Symmetries of the N=4 SYM S-matrix

TL;DR

This work analyzes the symmetries of the ${ m N}=4$ SYM S-matrix within a Cachazo–Svrček–Witten framework extended to loops. By deriving holomorphic-anomaly–driven corrections to the superconformal generators, the authors show that the full S-matrix can be formally superconformal invariant to all loops for IR-safe observables, using a new holomorphic-anomaly–friendly regularization. They construct a CSW-based generating functional for generalized MHV vertices and establish a recursive structure of corrected generators ${ar S}_{1\to1}$, ${ar S}_{1\to2}$, ${ar S}_{2\to1}$, and ${ar S}_{3\to0}$ that annihilate the S-matrix. A sub-MHV regularization is proposed to regulate IR divergences while preserving the $ar S$ symmetry, with the formal invariance expected to extend to dual conformal symmetry and possibly a Yangian structure. The results provide a coherent symmetry-based approach to all-loop scattering amplitudes in ${ m N}=4$ SYM and suggest new avenues for leveraging these symmetries in computations.

Abstract

Under the assumption of a CSW generalization to loop amplitudes in N=4 SYM, (1) We prove that, formally the S-matrix is superconformal invariant to any loop order, and (2) We argue that superconformal symmetry survives regularization. More precisely, IR safe quantities constructed from the S-matrix are superconformal covariant. The IR divergences are regularized in a new holomorphic anomaly friendly regularization. The CSW prescription is known to be valid for all tree level amplitudes and for one loop MHV amplitudes. In these cases, our formal results do not rely on any assumptions.

Figures (10)

• Figure 1: The cut of the one loop amplitude in the $t_1^{[m]}$ channel.
• Figure 2: The action of the superconformal generator $\bar{S}_ {1\to 1}^{(0)}$ on a one loop unitarity cut. The holomorphic anomalies set an internal momenta to be collinear to an external one, giving rise to a $n+1$ tree level amplitude with two collinear particles. We deduce that the correction to that generator must be of the form $\bar{S}^{(0)}_{2\to 1}$.
• Figure 3: The term in the CSW sum of ${\cal A}^{(0)}_{n+1}$ that has a cut in $t_1^{[m]}=(k_1+k_2+\dots+k_m)^2$, when integrated over the collinearity portion of leg $1'$ and leg $(n+1)'$.
• Figure 5: Result of the action of $\bar{S}_{1\to 1}$ on ${\mathbb S}^{(1)}$. The operator $\bar{S}_{1\to 1}$ goes through ${\cal L}$ thus acting on the MHV tree level generating function ${\mathbb S}^{(0)}$. From Bargheer:2009qu this gives rise to the action of $\bar{S}_{1\to 2}$ on ${\mathbb S}^{(0)}$. The two legs created by $\bar{S}_{1\to 2}$ can (a) be unrelated to the legs on which $\mathcal{L}$ acts, thus yielding terms which vanish for generic external momenta, (b) be acted upon by $\mathcal{L}$, giving a vanishing contribution due to the Grassmanian integration or (c,d) one of them can become an external leg while the other is acted upon by $\mathcal{L}$. The latter two contributions are identified with $\bar{S}_{2\to 1}$, (see figure \ref{['Blue']}).
• Figure 6: The combined action of the propagator inserting operator $\mathcal{L}$ and the leg splitting operator $\bar{S}_{1\to 2}$ gives rise to the leg joining operator $\bar{S}_{2\to 1}$.