Transcendentality and Crossing

TL;DR

The paper investigates how a nontrivial dressing phase in the planar N=4 SYM S-matrix modifies the universal scaling function f(g), while preserving Kotikov-Lipatov transcendentality. By formulating an integral equation for the scaling function with a generalized kernel, it shows four-loop corrections depend on the phase and explores a one-parameter family of solutions that interpolate between phase choices. A key result is the analytic continuation linking weak- and strong-coupling expansions of the dressing-phase coefficients c_{r,s}(g), with exact matching for the (2,3) case and a unified h_{r,s}(g/k) representation. The authors introduce Magic Kernels to identify a special set of coefficients that cancel or align odd-zeta contributions, deriving a closed-form expression and a crossing-symmetric dressing kernel. Their proposed coefficients agree with independent four-loop calculations and, if correct, offer a coherent AdS/CFT picture across coupling regimes, while outlining tests and open questions for exact integrability and finite-length operator dynamics.

Abstract

We discuss possible phase factors for the S-matrix of planar N=4 gauge theory, leading to modifications at four-loop order as compared to an earlier proposal. While these result in a four-loop breakdown of perturbative BMN-scaling, Kotikov-Lipatov transcendentality in the universal scaling function for large-spin twist operators may be preserved. One particularly natural choice, unique up to one constant, modifies the overall contribution of all terms containing odd zeta functions in the earlier proposed scaling function based on a trivial phase. Excitingly, we present evidence that this choice is non-perturbatively related to a recently conjectured crossing-symmetric phase factor for perturbative string theory on AdS_5xS^5 once the constant is fixed to a particular value. Our proposal, if true, might therefore resolve the long-standing AdS/CFT discrepancies between gauge and string theory.

Figures (4)

• Figure 1: Left: Plot of $h_{23}(z)$ with asymptotic value $8/3\pi$. Right: Plot of the partial sums in $c_{23}(g)$ with weak coupling expansion up to radius of convergence $g={\frac{1}{4}}$.
• Figure 2: Left: Plot of partial sums in $c_{2,3}(g)$ with asymptotic strong coupling expansion. Right: We have plotted $xc_{2,3}(1/x)$ instead to improve the view of the strong-coupling behavior.
• Figure 3: Top to bottom curve: Plot of the scaling functions $f_{\mathrm{m}}(g)$, $f_0(g)$ and $f(g)$ with $g={\frac{1}{4}}\sqrt{x}$.
• Figure 4: Contour $C$ and its decomposition (\ref{['eq:contour ']}).