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Investigating the Conformal Window of SU(N) Gauge Theories Thomas Pickup Wadham College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity 2011 Acknowledgements I would like to thank my supervisor, Dr Michael Teper, for his advice and support throughout my research. I would also like to thank my collab- orators Dr Francis Bursa, Dr Luigi Del Debbio, Dr Mads Frandsen, Dr Claudio Pica, and Liam Keegan. I acknowledge the support of an STFC studentship. Parts of this work were performed using the Darwin Super- computer of the University of Cambridge High Performance Computing Service, http://www.hpc.cam.ac.uk/, provided by Dell Inc. using Strate- gic Research Infrastructure Funding from the Higher Education Funding Council for England; computing resources funded by the University of Oxford and EPSRC; and the Horseshoe5 cluster at the supercomputing facility at the University of Southern Denmark (SDU) funded by a grant of the Danish Centre for Scientific Computing for the project Origin of Mass 2008/2009. Abstract In this thesis we are concerned with the existence of infrared fixed points and the conformal window for gauge theories with fermions. We are par- ticularly interested in those theories that are candidates for walking tech- nicolor. We discuss the background of technicolor and the techniques rel- evant to a theoretical understanding of the conformal window. Following this we extend the ideas of metric confinement and causal analyticity to theories with fermions in non-fundamental representations. We use these techniques to, respectively, provide a lower bound on the lower end of the conformal window and to provide a measure of perturbativity. As well as analytic calculations we use lattice techniques to investigate two particular candidate theories for walking technicolor { SU(2) with two adjoint fermions and with six fundamental fermions. We use Schr¨odinger Functional techniques to investigate the running of the theory across a wide range of scales. We measure both the running of the coupling and an estimator for the fermion mass anomalous dimension, γ. We find that both theories are consistent with an infrared fixed-point. However, paying particular attention to our error estimates, we are unable to absolutely confirm their existence. This is a not unexpected result for SU(2) with two adjoint fermions but is rather surprising for SU(2) with only six fun- damental fermions. In the region where we are consistent with a fixed point we find 0:05 < γ < 0:56 for SU(2) with two adjoint fermions and 0:135 < γ < 1:03 for SU(2) with six fundamental fermions. The measure- ment of γ for SU(2) with two adjoint fermions is the first determination of γ for any candidate theory of walking technicolor. Contents 1 Introduction 2 Technicolor 2.1 Electroweak Symmetry Breaking . 2.2 Technicolor . 2.2.1 QCD . 2.2.2 Coupling to Electroweak . 2.2.3 Naturalness . 2.3 Extended Technicolor . 2.4 Walking Technicolor . 2.4.1 Electroweak Precision Data . 2.5 Specific Theories . 2.5.1 SU(2) with Fundamental Fermions . 2.5.2 Minimal Walking Technicolor . 2.5.3 Next to Minimal Walking Technicolor . 2.6 Conclusion . 3 The Beta-Function and the Conformal Window 3.1 Introduction . 3.2 Beta-Function . 3.2.1 Definition and Computation . 3.2.2 Higher Orders . 3.3 Fixed Points . 3.3.1 Two-loop . 3.3.2 Banks-Zaks . 3.4 The Extent of the Conformal Window . 3.4.1 Schwinger-Dyson . 3.4.2 The ACS Conjecture . 3.4.3 NSVZ . 3.4.4 All-Orders Beta-Function . vii viii Contents 3.5 Four Loop MS Beta-Function . 3.5.1 Zerology . 3.5.2 The Sign of Beta-Function Coefficients . 3.5.3 Four-Loop Phenomenology . 3.6 Conclusions . 4 Delineating the conformal window 4.1 The conformal window . 4.2 Metric confinement . 4.3 Perturbation theory and analyticity . 4.3.1 Analyticity of the all-orders beta-function conjecture . 4.4 Comparison with lattice data and other methods . 