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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.

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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)