arXiv subjects

Sergey Derkachov

Publications and source records attributed to Sergey Derkachov.

5 recordsLinked to original sources

arXiv2021

Conformal quantum mechanics & the integrable spinning Fishnet

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 $Δ=2+iν$ for any left/right spins $\ell,\dot{\ell}$ of the particles. Such relations are interpreted in the language of Feynman diagrams as integral \emph{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 $(Δ_k,\ell_k, \dot{ \ell}_k)$ 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. 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 $\mathcal{N}=4$ and $\mathcal{N}=2$ supersymmetric theories.

hep-th↗

arXiv2021

Mirror channel eigenvectors of the $d$-dimensional fishnets

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 $L$ depend on a set of $L$ 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.

hep-th↗

arXiv2020

Exactly solvable single-trace four point correlators in $χ$CFT$_4$

In this paper we study a wide class of planar single-trace four point correlators in the chiral conformal field theory ($χ$CFT$_4$) arising as a double scaling limit of the $γ$-deformed $\mathcal{N}=4$ 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.

hep-th↗

arXiv2020

Exactly solvable magnet of conformal spins in four dimensions

We provide the eigenfunctions for a quantum chain of $N$ conformal spins with nearest-neighbor interaction and open boundary conditions in the irreducible representation of $SO(1,5)$ of scaling dimension $Δ= 2 - i λ$ and spin numbers $\ell=\dot{\ell}=0$. The spectrum of the model is separated into $N$ equal contributions, each dependent on a quantum number $Y_a=[ν_a,n_a]$ which labels a representation of the principal series. The eigenfunctions are orthogonal and we computed the spectral measure by means of a new star-triangle identity. Any portion of a conformal Feynmann diagram with square lattice topology can be represented in terms of separated variables, and we reproduce the all-loop "fishnet" integrals computed by B. Basso and L. Dixon via bootstrap techniques. We conjecture that the proposed eigenfunctions form a complete set and provide a tool for the direct computation of conformal data in the fishnet limit of the supersymmetric $\mathcal{N}=4\,$ Yang-Mills theory at finite order in the coupling, by means of a cutting-and-gluing procedure on the square lattice.

hep-th↗

arXiv2019

Basso-Dixon Correlators in Two-Dimensional Fishnet CFT

We compute explicitly the two-dimensional version of Basso-Dixon type integrals for the planar four-point correlation functions given by conformal fishnet Feynman graphs. These diagrams are represented by a fragment of a regular square lattice of power-like propagators, arising in the recently proposed integrable bi-scalar fishnet CFT. The formula is derived from first principles, using the formalism of separated variables in integrable SL(2,C) spin chain. It is generalized to anisotropic fishnet, with different powers for propagators in two directions of the lattice.

hep-th↗