Exploring Higher Dimensional Quantum Field Theories Through Fixed Points by Miss Rebecca M. Simms Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor of Philosophy Theoretical Physics Division Department of Mathematical Sciences University of Liverpool United Kingdom August 2018 Exploring Higher Dimensional Quantum Field Theories Through Fixed Points by Miss Rebecca M. Simms Abstract Renormalization was popularised in the 1940s following the appearance of non- sensical infinities in the calculation of the self-energy of the electron. Notably this led to Quantum Electrodynamics becoming a fully renormalizable quantum field theory. One useful tool that emerges from the technical aspects of renormal- ization is the Renormalization Group. In particular, the β-function defines the variation of the coupling constants with energy. The vanishing of the β-function at a particular value of the coupling is known as a fixed point, the location of which can be found using perturbation theory. Properties of quantum field the- ories such as ultraviolet behaviour can be studied using these fixed points. The calculation of two different types of fixed points forms the spine of this thesis. In Part I the d-dimensional Wilson-Fisher fixed point is used to connect scalar quantum field theories in different space-time dimensions. Specifically we look at dimensions greater than four and explore the property of universality through the Vasil'ev large N expansion. Different universality classes are examined, the first contains φ4 theory with O(N) symmetry while another incorporates O(N) O(m) × Landau-Ginzburg-Wilson theory. In the latter we perform a full fixed point sta- bility analysis and conformal window search which determines where conformal symmetry is present. Part I develops techniques that may later be applicable to calculations involving beyond the Standard Model physics including asymptotic safety, quantum gravity and emergent symmetries. Part II focuses on the non-trivial Banks-Zaks fixed point of four dimensional Quantum Chromodynamics. Using a variety of colour groups and representations we calculate the location of the fixed point and corresponding critical exponents to pinpoint exactly where the true value of the conformal window lies. Additionally a number of different renormalization schemes are used, including the momentum subtraction (MOM) and interpolating momentum subtraction (iMOM) schemes. This allows us to study where in the conformal window scheme dependence is most apparent. Both the Landau gauge and maximal abelian gauge are utilized to extend the analysis. Throughout this thesis we compare and contrast perturbative results with non-perturbative calculations such as those performed in lattice. Declaration I hereby declare that all work described in this thesis is the result of my own research activities unless reference to others is given. None of this material has been previously submitted to this or any other university. All work was carried out in the Theoretical Physics Division of the Department of Mathematical Sciences during the period of October 2014 to September 2018. iv Publication List Contributions from this work have been published in the following: [1] J.A. Gracey & R.M. Simms, Banks-Zaks Fixed Point Analysis in Momen- tum Subtraction Schemes, Phys. Rev. D91 (2015) no.8, 085037. [2] J.A. Gracey & R.M. Simms, Six Dimensional Landau-Ginzburg-Wilson The- ory, Phys. Rev. D95 (2017) no.2, 025029. [3] J.A. Gracey & R.M. Simms, Higher Dimensional Higher Derivative φ4 The- ory, Phys. Rev. D96 (2017) no.2, 025022. [4] J.A. Gracey & R.M. Simms, Renormalization of QCD in the Interpolating Momentum Subtraction Scheme at Three Loops, Phys. Rev. D97 (2018) no.8, 085016. v Acknowledgements The first person I wish to thank is my supervisor Prof. J.A. Gracey, without whom the work presented here would have been inconceivable. Thank you for the many useful discussions on books, travel, cricket and sometimes physics. I would certainly not have enjoyed the past four years as much without your en- couragement and enthusiasm. I am also grateful for the support provided by my secondary supervisor Prof. I. Jack. I would like to thank STFC and the University of Liverpool for the financial assistance throughout my PhD. During these four years I have been lucky enough to travel to many academic institutions around Europe. I thank the Galileo Galilei Institute for Theoretical Physics (GGI) in Florence, Italy for providing a productive and engaging workspace. I would like to thank the Abdus Salam International Centre for Theoretical Physics (ICTP) in Trieste, Italy for providing a fantastic conference program and for the financial support which allowed me to attend. I would also like to thank Humboldt Uni- versity in Berlin and Prof. Dr. D. Kreimer for the invitation to present my work and