Crossing symmetry including non planar diagrams in perturbative QFT

TL;DR

This work extends crossing symmetry in perturbative QFT from planar to non-planar diagrams by classifying non-planar topologies into trivial (one non-planar edge) and non-trivial (two or more). It employs Mizera's five-step analytic continuation framework, together with Landau equations and Schwinger-parameter (world-line) representations, to show that leading Landau singularities do not obstruct crossing even for non-planar graphs, with explicit 2-loop and 3-loop analyses. The author demonstrates crossing symmetry holds as two different limits of the same analytic function, T_{AB->CD} ≡ T_{Bar{C}→Dar{A}}, for non-planar diagrams, and provides general arguments that this extends to arbitrary loop order and arbitrary non-planar edge counts, alongside generalization to higher-point amplitudes. The results offer a concrete, diagrammatic validation of crossing in a broader perturbative setting and suggest avenues for connecting to string field theory and axiomatic frameworks. The approach emphasizes analytic structure, contour deformation, and the leading-Landau-singularity landscape to ensure the physical amplitude remains well-defined across crossing channels.

Abstract

We venture a proof of crossing symmetry for non-planar diagrams in perturbative QFT. For the planar diagrams a proof of crossing is available in the literature and our method closely follows the one depicted in that case. We classify the non-planar diagrams broadly into two types. For one of these types the proof is pretty straightforward and hence the result extends to all point all loop on-shell amplitudes. These are called the "trivial" cases while for the other type we find certain cases called the "non trivial" cases for which the proof is much more subtle. We present an explicit example of such a "non trivial" case at 3-loop order and argue how the proof of crossing symmetry holds true when all subtleties are taken into consideration. Based on this simple example we argue how the proof works out in general for these "non-trivial" cases at higher loop and with arbitrary number of non-planar edges.

Figures (11)

• Figure 1: Five steps to carry out the analytic continuation from one configuration to its crossed configuration i.e. from $AB\rightarrow CD$ to $B\bar{C}\rightarrow D\bar{A}$. (We acknowledge this figure to be an exact replica of figure 1 of Mizera:2021fap.)
• Figure 2: Flow of momenta in the direction of $p_1^{\pm}$ (blue) and $p_2^{\pm}$ (red). The side edges of the diagram always have non-zero components in both directions preventing the formation of singularities. (We acknowledge this figure to be an exact replica of figure 8 of Mizera:2021fap)
• Figure 3: Non planar diagram at 2-loop considered in Case 1.
• Figure 4: Spanning trees corresponding to $\lbrace e_1,e_4\rbrace,\lbrace e_1,e_6\rbrace$ and $\lbrace e_2,e_4\rbrace$ respectively. The direction of flow for each internal edge is the same as the original diagram depicted in figure \ref{['fig:3']}.
• Figure 5: Non planar diagram at 2-loop considered in Case 2.