Relation between photoionisation cross sections and attosecond time delays

Determination and interpretation of Wigner-like photoionisation delays is one of the most active fields of attosecond science. Previous results have suggested that large photoionisation delays are associated with structured continua, but a quantitative relation between photoionisation cross sections and time delays has been missing. Here, we derive a Kramers-Kronig-like relation between these quantities and demonstrate its validity for (anti)resonances. This new concept defines a topological analysis, which rationalises the sign of photoionisation delays and thereby sheds new light on a long-standing controversy regarding the sign of the photoionisation delay near the Ar 3s Cooper minimum. Our work bridges traditional photoionisation spectroscopy with attosecond chronoscopy and offers new methods for analysing and interpreting photoionisation delays.