The Large Hadron Collider (LHC) and the four major experiments set up along its 27 kilometers of circumference (ATLAS, CMS, ALICE and LHCb), have recently started to explore the high–energy frontier at √s = 8 TeV, and will move to even higher energy in just about 2 years. The aim of physics searches at LHC experiments was to complete the picture of the Standard Model (SM) of elementary particles with the discovery of the Higgs boson and to look for specific signatures of models extending the current understanding of particle interactions, at zero and non–zero temperature. In 2012, the official discovery of the Higgs boson, the only missing particle of the StandardModel, was announced by ATLAS and CMS. Other important results include the measurement of rare decay modes in heavy quarks systems, and indications of CP violation in charm decays by LHCb. Signatures of beyond the Standard Model (BSM) physics are currently being looked for in the experimental data, and this often requires the knowledge of quantities that can be computed only with non–perturbative methods. This thesis focuses on some possible extensions of the SM and the analysis of interesting physical observables, like masses or decay rates, calculated using non– perturbative lattice methods. The approach followed for the main part of this work is to model BSM theories as effective field theories defined on a lattice. This lattice approach has a twofold advantage: it allows us to explore non– renormalizable gauge theories by imposing an explicit gauge–invariant cutoff and it allows us to go beyond perturbative results in the study of strongly interacting systems. Some of the issues of the SM that we will try to address include, for example, the hierarchy problem and the origin of dynamical electroweak symmetry breaking (DEWSB). We investigate non–perturbatively the possibility that the lightness of the mass for an elementary scalar field in a four–dimensional quantum field theory might be due to a higher–dimensional gauge symmetry principle. This idea fits in the Gauge–Higgs unification approach to the hierarchy problem and the results we present extend what is known from perturbative expectations. Extra dimensional models are also often used to approach DEWSB. Another approach to DEWSB implies a new strongly interacting gauge sector that extends the SM at high energies and it is usually referred to as Technicolor. The phenomenological consequences of Technicolor can only be studied by non– perturbative methods at low energy since the theory is strongly coupled at large distances. We perform a comprehensive lattice study of fermionic and gluonic scalar bound states in one of the candidate theories for Technicolor BSM physics. We relate our findings to the nature of the newly discovered Higgs boson. New physics is also commonly believed to be hidden in the flavour sector of the SM. In this sector, lattice calculations of non–perturbative input parameters are needed in order to make precise predictions and extract signals of possible new physics. In particular, heavy quark physics on the lattice is still in development and it is important to understand the relevant discretisation errors. We describe a preliminary study of the mixing parameter of heavy–light mesons oscillations in a partially–quenched scenario, using staggered dynamical fermions and domain wall valence fermions.

It is expected that the number of flavors in a gauge theory plays an important role in model building for physics beyond the standard model. We study the phase structure of the 12 flavor case through lattice simulations using a Rational Hybrid Monte Carlo (RHMC) algorithm for different masses, betas, and volumes, to investigate the question of conformality for this number of flavors. In particular, we analyze the Fisher's zeroes, in the vicinity of the endpoint of a line of first order phase transitions. This is motivated by previous studies that show how the complex renormalization group (RG) flows can be understood by looking at the zeros. The pinching of the imaginary part of these zeros with respect to increasing volume provides information about a possible unconventional continuum limit. We also study the mass spectrum of a multiflavor linear sigma model with a splitting of fermion masses. The single mass linear sigma model successfully described a light sigma in accordance to recent lattice results. The extension to two masses predicts an unusual ordering of scalar masses, providing incentive for further lattice simulations with split quark mass.

Bibliography: p. 377-382.

