arXiv subjects
Publications and source records attributed to A. N. Manashov.
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.
Using the approach based on conformal symmetry we calculate the three-loop (NNLO) contribution to the evolution equation for flavor-nonsinglet leading twist operators in the $\overline{\text{MS}}$ scheme. The explicit expression for the three-loop kernel is derived for the corresponding light-ray operator in coordinate space. The expansion in local operators is performed and explicit results are given for the matrix of the anomalous dimensions for the operators up to seven covariant derivatives. The results are directly applicable to the renormalization of the pion light-cone distribution amplitude and flavor-nonsinglet generalized parton distributions.
We present a systematic study of higher-twist distribution amplitudes (DAs) of the B-meson which give rise to power-suppressed $1/m_B$ contributions to B-decays in final states with energetic light particles in the framework of QCD factorization. As the main result, we find that the renormalization group equations for the three-particle distributions are completely integrable in the large $N_c$ limit and can be solved exactly. General properties of the solutions are studied. We propose simple models for higher-twist DAs which satify all existing constraints and can be used in phenomenological studies.
We calculate the anomalous dimensions of higher-spin currents, both singlet and non-singlet, in the Gross-Neveu model at the $1/n^2$ order. It was conjectured that in the critical regime this model is dual to a higher-spin gauge theory on $AdS_4$. The $AdS/CFT$ correspondence predicts that the masses of higher-spin fields correspond to the scaling dimensions of the singlet currents in the Gross-Neveu model.
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.
QCD evolution equations in minimal subtraction schemes have a hidden symmetry: One can construct three operators that commute with the evolution kernel and form an $SL(2)$ algebra, i.e. they satisfy (exactly) the $SL(2)$ commutation relations. In this paper we find explicit expressions for these operators to two-loop accuracy going over to QCD in non-integer $d=4-2ε$ space-time dimensions at the intermediate stage. In this way conformal symmetry of QCD is restored on quantum level at the specially chosen (critical) value of the coupling, and at the same time the theory is regularized allowing one to use the standard renormalization procedure for the relevant Feynman diagrams. Quantum corrections to conformal generators in $d=4-2ε$ effectively correspond to the conformal symmetry breaking in the physical theory in four dimensions and the $SL(2)$ commutation relations lead to nontrivial constraints on the renormalization group equations for composite operators. This approach is valid to all orders in perturbation theory and the result includes automatically all terms that can be identified as due to a nonvanishing QCD $β$-function (in the physical theory in four dimensions). Our result can be used to derive three-loop evolution equations for flavor-nonsinglet quark-antiquark operators including mixing with the operators containing total derivatives. These equations govern, e.g., the scale dependence of generalized hadron parton distributions and light-cone meson distribution amplitudes.
We study higher twist distribution amplitudes for the SU_F(3) baryon octet. We identify independent functions for all baryons in the isospin symmetry limit and calculate the Wandzura-Wilczek contributions to the twist-4 and 5 distributions amplitudes.
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).
We find that the evolution equation for the three-particle quark-gluon B-meson light-cone distribution amplitude (DA) of subleading twist is completely integrable in the large $N_c$ limit and can be solved exactly. The lowest anomalous dimension is separated from the remaining, continuous, spectrum by a finite gap. The corresponding eigenfunction coincides with the contribution of quark-gluon states to the two-particle DA $ϕ_-(ω)$ so that the evolution equation for the latter is the same as for the leading-twist DA $ϕ_+(ω)$ up to a constant shift in the anomalous dimension. Thus, ``genuine'' three-particle states that belong to the continuous spectrum effectively decouple from $ϕ_-(ω)$ to the leading-order accuracy. In turn, the scale dependence of the full three-particle DA turns out to be nontrivial so that the contribution with the lowest anomalous dimension does not become leading at any scale. The results are illustrated on a simple model that can be used in studies of $1/m_b$ corrections to heavy-meson decays in the framework of QCD factorization or light-cone sum rules.
