Form factors and scattering amplitudes in N=4 SYM in dimensional and massive regularizations

TL;DR

The paper develops a regulator‑independent definition of the IR‑finite part of planar ${\cal N}=4$ SYM MHV amplitudes by factoring IR divergences into wedge functions and isolating a regulator‑independent hard function $\log H_n$. It shows that, for dimensional regularization and the common‑mass Higgs regulator, the finite part agrees up to known constants through two loops, and it extends the framework to a differential‑mass Higgs regulator using extended dual conformal symmetry. The authors derive all‑loop structure for the wedge functions and connect the finite part to the anomalous dual conformal Ward identity, thereby linking IR factorization, Regulator schemes, and dual conformal symmetry. While a full operator definition for the wedge function in the differential‑mass case remains open, the approach provides a coherent, regulator‑independent picture of IR behavior and offers potential insights for Wilson loops and beyond‑${\cal N}=4$ theories.

Abstract

The IR-divergent scattering amplitudes of N=4 supersymmetric Yang-Mills theory can be regulated in a variety of ways, including dimensional regularization and massive (or Higgs) regularization. The IR-finite part of an amplitude in different regularizations generally differs by an additive constant at each loop order, due to the ambiguity in separating finite and divergent contributions. We give a prescription for defining an unambiguous, regulator-independent finite part of the amplitude by factoring off a product of IR-divergent "wedge" functions. For the cases of dimensional regularization and the common-mass Higgs regulator, we define the wedge function in terms of a form factor, and demonstrate the regularization independence of the n-point amplitude through two loops. We also deduce the form of the wedge function for the more general differential-mass Higgs regulator, although we lack an explicit operator definition in this case. Finally, using extended dual conformal symmetry, we demonstrate the link between the differential-mass wedge function and the anomalous dual conformal Ward identity for the finite part of the scattering amplitude.

Figures (4)

• Figure 1: One- and two-loop form factor diagrams in dimensional regularization. All lines represent massless adjoint fields. The dot represents the insertion of ${\cal O} _{IJ}$.
• Figure 2: One- and two-loop form factor diagrams for the common-mass Higgs regulator. The solid/dashed lines represent massive/massless adjoint fields.
• Figure 3: Double-line version of the two-loop diagrams for the common-mass Higgs regulator. The solid/dotted lines represent fundamental fields of $U(M)/U(N)$.
• Figure 4: Two-mass-easy diagram corresponding to the integral $I^{\rm 2me}_{ij}$.