Searcharxiv⌕ Search

arXiv subjects

S. E. Derkachov

Publications and source records attributed to S. E. Derkachov.

At least 19 recordsLinked to original sources
arXiv2026

Conformal four-point ladder integrals in diverse dimensions and polylogarithms

In the paper, the family of conformal four-point ladder diagrams in arbitrary space-time dimensions is considered. We use the representation obtained via explicit calculation using the operator approach and conformal quantum mechanics to study their properties, such as symmetries, loop and dimensional shift identities. In even integer dimensions, latter allows one to reduce the problem to two-dimensional case, where the notable factorization holds. Additionally, for a specific choice of propagator powers, we show that the representation can be written in the form of linear combinations of classical polylogarithms (with coefficients that are rational functions) and explore the structure of the resulting expressions.

hep-th↗
arXiv2023

Evolution equations for quark-gluon distributions in multi-color QCD and open spin chains

We study the scale dependence of the twist-3 quark-gluon parton distributions using the observation that in the multi-color limit the corresponding QCD evolution equations possess an additional integral of motion and turn out to be effectively equivalent to the Schrodinger equation for integrable open Heisenberg spin chain model. We identify the integral of motion of the spin chain as a new quantum number that separates different components of the twist-3 parton distributions. Each component evolves independently and its scale dependence is governed by anomalous dimension given by the energy of the spin magnet. To find the spectrum of the QCD induced open Heisenberg spin magnet we develop the Bethe Ansatz technique based on the Baxter equation. The solutions to the Baxter equation are constructed using different asymptotic methods and their properties are studied in detail. We demonstrate that the obtained solutions provide a good qualitative description of the spectrum of the anomalous dimensions and reveal a number of interesting properties. We show that the few lowest anomalous dimensions are separated from the rest of the spectrum by a finite mass gap and estimate its value.

hep-ph↗
arXiv2023

Ladder and zig-zag Feynman diagrams, operator formalism and conformal triangles

We develop an operator approach to the evaluation of multiple integrals for multiloop Feynman massless diagrams. A commutative family of graph building operators $H_α$ for ladder diagrams is constructed and investigated. The complete set of eigenfunctions and the corresponding eigenvalues for the operators $H_α$ are found. This enables us to explicitly express a wide class of four-point ladder diagrams and a general two-loop propagator-type master diagram (with arbitrary indices on the lines) as Mellin-Barnes-type integrals. Special cases of these integrals are explicitly evaluated. A certain class of zig-zag four-point and two-point planar Feynman diagrams (relevant to the bi-scalar $D$-dimensional "fishnet" field theory and to the calculation of the $β$-function in $ϕ^4$-theory) is considered. The graph building operators and convenient integral representations for these Feynman diagrams are obtained. The explicit form of the eigenfunctions for the graph building operators of the zig-zag diagrams is fixed by conformal symmetry and these eigenfunctions coincide with the 3-point correlation functions in $D$-dimensional conformal field theories. By means of this approach, we exactly evaluate the diagrams of the zig-zag series in special cases. In particular, we find a fairly simple derivation of the values for the zig-zag multi-loop two-point diagrams for $D=4$. The role of conformal symmetry in this approach, especially a connection of the considered graph building operators with conformal invariant solutions of the Yang-Baxter equation is investigated in detail.

hep-th↗
arXiv2022

Elliptic hypergeometric function and $6j$-symbols for the SL(2,$\mathbb{C}$) group

We show that the complex hypergeometric function describing $6j$-symbols for $SL(2,\mathbb{C})$ group is a special degeneration of the $V$-function -- an elliptic analogue of the Euler-Gauss $_2F_1$ hypergeometric function. For this function, we derive mixed difference-recurrence relations as limiting forms of the elliptic hypergeometric equation and some symmetry transformations. At the intermediate steps of computations, there emerge a function describing the $6j$-symbols for the Faddeev modular double and the corresponding difference equations and symmetry transformations.

