Universality Class of O(N)O(N) Models

We point out that existing numerical data on the correlation length and magnetic susceptibility suggest that the two dimensional $O(3)$ model with standard action has critical exponent $\eta=1/4$, which is inconsistent with asymptotic freedom. This value of $\eta$ is also different from the one of the Wess-Zumino-Novikov-Witten model that is supposed to correspond to the $O(3)$ model at $\theta=\pi$.Comment: 8 pages, with 3 figures included, postscript. An error concerning the errors has been correcte

About the realization of chiral symmetry in QCD2

Two dimensional massless Quantum Chromodynamics presents many features which resemble those of the true theory. In particular the spectrum consists of mesons and baryons arranged in flavor multiplets without parity doubling. We analyze the implications of chiral symmetry, which is not spontaneously broken in two dimensions, in the spectrum and in the quark condensate. We study how parity doubling, an awaited consequence of Coleman's theorem, is avoided due to the dimensionality of space-time and confinement. We prove that a chiral phase transition is not possible in the theory.Comment: 9 pages, latex, ftuv/92-

Asymptotically Free Non-Abelian Gauge Theories With Fermions and Scalars As Alternatives to QCD

In this paper we construct non-Abelian gauge theories with fermions and scalars that nevertheless possess asymptotic freedom.The scalars are taken to be in a chiral multiplet transforming as $(2,2)$ under $SU(2)_L\otimes SU(2)_R$ and transforming as singlets under the colour SU(3) group. We consider two distinct scenarios, one in which the additional scalars are light and another in which they are heavier than half the Z-boson mass. It is shown that asymptotic freedom is obtained without requiring that all additional couplings keep fixed ratios with each other. It is also shown that both scenarios can not be ruled out by what are considered standard tests of QCD like R- parameter, g-2 for muons or deep inelastic phenomena. The light mass scenario is however ruled out by high precision Z-width data (and only by that one data).The heavy mass scenario is still viable and is shown to naturally pass the test of flavour changing neutral currents. It also is not ruled out by precision electroweak oblique parameters. Many distinctive experimental signatures of these models are also discussed.Comment: 37 pages in LATEX with 10 fig

Optimal designs for rational function regression

We consider optimal non-sequential designs for a large class of (linear and nonlinear) regression models involving polynomials and rational functions with heteroscedastic noise also given by a polynomial or rational weight function. The proposed method treats D-, E-, A-, and $\Phi_p$-optimal designs in a unified manner, and generates a polynomial whose zeros are the support points of the optimal approximate design, generalizing a number of previously known results of the same flavor. The method is based on a mathematical optimization model that can incorporate various criteria of optimality and can be solved efficiently by well established numerical optimization methods. In contrast to previous optimization-based methods proposed for similar design problems, it also has theoretical guarantee of its algorithmic efficiency; in fact, the running times of all numerical examples considered in the paper are negligible. The stability of the method is demonstrated in an example involving high degree polynomials. After discussing linear models, applications for finding locally optimal designs for nonlinear regression models involving rational functions are presented, then extensions to robust regression designs, and trigonometric regression are shown. As a corollary, an upper bound on the size of the support set of the minimally-supported optimal designs is also found. The method is of considerable practical importance, with the potential for instance to impact design software development. Further study of the optimality conditions of the main optimization model might also yield new theoretical insights.Comment: 25 pages. Previous version updated with more details in the theory and additional example