Higher Hochschild Homology,Topological Chiral Homology and Factorization Algebras

We study the higher Hochschild functor, factorization algebras and their relationship with topological chiral homology. To this end, we emphasize that the higher Hochschild complex is a functor sSet _∞ × CDGA _∞ where sSet and CDGA _∞ are the (∞,1)-categories of simplicial sets and commutative differential graded algebras, and give an axiomatic characterization of this functor. From the axioms, we deduce several properties and computational tools for this functor. We study the relationship between the higher Hochschild functor and factorization algebras by showing that, in good cases, the Hochschild functor determines a constant commutative factorization algebra. Conversely, every constant commutative factorization algebra is naturally equivalent to a Hochschild chain factorization algebra. Similarly, we study the relationship between the above concepts and topological chiral homology. In particular, we show that on their common domains of definition, the higher Hochschild functor is naturally equivalent to topological chiral homology. Finally, we prove that topological chiral homology determines a locally constant factorization algebra and, further, that this functor induces an equivalence between locally constant factorization algebras on a manifold and (local system of) E _n -algebras.     

Direct CP violation in hadronic B decays

There are different approaches for the hadronic B decay calculations, recently. In this paper, we upgrade three of them, namely factorization, QCD factorization and the perturbative QCD approach based on k _T factorization, using new parameters and full wave functions. Although they get similar results for many of the branching ratios, the direct CP asymmetries predicted by them are different, which can be tested by recent experimental measurements of B factories.     

Nonfactorization in D+→K0K+ Decays

We analyze D⁺→K⁰K⁺decay at the leading order, α s corrections with the QCD factorization approach and the soft-gluon effects with the light cone QCD sum rules. We find the prediction of naive factorization is far from the experimental data, and the QCD factorization result approaches to the experimental data. However, in QCD factorization method, if we consider the soft-gluon effects, then the result is in accordance with the experimental data well. Our calculation shows that the soft-gluon contributions which are firstly calculated in D meson nonleptonic decay are noticeable. So, it can't be neglected in the decay channel.     

Factorization in high energy hadron collisions

The concept of factorization is discussed for elastic diffraction scattering and diffraction dissociation of hadrons at high energy. In addition to the usual definition in terms of the t-channel, a natural definition of factorization in the s-channel is proposed and compared with the former. It is shown that s-channel factorization of all diffractive processes is consistent with the assumption that elastic scattering is identical to the shadow of the diffraction dissociation processes.     

The multiparticle S matrix in two-dimensional space-time models

The factorization of the multiparticle S matrix previously conjectured in various two-dimensional space-time models is established. It includes factorization equations needed for the determination of the two-body S matrix.     

Hard processes in general factorisation scheme

A framework for the formulation, in a general factorization scheme of QCD predictions for physical quantities involving parton distribution and/or fragmentation functions, is suggested. The main advantage of this method, based on the use of Jacobi polynomials and implying a novel parametrization of nonperturbative properties of hadrons, is that it allows an easy and straightforward transformation of the results of phenomenological analyses performed in arbitrary factorization scheme, into any other one. The study of nonsinglet nucleon structure functions furthermore demonstrates that the specification of the factorization convention, i.e. the way parton densities beyond the leading order are defined, may be as important as the choice of the factorization and hard scattering scalesM and μ.     

Landau problem with a general time-dependent electric field

The time evolution is studied for the Landau level problem with a general time dependent electric field E(t) in a plane perpendicular to the magnetic field. A general and explicit factorization of the time evolution operator is obtained with each factor having a clear physical interpretation. The factorization consists of a geometric factor (path-ordered magnetic translation), a dynamical factor generated by the usual time-independent Landau Hamiltonian, and a nonadiabatic factor that determines the transition probabilities among the Landau levels. Since the path-ordered magnetic translation and the nonadiabatic factor are, up to completely determined numerical phase factors, just ordinary exponentials whose exponents are explicitly expressible in terms of the canonical variables, all of the factors in the factorization are explicitly constructed. New quantum interference effects are implied by this result. The factorization is unique from the point of view of the quantum adiabatic theorem and provides a seemingly first rigorous demonstration of how the quantum adiabatic theorem (incorporating the Berry phase phenomenon) is realized when infinitely degenerate energy levels are involved. Since the factorization separates the effect caused by the electric field into a geometric factor and a nonadiabatic factor, it makes possible to calculate the nonadiabatic transition probabilities near the adiabatic limit. A formula for matrix elements that determines the mixing of the Landau levels for a general, nonadiabatic evolution is also provided by the factorization.     

Factorization and infrared properties of non-perturbative contributions to DIS structure functions

In this paper we present a new derivation of QCD factorization. We deduce the k _T and collinear factorizations for the DIS structure functions by consecutive reductions of a more general theoretical construction. We begin by studying the amplitude of forward Compton scattering off a hadron target, representing this amplitude as a set of convolutions of two blobs connected by the simplest, two-parton intermediate states. Each blob in the convolutions can contain both the perturbative and non-perturbative contributions. We formulate conditions for separating the perturbative and non-perturbative contributions and attributing them to the different blobs. After that the convolutions correspond to QCD factorization. Then we reduce this totally unintegrated (basic) factorization first to k _T -factorization and finally to collinear factorization. In order to yield a finite expression for the Compton amplitude, the integration over the loop momentum in the basic factorization must be free of both ultraviolet and infrared singularities. This obvious mathematical requirement leads to theoretical restrictions on the non-perturbative contributions (parton distributions) to the Compton amplitude and the DIS structure functions related to the Compton amplitude through the Optical Theorem. In particular, our analysis excludes the use of the singular factors x ^−a (with a>0) in the fits for the quark and gluon distributions because such factors contradict the integrability of the basic convolutions for the Compton amplitude. This restriction is valid for all DIS structure functions in the framework of both k _T -factorization and collinear factorization if we attribute the perturbative contributions only to the upper blob. The restrictions on the non-perturbative contributions obtained in the present paper can easily be extended to other QCD processes where the factorization is exploited.     

