UA(1) anomaly and $ \eta^{{\prime}}_{}$ -mass from an infrared singular quark-gluon vertex

The U _A(1) problem of QCD is inevitably tied to the infrared behaviour of quarks and gluons with its most visible effect being the -mass. A dimensional argument of Kogut and Susskind showed that the mixing of the pseudoscalar flavour-singlet mesons with gluons can provide a screening of the Goldstone pole in this channel if the full quark-quark interaction is strongly infrared singular as ∼ 1/k ⁴ . We investigate this idea using previously obtained results for the Landau gauge ghost and gluon propagator, together with recent determinations for the singular behaviour of the quark-gluon vertex. We find that, even with an infrared vanishing gluon propagator, the singular structure of the quark-gluon vertex for certain kinematics is apposite for yielding a non-zero screening mass.     

Infrared Features of Unquenched Lattice Landau Gauge QCD

The running coupling and the Kugo-Ojima parameter of unquenched lattice Landau gauge are simulated and compared with the continuum theory. Although the running coupling measured by the ghost and gluon dressing function is infrared suppressed, the running coupling has a maximum of α₀ ∼ 2 − 2.5 at around q = 0.5 GeV irrespective of the fermion actions (Wilson fermions and Kogut-Susskind (KS) fermions). The Kugo-Ojima parameter c which saturated to about 0.8 in quenched simulations becomes consistent with 1 in the MILC configurations produced with the use of the Asqtad action, after averaging the dependence on polarization directions caused by the asymmetry of the lattice. The presence of the correction factor 1 + c ₁/q ² in the running coupling depends on the lattice size and the sea quark mass. In the large lattice size and small sea quark mass, c ₁ is confirmed of the order of a few GeV. The MILC configuration of a = 0.09 fm suggests also the presence of dimension-4 condensates with a sign opposite to the dimension-2 condensates. The gluon propagator, the ghost propagator, and the running coupling are compared with recent pQCD results including an anomalous dimension of fields up to the four-loop level.     

Diffusion Limited Aggregation on a Cylinder

We consider the DLA process on a cylinder . It is shown that this process “grows arms”, provided that the base graph G has small enough mixing time. Specifically, if the mixing time of G is at most , the time it takes the cluster to reach the m ^th layer of the cylinder is at most of order . In particular we get examples of infinite Cayley graphs of degree 5, for which the DLA cluster on these graphs has arbitrarily small density. In addition, we provide an upper bound on the rate at which the “arms” grow. This bound is valid for a large class of base graphs G, including discrete tori of dimension at least 3. It is also shown that for any base graph G, the density of the DLA process on a G-cylinder is related to the rate at which the arms of the cluster grow. This implies that for any vertex transitive G, the density of DLA on a G-cylinder is bounded by 2/3.     

A Negative Mass Theorem for the 2-Torus

Let M be a closed surface. For a metric g on M, denote the area element by dA and the Laplace-Beltrami operator by Δ = Δ _g . We define the Robin mass m(p) at the point to be the value of the Green function G(p, q) at q = p after the logarithmic singularity has been subtracted off, and we define trace . This regularized trace can also be obtained by regularization of the spectral zeta function and is hence a spectral invariant which heuristically measures the total wavelength of the surface.We define the Δ-mass of (M, g) to equal , where is the Laplacian on the round sphere of area A. This scale invariant quantity is a non-trivial analog for closed surfaces of the ADM mass for higher dimensional asymptotically flat manifolds.In this paper we show that in each conformal class for the 2-torus, there exists a metric with negative Δ-mass. From this it follows that the minimum of the Δ-mass on is negative and attained by some metric . For this minimizing metric g, one gets a sharp logarithmic Hardy-Littlewood-Sobolev inequality and an Onofri-type inequality.We remark that if the flat metric in is sufficiently long and thin then the minimizing metric g is non-flat. The proof of our result depends on analyzing the ordinary differential equation which is equivalent to h′′ = 1 − 1/h. The solutions are periodic and we need to establish quite delicate, asymptotically sharp inequalities relating the period to the maximum value. The author was supported by the National Science Foundation #DMS-0302647.     