for the warm hospitality provided. My knowledge and love for physics has only grown on my travels and I thank everyone along the way who has supported my work. In particular I thank Henry for helping me find my way around Berlin. My thanks also go to Colin and Glyn for being brilliant office mates and to Matt and Alex for their support and friendship. I would like to acknowledge and thank the following people who have sup- ported me, not only during my PhD, but throughout my life. To my Mum and Dad, whose love and guidance are with me in whatever I pursue. To Holly, a pretty great little sister for all my complaining, thank you for your support and friendship. My heartfelt thanks go to my Nan and Auntie Julie for their support and unconditional love. Thanks also go to my Granddad, who I know would be proud of everything I have achieved. I would particularly like to thank Jack and Florence whose interest in my work and heart-warming kindness has been much appreciated. I am very grateful that you both have been part of my life these past couple of years. Finally, I am profoundly grateful to Paul for always having vi confidence in me even when I did not. Thank you for always believing in me and for your love, encouragement and patience. Above all else, thank you for being my best friend. The memories we make together I will treasure always. I love you all dearly. vii Contents Abstract iii Declaration iv Acknowledgements vi 1 Introduction 1 I Fixed Points of Scalar Quantum Field Theories 10 2 Background 11 2.1 Renormalizing Quantum Field Theory . 11 2.1.1 Dimensional Regularisation . 12 2.1.2 Renormalizing φ4 Theory in Four Dimensions . 13 2.1.3 Renormalization Schemes . 18 2.1.4 Weinberg's Theorem . 20 2.2 The Renormalization Group . 23 2.2.1 Critical Exponents and Universality . 28 2.2.2 Relation Between RG Functions and Renormalization Constants . 30 2.3 Computational Methods . 32 2.3.1 Conformal Integration . 33 2.3.2 The Large N Expansion . 38 2.3.3 Summary . 53 3 Ten Dimensional O(N) Scalar Field Theory 55 3.1 Introduction . 55 3.2 O(N) Symmetric Scalar Field Theories . 57 3.3 Ten Dimensions . 66 3.4 Calculation Techniques . 67 viii 3.4.1 Integral Reduction . 70 3.4.2 2-point Master Integrals . 75 3.4.3 Tarasov Method . 79 3.4.4 Master Integrals with Three Propagators . 82 3.4.5 Master Integrals with Four Propagators . 88 3.4.6 The Vacuum Bubble Expansion . 95 3.4.7 Renormalization . 100 3.5 Large N Checks and Results . 102 3.6 Discussion . 106 4 O(N) O(m) Landau-Ginzburg-Wilson Theory in Six × Dimensions 108 4.1 Introduction . 108 4.2 Landau-Ginzburg-Wilson Theory . 109 4.3 Six Dimensions . 113 4.4 Calculation Techniques . 115 4.4.1 Integral Reduction of the 2-point Function . 120 4.4.2 2-point Master Integrals . 127 4.4.3 3-point Interactions and the Insertion of a Propagator . 131 4.4.4 T abT acT bc and φiaφibT ab Interactions . 134 4.4.5 Renormalization . 136 4.5 Mass Mixing Matrix . 138 4.6 Results . 142 4.7 Large N Analysis . 155 4.8 Conformal Window Search . 160 4.9 Fixed Point Analysis . 169 4.10 Discussion . 183 5 Higher Derivative Higher Dimensional Quantum Field Theory 186 5.1 Introduction . 186 5.2 Critical Exponents of Higher Derivative Higher Dimensional Theories . 190 5.2.1 Calculation of NLO Diagrams . 194 5.2.2 Calculation of χ1 and η2 ................... 205 5.3 Perturbative Results for n =2.................... 208 5.4 Fixed Point Analysis . 212 5.5 Discussion . 216 ix II The Banks-Zaks Fixed Point of Quantum Chromo- dynamics 219 6 Background 220 6.1 Gauge Theories . 220 6.2 Quantum Chromodynamics . 222 6.2.1 The Banks-Zaks Fixed Point . 232 6.3 Renormalization Scheme Dependence . 235 7 Banks-Zaks Fixed Point Analysis in Momentum Subtraction Renormalization Schemes 238 7.1 Introduction . 238 7.2 Renormalization Group Functions . 239 7.3 Mass Operator Anomalous Dimension . 242 7.4 Results . 246 7.5 Discussion . 258 8 Banks-Zaks Fixed Point Analysis in Interpolating Momentum Subtraction Renormalization Schemes 260 8.1 Introduction . 260 8.2 Maximal Abelian Gauge . 262 8.3 Renormalization Group Functions . 265 8.4 Results . 273 8.5 Discussion . 282 9 Perspective 285 Appendix 291 A Master Integrals for Scalar Calculations 291 B Feynman Diagrams for O(N) O(m) Landau-Ginzburg-Wilson × Theory in Six Dimensions 299 C Reduction Relations for the 2-point Function in Landau-Ginzburg- Wilson Theory at Three Loops 304 D Banks-Zaks Fixed Points and Critical Exponents for MS, mMOM and MOMi Renormalization Schemes 307 E Banks-Zaks Fixed Points and Critical Exponents for iMOMi Renormalization Schemes 329 x F Renormalization Group Functions γA, γ and γc of QCD in MS Scheme 337 Bibliography 339 xi Chapter 1 Introduction The Standard Model of particle physics was developed in the latter half of the 20th century with contributions from many of the greatest physicists through- out history.