The next-to-next-to-leading order BFKL eigenvalue at odd conformal spin in planar N=4 super Yang-Mills: closed form, coefficient structure, and arithmetic

Technical Summary: The Next-to-Next-to-Leading Order BFKL Eigenvalue in Planar $\mathcal{N}=4$ SYM

Problem Statement
The paper addresses the calculation of the three-loop (next-to-next-to-leading order, NNLO) eigenvalue of the BFKL kernel in planar $\mathcal{N}=4$ super Yang–Mills theory. While the leading-order (LO) and next-to-leading-order (NLO) eigenvalues are known in closed form for arbitrary conformal spin $n$ and continuous variable $\nu$, the NNLO result had previously been accessible only via numerical evaluation of the Caron-Huot–Herranen integrand for specific spins or as closed forms restricted to specific lines in the $(n, \nu)$ plane (e.g., $n=0$, $n=1$, or $\nu=0$). The goal is to provide a closed-form expression for the eigenvalue $\chi_2(n, \nu)$ at arbitrary odd conformal spin $n$, expressed in terms of analytically continued nested harmonic sums and rational functions.

Methodology
The authors construct the solution through a rigorous, multi-stage procedure that avoids fitting and relies on exact rational arithmetic and Mellin transforms:

1. Master Generating Function: Rather than treating each spin $n$ in isolation, the authors resum the conformal spin dependence of the Caron-Huot–Herranen three-loop integrand into a single master generating function $F(x; \omega)$. This function is constructed from iterated integrals over a specific alphabet.
2. Mellin Extraction: The per-spin integrands are recovered as Taylor coefficients of the master function with respect to the spin-generating variable $\omega$. The authors extract the "atom tables" for each odd spin $n$ by performing a direct Mellin transform of the integrand $I_n(x)$. This yields a finite table of "atoms" (rational poles and binomial-alternating sums) with exact rational coefficients.
3. Reduction to Harmonic Sums: The atoms are reduced to nested harmonic sums using a set of one-variable identities. This involves:
   • Tower Reductions: Converting bare sums into polygamma functions and harmonic sums.
   • Kernel Closures: Reducing 16 specific "nested-Lerch" kernels (dressed sums) to closed forms involving nested harmonic sums, constants ($\zeta_k, \ln 2, \text{Li}_4(1/2)$), and rational functions. Eleven of these closures are proven analytically; five are confirmed numerically to high precision.
4. Coefficient Structure Analysis: The coefficients of the resulting harmonic sums are analyzed. They are found to be rational multiples of $1, \pi^2,$ and $\zeta^3$. The authors identify that approximately 85% of the coefficient slots close in an "elementary" form (rational functions of ladder distances), while the remaining 15% form a "residual" layer governed by five specific kernels.
5. Transcendental Collapse: The residual layer, which initially appears as a complex sum over a ladder of shifted arguments $z+k$, is shown to collapse onto two depth-one polygamma transcendentals: $\psi(z+1) + \gamma_E$ and $\Phi(-1, 2, z+1)$ (a Lerch transcendent related to the trigamma function).

Key Contributions and Results

• Closed Form for Odd Spins: The paper presents the eigenvalue $\chi_2(n, \nu)$ for any odd conformal spin $n$ in a closed form:
  $$\chi_2(n, \nu) = \frac{1}{4} [F_n(z) + F_n(\bar{z})]$$
  where $F_n(z)$ is a finite combination of analytically continued nested harmonic sums evaluated on a ladder of integer-shifted arguments $z+k$ (from $z$ down to the reflected point $-\bar{z}$), rational functions, and constants.
• Exact Atom Tables: For every odd $n \ge 3$, the authors provide a driver script that regenerates the exact atom table directly from the Caron-Huot–Herranen integrand. These tables are verified to be byte-identical to previously known results for low spins and are provided for spins up to $n=91$.
• Coefficient Classification:
  • Elementary Slots: 40 out of 47 coefficient slots close in elementary forms (rational combinations of harmonic sums in the ladder distances).
  • Exceptional Kernels: The remaining 7 slots reduce to 5 independent kernels. One of these satisfies a verified holonomic recurrence (P-recursive) on holdout data.
• Transcendental Economy: The complex ladder sum of the residual kernels collapses exactly onto two depth-one transcendentals, $\psi(z+1)$ and $\Phi(-1, 2, z+1)$, plus a rational remainder. This reduces the functional complexity significantly.
• Arithmetic Obstruction: The authors identify a "finite prime law": the denominators of the raw exceptional-grid coefficients recruit a new prime factor $p$ at each spin $n = p+2$ whenever $n-2$ is prime. Based on this, they conjecture that the generating data for these coefficients cannot be algebraic or globally bounded, though they remain consistent with holonomy.
• Validation:
  • The closed form reproduces independent Quantum Spectral Curve (QSC) intercepts ($\nu=0$) for all odd $n$ up to 91, including 26 "out-of-sample" spins not used in the construction.
  • The result matches the known $n=1$ block and vanishes at $n=1, \nu=0$ as required.
  • The construction preserves hermitian separability (purely additive separation of $z$ and $\bar{z}$) for odd spins.

Significance and Claims

The paper claims to realize the structural expectation of the "reflection-identity program," demonstrating that the three-loop eigenvalue at arbitrary odd conformal spin resides within the class of analytically continued nested harmonic sums.

• Structural Insight: The result explicitly reveals the analytic structure (pole structure in $\nu$, transcendental weight, and separability) that numerical spectra leave implicit. It confirms that the eigenvalue is built from the same function class as lower orders but evaluated along a "ladder" of shifted arguments connecting $z$ and its reflection.
• Exactness vs. Fitting: The authors emphasize that the decomposition and coefficients are derived via exact Mellin extraction and algebraic reduction, not fitted to numerical data. The only conjectural element is the identification of the master generating function with the Caron-Huot–Herranen integrand for all spins (Conjecture 1); however, the per-spin tables used for the final result are derived directly from the integrand without relying on this master function.
• Limitations: The paper explicitly restricts its closed-form results to odd conformal spins. Even spins are conjectured to be contained in the same master function but involve half-integer ladder shifts and cyclotomic content, which are left for future work.
• Negative Results: The paper reports three negative findings as results in their own right:
  1. No bivariate rational master exists for the pole-order-one rational slot (Conjecture 3).
  2. The low-order D-finiteness of the raw grids is mixed (some admit recurrences, others do not).
  3. The onset of certain kernel contributions is gauge-dependent, with only the $K_{cw3}$ onset at $n=19$ being reproducible across gauges.

The work provides a complete, reproducible, and exact description of the NNLO BFKL eigenvalue for the odd-spin sector of planar $\mathcal{N}=4$ SYM, serving as a boundary condition for any amplitude-side determination and a candidate for the highest-weight part of the corresponding QCD eigenvalue.