Three-loop onset of the wormhole length in DSSYK

Abstract

The length of the Einstein-Rosen bridge in sine-dilaton gravity at disk level equals the Krylov spread complexity of the dual double-scaled SYK (DSSYK) model, with the double-scaling parameter $λ$ controlling the semiclassical expansion. We compute the onset value, $L_{0}$, in the thermofield double state at $t=0$ and arbitrary inverse temperature $β$, through three loops and in closed form, extending the known one-loop result. The quantity $L_{0}$ represents the preparation complexity of the initial thermal Hartle-Hawking state. In DSSYK, $L_{0}$ is $λ$ times the average chord number of the thermal state, providing a microscopic, unambiguous determination of an additive constant otherwise fixed only by a choice of holographic renormalization scheme. The two-loop contribution follows from a saddle point evaluation of the DSSYK two-point function at coincident insertion points, where individually divergent contributions cancel only in their sum. At three loops, we bypass the saddle point analysis using an exact recursion relation that generates long high-temperature expansions at low cost, fixing and testing a proposed Ansatz against many independent data points. Finally, in the low-temperature regime each loop order contributes one further power of $β$, reorganizing the semiclassical series into an expansion in the Schwarzian coupling $λβ$, whose leading coefficient we check against an independent one-loop Schwarzian computation. The same methods, applied to the length variance and third-order cumulant at $t=0$, give their semiclassical expansion through two loops.