4.4.1 SU(2) and SU(3) theories with fundamental flavours . 4.4.2 Two flavor SU(2) adjoint theories . 4.4.3 Two flavor SU(3) sextet theory . 4.5 Conclusions . 5 Lattice Gauge Theory 5.1 Introduction . 5.2 Formulation . 5.2.1 Action . 5.2.2 Monte Carlo . 5.3 Fermions . 5.3.1 Formulation . 5.3.2 HMC . 5.4 Schr¨odingerFunctional . 5.4.1 Coupling . 5.4.2 Step Scaling . 5.4.3 Fermions . 5.5 Conclusion . 6 SU(2) with two Adjoint Fermions 6.1 Introduction . 6.1.1 Technicolor . 6.1.2 Minimal Walking Technicolor . 6.1.3 Simulation . 6.2 Lattice formulation . Contents ix 6.2.1 Adjoint Fermions . 6.2.2 SF . 6.2.3 Lattice parameters . 6.3 Evidence for fixed points . 6.4 Results for the coupling . 6.4.1 Raw Data . 6.4.2 SF Coupling . 6.4.3 Fitting the Data . 6.4.4 Step Scaling Function . 6.5 Running mass . 6.5.1 Raw Data . 6.5.2 Pseudoscalar Renormalisation Constant . 6.6 Conclusions . 7 Mass anomalous dimension in SU(2) with six fundamental fermions 7.1 Introduction . 7.2 SF formulation . 7.2.1 Lattice parameters . 7.3 Results for the coupling . 7.4 Running mass . 7.5 Conclusions . 8 Outlook A Group Theoretical Factors and Beta-Functions A.1 Group Theory . A.2 Beta-Function . B Systematic Error Analysis B.1 Coupling error analysis . B.2 Mass error analysis . References List of Figures 2.1 The triangle diagram contributing to the Adler-Bell-Jackiw anomaly . 2.2 Feynman diagram for the sm fermion mass term in the etc framework. 2.3 Feynman diagrams for K0, K0 mixing via a, (SM) box diagrams and b, etc.................................... 2.4 The beta-function and running coupling for a walking theory. 2.5 Feynman diagram for the leading order term in W 3 B polarisation amplitude. 3.1 One-loop diagrams contributing to the beta-function . 3.2 The allowed forms of the two-loop beta-function. 3.3 The dressed fermion propagator . 3.4 Diagrams for the rainbow and ladder approximations . 3.5 The Conformal window of SU(N) gauge theories from the ladder ap- proximation. 3.6 Topologies of the zeros of the four-loop ms β(α)............ 3.7 Topologies of the zeros of the four-loop ms β(g)............ 3.8 Comparing the location of fixed points of the two three and four-loop beta functions for SU(3) with fundamental fermions . 3.9 The order of sign changes in the beta-function coefficients . 3.10 The appearance of a pair of fixed points in SU(3) fundamental. 4.1 Conformal windows for SU theories with Dirac fermions . 4.2 Conformal windows for SO theories with Dirac fermions . 4.3 The maximal value of the two-loop coupling Ncx(Q) in the complex j j plane as a function of Nf ......................... 4.4 The maximal value of the two-loop coupling x(Q) in the complex j j plane as a function of Q2 =Λ2 ...................... j j 5.1 Gauge invariant loops including the plaquette . 5.2 The plaquettes comprising the clover term. x List of Figures xi 6.1 Example plot of dS/dη........................... 6.2 Comparing dS/dη at different L...................... 6.3 Example results for the average plaquette, MP CAC and ZP ...... 6.4 Data for the running coupling as computed from lattice simulations of the Schr¨odingerfunctional. 6.5 Comparing results to previous simulations . 6.6 Raw results for Σ(u; 4=3; 6)=u and Σ(u; 3=2; 8)=u............ 6.7 Results for the lattice step-scaling function Σ(4=3; u; a=L)....... 6.8 The relative step-scaling function σ(u)=u obtained with a both a con- stant and linear continuum extrapolation. 6.9 Example Markov Chain for ZP using unit boundary conditions . 6.10 Comparing ZP with sf and unit boundary conditions . 6.11 Example results for the average plaquette, dS/dη and MP CAC ..... 6.12 Data for the renormalisation constant ZP as computed from lattice simulations of the Schr¨odingerfunctional. 6.13 Results for the lattice step-scaling function ΣP (u; a=L). 6.14 The step-scaling function for the running mass σP (u), using a linear continuum extrapolation. ..
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