General theory of renormalization of gauge theories in nonlinear gauges

We discuss the general theory of renormalization of unbroken gauge theories in the nonlinear gauges in which the gauge-fixing term is of the form We show that higher loop renormalization modifiesfα [A] to contain ghost terms of the form and show how the corresponding ghost terms are deduced fromfα [A, c, c] uniquely. We show that the theory can be renormalized while preserving a modified form of BRS invariance by multiplicative and independent renormalizations onA, c, g, η, ζ, τ. We briefly discuss the independence of the renormalized S-matrix from η,ζ, τ.     

One-and-a-Half Quantum de Finetti Theorems

When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation of the unitary group U(d), for highest weights μ, ν and μ + ν. Let ξ_μ be the state obtained by tracing out U _ν. Then ξ_μ is close to a convex combination of the coherent states , where and is the highest weight vector in U _μ. For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our “half” theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general.     

Moment Bounds for the Smoluchowski Equation and their Consequences

We prove bounds on moments of the Smoluchowski coagulation equations with diffusion, in any dimension d ≥ 1. If the collision propensities α(n, m) of mass n and mass m particles grow more slowly than , and the diffusion rate is non-increasing and satisfies for some b ₁ and b ₂ satisfying 0 ≤ b ₂ < b ₁ < ∞, then any weak solution satisfies for every and T ∈(0, ∞), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions are sufficient to ensure uniqueness of a weak solution and its conservation of mass. This work was performed while A.H. held a postdoctoral fellowship in the Department of Mathematics at U.B.C. This work is supported in part by NSF grant DMS0307021.     

Propagator of the Lattice Domain Wall Fermion and the Staggered Fermion

We calculate the propagator of the domain wall fermion (DWF) of the RBC/UKQCD collaboration with 2 + 1 dynamical flavors of 16³ × 32 × 16 lattice in Coulomb gauge, by applying the conjugate gradient method. We find that the fluctuation of the propagator is small when the momenta are taken along the diagonal of the 4-dimensional lattice. Restricting momenta in this momentum region, which is called the cylinder cut, we compare the mass function and the running coupling of the quark-gluon coupling α _s,g1(q) with those of the staggered fermion of the MILC collaboration in Landau gauge. In the case of DWF, the ambiguity of the phase of the wave function is adjusted such that the overlap of the solution of the conjugate gradient method and the plane wave at the source becomes real. The quark-gluon coupling α _s,g1(q) of the DWF in the region q > 1.3 GeV agrees with ghost-gluon coupling α _s (q) that we measured by using the configuration of the MILC collaboration, i.e., enhancement by a factor (1 + c/q ²) with c ≃ 2.8 GeV² on the pQCD result. In the case of staggered fermion, in contrast to the ghost-gluon coupling α _s (q) in Landau gauge which showed infrared suppression, the quark-gluon coupling α _s,g1(q) in the infrared region increases monotonically as q→ 0. Above 2 GeV, the quark-gluon coupling α _s,g1(q) of staggered fermion calculated by naive crossing becomes smaller than that of DWF, probably due to the complex phase of the propagator which is not connected with the low energy physics of the fermion taste. An erratum to this article can be found at     

Spectral Measure of Heavy Tailed Band and Covariance Random Matrices

We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N × N symmetric matrix whose (i, j) entry is , where (x _ij , 1 ≤ i ≤ j < ∞) is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an α-stable law, , and σ is a deterministic function. For random diagonal D _N independent of and with appropriate rescaling a _N , we prove that converges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries. Supported in part by a Discovery grant from the Natural Sciences and Engineering Research Council of Canada and a University of Saskatchewan start-up grant. Research partially supported by NSF grant #DMS-0806211.     

