QCD sum rules,the spontaneous breakdown of chiral symmetry and short-distance behaviour in lattice gauge theories

We analyze the behaviour that correlation functions ought to have on the lattice in order to reproduce QCD sum rules in the continuum limit. We formulate a set of relations between lattice correlation functions of meson operators at small time separation and the quark condensates responsible for spontaneous breakdown of chiral symmetery. We suggest that the degree to which such relations are satisfied will provide a set of consistency checks on the ability of lattice Monte Carlo simulations to reproduce the correct spontaneous chiral symmetry breaking of the continuum theory.     

On the linear chain with arbitrary masses and force constants. II. Ergodic properties

The ergodic properties of linear and quadratic phase functions of the classical linear chain are studied for the uniform statistical distributions on the energy surface and in the manifold belonging to fixed values of the energy and the total momentum. This is done for the finite chain by using the time dependent correlation functions studied in a previous paper¹). The thermodynamic limit is also discussed. As an example, sufficient conditions on the masses and force constants are given to ensure that the kinetic energy of a certain particle remains nonergodic in the thermodynamic limit, the conditions defining a non-exceptional set of chains.     

The extended Hubbard model in the ionic limit

In this paper, we study the Hubbard model with intersite Coulomb interaction in the ionic limit (i.e. no kinetic energy). It is shown that this model is isomorphic to the spin-1 Ising model in presence of a crystal field and an external magnetic field. We show that for such models it is possible to find, for any dimension, a finite complete set of eigenoperators and eigenvalues of the Hamiltonian. Then, the hierarchy of the equations of motion closes and analytical expressions for the relevant Green's functions and correlation functions can be obtained. These expressions are formal because these functions depend on a finite set of unknown parameters, and only a set of exact relations among the correlation functions can be derived. In the one-dimensional case we show that by means of algebraic constraints it is possible to obtain extra equations which close the set and allow us to obtain a complete exact solution of the model. The behavior of the relevant physical properties for the 1D system is reported.     

Pfaffian pairing wave functions in electronic-structure quantum Monte Carlo simulations

We investigate the accuracy of trial wave functions for quantum Monte Carlo based on Pfaffian functional form with singlet and triplet pairing. Using a set of first row atoms and molecules we find that these wave functions provide very consistent and systematic behavior in recovering the correlation energies on the level of 95%. In order to get beyond this limit we explore the possibilities of multi-Pfaffian pairing wave functions. We show that a small number of Pfaffians recovers another large fraction of the missing correlation energy comparable to the larger-scale configuration interaction wave functions. We also find that Pfaffians lead to substantial improvements in fermion nodes when compared to Hartree-Fock wave functions.     

Unitarity relation for deep inelastic scattering

Saturating the multiparticle states in the unitarity relation for virtual Compton scattering by two-particle states consisting of an infinite set of high mass vector mesons and a nucleon leads to a non-linear integral equation the solution of which determines the behaviour in the scaling limit and the non-forward scaling functions for deep inelastic scattering.     

关于光学系统中空间关联函数的位相恢复问题

本文把文献[1]中提出的振幅和位相恢复问题的一般表述用于处理空间关联函数位相的恢复问题,类似于文献[1],得到了一组确定位相的方程,用该方程组和迭代求解法,可以求解空间关联函数的位相,但本文不假定传输矩阵R是幺正的。可以证明,如果R是幺正矩阵,在相干极限下,本文的方程组回到文献[1]中的相应方程。 关键词：     

The Hierarchy Principle and the Large Mass Limit of the Linear Sigma Model

In perturbation theory we study the matching in four dimensions between the linear sigma model in the large mass limit and the renormalized nonlinear sigma model in the recently proposed flat connection formalism. We consider both the chiral limit and the strong coupling limit of the linear sigma model. Our formalism extends to Green functions with an arbitrary number of pion legs, at one loop level, on the basis of the hierarchy as an efficient unifying principle that governs both limits. While the chiral limit is straightforward, the matching in the strong coupling limit requires careful use of the normalization conditions of the linear theory, in order to exploit the functional equation and the complete set of local solutions of its linearized form.     

On the thermodynamic size limit of nanowires grown by the vapor-liquid-solid process

For nanowires grown by the vapor-liquid-solid (VLS) process, expressions of the thermodynamically allowed minimum sizes of the nanowire and the liquid phase droplet as functions of the relevant thermodynamic variables have been obtained using Si nanowires (SiNW) grown from metal-silicon (M-Si) systems as the model case. In these expressions the binary nature of the M-Si system, which involves four phases of materials, is accounted for. The liquid droplet minimum size is determined by a unique set of the external M and Si vapor phase pressure values. The SiNW minimum size expression contains two contributions, one due to the liquid droplet composition and one due to the droplet size. These expressions do not predict a limit on the attainable VLS SiNW minimum size, implying ever smaller SiNW can be grown until reaching some growth kinetic limit which is presently unknown. A set of size data of the smallest experimentally grown SiNW appears to have approached an effective limit set by the liquid composition. PACS 81.07b; 61.46+w; 61.25Mv     

Dynamics of Many-Body Correlation Green's Functions Ⅱ. Equal Time Limit

Under equal time limit,it is shown that the dynamic equations of the n-body correlation Green's functions G(n)c leads to the set of equations for time evolution of the n-body correlations c_n and the many-body correlation Green's functions and their equations of motion are independent of the time order.     

Reflection positivity and graviton doubling in Euclidean lattice gravity

Lattice Euclidean quantum gravity, formulated in analogy to the usual gauge theories, is considered. We prove that positivity is satisfied only for a special set of quantities which, however, in the continuum limit furnish the expectation values of the closed loops and correlation functions thereof. We work out the perturbative limit by expanding around a flat background in order to examine the particle content of the theory. A doubling phenomenon (analogous to the doubling of lattice chiral fermions) appears; such a phenomenon is shown to be of general nature for a class of lattice formulations of gravity.     

