Wolfdieter Lang

Perturbation Calculations for the Scalar Multiplet in a Superfield Formulation

Based on the superfield propagators for the scalar multiplet, we investigate the occurence of divergence cancellations in supersymmetric models. This cancellation mechanism is governed by the Ward identity which has an integral form in the present formulation. The power counting rules for the actual divergence of diagrams in the models with a φ3 and a φ4 self-interaction are given. Finally we present a systematic treatment of the φ4 model up to the two-loop level. The structure of the necessary counter terms is specified and the appearance of negative norm states is indicated.

16/16 supergravity coupled to matter: the low energy limit of the superstring

Abstract We analyze irreducible, N = 1 supergravity theories with 16 fermionic degrees of freedom. The lagrangians for pure 16 16 supergravity, and for 16 16 supergravity coupled to arbitrary chiral superfields are constructed. These theories are shown to have natural SU(1,1) non-compact symmetry. The low energy field theory limit of the superstring is conjectured to be of this type.

On the effective potential for the scalar multiplet in the supersymmetric φ3 model

Abstract The one-loop effective potential for the scalar multiplet in the supersymmetric model with a φ 3 self-interaction is evaluated, and the vacuum stability is discussed. It is shown that the asymmetric vacua give rise to an imaginary part of the effective potential; consequently they are unstable. Our result also indicates that the massless degenerate solution is stable against infrared divergences.

On sums of powers of zeros of polynomials

Abstract Due to Girards (sometimes called Warings) formula the sum of the rth power of the zeros of every one variable polynomial of degree N, PN(x), can be given explicitly in terms of the coefficients of the monic P N (x) polynomial. This formula is closely related to a known N − 1 variable generalisation of Chebyshevs polynomials of the first kind, Tr(N − 1). The generating function of these power sums (or moments) is known to involve the logarithmic derivative of the considered polynomial. This entails a simple formula for the Stieltjes transform of the distribution of zeros. Perron-Stieltjes inversion can be used to find this distribution, e.g., for N → ∞. Classical orthogonal polynomials are taken as examples. The results for ordinary Chebyshev TN(x) and UN(x) polynomials are presented in detail. This will correct a statement about power sums of zeros of Chebyshevs T-polynomials found in the literature. For the various cases (Jacobi, Laguerre, Hermite) these moment generating functions provide solutions to certain Riccati equations.

Current in supersymmetric gauge theories

Abstract A construction principle for the supercurrent emphasizing its transformation property under the superconformal group is applied to the supersymmetric extension of massless gauge theories. A suggestion for a superspace current containing all improved currents of the superconformal group is made. This current is obtained from the supercurrent as a certain moment involving the superspace transformation parameters of the superconformal group. The connection to Noethers theorem in superspace is pointed out. The one-loop anomalies for typical on-shell matrix elements of the components of the supercurrent are calculated, and their supertransformation behaviour is elucidated.

Potentials of N = 1 supergravity couplings of scalar multiplets

Abstract Couplings of chiral superfields to supergravities with various content of auxiliary fields are considered. Potentials and kinetic terms of the scalar fields are computed.

Investigation of a non-renormalizable Lagrangian model invariant under supertransformations

Abstract A non-renormalizable Lagrangian model invariant under supertransformations is studied in the one-loop approximation, and a remarkable cancellation of divergences is found.

On the characteristic polynomials of Fibonacci chains

Special diatomic linear chains with elastic nearest-neighbour interaction and the two masses distributed according to the binary Fibonacci sequence are studied.

The Wythoff and the Zeckendorf Representations of Numbers are Equivalent

The quintessence of many application of Fibonacci numbers is the binary substitution sequence 1→10, 0→1. The infinite sequence generated this way is self-similar and quasiperiodic. See refs. [7, 16] for details on this rabbit or golden sequences. It is intimately related to Wythoffs complementary sequences which cover the natural numbers ([·] is the greatest integer function)