Four-quark states and nucleon-antinucleon annihilation within the quark model with QCD vacuum-induced interaction

The spectrum of , J^p=0⁺, 2⁺ mesons is discussed. We have shown that due to instanton-induced forces the physical states are strong mixtures of theSU _f (3) group basis states. The cross-sections for annihilation of the system into mesons are obtained. Thea ₀(980) meson is considered as meson consisting of 9 _f and 36 _f plets. The branchings are also predicted for the cross-sections for production of thea ₀(980) and tensor mesons in annihilation.     

Distribution of Particles Which Produces a “Smart” Material

If A _q(β, α, k) is the scattering amplitude, corresponding to a potential , where D⊂ℝ³ is a bounded domain, and is the incident plane wave, then we call the radiation pattern the function , where the unit vector α, the incident direction, is fixed, β is the unit vector in the direction of the scattered wave, and k>0, the wavenumber, is fixed. It is shown that any function , where S ² is the unit sphere in ℝ³, can be approximated with any desired accuracy by a radiation pattern: , where ∊ >0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ∊, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles D _m⊂ D, 1≤ m≤ M, distributed in an a priori given bounded domain D⊂ℝ³. The geometrical shape of a small particle D _m is arbitrary, the boundary S _m of D _m is Lipschitz uniformly with respect to m. The wave number k and the direction α of the incident upon D plane wave are fixed. It is shown that a suitable distribution of the above particles in D can produce the scattering amplitude , at a fixed k>0, arbitrarily close in the norm of L ²(S ²× S ²) to an arbitrary given scattering amplitude f(α ', α), corresponding to a real-valued potential q∊ L ²(D), i.e., corresponding to an arbitrary refraction coefficient in D. MSC: 35J05, 35J10, 70F10, 74J25, 81U40, 81V05, 35R30. PACS: 03.04.Kf.     

Phase Transition in the 1d Random Field Ising Model with Long Range Interaction

We study one–dimensional Ising spin systems with ferromagnetic, long–range interaction decaying as n ^−2+α , , in the presence of external random fields. We assume that the random fields are given by a collection of symmetric, independent, identically distributed real random variables, gaussian or subgaussian. We show, for temperature and strength of the randomness (variance) small enough, with IP = 1 with respect to the random fields, that there are at least two distinct extremal Gibbs measures. Supported by: GDRE 224 GREFI-MEFI, CNRS-INdAM. P.P was also partially supported by INdAM program Professori Visitatori 2007; M.C and E.O were partially supported by Prin07: 20078XYHYS.     

Large number hypothesis and the matter-dominated universe

Dirac’s large number hypothesis (LNH), in the formG/G ₀=HH ₀ ⁻¹ , is applied to the matter-dominated cosmological era, using the framework of the scale covariant theory (Canuto et al., 1977). We obtain explicit expressions forR andβ _a as functions ofR _E , whereR andR _E are the scale factors of the cosmological Robertson-Walker metric, expressed in atomic and gravitational units, respectively, andβ _a is the ratio between the rates of gravitational and atomic clocks. The parameters in these expressions are , the deceleration parameter in gravitational units, and (t ₀)H ₀ ⁻¹ where (t ₀) is the present epoch value of the derivative ofβ _a with respect to atomic time. We find that a necessary condition for the LNH to be compatible with a Robertson-Walker model is that (t ₀)H ₀ ⁻¹ ≥ ₂ ¹ . The only experimental values for (t ₀) available at present are those based on the lengthening of the Moon’s period of revolution around the Earth, suggesting 0.86≥ (t ₀)H ₀ ⁻¹ ≥0.21; the more promising technique of radar ranging to the inner planets has not yet produced a value for (t ₀). Using the lunar data, it follows that 0≤ ≲0.42 corresponding to an open universe (k=−1). Closed models (k=1, >1/2) are not compatible with the LNH since the required values of (t ₀)H ₀ ⁻¹ are more than an order of magnitude above the observational upper limit. Presented at the Dirac Symposium, Loyola University, New Orleans, 1981.     

