Preprint typeset in JHEP style - PAPER VERSION Instantons and large N An introduction to non-perturbative methods in QFT Marcos Mari˜no D´epartement de Physique Th´eorique et Section de Math´ematiques, Universit´ede Gen`eve, Gen`eve, CH-1211 Switzerland [email protected] Abstract: Lecture notes for a course on non-perturbative methods in QFT. Contents 1. Introduction 2 2. Instantons in quantum mechanics 4 2.1 QM as a one-dimensional field theory 4 2.2 Unstable vacua in quantum mechanics 8 2.3 A toy model integral 9 2.4 Path integral around an instanton in QM 12 2.5 Calculation of functional determinants I: solvable models 18 2.6 Calculation of functional determinants II: Gelfand–Yaglom method 22 2.7 Instantons in the double well 29 2.8 Multi-instantons in the double well 32 2.9 The dilute instanton approximation 36 2.10 Beyond the dilute instanton approximation 37 3. Unstable vacua in QFT 40 3.1 Bounces in scalar QFT 40 3.2 The fate of the false vacuum 45 3.3 Instability of the Kaluza–Klein vacuum 45 4. Large order behavior and Borel summability 49 4.1 Perturbation theory at large order 49 4.2 The toy model integral, revisited 50 4.3 The anharmonic oscillator 52 4.4 Asymptotic expansions and Borel resummation 55 4.5 Borel transforms and large order behavior 60 4.6 Instantons and large order behavior in quantum theory 62 4.6.1 Stable vacua 63 4.6.2 Unstable vacua 63 4.6.3 Complex instantons 64 4.6.4 Cancellation of nonperturbative ambiguities 65 5. Nonperturbative aspects of gauge theories 67 5.1 Conventions and basics 67 5.2 Topological charge and θ vacua 68 5.3 Instantons in Yang–Mills theory 71 5.4 Instantons and theta vacua 76 5.5 Renormalons 79 6. Instantons, fermions and supersymmetry 84 6.1 Instantons in supersymmetric quantum mechanics 84 6.1.1 General aspects 84 6.1.2 Supersymmetry breaking 86 6.1.3 Instantons and fermionic zero modes 88 6.2 Fermions and anomalies in Yang–Mills theory 94 – 1 – 7. Sigma models at large N 98 7.1 The O(N) non-linear sigma model 98 N 1 7.2 The P − sigma model 102 7.2.1 The model and its instantons 102 7.2.2 The effective action at large N 105 7.2.3 Topological susceptibility at large N 107 8. The 1/N expansion in QCD 109 8.1 Fatgraphs 109 8.2 Large N rules for correlation functions 114 8.3 QCD spectroscopy at large N: mesons and glueballs 117 8.4 Baryons at large N 118 8.5 Analyticity in the 1/N expansion 120 8.6 Large N instantons 123 9. A solvable toy model: large N matrix quantum mechanics 125 9.1 Defining the model. Perturbation theory 125 9.2 Exact ground state energy in the planar approximation 127 9.3 Excited states, or glueball spectrum 131 9.4 Some examples 132 9.5 Large N instantons in matrix quantum mechanics 134 9.6 Adding fermions, or meson spectrum 136 10. Applications in QCD 141 10.1 Chiral symmetry and chiral symmetry breaking 141 10.2 The U(1) problem 145 10.3 The U(1) problem at large N. Witten–Veneziano formula 146 A. Polology and spectral representation 149 B. Chiral Lagrangians 151 C. Effective action for large N sigma models 156 1. Introduction A nonperturbative effect in QFT or QM is an effect which can not be seen in perturbation theory. In these notes we will study two types of nonperturbative effects. The first type is due to instantons, i.e. to nontrivial solutions to the classical equations of motion. If g is the coupling constant, these effects have the dependence A/g e− . (1.1) Notice that this is small if g is small, but on the other hand it is completely invisible in perturbation theory, since it displays an essential singularity at g = 0. – 2 – Figure 1: Two quantum-mechanical potentials where instanton effects change qualitatively our understanding of the vacuum structure. Instanton effects are responsible of one of the most important quantum-mechanical effect: tunneling through a potential barrier. This effect changes qualitatively the structure of the quantum vacuum. In a potential with a perturbative ground state degeneracy, like the one shown on the l.h.s. of Fig. 1, tunneling effects lift the degeneracy. There a single ground state, and the energy difference between the ground state and the first excited state is an instanton effect of the form (1.1), A/g E (g) E (g) e− . (1.2) 1 − 0 ∼ In a potential with a metastable vacuum, like the one shown in the r.h.s. of Fig. 1, the perturbative vacuum obtained by small