Guises and disguises of quadratic divergences

In this contribution, we present a new perspective on the control of quadratic divergences in quantum field theory, in general, and in the Higgs naturalness problem, in particular. Our discussion is essentially based on an approach where UV divergences are parameterized, after being reduced to basic divergent integrals (BDI) in one internal momentum, as functions of a cutoff and a renormalization group scale λ$\lambda$. We illustrate our proposal with well-known examples, such as the gluon vacuum self energy of QCD and the Higgs decay in two photons within this approach. We also discuss frameworks in effective low-energy QCD models, where quadratic divergences are indeed fundamental.     

Naturalness and Theoretical Constraints on the Higgs Boson Mass

Arbitrary regularization dependent parameters in Quantum Field Theory are usually fixed on symmetry or phenomenology grounds. We verify that the quadratically divergent behavior responsible for the lack of naturalness in the Standard Model (SM) is intrinsically arbitrary and regularization dependent. While quadratic divergences are welcome for instance in effective models of low energy QCD, they pose a problem in the SM treated as an effective theory in the Higgs sector. Being the very existence of quadratic divergences a matter of debate, a plausible scenario is to search for a symmetry requirement that could fix the arbitrary coefficient of the leading quadratic behavior to the Higgs boson mass to zero. We show that this is possible employing consistency of scale symmetry breaking by quantum corrections. Besides eliminating a fine-tuning problem and restoring validity of perturbation theory, this requirement allows to construct bounds for the Higgs boson mass in terms of $\delta m^{2}/m^{2}_{H}$ (where m _H is the renormalized Higgs mass and δm ² is the 1-loop Higgs mass correction). Whereas $\delta m^{2}/m^{2}_{H}<1$ (perturbative regime) in this scenario allows the Higgs boson mass around the current accepted value, the inclusion of the quadratic divergence demands $\delta m^{2}/m^{2}_{H}$ arbitrarily large to reach that experimental value.     

Estimates for the eigenvalues of Hill's equation and applications for the eigenvalues of the laplacian on toroidal surfaces

In this paper we give upper and lower bounds for each eigenvalue λ _n of Hill's differential equation. We apply the results to toroidal surfaces of revolution in order to get estimates for the eigenvalues of the Laplacian in terms of curvature expressions; they are sharp for the flat torus. As an example, we investigate the standard torus in IR³; here, the bounds depend on the radii only.     

Euclidean algorithms are Gaussian

We obtain a central limit theorem for a general class of additive parameters (costs, observables) associated to three standard Euclidean algorithms, with optimal speed of convergence. We also provide very precise asymptotic estimates and error terms for the mean and variance of such parameters. For costs that are lattice (including the number of steps), we go further and establish a local limit theorem, with optimal speed of convergence. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various other techniques: Dirichlet series, Perron's formula, quasi-powers theorems, and the saddle-point method. Such dynamical analyses had previously been used to perform the average-case analysis of algorithms. For the present (dynamical) analysis in distribution, we require estimates on transfer operators when a parameter varies along vertical lines in the complex plane. To prove them, we adapt techniques introduced recently by Dolgopyat in the context of continuous-time dynamics (Ann. Math. 147 (1998) 357).     

Synthesis of new amphiphilic and lypophilic polymer networks containing 2‐methyl‐ and 2‐nonyl‐2‐oxazoline by the macroinitiator method

