Massive fishnets

Abstract

Recently, infinite families of massive Feynman integrals were found to feature an unexpected Yangian symmetry. In the massless case, similar integrability properties are understood via the interpretation of individual Feynman integrals as correlators in the massless fishnet theory introduced by Gürdoğan and Kazakov. Here we seek for an analogous interpretation of the integrability of massive Feynman integrals. We contrast two approaches to define simple massive quantum field theories in four dimensions. First, we discuss spontaneous symmetry breaking in the massless bi-scalar fishnet theory. We then propose an alternative route to a massive fishnet theory by taking a double-scaling limit of $$ \mathcal{N} $$ N = 4 SYM theory on the Coulomb branch. Both approaches lead to a massive extension of the massless fishnet theory, differing in how masses enter into the propagators. In the latter theory, planar off-shell amplitudes are in one-to-one correspondence with precisely those massive Feynman integrals that were shown to be invariant under the Yangian. This suggests a re-investigation of Coulomb branch $$ \mathcal{N} $$ N = 4 SYM theory with regard to integrability. Finally, we demonstrate that in the case of spontaneous symmetry breaking, the original conformal symmetry leads to soft theorems for scattering amplitudes in the broken phase.

Fishnet four-point integrals: integrable representations and thermodynamic limits

“…Doing so, one projects over states carrying arbitrary scaling dimension ∆ along the radial direction. 13 Fishnets of this type were considered in the thermodynamic limit in ref. [22], for certain states, and their free energy was calculated for a large range of scaling dimensions ∆ using the Thermodynamic Bethe Ansatz (TBA) equations of the fishnet theory.…”

“…This web of theories allows us to shed light on properties of a broad family of higher-loop planar integrals, with regular iterative structures, the so-called fishnet graphs. These diagrams are endowed with integrable structures, such as Yangian symmetry [10][11][12][13], which put severe constraints on their analytic expressions and facilitate their determination.…”

“…They are also at odds with common beliefs about universality of large-order behaviour of Feynman diagrams[68,69] 13. Strictly speaking, ∆ = 2 + iν with ν ∈ R for principal series representations and other values of ∆ are reached by analytic continuation.…”

Massive Integrability: From Fishnet Theories to Feynman Graphs and Back

“…Via their interpretation as correlation functions in the so-called fishnet theories, these integrable structures trace back to the integrability of massless planar N =4 super Yang-Mills (SYM) theory [1][2][3][4]. In the following, we review the generalization to the massive case [5][6][7], which corresponds to the first findings of integrable structures in massive quantum field theory in four dimensions.…”

Yangian bootstrap for massive Feynman integrals

“…This double scaling limit yields a massless fishnet theory [17], whose correlators or scattering amplitudes are in one-to-one correspondence with individual Feynman graphs of fishnet structure. Similarly, a massive fishnet theory can be obtained from a double-scaling limit of N = 4 SYM theory on the Coulomb branch, allowing to identify massive Feynman integrals with Yangian invariant scattering amplitudes [18]. Also the massive version of the Yangian can be understood as the closure of massive dual conformal symmetry and a novel massive extension of ordinary conformal symmetry [3].…”