Alexander N. Manashov

Baryon distribution amplitudes in QCD

Abstract We develop a new theoretical framework for the description of leading twist light-cone baryon distribution amplitudes which is based on integrability of the helicity λ = 3 2 evolution equation to leading logarithmic accuracy. A physical interpretation is that one can identify a new ‘hidden’ quantum number which distinguishes components in the λ = 3 2 distribution amplitudes with different scale dependence. The solution of the corresponding evolution equation is reduced to a simple three-term recurrence relation. The exact analytic solution is found for the component with the lowest anomalous dimension for all moments N, and the WKB-type expansion is constructed for other levels, which becomes asymptotically exact at large N. Evolution equations for the λ = 1 2 distribution amplitudes (e.g. for the nucleon) are studied as well. We find that the two lowest anomalous dimensions for the λ = 1 2 operators (one for each parity) are separated from the rest of the spectrum by a finite ‘mass gap’. These special states can be interpreted as scalar diquarks.

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.Abstract 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.

Baxter Bbb Q-operator and separation of variables for the open SL(2,Bbb R) spin chain

We construct the Baxter -operator and the representation of the Separated Variables (SoV) for the homogeneous open SL(2,) spin chain. Applying the diagrammatical approach, we calculate Sklyanins integration measure in the separated variables and obtain the solution to the spectral problem for the model in terms of the eigenvalues of the -operator. We show that the transition kernel to the SoV representation is factorized into the product of certain operators each depending on a single separated variable. As a consequence, it has a universal pyramid-like form that has been already observed for various quantum integrable models such as periodic Toda chain, closed SL(2,) and SL(2,) spin chains.We construct the Baxter Q-operator and the representation of the Separated Variables (SoV) for the homogeneous open SL(2,R) spin chain. Applying the diagrammatical approach, we calculate Sklyanins integration measure in the separated variables and obtain the solution to the spectral problem for the model in terms of the eigenvalues of the Q-operator. We show that the transition kernel to the SoV representation is factorized into the product of certain operators each depending on a single separated variable. As a consequence, it has a universal pyramid-like form that has been already observed for various quantum integrable models such as periodic Toda chain, closed SL(2,R) and SL(2,C) spin chains.

Superconformal operators in N=4 superYang-Mills theory

We construct, in the framework of the N=4 SYM theory, a supermultiplet of twist-two conformal operators and study their renormalization properties. The components of the supermultiplet have the same anomalous dimension and enter as building blocks into multi-particle quasipartonic operators. The latter are determined by the condition that their twist equals the number of elementary constituent fields from which they are built. A unique feature of the N=4 SYM is that all quasipartonic operators with different SU(4) quantum numbers fall into a single supermultiplet. Among them there is a subsector of the operators of maximal helicity, which has been known to be integrable in the multi-color limit in QCD, independent of the presence of supersymmetry. In the N=4 SYM theory, this symmetry is extended to the whole supermultiplet of quasipartonic operators and the one-loop dilatation operator coincides with a Hamiltonian of integrable SL(2|4) Heisenberg spin chain.

Separation of variables for the quantum SL(2,Bbb R) spin chain

We construct representation of the Separated Variables (SoV) for the quantum SL(2,R) Heisenberg closed spin chain and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the samepyramid diagramas appeared before in the SoV representation for the SL(2,C) spin magnet. We argue that this kernel is given by the product of the Baxter Q-operators projected onto a special reference state.We construct a representation of the Separated Variables (SoV) for the quantum SL(2,) Heisenberg closed spin chain following the Sklyanins approach and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same ``pyramid diagram as appeared before in the SoV representation for the SL(2,) spin magnet. We argue that this kernel is given by the product of the Baxter -operators projected onto a special reference state.

Gluon contribution to the structure function g2(x,Q2)

We calculate the one-loop twist-3 gluon contribution to the flavor-singlet structure function g2(x,Q2) in polarized deep-inelastic scattering and find that it is dominated by the contribution of the three-gluon operator with the lowest anomalous dimension (for each moment N). The similar property was observed earlier for the nonsinglet distributions, although the reason is in our case different. The result is encouraging and suggests a simple evolution pattern of g2(x,Q2) in analogy with the conventional description of twist-2 parton distributions.We calculate the one-loop twist-3 gluon contribution to the flavor-singlet structure function g_2(x,Q^2) in polarized deep-inelastic scattering and find that it is dominated by the contribution of the three-gluon operator with the lowest anomalous dimension (for each moment N). The similar property was observed earlier for the nonsinglet distributions, although the reason is in our case different. The result is encouraging and suggests a simple evolution pattern of g_2(x,Q^2) in analogy with the conventional description of twist-2 parton distributions.

Noncompact Heisenberg spin magnets from high-energy QCD III. Quasiclassical approach

The exact solution of the noncompact SL(2,C) Heisenberg spin magnet reveals a hidden symmetry of the energy spectrum. To understand its origin, we solve the spectral problem for the model within quasiclassical approach. In this approach, the integrals of motion satisfy the Bohr-Sommerfeld quantization conditions imposed on the orbits of classical motion. In the representation of the separated coordinates, the latter wrap around a Riemann surface defined by the spectral curve of the model. A novel feature of the obtained quantization conditions is that they involve both the alpha- and beta-periods of the action differential on the Riemann surface, thus allowing us to find their solutions by exploring the full modular group of the spectral curve. We demonstrate that the quasiclassical energy spectrum is in a good agreement with the exact results.Abstract The exact solution of the noncompact SL(2, C ) Heisenberg spin magnet reveals a hidden symmetry of the energy spectrum. To understand its origin, we solve the spectral problem for the model within quasiclassical approach. In this approach, the integrals of motion satisfy the Bohr–Sommerfeld quantization conditions imposed on the orbits of classical motion. In the representation of the separated coordinates, the latter wrap around a Riemann surface defined by the spectral curve of the model. Axa0novel feature of the obtained quantization conditions is that they involve both the α - and β -periods of the action differential on the Riemann surface, thus allowing us to find their solutions by exploring the full modular group of the spectral curve. We demonstrate that the quasiclassical energy spectrum is in a good agreement with the exact results.

Noncompact SL(2,ℝ) spin chain

We consider the integrable spin chain model — the noncompact SL(2,) spin magnet. The spin operators are realized as the generators of the unitary principal series representation of the SL(2,) group. In an explicit form, we construct -matrix, the Baxter -operator and the transition kernel to the representation of the Separated Variables (SoV). The expressions for the energy and quasimomentum of the eigenstates in terms of the Baxter -operator are derived. The analytic properties of the eigenvalues of the Baxter operator as a function of the spectral parameter are established. Applying the diagrammatic approach, we calculate Sklyanins integration measure in the separated variables and obtain the solution to the spectral problem for the model in terms of the eigenvalues of the -operator. We show that the transition kernel to the SoV representation is factorized into a product of certain operators each depending on a single separated variable.

SL(2,C) Gustafson Integrals

It was shown recently that many of the Gustafson integrals appear in studies of the

Superconformal operators inN=4super-Yang-Mills theory

mathrm{SL}(2,mathbb{R})