Gluing Ladders into Fishnets

TL;DR

This work targets conformal four-point fishnet diagrams in planar $\phi^4$ theory at weak coupling by uniting integrability with Steinmann analyticity. The authors conjecture a determinantal representation: $I_{m,n} = \det M$ with $M_{ij} = c_{ij} L_{n-m-1+i+j}$ and $\Phi_{m,n}(u,v) = [\frac{(1-z)(1-\bar{z})}{z-\bar{z}}]^m I_{m,n}(z,\bar{z})$, where $L_p$ are ladder integrals of weight $2mn$. They derive two integral representations from flux-tube and BMN pictures, and show that the Steinmann constraints force a unique determinant structure that cancels double discontinuities, validating the conjecture up to substantial $(m,n)$ and guiding future proofs. The results offer a compact, structurally transparent description of a broad class of conformal integrals and hint at deep connections to the amplitude bootstrap program and higher-point generalizations.

Abstract

We use integrability at weak coupling to compute fishnet diagrams for four-point correlation functions in planar $φ^4$ theory. The results are always multi-linear 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.

Figures (4)

• Figure 1: Fishnet diagram in $\phi^4$ theory and its dual off-shell (color-ordered) scattering amplitude.
• Figure 2: The correlator can be put inside a null square Wilson loop, with $x_{4}^{\mu} = n^{\mu}, x^{\mu}_{2} = \bar{n}^{\mu}, \bar{n}x_{3} = -e^{2\sigma_{1}}, nx_{1} = -e^{-2\sigma_{2}}$, and $n^2 = \bar{n}^2 = 0, n\cdot \bar{n} = 1$. Moving $\phi_{1,2}$ along the edges is the same as changing the cross ratios $z = -e^{2\sigma_{1}}$ and $\bar{z} = -e^{-2\sigma_{2}}$.
• 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 $\Box /g^2 = -z\bar{z}\partial_{z}\partial_{\bar{z}}/g^2$ or, equivalently, by introducing the form factor $(u^2+a^2/4)/g^2$ in rapidity space. A similar rule was used Basso2013vsaBelitsky2014slaBasso2015rta to transpose between MHV and NMHV 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 "NMHV" degree, or number of scalars at the cusp. This is readily seen to correct the mismatch between $\ell$ and $n$.