In this thesis I explore the physical effects of improved actions combined with improved operators in the framework of lattice QCD. All calculations are done in the quenched approximation, that is, when all of the dynamical fermion interactions have been suppressed by setting the determinant of the fermion matrix to a constant. The thesis first briefly introduces lattice QCD to familiarize the reader with the basic concepts. It then describes the common numerical procedures used. It is made up of three major sections. The first is the exploration of gauge field configurations and the study of the role of instantons in lattice QCD. In this work the Wilson gauge action and a standard 1 loop topological charge operator are used to determine the relative rates of standard cooling and smearing algorithms in pure SUc(3)-color gauge theory. I consider representative gauge field configurations on 16³ × 32 lattices at β = 5.70 and 24³ × 36 lattices at β = 6.00. I find the relative rate of variation in the action and topological charge under various algorithms may be succinctly described in terms of simple formulae ¹. The results are in accord with recent suggestions from fat-link perturbation theory. This work is then extended to O(a²)-improved gauge action and O(a²)-improved operators ². In particular, an O(a²)-improved version of APE smearing is motivated by considerations of smeared link projection and cooling. The extent to which the established benefits of improved cooling carry over to improved smearing is critically examined. I consider representative gauge field configurations generated with an O(a²)-improved gauge field action on 16³ × 32 lattices at β = 4.38 and 24³ × 36 lattices at β = 5.00 having lattice spacings of 0.165(2) fm and 0.077(1) fm respectively. While the merits of improved algorithms are clearly displayed for the coarse lattice spacing, the fine lattice results put the various algorithms on a more equal footing and allow a quantitative calibration of the smoothing rates for the various algorithms. I find that the relative rate of variation in the action may also be described in terms of simple calibration formulae for O(a²)-improvement which accurately describes the relative smoothness of the gauge field configurations at a microscopic level. In the second section the first calculation of the gluon propagator using an O(a²)- improved action with the corresponding O(a²)-improved Landau gauge fixing ³ condition is presented ⁴. The gluon propagator obtained from the improved action and improved Landau gauge condition is compared with earlier unimproved results on similar physical lattice volumes of 3.2³ × 6.4⁴ fm. It is found that there is good agreement between the improved propagator calculated on a coarse lattice with lattice spacing a = 0.35 fm and the unimproved propagator calculated on a fine lattice with spacing a = 0.10 fm. This motivated us to calculate the gluon propagator on a coarse very large-volume lattice of 5.6³ × 11.2⁴fm. The infrared behavior observed in previous studies is confirmed. The gluon propagator is enhanced at intermediate momenta and suppressed at infrared momenta. The observed infrared suppression of the Landau gauge gluon propagator is not a finite volume effect. This work is then extended to a variety of lattices with spacing ranging from a = 0.17 to a = 0.4 fm ⁵ to further explore finite volume and discretization effects. In this work a technique previously used for minimizing lattice artifacts, known as “tree-level correction”, has also been extended. It is demonstrated that by using tree-level correction, determined by the tree-level behavior of the action being considered, it is possible to obtain scaling behavior over a very wide range of momenta and lattice spacings. This makes it possible to explore the infinite volume and continuum limits of the Landau-gauge gluon propagator. As a final part of this thesis I present the first results for the quark propagator using an Overlap fermionic quark action ⁶. I compare the results with those obtained from the standard Wilson fermion. The overlap quark action is O(a)-improved compared with the Wilson fermion. This action realizes exact chiral symmetry on the lattice unlike the Wilson fermion and it demonstrates that the fastest way forward in this field is with improved lattice operators. The idea of studying improved actions in lattice gauge theory was suggested to me by A/Prof. Anthony G. Williams during the “Nonperturbative Methods in Quantum Field Theory” workshop in early February 1998. Initially it was suggested to me that a calculation of the gluon propagator using improved action on large volumes, following a study just done with standard gauge action in Ref. [62]. The point of interest was to study the effect an improved gauge field action would have on the gluon propagator. This study would then be extended to quark actions. In the meantime when generating gauge field configurations using a computer code written in Fortran 77 (provided by Dr. Derek B. Leinweber), it occurred to me that it would be good to explore the content of these gauge field configurations. In order to do realistic calculations on large lattices we needed a gauge field configuration generator that would run on our CM5 computer and so Connection Machine Fortran (CMF) became the adopted language. I started writing the computer code to generate the gauge field configuration in the SUc(2) with the help of Dr. Derek B. Leinweber, who introduced me to the basic concepts in lattice QCD. I then extended this code to the SUc(3) gauge group. This is commonly known as the standard Wilson gauge action. After investigating with some of the optimization possibilities, I moved on to code an O(a²)-improved gauge action. The code uses a masking procedure for the link update. I have generalized the masking procedure for any planar gauge field action in SUc(N), Ref. [18]. From there it was very obvious that by applying a continuous repetition of some sections of code that I written, that some bigger Wilson loops could easily be included in the action and hence highly improved actions could be easily constructed. The only difficulty was to calculate the improvement coefficients. I then moved on to study smearing algorithms. I adapted the gauge field configuration code to a cooling and a 1 × 2 and 2 × 1 improved cooling code in which we inserted higher order loop operators. This was the tool used to explore gauge field configurations and their topological structures. Once the short range quantum fluctuations are removed it is possible to see instantons. Instantons are believed to play a crucial role in the spontaneous chiral symmetry breaking mechanism. We improved the topological charge operator from the clover term to an (1 × 2 and 2 × 1) O(a²)–improved topological charge operator (see Appendices, Sections E.16 and E.17). This code was subsequently adapted by Sundance Bilson-Thompson so that he could insert higher order loops. I have also inserted my O(a²)–improved operator to construct an O(a²)–improved smearing algorithm. Using these tools I have calibrated the relative rates of cooling and smearing. Another piece of work on gauge fixing, reviewed in Chapter 8, was led by Dr. Patrick O. Bowman, Ref. [63]. There I supplied the gauge field configurations and checked some of the analytical work. For the gluon propagator work I supplied all of the lattice configurations with the exception of the 32³ × 64 used in Ref. [62]. The analysis was primarily carried out by Dr. Patrick O. Bowman and partly inspired by the one carried out in hep-lat/0106023. While this gluon propagator work is not being presented here as my own Ph. D. qualifying work, I am a co author on the subsequent papers and so I have therefore decided to include a review of this work in Chapter 9. I have also made some contribution in the construction of the Fat–link quark action (with and without the clover term) developed by James M. Zanotti. These contributions involve the code for the Reunitarization of the smeared links, Appendix E.21. Because of the code developed for the improved lattice definition of the Fµν(x) term I have also made some contribution to the Fat–link clover quark action although I will not discuss about this work in the following thesis. My main contribution for the overlap quark propagator study was in the analysis of the propagator data. The overlap propagators were generated by Dr. Jianbo Zhang and the research was also carried out in collaboration with A/Prof. Anthony G. Williams and Dr. Derek B. Leinweber. The quark propagators for the Wilson fermion were generated by a computer code parallelized by James M. Zanotti and originally written by Prof. Frank X. Lee. The anisotropic lattice code has not been used in any calculations yet although it has been tested and verified. The code was extended from the isotropic improved generator code in SUc(3). After a literature search, we decided to implement the action described in Ref. [31] for the anisotropic Wilson action and in Ref. [11, 32] for the improved anisotropic case. Apart from the work on the gauge fixing and the gluon propagator, done in collaboration with Dr. Patrick O. Bowman, and which for completeness is briefly reviewed in Chapters 8 and 9 respectively, this thesis contains no material which has been accepted for the award of any other degree or diploma in any university or other institution and to the best of knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being available for loan and photocopying. Fr´ed´eric D. R. Bonnet Date: 20th of September 2001. ____________ [Footnotes]: ¹F. D. R. Bonnet, P. Fitzhenry, D. B. Leinweber, M. R. Stanford & A. G. Williams, Phys. Rev. D 62, 094509 (2000) [hep lat/0001018]. ²F. D. R. Bonnet, D. B. Leinweber, A. G. Williams & J. M. Zanotti, Submitted to Phys. Rev. D. [hep-lat/0106023]. ³F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, D. G. Richards & A. G. Williams, Aust. J. Phys. 52, 939 (1999). ⁴F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber & A. G. Williams, Infrared behavior of the gluon propagator on a large volume lattice, Phys. Rev. D 62, 051501, (2000). ⁵F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, A. G. Williams & J. M. Zanotti, Infinite volume and continuum limits of the landau gauge gluon propagator, Phys. Rev. D 64, 034501 (2001) [hep-lat/0101013]. ⁶F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, A. G. Williams & J. Zhang, Overlap Propagator in Landau Gauge, to be Submitted to Phys. Rev. D. / Thesis (Ph.D.) -- University of Adelaide, School of Chemistry and Physics, 2002

