OPE for all Helicity Amplitudes

Abstract

We extend the Operator Product Expansion (OPE) for scattering amplitudes in planar N=4 SYM to account for all possible helicities of the external states. This is done by constructing a simple map between helicity configurations and so-called charged pentagon transitions. These OPE building blocks are generalizations of the bosonic pentagons entering MHV amplitudes and they can be bootstrapped at finite coupling from the integrable dynamics of the color flux tube. A byproduct of our map is a simple realization of parity in the super Wilson loop picture.

Figures (7)

• Figure 1: a) The pentagon transitions are the building blocks of null polygonal Wilson loops. They represent the transition $\psi\rightarrow\psi'$ undergone by the flux-tube state as we move from one square to the next in the OPE decomposition. This breaking into squares is univocally defined by specifying the middle (or inner dashed) edge of the pentagon to be $Z_\text{middle}\propto\langle j-2,j,j+2,j-1\rangle Z_{j+1}-\langle j-2,j,j+2,j+1\rangle Z_{j-1}$. b) In the OPE-friendly labelling of edges, adopted in this paper, the middle edge of the $j$-th pentagon ends on the $j$-th edge. As a result, the very bottom edge is edge $-1$ while the very top one is edge $n-2$. The map between the OPE index $j$ and the more common cyclic index $j_\text{cyc}$ reads $j_\text{cyc}=\frac{3}{4}-\frac{1}{4} (-1)^j (2 j+3)$ mod $n$.
• Figure 2: We study the conformally invariant and finite ratio $\mathcal{W}$ introduced in short. It is obtained by dividing the expectation value of the super Wilson loop by all the pentagons in the decomposition and by multiplying it by all the middle squares. The twistors that define these smaller pentagons are either the twistors of the original polygon (an heptagon in this figure) or the middle twistors described in figure \ref{['pentagontransition']}.a (in the above figure there are three distinct middle twistors, for instance).
• Figure 3: a) Any square in the OPE decomposition stands for a transition from the state at its bottom ($\psi_\text{bottom}$) to the state at its top ($\psi_\text{top}$). This transition is generated by a conformal symmetry of the right and left edges of that square (conjugate to the flux time $\tau$). b) Similarly, the super pentagon $\mathbb P$ represents a transition from the state at its bottom to the state at its top. In the fermionic $\chi$-directions, this transition is generated by a super-conformal symmetry of the $(j-1)$-th, $j$-th and $(j+1)$-th edges in this figure.
• Figure 4: Leading OPE contribution to the NMHV octagon component ${\cal P}_1 \circ {\cal P}_2 \circ {\cal P}_3 \circ {\cal P}_4 = \frac{\partial}{\partial \chi_1^1} \frac{\partial}{\partial \chi_2^2} \frac{\partial}{\partial \chi_3^3} \frac{\partial}{\partial \chi_4^4} \mathcal{W}$. For this component, each of the four pentagons in the octagon decomposition carries one unit of $R$-charge and fermion number. From the flux tube point of view, this corresponds to the sequence of transitions in equation (\ref{['sequance']}).
• Figure 5: A remarkable feature of our construction is that the inverse map turns out to be local. Namely, charging edge $j$ is done by charging the five pentagons touching this edge and these five pentagons alone. We notice in particular that the two outermost pentagons in this neighbourhood, which are shown in green above, are touching the endpoints of edge $j$ only.