quantum fluctuations around this metastable vac- uum will eventually decay. This means that the ground state energy has a small imaginary part, A/g E (g)=Re E (g)+iIm E (g), Im E (g) e− (1.3) 0 0 0 0 ∼ which also has the dependence on g typical of an instanton effect. Some of these instanton effects appear as well in quantum field theories, and they are an important source of information about the dynamics of these theories. However, there are many important strong coupling phenomena in QFT, like confinement and chiral symmetry breaking in QCD, which can not be explained in a satisfactory way in terms of instantons. We should warn the reader that this is a somewhat polemical statement, since for example practicioners of the instanton liquid approach claim that they can explain many aspects of nonperturbative QCD with a semi-phenomenological model based on instanton physics (see [73] for a review). Some aspects of this debate were first pointed out by Witten in his seminal paper [86], and the debate is still going on (see for example [46]). A different type of nonperturbative method in QFT is based on resumming an infinite subset of diagrams in perturbation theory. This is nonperturbative in the sense that, typically, the effects that one discovers in this way cannot be seen at any finite order of perturbation theory. As an illustration of this, taken from [87], consider the following series: (log g)2 (log g)3 f (g)= g g log g + g g + (1.4) 0 − 2 − 6 · · · We see that, order by order in perturbation theory, one has the property lim f0(g) = 0. (1.5) g 0 → – 3 – However, each term vanishes more slowly than the one before, and taking into account all the terms in the series one finds f0(g) = 1. Therefore, the property (1.5), which holds at any order in perturbation theory, is not a property of the full resummed series, which satisfies instead lim f0(g) = 0. (1.6) g 0 6 → In this sense, the result (1.6) should be also regarded as a nonperturbative effect. Notice that, in this approach, one does not consider a different saddle-point in the path integral, as in instanton physics. Rather, one resums an infinite number of terms in the perturbative series around the conventional vacuum. The most powerful nonperturbative method of this type is probably the 1/N expansion of gauge theories [77], where one re-organizes the set of diagrams appearing in perturbation theory according to their dependence on the number N of degrees of freedom. In these notes we give a pedagogical introduction to these two methods, instantons and large N. We will present general aspects of these methods and we will illustrate them in exactly solvable models. 2. Instantons in quantum mechanics 2.1 QM as a one-dimensional field theory We first recall that the ground state energy of a quantum mechanical system in a poten- tial W (q) can be extracted from the small temperature behavior of the thermal partition function, βH(β) Z(β) = tr e− , (2.1) as 1 E = lim log Z(β). (2.2) − β β →∞ In the path integral formulation, S(q) Z(β)= [q(t)]e− , (2.3) D Z where S(q) is the action of the Euclidean theory, β/2 1 S(q)= dt (q ˙(t))2 + W (q(t)) (2.4) β/2 2 Z− and the path integral is over periodic trajectories q( β/2) = q(β/2). (2.5) − We note that the Euclidean action can be regarded as an action in Lagrangian mechanics, β/2 1 S(q)= dt (q ˙(t))2 V (q) (2.6) β/2 2 − Z− where the potential is V (q)= W (q), (2.7) − – 4 – i.e. it is the inverted potential of the original problem. It is possible to compute the ground state energy by using Feynman diagrams. We will assume that the potential W (q) is of the form m2 W (q)= q2 + W (q) (2.8) 2 int where Wint(q) is the interaction term. Then, the path integral defining Z can be computed in standard Feynman perturbation theory by expanding in Wint(q). We will actually work in the limit in which β , since in this limit many features are simpler, like for example →∞ the form of the propagator. In this limit, the free energy will be given by β times a β- independent constant, as follows from (2.2). In order to extract the ground state energy we have to take into account the following 1. Since we have to consider F (β) = log Z(β), only connected bubble diagrams con- tribute. 2. The standard Feynman rules in position space will lead to n integrations, where n is the number of vertices in the diagram.