The Path-Integral Approach to Spontaneous Symmetry Breaking arXiv:0810.1412v2 [hep-ph] 7 Jan 2009 The front cover shows quantum mechanics (symbolized by Schr¨odinger’s cat) playing with the Higgs-field in the Mexican hat potential. The back cover shows the Maxwell construction of the Mexican hat potential, which is obtained in the path-integral approach to the N =2 linear sigma model. I wish to thank Pepper the cat for her cooperation in the making of the cover. The Path-Integral Approach to Spontaneous Symmetry Breaking Een wetenschappelijke proeve op het gebied der Natuurwetenschappen, Wiskunde en Informatica Proefschrift ter verkrijging van de graad van doctor aan de Radboud Universiteit Nijmegen op gezag van de rector magnificus prof. mr. S.C.J.J. Kortmann, volgens besluit van het College van Decanen in het openbaar te verdedigen op dinsdag 3 februari 2009 om 15.30 uur precies door Marcel Theodorus Maria van Kessel geboren op 19 maart 1980 te Tegelen Promotores: Prof. dr. R.H.P. Kleiss Prof. dr. E.N. Argyres (NCSR Demokritos Athens) Manuscriptcommissie: Prof. dr. S. de Jong Prof. dr. E. Laenen (NIKHEF Amsterdam) Prof. dr. C. Papadopoulos (NCSR Demokritos Athens) Het werk beschreven in dit proefschrift maakt deel uit van het onderzoeksprogramma van de Stichting voor Fundamenteel Onderzoek der Materie (FOM), die financieel wordt gesteund door de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). ISBN 978-90-9023799-2 Contents 1 Introduction 1 1.1 SpontaneousSymmetryBreaking . 1 1.2 ThePathIntegral................................. 2 1.3 Literature ..................................... 3 1.4 OutlineofthisThesis............................... 5 2 The Effective Action 7 2.1 Definition ..................................... 7 2.2 TheMeaningoftheEffectiveAction .. .. 8 2.3 TheArgumentforConvexity. 10 2.4 TheEffectivePotential.............................. 11 3 The N =1 LSM: The Canonical Approach 13 3.1 Green’sFunctions................................. 13 3.2 TheEffectivePotential.............................. 19 3.2.1 d =1.................................... 20 3.2.2 d =4.................................... 22 3.3 Instantons..................................... 23 4 The N =1 LSM: The Path-Integral Approach 25 4.1 TheEffectivePotential.............................. 25 4.2 TheGreen’sFunctions .............................. 29 4.3 The Green’s Functions Near Another Particle . .. 31 5 The N =1 LSM: Fixing the Paths 33 5.1 TheFreeTheorywithFixedPaths . .. .. 33 5.1.1 TheGreen’sFunctions .......................... 35 5.1.2 TheEffectiveAction ........................... 35 5.1.3 Conclusions ................................ 36 5.2 The N =1LinearSigmaModel ......................... 38 5.2.1 The Classical Solutions . 40 5.2.2 TheClassicalAction ........................... 45 5.2.3 ThePathIntegral............................. 47 5.2.4 TheDivergences ............................. 51 5.2.5 The Alternative Effective Potential . 57 i ii CONTENTS 5.2.6 TheGreen’sFunctions .......................... 58 6 The N =2 LSM: The Canonical Approach 61 6.1 Green’sFunctions................................. 61 6.2 TheEffectivePotential.............................. 72 6.2.1 ZeroDimensions ............................. 74 6.2.2 Calculating The Effective Potential . 76 6.2.3 d = 1 And d =2 ............................. 77 6.2.4 d =4.................................... 77 7 The N =2 LSM: The Path-Integral Approach I 79 7.1 Green’sFunctions................................. 80 7.2 TheEffectivePotential.............................. 84 7.2.1 Including One Minimum . 86 7.2.2 Including All Minima . 88 8 Path Integrals in Polar Variables 91 8.1 Introduction.................................... 91 8.2 AConjecture ................................... 92 8.3 TheShiftedToyModel.............................. 94 8.3.1 CartesianResults............................. 94 8.3.2 PolarResults ............................... 95 8.4 TheArctangentToyModel ........................... 105 8.4.1 CartesianResults ............................. 105 8.4.2 PolarResults ............................... 107 8.5 ProofoftheConjecture............................. 115 8.5.1 AnExample................................ 129 8.5.2 The Jacobian and w-Loops........................ 130 8.5.3 The Dimensional Regularization Scheme . 132 8.6 The1-DimensionalCase ............................. 132 8.7 A 1-Dimensional Illustration . 138 8.7.1 r(x) ................................... 139 h i 8.7.2 The ϕ1-Propagator ............................ 146 9 The N =2 LSM: The Path-Integral Approach II 151 9.1 Green’sFunctions................................. 152 9.1.1 η- And w-Green’s-Functions . 155 9.1.2 The ϕ-Green’s-Functions. 157 9.1.3 Schwinger-DysonCheck . 159 9.1.4 The Canonical ϕ1-Propagator ...................... 