math-ph↗
arXiv2018

On the 6j-symbols for SL(2,C) group

We study 6j-symbols, or Racah coefficients for tensor products of infinite-dimensional unitary principal series representations of the group SL(2,C). These symbols were constructed earlier by Ismagilov and we rederive his result (up to some slight difference associated with equivalent representations) using the Feynman diagrams technique. The resulting 6j-symbols are expressed either as a triple integral over complex plane with a symmetric kernel or as an infinite bilateral sum of integrals of the Mellin-Barnes type.

math-ph↗
arXiv2017

Spin Chains and Gustafson's Integrals

Gustafson's integrals are multidimensional generalizations of the classical Mellin-Barnes integrals. We show that some of these integrals arise from relations between matrix elements in Sklyanin's representation of Separated Variables in spin chain models. We also present several new integrals.

math-ph↗
arXiv2016

Gustafson integrals for $SL(2,\mathbb{C})$ spin magnet

It was observed recently that the multidimensional Mellin--Barnes integrals (Gustafson's integrals) arise naturally in studies of the $SL(2,R)$ spin chain models. We extend this analysis to the noncompact $SL(2,\mathbb{C})$ spin magnets and obtain integrals which generalize Gustafson's integrals to the complex case.

math-ph↗
arXiv2015

New elliptic solutions of the Yang-Baxter equation

We consider finite-dimensional reductions of an integral operator with the elliptic hypergeometric kernel describing the most general known solution of the Yang-Baxter equation with a rank 1 symmetry algebra. The reduced R-operators reproduce at their bottom the standard Baxter's R-matrix for the 8-vertex model and Sklyanin's L-operator. The general formula has a remarkably compact form and yields new elliptic solutions of the Yang-Baxter equation based on the finite-dimensional representations of the elliptic modular double. The same result is also derived using the fusion formalism.

math-ph↗
arXiv2015

Iterative construction of eigenfunctions of the monodromy matrix for SL(2,C) magnet

Eigenfunctions of the matrix elements of the monodromy matrix provide a convenient basis for studies of spin chain models. We present an iterative method for constructing the eigenfunctions in the case of the SL(2,C) spin chains. We derived an explicit integral representation for the eigenfunctions and calculated the corresponding scalar products (Sklyanin's measure).

math-ph↗
arXiv2015

Finite-dimensional representations of the elliptic modular double

We investigate the kernel space of an integral operator M(g) depending on the "spin" g and describing an elliptic Fourier transformation. The operator M(g) is an intertwiner for the elliptic modular double formed from a pair of Sklyanin algebras with the parameters $η$ and $τ$, Im$ τ>0$, Im$η>0$. For two-dimensional lattices $g=nη+ mτ/2$ and $g=1/2+nη+ mτ/2$ with incommensurate $1, 2η,τ$ and integers $n,m>0$, the operator M(g) has a finite-dimensional kernel that consists of the products of theta functions with two different modular parameters and is invariant under the action of generators of the elliptic modular double.

math.QA↗
arXiv2014

Integrability of the evolution equations for heavy-light baryon distribution amplitudes

We consider evolution equations describing the scale dependence of the wave function of a baryon containing an infinitely heavy quark and a pair of light quarks at small transverse separations, which is the QCD analogue of the helium atom.The evolution equations depend on the relative helicity of the light quarks. For the aligned helicities, we find that the equation is completely integrable, that is it has a nontrivial integral of motion, and obtain exact analytic expressions for the eigenfunctions and the anomalous dimensions. The evolution equation for anti-aligned helicities contains an extra term that breaks integrability and creates a "bound state" with the anomalous dimension separated from the rest of the spectrum by a finite gap. The corresponding eigenfunction is found using numerical methods. It describes the momentum fraction distribution of the light quarks in, e.g., $Λ_b$-baryon at large scales.

