curve in the space of allowed S-matrices can be determined analytically in terms of the $(2,2n+1)$-torus Alexander polynomial. The agreement with the pion S-matrix bootstrap is impressive. The global bounds are also derived using Geometric Function Theory (GFT) techniques and shown to be identical.