161 9.2 1-Dimensional Calculation . 162 9.2.1 r(x) ................................... 163 h i 9.2.2 The ϕ1-Propagator ............................ 165 9.2.3 Recovering(9.56) ............................. 169 9.3 TheEffectivePotential.............................. 170 CONTENTS iii 10 Conclusions 177 A Standard Integrals 179 A.1 d =1........................................ 179 A.2 d =4........................................ 180 Summary 185 Samenvatting 187 Curriculum Vitae 189 Acknowledgements 191 iv CONTENTS Chapter 1 Introduction For the last couple of decades a theory that goes by the unassuming name of ‘The Standard Model’, has been the generally accepted theory of fundamental physics. This Standard Model has been very successful in describing experiments in particle physics. All particles that were theoretically predicted by it have been detected, except for one: the Higgs particle. By now this Higgs particle, often called ‘the holy grail of high-energy physics’, has become so important that billions of euros are spent to build large particle colliders, hoping to produce these Higgs particles. In Europe the LHC is being built, mainly for this purpose, and this 27 km long accelerator is expected to become operational in 2008. Knowing this it is clear that the Higgs sector of the Standard Model is very important and interesting. The Higgs mechanism was proposed in the 60’s by Brout and Englert [1], Higgs [2, 3] and Guralnik, Hagen and Kibble [4] to give masses to the gauge bosons and the fermions, while keeping the theory renormalizable. The main feature of this Higgs mechanism is the mechanism of spontaneous symmetry breaking (SSB), which was introduced into quantum field theory by Nambu [5, 6], in analogy to the BCS theory of superconductivity. This mechanism of SSB will be the main topic of this thesis. A nice introduction to SSB and the Higgs mechanism can be found in a review article by Bernstein [7]. 1.1 Spontaneous Symmetry Breaking How does SSB work in quantum field theory, and what is it? The canonical approach to SSB, which one finds in most textbooks (e.g. [8, 9, 10]), is as follows. One starts with a (bare) Lagrangian, obeying some symmetry in the fields (e.g. reflection or rotational symmetry), of which the bare (classical) potential has more than one minimum. The most common, and most important, example is the ‘Mexican hat’ potential. This means that the set of minima must also obey the symmetry, which means again that in any given minimum the fields cannot all be zero. Writing all fields into the single vector ϕ we have at the minima: ϕ = 0. Therefore the classical lowest energy states, or vacua, are degenerate and have a non-zero6 field value, ϕVAC = 0. In a quantum field theory the lowest energy state, or vacuum VAC , should be calculated6 from the Schr¨odinger equation: | i H VAC = E VAC . (1.1) | i VAC| i 1 2 CHAPTER 1. INTRODUCTION Clearly, because of the very complicated form of the Hamiltonian H in a quantum field theory, this equation can not be solved. Inspired by the classical minimum-energy states, one therefore postulates that also the quantum vacuum is degenerate, and that: VAC ϕ VAC =0 . (1.2) h | | i 6 So there are multiple vacuum states. But we can only live in one of these, and nature has chosen one of these vacuum states. Which one has been chosen, cannot be determined, and is therefore unimportant, because all theories built on one of these states have exactly the same physics. This is called spontaneous symmetry breaking, i.e. the vacuum state of the theory does not have the same symmetry as the Lagrangian. So the dynamics of the theory obey a certain symmetry, which is not respected by the vacuum state. Having postulated (1.2) one can then derive, via the equations of motion, the Schwinger- Dyson equations and the Feynman rules, that this gives a mass-like term for all particles coupling to the (Higgs) field ϕ. The fluctuation in this (Higgs) field around the constant value it has in the chosen vacuum is the Higgs particle. After this one can calculate all Green’s functions of the theory. Also one can construct the 1PI Green’s functions and sum them, in the appropriate way, to obtain the effective potential. As we shall see this effective potential comes out to be complex and can be non- convex in certain domains. This is the well known convexity problem, i.e. the canonical perturbative calculation gives a non-convex effective potential, whereas general arguments show that this effective potential is convex. The precise meaning of convex will be discussed in chapter 2. Also its convexity will be proven there. 1.2 The Path Integral Now this mechanism of SSB can also be studied from the viewpoint of the path integral. We know that the path-integral approach, by Feynman, is just