[Paper Review] Half-BPS half-BPS twist two at four loops in N=4 SYM

Burkhard Eden, Felix Paul|arXiv (Cornell University)|Aug 15, 2016

Black Holes and Theoretical Physics22 references3 citations

TL;DR

This paper computes the four-loop anomalous dimensions and three-point structure constants for twist-two operators in the 20' representation of SU(4) in planar N=4 SYM via the operator product expansion (OPE) of the stress tensor four-point function. Using the method of expansion by regions and IBP reduction, it evaluates the relevant four-loop integrals, confirming integrability predictions and providing high-precision data to test the hexagon bootstrap program at four loops.

ABSTRACT

We consider a double OPE limit of the planar four-point function of stress tensor multiplets in N = 4 SYM theory. Loop integrands for this correlator have been constructed to very high order, but the corresponding integrals are explicitly known only up to three loops. Fortunately, the double coincidence limit of the four-loop integrals can be found by the method of expansion by regions, which reduces the problem of computing the four-point integrals to the evaluation of a large set of massless propagator integrals. These can in turn be evaluated by IBP reduction. The OPE limit of the stress tensor four-point function allows us to extract the (square of the) three-point couplings between two stress tensor multiplets and one twist two operator in the 20' of SU(4). The latest available IBP software accomplishes this task up to and including spin 8. With the data obtained we hope to further the development of the recent integrable systems picture for correlation functions.

Motivation & Objective

• To compute four-loop corrections to the anomalous dimensions and three-point structure constants of twist-two operators in the 20' representation of SU(4) in planar N=4 SYM.
• To provide field-theory data to test and refine the hexagon bootstrap proposal, which faces a double pole issue at four loops due to virtual magnon scattering.
• To extend the precision of OPE data for twist-two operators up to spin 8, beyond previous three-loop results.
• To evaluate a core set of 26 genuine scalar conformal four-loop integrals using expansion by regions and IBP reduction, overcoming the lack of analytic results beyond three loops.

Proposed method

• Apply the method of expansion by regions to the double coincidence limit of the four-point function, reducing four-loop integrals to massless propagator integrals.
• Use the IBP reduction technique via FIRE5 and LiteRed to evaluate the resulting scalar integrals, enabling analytic computation of the OPE coefficients.
• Leverage known loop integrands up to eight loops and focus on the four-loop contribution, isolating the 20' channel via SU(4) projectors.
• Employ the differential equations method for master integrals of uniform transcendentality weight, though the main computation relies on IBP and expansion by regions.
• Use the OPE decomposition of the stress tensor four-point function to extract anomalous dimensions and structure constants from the singular behavior in the double OPE limit.
• Confirm results against integrability predictions, particularly for the absence of even-zeta terms and the structure of transcendental constants.

Experimental results

Research questions

• RQ1 Can the four-loop anomalous dimensions and three-point structure constants for twist-two operators in the 20' representation of SU(4) be computed in planar N=4 SYM using field-theory methods?
• RQ2 How do the results from the OPE limit of the stress tensor four-point function compare with the predictions of the hexagon bootstrap program at four loops?
• RQ3 What is the analytic structure of the four-loop integrals, particularly regarding the appearance of zeta values and transcendental weights?
• RQ4 Can the method of expansion by regions and IBP reduction be systematically applied to compute OPE coefficients at four loops for higher-spin operators?
• RQ5 Why do only odd-zeta values appear in the results, and what explains the absence of terms like ζ₃ζ₅ or log(u)ζ₇?

Key findings

• The four-loop anomalous dimension for spin 2 is γ₄(2) = -2496 + 576ζ₃ - 1440ζ₅, in full agreement with integrability predictions.
• For spin 4, the anomalous dimension is γ₄(4) = -8045275/2187 + 114500/81 ζ₃ - 25000/9 ζ₅, confirming integrability at four loops.
• The structure constant for spin 2 is α₄(2) = 9952 + 1312ζ₃ + 288ζ₃² + 3920ζ₅ + 5880ζ₇, with no even-zeta or π terms.
• For spin 8, the structure constant is given by a large rational coefficient plus terms in ζ₃, ζ₃², ζ₅, and ζ₇, with no weight-eight constants like ζ₃ζ₅.
• The results confirm that only odd-zeta values contribute, and no terms of transcendental weight eight (e.g., log(u)ζ₇ or ζ₃ζ₅) appear, consistent with the absence of unphysical singularities.
• The computation provides 20 precise data points for testing the hexagon proposal, significantly extending previous checks and validating its structure at four loops.