Nonperturbative effects on factorization in high-momentum processes

It is pointed out that the usual demonstrations within QCD perturbation theory of factorization at high-momentum transfers are incomplete when nonperturbative instanton effects are incorporated in a dilute gas approximation. The apparent violation of factorization does not vanish at large Q².     

The Virasoro vertex algebra and factorization algebras on Riemann surfaces

This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello–Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta–gamma system using the method of effective BV quantization.     

Generalized Operator Yang-Baxter Equations,Integrable ODEs and Nonassociative Algebras

Abstract

Reductions for systems of ODEs integrable via the standard factorization method (the Adler-Kostant-Symes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such reductions, operator Yang-Baxter equations, and some kinds of non-associative algebras are established.     

On factorization, quark counting and vector dominance

Using an eikonal structure for the scattering amplitude, Block and Kaidalov [1] have derived factorization theorems for nucleon-nucleon, and scattering at high energies, using only some very general assumptions. We present here an analysis giving experimental confirmation for factorization of cross sections, nuclear slope parameters B and -values (ratio of real to imaginary portion of forward scattering amplitudes), showing that: – the three factorization theorems [1] hold, – the additive quark model holds to , – and vector dominance holds to better than . Received: 6 November 2001 / Revised version: 29 November 2001 / Published online: 8 February 2002     

On transverse-momentum dependent light-cone wave functions of light mesons

Transverse-momentum dependent (TMD) light-cone wave functions of a light meson are important ingredients in the TMD QCD factorization of exclusive processes. This factorization allows one conveniently resume Sudakov logarithms appearing in collinear factorization. The TMD light-cone wave functions are not simply related to the standard light-cone wave functions in collinear factorization by integrating them over the transverse momentum. We explore relations between TMD light-cone wave functions and those in the collinear factorization. Two factorized relations can be found. One is helpful for constructing models for TMD light-cone wave functions, and the other can be used for resummation. These relations will be useful to establish a link between two types of factorization.     

NLO semi-inclusive Drell–Yan cross-section in quantum chromodynamics as a factorization analyzer

We evaluate in perturbative QCD the semi-inclusive Drell–Yan cross-section for the production of a single hadron accompanying the lepton pair. We demonstrate to one loop level a collinear factorization formula within the fracture functions approach. We propose such a process as a factorization analyzer in hadronic collisions. Phenomenological implications at the hadron colliders are briefly discussed.     

Factorization properties of birational mappings

We analyse birational mappings generated by transformations on q × q matrices which correspond respectively to two kinds of transformations: the matrix inversion and a permutation of the entries of the q × q matrix. Remarkable factorization properties emerge for quite general involutive permutations.

It is shown that factorization properties do exist, even for birational transformations associated with noninvolutive permutations of entries of q × q matrices, and even for more general transformation which are rational transformations but no longer birational. The existence of factorization relations independent of q, the size of the matrices, is underlined.

The relations between the polynomial growth of the complexity of the iterations, the existence of recursions in a single variable and the integrability of the mappings, are sketched for the permutations yielding these properties.

All these results show that permutations of the entries of the matrix yielding factorization properties are not so rare. In contrast, the occurrence of recursions in a single variable, or of the polynomial growth of the complexity are, of course, less frequent but not completely exceptional.     

Factorization approach to superintegrable systems: Formalism and applications

The factorization technique for superintegrable Hamiltonian systems is revisited and applied in order to obtain additional (higher-order) constants of the motion. In particular, the factorization approach to the classical anisotropic oscillator on the Euclidean plane is reviewed, and new classical (super) integrable anisotropic oscillators on the sphere are constructed. The Tremblay–Turbiner–Winternitz system on the Euclidean plane is also studied from this viewpoint.

     

Factorization and escorting in the game-theoretical approach to non-extensive entropy measures

The game-theoretical approach to non-extensive entropy measures of statistical physics is based on an abstract measure of complexity from which the entropy measure is derived in a natural way. A wide class of possible complexity measures is considered and a property of factorization, apparently related to escorting, is investigated. It is shown that only those complexity measures which are connected with Tsallis entropy have the factorization property.     

Darboux transformations for a system of coupled discrete Schrödinger equations

Darboux transformations and a factorization procedure are presented for a system of coupled finite-difference Schrödinger equations. The conformity between generalized Darboux transformations and the factorization method is established. Factorization chains and consequences of Darboux transformations are obtained for a system of coupled discrete Schrödinger equations. The proposed approach permits constructing a new series of potential matrices with known spectral characteristics for which coupled-channel discrete Schrödinger equations have exact solutions.