Undetermined parameter in matching sub‑subleading trajectories at two loops

Determine the constant b controlling the one‑loop residue structure in the small‑\xi expansion of sub‑subleading Regge trajectories by computing higher‑order data (e.g., additional perturbative terms) sufficient to fix it independently.

Background

When matching the two‑loop behavior of sub‑subleading Regge trajectories inferred from cusp data to the Regge‑limit amplitude, an undetermined parameter b appears in the small‑\xi expansion of residues.

The authors show consistency with known results if b takes a particular value, but independent determination requires additional data, motivating further perturbative calculations.

References

We don't have enough data to compute b independently, but we can check that, by imposing \begin{equation} j_{2,1}{(\text{2-loop})}(0) \equiv 18 - 8 \sqrt{5} \quad \Rightarrow \quad b = -2-\frac{82}{\sqrt{5}\,, \end{equation} then automatically $j_{2,2}{(\text{2-loop})}(0) = 18 + 8 \sqrt{5}$ as expected from eq:Gamma_cusp2_two_loop.

— From Partons to Strings: Scattering on the Coulomb Branch of $\mathcal{N}=4$ SYM (2510.19909 - Alday et al., 22 Oct 2025) in Section 3.5 (Checks of the Regge/cusp correspondence)