Gluing Ladder Feynman Diagrams into Fishnets

We use integrability at weak coupling to compute fishnet diagrams for four-point correlation functions in planar ${\varphi }^4$ theory. The results are always multilinear combinations of ladder integrals, which are in turn built out of classical polylogarithms. The Steinmann relations provide a powerful constraint on such linear combinations, leading to a natural conjecture for any fishnet diagram as the determinant of a matrix of ladder integrals.

Figure 1

Fishnet diagram in ${\varphi }^4$ theory and its dual off-shell (color-ordered) scattering amplitude.

Figure 2

The correlator can be put inside a null square Wilson loop, with $x_4^{\mu }=n^{\mu }$, $x_2^{\mu }={\overline{n}}^{\mu }$, $\overline{n}{x}_3=-e^{2{\sigma }_1}$, $n{x}_1=-e^{-2{\sigma }_2}$, and $n^2={\overline{n}}^2=0$, $n\overline{n}=1$. Moving ${\varphi }_{1,2}$ along the edges is the same as changing the cross ratios $z=-e^{2{\sigma }_1}$ and $\overline{z}=-e^{-2{\sigma }_2}$.

Figure 3

We can decompose the correlator using triangles (also known as hexagons). The red beam is made of $m$ magnons, produced on the bottom triangle and absorbed on the top one. The correlator is the scalar product between the two wave functions.

Figure 4

By supersymmetry, the measure ${\mu }_a(u)$ describes both the one-loop gluon diagram, on the left, and the one-loop scalar cross diagram, on the right. To get a free propagator, we must deconvolute the scalar diagram, by acting with the box operator $\square /g^2=-z\overline{z}{\partial }_z{\partial }_{\overline{z}}/g^2$ or, equivalently, by introducing the form factor $(u^2+a^2/4)/g^2$ in rapidity space. A similar rule was used [2, 35, 36] to transpose between maximally helicity violating and next-to-maximally helicity violating amplitudes, in the flux tube picture. In general, the conversion is achieved through inclusion of a factor $[(u^2+a^2/4)/g^2{]}^m$, per excitation, where $m$ is the next-to-maximally helicity violating degree, or number of scalars at the cusp. This is readily seen to correct the mismatch between $\ell$ and $n$.