The Checkerboard conformal field theory is an interesting representative of a large class of non-unitary, logarithmic Fishnet CFTs (FCFT) in arbitrary dimension which have been intensively studied in the last years. Its planar Feynman graphs have the structure of a regular square lattice with checkerboard colouring. Such graphs are integrable since each coloured cell of the lattice is equal to an R-matrix in the principal series representations of the conformal group. We compute perturbatively and numerically the anomalous dimension of the shortest single-trace operator in two reductions of the Checkerboard CFT: the first one corresponds to the Fishnet limit of the twisted ABJM theory in 3D, whereas the spectrum in the second, 2D reduction contains the energy of the BFKL Pomeron. We derive an analytic expression for the Checkerboard analogues of Basso-Dixon 4-point functions, as well as for the class of Diamond-type 4-point graphs with disc topology. The properties of the latter are studied in terms of OPE for operators with open indices. We prove that the spectrum of the theory receives corrections only at even orders in the loop expansion and we conjecture such a modification of Checkerboard CFT where quantum corrections occur only with a given periodicity in the loop order.

Russia

Mikhail Alfimov

Israel

Gwenaël Ferrando

France

Vladimir Kazakov

Canada

Enrico Olivucci

Checkerboard CFT

This work presents the building-blocks of an integrability-based representation for multi-point Fishnet Feynman integrals with any number of loops. Such representation relies on the quantum separation of variables (SoV) of a non-compact spin-chain with symmetry SO(1, 5) explained in the first paper of this series. The building-blocks of the SoV representation are overlaps of the wave-functions of the spin-chain excitations inserted along the edges of a triangular tile of Fishnet lattice. The zoology of overlaps is analyzed along with various worked out instances in order to achieve compact formulae for the generic triangular tile. The procedure of assembling the tiles into a Fishnet integral is presented exhaustively. The present analysis describes multi-point correlators with disk topology in the bi-scalar limit of planar γ-deformed   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM theory, and it verifies some conjectural formulae for hexagonalisation of Fishnets CFTs present in the literature. The findings of this work are suitable of generalization to a wider class of Feynman diagrams.

Hexagonalization of Fishnet integrals. Part II. Overlaps and multi-point correlators

We study fishnet Feynman diagrams defined by a certain triangulation of a planar <math><mrow><mi>n</mi></mrow></math>-gon, with massless scalars propagating along and across the cuts. Our solution theory uses the technique of separation of variables, in combination with the theory of symmetric polynomials and Mellin space. The <math><mi>n</mi></math>-point split-ladders are solved by a recursion where all building blocks are made fully explicit. In particular, we find an elegant formula for the coefficient functions of the light-cone leading logs. When the diagram grows into a fishnet, we obtain new results exploiting a Cauchy identity decomposition of the measure over separated variables. This leads to an elementary proof of the Basso-Dixon formula at 4-points, while at <math><mi>n</mi></math>-points it provides a natural operator product expansion-like stratification of the diagram. Finally, we propose an independent approach based on “stampede” combinatorics to study the light-cone behavior of the diagrams as the partition function of a certain vertex model.

Spain

Francesco Aprile

Multipoint fishnet Feynman diagrams: Sequential splitting

We propose a broad class of d-dimensional conformal field theories of SU(N) adjoint scalar fields generalising the 4d Fishnet CFT (FCFT) discovered by Ö. Gürdogan and one of the authors as a special limit of γ-deformed   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM theory. In the planar N → ∞ limit the FCFTs are dominated by the “fishnet” planar Feynman graphs. These graphs are explicitly integrable, as was shown long ago by A. Zamolodchikov. The Zamolodchikov’s construction, based on the dual Baxter lattice (straight lines on the plane intersecting at arbitrary slopes) and the star-triangle identities, can serve as a “loom” for “weaving” the Feynman graphs of these FCFTs, with certain types of propagators, at any d. The Baxter lattice with M different slopes and any number of lines parallel to those, generates an FCFT consisting of M (M – 1) fields and a certain number of chiral vertices of different valences with distinguished couplings. These non-unitary, logarithmic CFTs enjoy certain reality properties for their spectrum due to a symmetry similar to the PT-invariance of non-hermitian hamiltonians proposed by C. Bender and S. Boettcher. We discuss in more detail the theories generated by a loom with M = 2, 3, 4, and the generalisation of the loom FCFTs for spinning fields in 4d.

The loom for general fishnet CFTs

We consider correlation functions of single trace operators approaching the cusps of null polygons in a double-scaling limit where so-called cusp times <math><msubsup><mi>t</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><msup><mi>g</mi><mn>2</mn></msup><mi>log</mi><msubsup><mi>x</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>i</mi></mrow><mn>2</mn></msubsup><mi>log</mi><msubsup><mi>x</mi><mrow><mi>i</mi><mo>,</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup></math> are held fixed and the ‘t Hooft coupling is small. With the help of stampedes, symbols, and educated guesses, we find that any such correlator can be uniquely fixed through a set of coupled lattice PDEs of Toda type with several intriguing novel features. These results hold for most conformal gauge theories with a large number of colors, including planar <math><mi>N</mi><mo>=</mo><mn>4</mn></math> SYM.