hep-ph↗
arXiv2014

Quantum mechanics of null polygonal Wilson loops

Scattering amplitudes in maximally supersymmetric gauge theory are dual to super-Wilson loops on null polygonal contours. The operator product expansion for the latter revealed that their dynamics is governed by the evolution of multiparticle GKP excitations. They were shown to emerge from the spectral problem of an underlying open spin chain. In this work we solve this model with the help of the Baxter Q-operator and Sklyanin's Separation of Variables methods. We provide an explicit construction for eigenfunctions and eigenvalues of GKP excitations. We demonstrate how the former define the so-called multiparticle hexagon transitions in super-Wison loops and prove their factorized form suggested earlier.

hep-th↗
arXiv2013

Dual conformal symmetry on the light-cone

We study the properties of conformal operators in the SL(2) sector of planar N=4 SYM and its supersymmetric SL(2|2) extension. The correlation functions of these operators and their form factors with respect to asymptotic on-shell states are determined in the appropriate limit by two different polynomials which can be identified as eigenstates of the dilatation operator in the coordinate and momentum representations, respectively. We argue that, in virtue of integrability of the dilatation operator, the two polynomials satisfy a duality relation -- they are proportional to each other upon an appropriate identification of momenta and coordinates. Combined with the conventional N=4 superconformal symmetry, this leads to the dual superconformal symmetry of the dilatation operator. We demonstrate that this symmetry is powerful enough to fix the eigenspectrum of the dilatation operator to the lowest order in the coupling. We use the relation between the one-loop dilatation operator and Heisenberg spin chain to show that, to lowest order in the coupling, the dual symmetry is generated by the Baxter Q-operator in the limit of large spectral parameter.

hep-th↗
arXiv2013

Yang-Baxter equation, parameter permutations, and the elliptic beta integral

We construct an infinite-dimensional solution of the Yang-Baxter equation (YBE) of rank 1 which is represented as an integral operator with an elliptic hypergeometric kernel acting in the space of functions of two complex variables. This R-operator intertwines the product of two standard L-operators associated with the Sklyanin algebra, an elliptic deformation of sl(2)-algebra. It is built from three basic operators $\mathrm{S}_1, \mathrm{S}_2$, and $\mathrm{S}_3$ generating the permutation group of four parameters $\mathfrak{S}_4$. Validity of the key Coxeter relations (including the star-triangle relation) is based on the elliptic beta integral evaluation formula and the Bailey lemma associated with an elliptic Fourier transformation. The operators $\mathrm{S}_j$ are determined uniquely with the help of the elliptic modular double.

math-ph↗
arXiv2013

3j-symbol for the modular double of SL_q(2,R) revisited

Modular double of quantum group SL_q(2,R) with |q|=1 has a series of selfadjoint irreducible representations parameterized by s. Ponsot and Teschner considered a decomposition of the tensor product of two representations into irreducibles. In our paper we give more detailed derivation and some new proofs.

math-ph↗
arXiv2011

Noncompact sl(N) spin chains: BGG-resolution, Q-operators and alternating sum representation for finite dimensional transfer matrices

We study properties of transfer matrices in the sl(N) spin chain models. The transfer matrices with an infinite dimensional auxiliary space are factorized into the product of N commuting Baxter Q-operators. We consider the transfer matrices with auxiliary spaces of a special type (including the finite dimensional ones). It is shown that they can be represented as the alternating sum over the transfer matrices with infinite dimensional auxiliary spaces. We show that certain combinations of the Baxter Q-operators can be identified with the Q-functions which appear in the Nested Bethe Ansatz.

nlin.SI↗
arXiv2009

Factorization of R-matrix and Baxter Q-operators for generic sl(N) spin chains

We develop an approach for constructing the Baxter Q-operators for generic sl(N) spin chains. The key element of our approach is the possibility to represent a solution of the the Yang Baxter equation in the factorized form. We prove that such a representation holds for a generic sl(N) invariant R-operator and find the explicit expression for the factorizing operators. Taking trace of monodromy matrices constructed of the factorizing operators one defines a family of commuting (Baxter) operators on the quantum space of the model. We show that a generic transfer matrix factorizes into the product of N Baxter Q-operators and discuss an application of this representation for a derivation of functional relations for transfer matrices.

nlin.SI↗