A level set method for the semiclassical limit of the Schrödinger equation with discontinuous potentials

We propose a level set method for the semiclassical limit of the Schrödinger equation with discontinuous potentials. The discontinuities in the potential corresponds to potential barriers, at which incoming waves can be partially transmitted and reflected. Previously such a problem was handled by Jin and Wen using the Liouville equation – which arises as the semiclassical limit of the Schrödinger equation – with an interface condition to account for partial transmissions and reflections (S. Jin, X. Wen, SIAM J. Num. Anal. 44 (2006) 1801–1828). However, the initial data are Dirac-delta functions which are difficult to approximate numerically with a high accuracy. In this paper, we extend the level set method introduced in (S. Jin, H. Liu, S. Osher, R. Tsai, J. Comp. Phys. 210 (2005) 497–518) for this problem. Instead of directly discretizing the Delta functions, our proposed method decomposes the initial data into finite sums of smooth functions that remain smooth in finite time along the phase flow, and hence can be solved much more easily using conventional high order discretization schemes.     

The replica method and Toda lattice equations for QCD3

We consider the ε-regime of QCD in 3 dimensions. It is shown that the leading term of the effective partition function satisfies a set of Toda lattice equations, recursive in the number of flavors. Taking the replica limit of these Toda equations allows us to derive the microscopic spectral correlation functions for the QCD Dirac operator in 3 dimensions. For an even number of flavors we reproduce known results derived using other techniques. In the case of an odd number of flavors the theory has a severe sign problem, and we obtain previously unknown microscopic spectral correlation functions.     

Fractal Strings and Multifractal Zeta Functions

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.     

On solving coupled equations for scattering problems under the many channel and short wavelength conditions of heavy ion collisions

An iterative method for solving coupled equations in scattering problems is presented. Radical wave functions are expressed in terms of amplitudes on regular and outgoing components. This is analogous to the semi-quantal approach and gives many of its convenient features for heavy ion collisions. One solves directly for the scattering matrix without generating a complete set of solutions. The amplitudes are smoothly varying compared to the total wave functions. The iteration is non-perturbative, as each step contains the interaction to all orders. The convergence improves as the wavelength of relative motion decreases. In the short wavelength limit only one iteration is required.     

Accurately solving the electronic Schrödinger equation of atoms and molecules using explicitly correlated (r 12-)MR-CI IV. The helium dimer (He 2)

The helium dimer interaction potential is computed using the recently proposed (explicitly correlated) r ₁₂-MR-ACPF (averaged coupled-pair functional) method and a [11s8p6d5f4g] basis set. With an MR-ACPF ansatz that contains 121 references we obtain interaction energies that are close to full CI. In a smaller reference space containing 9 functions, however, even by successively adding [3h] and [2i] functions to the basis set mentioned above, the basis set limit could not be reached. While convergence to the basis set limit is slow, it nevertheless is monotonic and therefore allows for extrapolation to the limit. We obtain basis set corrections at R = 4 a ₀ and 5.6 a ₀ which we further extend to all distances and which we apply to the potential energy curve mentioned above. From our calculations, we conclude that a very recent potential which has been calculated using the SAPT (symmetry adapted perturbation theory) method, and which previously was assumed to be the most accurate available, is insufficiently repulsive at short distances. We correct our extrapolated potential for retardation and finally calculate the expectation value of the interatomic distance (?R?) and dissociation energy (D ₀) by solving the Schrödinger equation of the vibrating ⁴He₂. Our results (?R? = 41 ± 13 Å and D ₀ = 2.2 ± 1.0 mK) are in acceptable agreement with very recent calculations in the literature, but they disagree with a recent experiment.     

Small-distance-behaviour analysis and Wilson expansions

A previously described method to obtain the asymptotic forms of vertex functions at large momenta is, with the help of Wilson operator product expansion formulas, extended to momenta where the vertex functions of the zero-mass theory underlying the asymptotic forms are infrared singular. To obtain from asymptotic forms information on asymptotic behaviour requires assumptions on the behaviour of the zero-mass theory in the limit of infinite dilatation. One particular set of assumptions is discussed and found to pass a simple consistency test; this set of assumptions leads to power laws, or slight modifications thereof, with coupling-constant-independent exponents. The detailed discussion is given for the ⁴ model.     

On the homogeneization of transverse gas discharges by preionization

Detailed data of the minimum preionization electron density (MPED)n ₀ necessary to initiate a homogeneous discharge mode in transverse gas discharges are computed. An upper limit for the inductance of the discharge circuit is set by the power balance during breakdown. The results are presented as functions of generalized parameters. Applicability of the computations is proved by experimental observations.     

Hadamard design and artificial neural nets

Hadamard theory is shown to play an important role in the generation of Boolean decision functions, a fundamental tool in the field of artificial neural network design. Based on a group-theoretic introduction of a complete set of Hadamard vectors, whose matrices are of the order of a power of two, we classify subsets according to the degree of their linear dependence. We show in the thermodynamic limit that essentially the whole Hadamard space is occupied by representatives with defect not exceeding two or three.     

Disk level S-matrix elements at eikonal Regge limit

We examine the calculation of the color-ordered disk level S-matrix element of massless scalar vertex operators for the special case that some of the Mandelstam variables for which there are no open string channel in the amplitude, are set to zero. By explicit calculation we show that the string form factors in the 2n-point functions reduce to one at the eikonal Regge limit.