Thesis (Ph.D.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 2002? / Bibliography: p. 377-382.

Thesis (Ph.D.)--Boston University / In this dissertation I use lattice gauge theory to study models of electroweak symmetry breaking that involve new strong dynamics. Electroweak symmetry breaking (EWSB) is the process by which elementary particles acquire mass. First proposed in the 1960s, this process has been clearly established by experiments, and can now be considered a law of nature. However, the physics underlying EWSB is still unknown, and understanding it remains a central challenge in particle physics today. A natural possibility is that EWSB is driven by the dynamics of some new, strongly-interacting force. Strong interactions invalidate the standard analytical approach of perturbation theory, making these models difficult to study. Lattice gauge theory is the premier method for obtaining quantitatively-reliable, nonperturbative predictions from strongly-interacting theories. In this approach, we replace spacetime by a regular, finite grid of discrete sites connected by links. The fields and interactions described by the theory are likewise discretized, and defined on the lattice so that we recover the original theory in continuous spacetime on an infinitely large lattice with sites infinitesimally close together. The finite number of degrees of freedom in the discretized system lets us simulate the lattice theory using high-performance computing. Lattice gauge theory has long been applied to quantum chromodynamics, the theory of strong nuclear interactions. Using lattice gauge theory to study dynamical EWSB, as I do in this dissertation, is a new and exciting application of these methods. Of particular interest is non-perturbative lattice calculation of the electroweak S parameter. Experimentally S ~ -0.15(10), which tightly constrains dynamical EWSB. On the lattice, I extract S from the momentum-dependence of vector and axial-vector current correlators. I created and applied computer programs to calculate these correlators and analyze them to determine S. I also calculated the masses and other properties of the new particles predicted by these theories. I find S > 0.1 in the specific theories I study. Although this result still disagrees with experiment, it is much closer to the experimental value than is the conventional wisdom S > 0.3. These results encourage further lattice studies to search for experimentally viable strongly-interacting theories of EWSB.

Discutimos um modelo de calibre com simetria Z (N) na rede, sendo as variáveis dinâmicas definidas em faces de cubos. Mostramos a dualidade com um sistema de spins Z (N) em quatro dimensões e a autodualidade em seis dimensões para este modelo, utilizando o formalismo da matriz de transferência. Analisamos as funções de correlação invariantes por transformações de calibre, constatando os decaimentos exponenciais com o volume (para altas temperaturas e d &#8805 3) e com a área (para baixas temperaturas e d > 3). Para três dimensões, o modelo não apresenta transição de fase sendo exatamente solúvel. Estudamos também a versão U (1) do modelo e mostramos sua equivalência com uma teoria de campos clássica livre na região de baixas temperaturas / We discussus a model with a Z (N) gauge symmetry on a lattice, the dynamical variables being defined on faces of cubes. The duality with a Z (N) spin system in four dimensions and the selfduality in six dimensions is shown for this model, using the transfer matrix formalism. The gauge invariant correlation functions have been analysed and we verify their exponential decay with volume (at high temperatures and d &#8805 3) and with the área (at low temperatures and d > 3). For three dimensions, the model exhibits no phase transition, being exactly soluble. We also study a U (I) version o four model and show its equivalence with a free classical field theory in the low temperature region