Harmonic Bilocal Fields Generated by Globally Conformal Invariant Scalar Fields

The twist two contribution in the operator product expansion of for a pair of globally conformal invariant, scalar fields of equal scaling dimension d in four space–time dimensions is a field V ₁ (x₁, x₂) which is harmonic in both variables. It is demonstrated that the Huygens bilocality of V ₁ can be equivalently characterized by a “single–pole property” concerning the pole structure of the (rational) correlation functions involving the product . This property is established for the dimension d = 2 of . As an application we prove that any system of GCI scalar fields of conformal dimension 2 (in four space–time dimensions) can be presented as a (possibly infinite) superposition of products of free massless fields.     

Infrared gluon and ghost propagator exponents from lattice QCD

The compatibility of the pure power law infrared solution of QCD and lattice data for the gluon and ghost propagators in Landau gauge is discussed. For the gluon propagator, the lattice data are well described by a pure power law with an infrared exponent κ∼0.53, in the Dyson–Schwinger notation. κ is measured using a technique that suppresses finite volume effects. This value is consistent with a vanishing zero momentum gluon propagator, in agreement with the Gribov–Zwanziger confinement scenario. For the ghost propagator, the lattice data seem not to follow a pure power law, at least for the range of momenta accessed in our simulation.     

Effect of a magnetic field on the yield point of NaCl crystals

A constant magnetic field is found to have a substantial effect on the macroplasticity of NaCl crystals when they are being actively strained at a constant rate during magnetic treatment. We have measured the dependence of the yield point σ _y on the magnetic induction B=0–0.48 T and the strain rate . It is shown that this magnetic effect has a threshold character and is observed only for B>B _c , where B _c grows with increasing as . The lower the strain rate , the larger the relative decrease in the yield point σ _y (B)/σ _y (0) at fixed field B>B _c . At small enough strain rates the threshold field B _c ceases to depend on and goes constant. A theoretical model is proposed which is in good agreement with the observed regularities. The model is based on the competition between thermally activated and magnetically stimulated depinning of dislocations from paramagnetic impurity centers. Zh. éksp. Teor. Fiz. 115, 951–958 (March 1999)     

Role of Fe,Mn, and Zn ions as dopants on the electrical conductivity behavior of sodium phosphate glass

Sodium phosphate glass undoped and doped with different concentrations of chlorides of iron, manganese, and zinc were prepared by melt quenching. The synthesized glasses were characterized by elemental analysis, X-ray diffraction, infrared (IR) spectroscopy, differential scanning calorimetry, and electrical conductivity studies. The undoped sodium phosphate glass (Na₂O–P₂O₅) has low electrical conductivity σ compared to all doped glasses except for 10% FeCl₃-doped samples for which σ is found to be the lowest, and the trend is

On the Global Wellposedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations (NS _ν) with initial data in the scaling invariant Besov space, here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations (ANS _ν), where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, and Then with initial data in the scaling invariant space we prove the global wellposedness for (ANS _ν) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (ANS _ν) with high oscillatory initial data (1.2).     

The Spectrum of Heavy Tailed Random Matrices

Let X _N be an N → N random symmetric matrix with independent equidistributed entries. If the law P of the entries has a finite second moment, it was shown by Wigner [14] that the empirical distribution of the eigenvalues of X _N , once renormalized by , converges almost surely and in expectation to the so-called semicircular distribution as N goes to infinity. In this paper we study the same question when P is in the domain of attraction of an α-stable law. We prove that if we renormalize the eigenvalues by a constant a _N of order , the corresponding spectral distribution converges in expectation towards a law which only depends on α. We characterize and study some of its properties; it is a heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero. This work was partially supported by Miller institute for Basic Research in Science, University of California Berkeley.