Symmetry-based matrix factorization

AbstractWe present a method for factoring a given matrix M into a short product of sparse matrices, provided that M has a suitable “symmetry”. This sparse factorization represents a fast algorithm for the matrix–vector multiplication with M. The factorization method consists of two essential steps. First, a combinatorial search is used to compute a suitable symmetry of M in the form of a pair of group representations. Second, the group representations are decomposed stepwise, which yields factorized decomposition matrices and determines a sparse factorization of M. The focus of this article is the first step, finding the symmetries. All algorithms described have been implemented in the library AREP. We present examples for automatically generated sparse factorizations—and hence fast algorithms—for a class of matrices corresponding to digital signal processing transforms including the discrete Fourier, cosine, Hartley, and Haar transforms

Chiral-odd pion generalized parton distributions beyond leading twist

We define in a systematic way, based on the light-cone collinear factorization method, the chiral-odd generalized parton distributions (GPDs) of a pseudoscalar hadron (such as the pi0) up to twist 5. For that, we introduce the relevant matrix elements for 2-parton non-local operators, as well as matrix elements for 3-parton non-local correlators. Their detailed parametrization is fixed based on parity, charge conjugation and time reversal invariance. This leads to the introduction of 20 real GPDs, whose symmetry properties are explicitely given. The reduction of these GPDs to a minimal set is performed by the use of constraints provided by QCD equations of motion and rotation on the light-cone. We show that these 20 GPDs can be expressed through 8 GPDs which satisfy 4 integral sum rules. A surprising outcome of this analysis is the fact that, when assuming the vanishing of 3-parton correlators, as in the so-called Wandzura-Wilczek approximation, the whole set of GPDs vanishes.Comment: 27 pages, 4 figures, 1 tabl

Quark and Pole Models of Nonleptonic Decays of Charmed Baryons

Quark and pole models of nonleptonic decays of charmed baryons are analysed from the point of view of their symmetry properties. The symmetry structure of the parity conserving amplitudes that corresponds to the contribution of the ground-state intermediate baryons is shown to differ from the one hitherto employed in the symmetry approach. It is pointed out that the "subtraction" of sea quark effects in hyperon decays leads to an estimate of $W$-exchange contributions in charmed baryon decays that is significantly smaller than naively expected on the basis of $SU(4)$. An $SU(2)_{W}$ constraint questioning the reliability of the factorization technique is exhibited. Finally, a successful fit to the available data is presented.Comment: 25 pages, LATEX, 1643/PH IFJ Krako

Nested algebraic Bethe ansatz for open GL(N) spin chains with projected K-matrices

We consider an open spin chain model with GL(N) bulk symmetry that is broken to GL(M) x GL(N-M) by the boundary, which is a generalization of a model arising in string/gauge theory. We prove the integrability of this model by constructing the corresponding commuting transfer matrix. This construction uses operator-valued "projected" K-matrices. We solve this model for general values of N and M using the nested algebraic Bethe ansatz approach, despite the fact that the K-matrices are not diagonal. The key to obtaining this solution is an identity based on a certain factorization property of the reduced K-matrices into products of R-matrices. Numerical evidence suggests that the solution is complete.Comment: 27 pages; v2: minor change; v3: references adde

Factorization and Criticality in the Anisotropic XY Chain via Correlations

In this review, we discuss the zero and finite temperature behavior of various bipartite quantum and total correlation measures, the skew information-based quantum coherence, and the local quantum uncertainty in the thermal ground state of the one-dimensional anisotropic XY model in transverse magnetic field. We compare the ability of considered measures to correctly detect or estimate the quantum critical point and the non-trivial factorization point possessed by the spin chain.Comment: 29 pages, 8 figures. A review paper accepted for publication in the special issue Entanglement Entropy in the journal Entrop

Non-Leptonic Weak Decays of B Mesons

We present a detailed study of non-leptonic two-body decays of B mesons based on a generalized factorization hypothesis. We discuss the structure of non-factorizable corrections and present arguments in favour of a simple phenomenological description of their effects. To evaluate the relevant transition form factors in the factorized decay amplitudes, we use information extracted from semileptonic decays and incorporate constraints imposed by heavy-quark symmetry. We discuss tests of the factorization hypothesis and show how unknown decay constants may be determined from non-leptonic decays. In particular, we find f_{Ds}=(234+-25) MeV and f_{Ds*}=(271+-33) MeV.Comment: two references added and one entry in Table 9 corrected; to appear in the Second Edition of "Heavy Flavours", edited by A.J. Buras and M. Lindner (World Scientific, Singapore