New amphiphilic and lypophilic polymer networks were obtained by the copolymerization of 2‐methyl‐2‐oxazoline (MeOXA), and/or 2‐nonyl‐2‐oxazoline (NoOXA) and 2,2′‐tetramethylenebis(2‐oxazoline) (BisOXA), respectively, initiating the copolymerization by random copolymers of chloromethylstyrene and methyl methacrylate or of chloromethylstyrene and styrene (macroinitiator method). Potassium iodide was used as an activator agent and the reaction was carried out in benzonitrile at 110 °C. In general, the polymer gels were obtained with a yield of 62 to 88%. The networks were characterized by high‐resolution magic angle spinning (HRMAS) NMR spectroscopy and by its absorption of polar and nonpolar solvents. In the case of amphiphilic polymer networks, the absorption of solvents depends on the molar ratio of 2‐methyl‐ to 2‐nonyl‐2‐oxazoline inside the polymer network favoring the absorption of polar solvents with a higher content of 2‐methyl‐2‐oxazoline. These gels showed a maximal swelling degree of 13 mL of water, 20 mL of methanol, and 13 mL of chloroform, respectively, per g of polymer. The lypophilic polymer networks containing only 2‐nonyl‐2‐oxazoline showed a maximal swelling degree of 8 mL of toluene, 14 mL of chloroform, and 2 mL of methanol, respectively, per g of the lypophilic network. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 122–128, 2005     

Nitroxide‐mediated homo‐ and block copolymerization of styrene and multifunctional acryl‐ and methacryl derivatives

The ability of different alkoxyamines ( I1 , I2 , I3 , I4 , and I5 ) to initiate controlled radical polymerization of styrene was evaluated. Among them, 2‐hydroxymethyl‐2‐[(2‐methyl‐1‐phenyl‐propyl)‐(1‐phenyl‐ethoxy)‐amino]‐propane‐1,3‐diol ( I5 ) gave the highest polymerization rate of styrene, and the best control over the molecular weight and the molecular weight distribution of polystyrene. Kinetic studies confirmed that with initiator I5 the polymerization of styrene proceeded in a controlled way. The controlled radical homopolymerization of multifunctional acryl‐ and methacryl derivatives using initiator I5 could not be realized as demonstrated by the high polydispersities (PD) obtained. However, it was possible to polymerize multifunctional acryl‐ and methacryl derivatives using a polystyrene macroinitiator ( Pst ) and, thus, novel amphiphilic block copolymers with a narrow molecular weight distribution were obtained. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 1873–1882, 2005     

An analytical approach to global optimization

Global optimization problems with a few variables and constraints arise in numerous applications but are seldom solved exactly. Most often only a local optimum is found, or if a global optimum is detected no proof is provided that it is one. We study here the extent to which such global optimization problems can be solved exactly using analytical methods. To this effect, we propose a series of tests, similar to those of combinatorial optimization, organized in a branch-and-bound framework. The first complete solution of two difficult test problems illustrates the efficiency of the resulting algorithm. Computational experience with the programbagop, which uses the computer algebra systemmacsyma, is reported on. Many test problems from the compendiums of Hock and Schittkowski and others sources have been solved.The research of the first and the third authors has been supported by AFOSR grants #0271 and #0066 to Rutgers University. Research of the second author has been supported by NSERC grant #GP0036426 and FCAR grants #89EQ4144 and #90NC0305.     

Cord-slope form of Taylor's expansion in univariate global optimization

Interval arithmetic and Taylor's formula can be used to bound the slope of the cord of a univariate function at a given point. This leads in turn to bounding the values of the function itself. Computing such bounds for the function, its first and second derviatives, allows the determination of intervals in which this function cannot have a global minimum. Exploiting this information together with a simple branching rule yields an efficient algorithm for global minimization of univariate functions. Computational experience is reported.The first and second authors have been supported by FCAR (Fonds pour la Formation de Chercheurs et l'Aide à la Recherche) Grant 92EQ1048 and AFOSR Grant 90-0008 to Rutgers University. The first author has also been supported by NSERC (Natural Sciences and Engineering Research Council of Canada) Grant to HEC and NSERC Grant GP0105574. The second author has been supported by NSERC Grant GP0036426, FCAR Grant 90NC0305, and a NSF Visiting Professorship for Women in Science at Princeton University. Work of the third author was done in part while he was a graduate student at the Department of Mathematics, Rutgers University, New Brunswick, New Jersey, USA and during a visit to GERAD, June–August 1991.