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Bootstrapping a Two-Loop Four-Point Form Factor

Yuanhong Guo, Lei Wang, Gang Yang

TL;DR

The authors solve for the two-loop four-point form factor of a length-3 half-BPS operator in planar ${\cal N}=4$ SYM by bootstrapping an ansatz built from a uniform-transcendentality master-integral basis and constraining the coefficients with IR divergences, collinear factorization, spurious-pole cancellation, and a targeted unitarity cut. The resulting analytic result is presented in both the symbol and Goncharov polylogarithm representations, with a finite remainder that exhibits expected collinear behavior and cancellation of nonphysical letters. A key innovation is performing the bootstrap directly on master integrals rather than starting from pure symbols, enabling explicit control over IR and unitarity constraints and extending to ${\cal O}(\epsilon)$ terms. The method yields compact, fully analytic expressions and demonstrates consistency with numerical checks, offering a path to general observables in gauge theories and insight into maximal transcendentality.

Abstract

We compute the two-loop four-point form factor of a length-3 half-BPS operator in planar N=4 SYM, which belongs to the class of two-loop five-point scattering observables with one off-shell color-singlet leg. A new bootstrapping strategy is developed to obtain this result by starting with an ansatz expanded in terms of master integrals and then solving the master coefficients via various physical constraints. We find that consistency conditions of infrared divergences and collinear limits, together with the cancellation of spurious poles, can fix a significant part of the ansatz. The remaining degrees of freedom can be fixed by one simple type of two-double unitarity cut. Full analytic results in terms of both symbol and Goncharov polylogarithms are provided.

Bootstrapping a Two-Loop Four-Point Form Factor

TL;DR

The authors solve for the two-loop four-point form factor of a length-3 half-BPS operator in planar SYM by bootstrapping an ansatz built from a uniform-transcendentality master-integral basis and constraining the coefficients with IR divergences, collinear factorization, spurious-pole cancellation, and a targeted unitarity cut. The resulting analytic result is presented in both the symbol and Goncharov polylogarithm representations, with a finite remainder that exhibits expected collinear behavior and cancellation of nonphysical letters. A key innovation is performing the bootstrap directly on master integrals rather than starting from pure symbols, enabling explicit control over IR and unitarity constraints and extending to terms. The method yields compact, fully analytic expressions and demonstrates consistency with numerical checks, offering a path to general observables in gauge theories and insight into maximal transcendentality.

Abstract

We compute the two-loop four-point form factor of a length-3 half-BPS operator in planar N=4 SYM, which belongs to the class of two-loop five-point scattering observables with one off-shell color-singlet leg. A new bootstrapping strategy is developed to obtain this result by starting with an ansatz expanded in terms of master integrals and then solving the master coefficients via various physical constraints. We find that consistency conditions of infrared divergences and collinear limits, together with the cancellation of spurious poles, can fix a significant part of the ansatz. The remaining degrees of freedom can be fixed by one simple type of two-double unitarity cut. Full analytic results in terms of both symbol and Goncharov polylogarithms are provided.

Paper Structure

This paper contains 13 sections, 64 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Ansatz of the form factor
  3. Physical constraints
  4. Solving the ansatz
  5. Full form factor and remainder
  6. Discussion
  7. Supplemental Material
  8. A. One-loop form factor
  9. B. A brief review of symbol
  10. C. Symbol letters
  11. D. Collinear limit of letters
  12. F. Kinematics and evaluation of GPL functions
  13. G. Numerical check for the collinear limit and spurious pole cancellation

Figures (3)

  • Figure 1: Topologies of the maximal number of propagators where the blue leg carries off-shell momentum $q$ and on-shell leg configurations are $(p_i, p_j, p_k, p_l) \in$ cyclic$(p_1, p_2, p_3, p_4)$.
  • Figure 2: Dual periodic Wilson line configuration for the four-point form factor in momentum twistor space.
  • Figure 3: Figures (a)--(c) are master integrals related to remaining 10 free parameters. All of them can be determined by the unitarity cuts in Fig. (d).