Leading-twist operators have a remarkable property that their divergence vanishes in a free theory. Recently it was suggested that this property can be used for an alternative technique to calculate anomalous dimensions of leading-twist operators and allows one to gain one order in perturbation theory so that, i.e., two-loop anomalous dimensions can be calculated from one-loop Feynman diagrams, etc. In this work we study feasibility of this program on a toy-model example of the $φ^3$ theory in six dimensions. Our conclusion is that this approach is valid, although it does not seem to present considerable technical simplifications as compared to the standard technique. It does provide one, however, with a very nontrivial check of the calculation as the structure of the contributions is very different.
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.
QCD evolution equations in $\text{MS}$-like schemes can be recovered from the same equations in a modified theory, QCD in non-integer $d=4-2ε$ dimensions, which enjoys exact scale and conformal invariance at the critical point. Restrictions imposed by the conformal symmetry of the modified theory allow one to obtain complete evolution kernels in integer (physical) dimensions at the given order of perturbation theory from the spectrum of anomalous dimensions added by the calculation of the special conformal anomaly at one order less. We use this technique to derive two-loop evolution equations for flavor-nonsinglet quark-antiquark light-ray operators that encode the scale dependence of generalized hadron parton distributions.
The existing QCD predictions for the Deeply Virtual Compton Scattering (DVCS) depend on the convention used for the skewedness parameter and on the reference frame used to define helicity amplitudes. These ambiquities are formally power-suppressed but numerically significant. They are cancelled by finite-$t$ and target mass corrections that have been calculated recently to the $1/Q^2$ accuracy. It turns out that these corrections can be minimized, at least for unpolarized observables, by choosing a specific reference frame where longitudinal directions are defined by the photon momenta.
QCD in non-integer d=4-2 epsilon space-time dimensions possesses a nontrivial critical point and enjoys exact scale and conformal invariance. This symmetry imposes nontrivial restrictions on the form of the renormalization group equations for composite operators in physical (integer) dimensions and allows to reconstruct full kernels from their eigenvalues (anomalous dimensions). We use this technique to derive two-loop evolution equations for flavor-nonsinglet quark-antiquark light-ray operators that encode the scale dependence of generalized hadron parton distributions and light-cone distribution amplitudes in the most compact form.
The Lange-Neubert evolution equation describes the scale dependence of the wave function of a meson built of an infinitely heavy quark and light antiquark at light-like separations, which is the hydrogen atom problem of QCD. It has numerous applications to the studies of B-meson decays. We show that the kernel of this equation can be written in a remarkably compact form, as a logarithm of the generator of special conformal transformation in the light-ray direction. This representation allows one to study solutions of this equation in a very simple and mathematically consistent manner. Generalizing this result, we show that all heavy-light evolution kernels that appear in the renormalization of higher-twist B-meson distribution amplitudes can be written in the same form.
We derive the higher twist (four and five) nucleon distribution amplitudes in the Wandzura-Wilczek approximation. Our method is based on an analysis of the conformal expansion of nonlocal operators in the spinor formalism.
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.
We study implications of exact conformal invariance of scalar quantum field theories at the critical point in non-integer dimensions for the evolution kernels of the light-ray operators in physical (integer) dimensions. We demonstrate that all constraints due the conformal symmetry are encoded in the form of the generators of the collinear sl(2) subgroup. Two of them, S_- and S_0, can be fixed at all loops in terms of the evolution kernel, while the generator of special conformal transformations, S_+, receives nontrivial corrections which can be calculated order by order in perturbation theory. Provided that the generator S_+ is known at the k-1 loop order, one can fix the evolution kernel in physical dimension to the k-loop accuracy up to terms that are invariant with respect to the tree-level generators. The invariant parts can easily be restored from the anomalous dimensions. The method is illustrated on two examples: The O(n)-symmetric phi^4 theory in d=4 to the three-loop accuracy, and the su(n) matrix phi^3 theory in d=6 to the two-loop accuracy. We expect that the same technique can be used in gauge theories e.g. in QCD.