Discutimos um modelo de calibre com simetria Z (N) na rede, sendo as variáveis dinâmicas definidas em faces de cubos. Mostramos a dualidade com um sistema de spins Z (N) em quatro dimensões e a autodualidade em seis dimensões para este modelo, utilizando o formalismo da matriz de transferência. Analisamos as funções de correlação invariantes por transformações de calibre, constatando os decaimentos exponenciais com o volume (para altas temperaturas e d &#8805 3) e com a área (para baixas temperaturas e d > 3). Para três dimensões, o modelo não apresenta transição de fase sendo exatamente solúvel. Estudamos também a versão U (1) do modelo e mostramos sua equivalência com uma teoria de campos clássica livre na região de baixas temperaturas / We discussus a model with a Z (N) gauge symmetry on a lattice, the dynamical variables being defined on faces of cubes. The duality with a Z (N) spin system in four dimensions and the selfduality in six dimensions is shown for this model, using the transfer matrix formalism. The gauge invariant correlation functions have been analysed and we verify their exponential decay with volume (at high temperatures and d &#8805 3) and with the área (at low temperatures and d > 3). For three dimensions, the model exhibits no phase transition, being exactly soluble. We also study a U (I) version o four model and show its equivalence with a free classical field theory in the low temperature region

In this thesis, we study the Fisher's zeros in lattice gauge theory. The analysis of singularities in the complex coupling plane is an important tool to understand the critical phenomena of statistical models. The Fisher's zero structure characterizes the scaling properties of the underlying models and has a strong influence on the complex renormalization group transformation flows in the region away from both the strong and weak coupling regimes. By reconstructing the density of states, we try to develop a systematical method to investigate these singularities and we apply the method to SU(2) and U(1) lattice gauge models with a Wilson action in the fundamental representation. We first take the perturbative approach. By using the saddle point approximation, we construct the series expansions of the density of states in both of the strong and weak regimes from the strong and weak coupling expansions of the free energy density. We analyze the SU(2) and U(1) models. The expansions in the strong and weak regimes for the two models indicate both possess finite radii of convergence, suggesting the existence of complex singularities. We then perform the numerical calculations. We use Monte Carlo simulations to construct the numerical density of states of the SU(2) and U(1) models. We also discuss the convergence of the Ferrenberg-Swendsen's method which we use for the SU(2) model and propose a practical method to find the initial values that improve the convergence of the iterations. The strong and weak series expansions are in good agreement with the numerical results in their respective limits. The numerical calculations also enable the discussion of the finite volume effects which are important to the weak expansion. We calculate the Fisher's zeros of the SU(2) and U(1) models at various volumes using the numerical entropy density functions. We compare different methods of locating the zeros. By the assumption of validity of the saddle point approximation, we find that the roots of the second derivative of the entropy density function have an interesting relation with the actual zeros and may possibly reveal the scaling property of the zeros. Using the analytic approximation of the numerical density of states, we are able to locate the Fisher's zeros of the SU(2) and U(1) models. The zeros of the SU(2) stabilize at a distance from the real axis, which is compatible with the scenario that a crossover instead of a phase transition is expected in the infinite volume limit. In contrast, with the precise determination of the locations of Fisher's zeros for the U(1) model at smaller lattice sizes L=4, 6 and 8, we show that the imaginary parts of the zeros decrease with a power law of L-3.07 and pinch the real axis at β= 1.01134, which agrees with results using other methods. Preliminary results at larger volumes indicate a first-order transition in the infinite volume limit.

The analysis of the error of perturbative series by comparing it to the exact solution is an important tool to understand the non-perturbative physics of statistical models. For some toy models, a new method can be used to calculate higher order weak coupling expansion and modified perturbation theory can be constructed. However, it is nontrivial to generalize the new method to understand the critical behavior of high dimensional spin and gauge models. Actually, it is a big challenge in both high energy physics and condensed matter physics to develop accurate and efficient numerical algorithms to solve these problems. In this thesis, one systematic way named tensor renormalization group method is discussed. The applications of the method to several spin and gauge models on a lattice are investigated. theoretically, the new method allows one to write an exact representation of the partition function of models with local interactions. E.g. O(N) models, Z2 gauge models and U(1) gauge models. Practically, by using controllable approximations, results in both finite volume and the thermodynamic limit can be obtained. Another advantage of the new method is that it is insensitive to sign problems for models with complex coupling and chemical potential. Through the new approach, the Fisher's zeros of the 2D O(2) model in the complex coupling plane can be calculated and the finite size scaling of the results agrees well with the Kosterlitz-Thouless assumption. Applying the method to the O(2) model with a chemical potential, new phase diagram of the models can be obtained. The structure of the tensor language may provide a new tool to understand phase transition properties in general.