Low dimensional supersymmetric field theories on the lattice

In der vorliegenden Arbeit werden verschiedene supersymmetrische Modelle in einer und zwei Raumzeit-Dimensionen untersucht, welche wesentliche Bestandteile von realistischeren Theorien, wie z.B. dem minimalen supersymmetrischen Standardmodell, beinhalten. Durch die separate Untersuchung der einzelnen Aspekte ist es möglich die Vor- und Nachteile der jeweils verwendeten Gittermethoden herauszuarbeiten. Zusätzlich erlaubt die niedrige Dimensionalität sehr präzise numerische Studien, welche konzeptuelle und technische Probleme bei der Behandlung von supersymmetrischen Theorien auf dem Gitter aufdecken können. Am Beginn der Untersuchung von supersymmetrischen Theorien auf dem Gitter steht das pädagogische Beispiel einer supersymmetrischen Quantenmechanik mit dynamisch gebrochener Supersymmetrie. Hieran wird die grundlegende Anwendbarkeit von Gittermethoden auf Theorien mit dynamisch gebrochener Supersymmetrie verifiziert. Am N=2 Wess-Zumino-Modell in 1+1 Dimensionen werden fünf verschiedene Gitterformulierungen verglichen, von denen drei eine explizite Realisierung eines Teils der vollen Supersymmetrie auf dem Gitter darstellen. Die Durchführung von hochpräzisen Messungen stellt selbst in zweidimensionalen Theorien eine große numerische Aufgabe dar. Daher werden die algorithmischen Verbesserungen, die im Verlaufe dieser Arbeit benutzt wurden, am Beispiel des N=2 Wess-Zumino-Modells exemplarisch dargestellt. Als Minimalversion einer supersymmetrischen Feldtheorie mit supersymmetriebrechendem Phasenübergang wird das N=1 Wess-Zumino-Modell in 1+1 Dimensionen analysiert. Die letzte Modellklasse dieser Arbeit bilden (supersymmetrische) nichtlineare Sigma-Modelle. Zunächst wird die Instantonen-Struktur von bosonischen nichtlinearen CP(N)-Sigma-Modellen mit getwisteten Randbedingungen konstruiert. Die Arbeit schließt mit einer Analyse des supersymmetrischen nichtlinearen O(3)-Sigma-Modells auf dem Gitter

Casimir Scaling and String Breaking in G(2) Gluodynamics

We study the potential energy between static charges in G(2) gluodynamics in three and four dimensions. Our work is based on an efficient local hybrid Monte-Carlo algorithm and a multi-level L\"uscher-Weisz algorithm with exponential error reduction to accurately measure expectation values of Wilson- and Polyakov loops. Both in three and four dimensions we show that at intermediate scales the string tensions for charges in various G(2)-representations scale with the second order Casimir. In three dimensions Casimir scaling is confirmed within one percent for charges in representations of dimensions 7, 14, 27, 64, 77, 77', 182 and 189 and in 4 dimensions within 5 percent for charges in representions of dimensions 7, 14, 27 and 64. In three dimensions we detect string breaking for charges in the two fundamental representations. The scale for string breaking agrees very well with the mass of the created pair of glue-lumps.Comment: 20 pages, 17 figure

Two-Dimensional Wess-Zumino Models at Intermediate Couplings

We consider the two-dimensional N=(2,2) Wess-Zumino model with a cubic superpotential at weak and intermediate couplings. Refined algorithms allow for the extraction of reliable masses in a region where perturbation theory no longer applies. We scrutinize the Nicolai improvement program which is supposed to guarantee lattice supersymmetry and compare the results for ordinary and non-standard Wilson fermions with those for SLAC derivatives. It turns out that this improvement completely fails to enhance simulations for Wilson fermions and only leads to better results for SLAC fermions. Furthermore, even without improvement terms the models with all three fermion species reproduce the correct values for the fermion masses in the continuum limit.Comment: 15 pages, 18 figure

Phase Structure of Z(3)-Polyakov-Loop Models

We study effective lattice actions describing the Polyakov loop dynamics originating from finite-temperature Yang-Mills theory. Starting with a strong-coupling expansion the effective action is obtained as a series of Z(3)-invariant operators involving higher and higher powers of the Polyakov loop, each with its own coupling. Truncating to a subclass with two couplings we perform a detailed analysis of the statistical mechanics involved. To this end we employ a modified mean field approximation and Monte Carlo simulations based on a novel cluster algorithm. We find excellent agreement of both approaches concerning the phase structure of the theories. The phase diagram exhibits both first and second order transitions between symmetric, ferromagnetic and anti-ferromagnetic phases with phase boundaries merging at three tricritical points. The critical exponents nu and gamma at the continuous transition between symmetric and anti-ferromagnetic phases are the same as for the 3-state Potts model.Comment: 20 pages, 22 figure

Supersymmetry Breaking in Low Dimensional Models

We analyse supersymmetric models that show supersymmetry breaking in one and two dimensions using lattice methods. Starting from supersymmetric quantum mechanics we explain the fundamental principles and problems that arise in putting supersymmetric models onto the lattice. We compare our lattice results (built upon the non-local SLAC derivative) with numerically exact results obtained within the Hamiltonian approach. A particular emphasis is put on the discussion of boundary conditions. We investigate the ground state structure, mass spectrum, effective potential and Ward identities and conclude that lattice methods are suitable to derive the physical properties of supersymmetric quantum mechanics, even with broken supersymmetry. Based on this result we analyse the two dimensional N=1 Wess-Zumino model with spontaneous supersymmetry breaking. First we show that (in agreement with earlier analytical and numerical studies) the SLAC derivative is a sensible choice in the quenched model, which is nothing but the two dimensional phi^4 model. Then, we present the very first computation of a renormalised critical coupling for the complete supersymmetric model. This calculation makes use of Binder cumulants and is supported by a direct comparison to Ward identity results, both in the continuum and infinite volume limit. The physical picture is completed by masses at two selected couplings, one in the supersymmetric phase and one in the supersymmetry broken phase. Signatures of the Goldstino in the fermionic correlator are clearly visible in the broken case.Comment: 33 pages, 28 figure