Mirror channel eigenvectors of the d-dimensional fishnets

Abstract

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 (uk, lk ), 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.

Conformal Triangles and Zig-Zag Diagrams

“…The construction of the complete basis of corresponding eigenfunctions in two-dimensional fishnet theory is performed in [22] by using the methods of the theory of integrable spin chains. These results were generalized to the case of four-dimensional fishnet theory in [23,24] and in [25] to the case of D-dimensional fishnet theory [29]. The typical Feynman diagrams in the fishnet CFT posses special iterative structure and are constructed by using various graph-building operators [27,28,29,16,30].…”

“…Our approach is partially inspired by papers [22,23,24,25] devoted to the representation of separated variables for the Basso-Dixon integrals [26]. B. Basso and L. Dixon originally proposed in [26] a nice explicit determinant formula for some family of Feynman diagrams in D = 4 fishnet conformal field theories (CFT) [27,28].…”

The SAGEX Review on Scattering Amplitudes, Chapter 9: Integrability of Amplitudes in Fishnet Theories

“…This non-compact spin chain perspective is very fruitful, and it helps to reveal integrability of the conformal fishnet theory (1) at the level of individual Feynman graphs [41]. In particular, it allows for evaluating exactly a number of four-point correlation functions [16,18] and multi-loop Feynman integrals [42,43] by solving the spectral problem for the relevant quantum mechanical spin-chain model [44]. Here we concentrate on the Yangian symmetry implications of the spin-chain perspective.…”

Form-factors and complete basis of observables via separation of variables for higher rank spin chains

“…In particular, the functional SoV approach allows one to naturally compute a family of diagonal form factors xΨ|B p Î|Ψy, where p is some parameter of the model and Î is an integral of motion, using standard quantum mechanical perturbation theory arguments [14,19,20]. Interesting quantities which can be extracted using this approach are the formfactors of various local operators, including a family of non-trivial Feynman diagrams [19] in 1 The operator-based SoV construction has also been used to compute Basso-Dixon correlators in 2D fishnet CFT [15] and has recently seen remarkable extensions to the 4D setting [16][17][18]. The crucial difference with our approach is that in those papers the SoV is applied in the "mirror" channel.…”