Superspace calculation of the three-loop dilatation operator ofN=4supersymmetric Yang-Mills theory

Abstract

We derive the three-loop dilatation operator of the flavor SUð2Þ subsector of N ¼ 4 supersymmetric Yang-Mills theory in the planar limit by a direct Feynman diagram calculation in N ¼ 1 superspace. The transcendentality three contributions which appear in intermediate steps cancel among each other, leaving a rational result which confirms the predictions from integrability. We derive finiteness conditions that allow us to avoid the explicit evaluation of entire classes of Feynman graphs and also yield constraints on the D-algebra manipulations. Based on these results, we discover universal cancellation mechanisms. As a check for the consistency of our result, we verify the cancellation of all higher-order poles.

Integrability inN=2superconformal gauge theories

“…This crucial step of our argument is taken in Sections 6.1 and 6. due to the non-renormalization theorem of [23,24] that we discuss in Section 6.2. The first of the points above, "the choice of the sector" refers to the fact that gauge-invariant local operators O ∈ SU(2, 1|2) are made only out of fields in the vector multiplet and have the spinor indices α always in the symmetric representation of the SU(2) α ∈ SU(2, 2|2).…”

“…These vertices do not contribute due to planarity (they cannot be Wick-contracted with operators of the SU(2, 1|2) sector), the choice of the sector and Lorentz invariance, • of the new vertices that appear in the effective action for the first time at two loops δΓ (2) new , the only one that can be planarly contracted to O ∈ SU(2, 1|2) is shown in Fig. 8 and it will not contribute due to the non-renormalization theorem of [23,24] that we will further discuss in Section 6.2 and in Appendix A.3.…”

“…It is important to realize that for the SU(2, 1|2) sector Hamiltonian there are no bare skeleton diagrams made out of vertices coming from the superpotential! This is in strict distinction with the SU(2) sector of N = 4 SYM presented in [24] and the deformed SU(2) sector of the N = 2 interpolating theory presented in [11] that is made out of hyper-multiplets. The one-loop divergences in the self-energy diagrams are, for any superconformal gauge theory [11], identical to the ones in N = 4.…”

“…Disconnected diagrams (after stripping off the composite operators) do not contribute to the Hamiltonian, because their overall divergences only contain higher degree poles (1/ε n with n > 1) [24]. The contents of the rest of paper are as follows.…”

Cutting through form factors and cross sections of non-protected operators in N = 4 $$ \mathcal{N}=4 $$ SYM

“…4 The Konishi anomalous dimension γK is currently known up to five loops from field theory calculations and up to ten loops from the conjectured integrability. The three-loop result was conjectured in [50] and confirmed in [51,52]. The four-loop result was determined by calculating the wrapping corrections to the integrability-based asymptotic dilatation operator in [53,54] and by a computer-based direct calculation in [55].…”