Brazil

Pedro Vieira

Null Polygons in Conformal Gauge Theory

Some quantities in quantum field theory are dominated by so-called leading logs and can be re-summed to all loop orders. In this work we introduce a notion of stampede which is a simple time-evolution of a bunch of particles which start their life in a corner — on the very right say — and hop their way to the opposite corner — on the left — through the repeated action of a quantum Hamiltonian. Such stampedes govern leading logs quantities in certain quantum field theories. The leading euclidean OPE limit of correlation functions in the fishnet theory and null double-scaling limits of correlators in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM are notable examples. As an application, we use these results to extend the beautiful bootstrap program of Coronado [1] to all octagons functions with arbitrary diagonal bridge length.

Stampedes I: fishnet OPE and octagon Bootstrap with nonzero bridges

We present a basis of eigenvectors for the graph building operators acting along the mirror channel of planar fishnet Feynman integrals in d-dimensions. The eigenvectors of a fishnet lattice of length N depend on a set of N quantum numbers (u  k  , l  k  ), each associated with the rapidity and bound-state index of a lattice excitation. Each excitation is a particle in (1 + 1)-dimensions with O(d) internal symmetry, and the wave-functions are formally constructed with a set of creation/annihilation operators that satisfy the corresponding Zamolodchikovs-Faddeev algebra. These properties are proved via the representation, new to our knowledge, of the matrix elements of the fused R-matrix with O(d) symmetry as integral operators on the functions of two spacetime points. The spectral decomposition of a fishnet integral we achieved can be applied to the computation of Basso-Dixon integrals in higher dimensions.

Sergey Derkachov

Mirror channel eigenvectors of the d-dimensional fishnets

In this paper we consider a conformal invariant chain of L sites in the unitary irreducible representations of the group SO(1, 5). The k-th site of the chain is defined by a scaling dimension ∆ k  and spin numbers   <math> <mfrac> <msub> <mi>ℓ</mi> <mi>k</mi> </msub> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <msub> <mi>ℓ</mi> <mi>k</mi> </msub> <mn>2</mn> </mfrac> </math>  $$ \frac{\ell_k}{2},\frac{\ell_k}{2} $$  The model with open and fixed boundaries is shown to be integrable at the quantum level and its spectrum and eigenfunctions are obtained by separation of variables. The transfer matrices of the chain are graph-builder operators for the spinning and inhomogeneous generalization of squared-lattice “fishnet” integrals on the disk. As such, their eigenfunctions are used to diagonalize the mirror channel of the Feynman diagrams of Fishnet conformal field theories. The separated variables are interpreted as momentum and bound-state index of the mirror excitations of the lattice: particles with SO(4) internal symmetry that scatter according to an integrable factorized   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$ -matrix in (1 + 1) dimensions

Hexagonalization of Fishnet integrals. Part I. Mirror excitations

In this paper we consider systems of quantum particles in the 4d Euclidean space which enjoy conformal symmetry. The algebraic relations for conformal-invariant combinations of positions and momenta are used to construct a solution of the Yang-Baxter equation in the unitary irreducibile representations of the principal series ∆ = 2 + iν for any left/right spins ℓ,   <math> <mover> <mi>ℓ</mi> <mo>̇</mo> </mover> </math>  $$ \dot{\ell} $$  of the particles. Such relations are interpreted in the language of Feynman diagrams as integral star-triangle identites between propagators of a conformal field theory. We prove the quantum integrability of a spin chain whose k-th site hosts a particle in the representation (∆ k  , ℓ  k  ,   <math> <mover> <mi>ℓ</mi> <mo>̇</mo> </mover> </math>  $$ \dot{\ell} $$   k ) of the conformal group, realizing a spinning and inhomogeneous version of the quantum magnet used to describe the spectrum of the bi-scalar Fishnet theories [1]. For the special choice of particles in the scalar (1, 0, 0) and fermionic (3/2, 1, 0) representation the transfer matrices of the model are Bethe-Salpeter kernels for the double-scaling limit of specific two-point correlators in the γ-deformed   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 and   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supersymmetric theories.

Conformal quantum mechanics &amp; the integrable spinning Fishnet

In this paper we study a wide class of planar single-trace four point correlators in the chiral conformal field theory (χCFT4) arising as a double scaling limit of the γ-deformed   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM theory. In the planar (t’Hooft) limit, each of such correlators is described by a single Feynman integral having the bulk topology of a square lattice “fishnet” and/or of an honeycomb lattice of Yukawa vertices. The computation of this class of Feynmann integrals at any loop is achieved by means of an exactly-solvable spin chain magnet with SO(1, 5) symmetry. In this paper we explain in detail the solution of the magnet model as presented in our recent letter and we obtain a general formula for the representation of the Feynman integrals over the spectrum of the separated variables of the magnet, for any number of scalar and fermionic fields in the corresponding correlator. For the particular choice of scalar fields only, our formula reproduces the conjecture of B. Basso and L. Dixon for the fishnet integrals.