September 2021 Basics of Thermal Field Theory A Tutorial on Perturbative Computations 1 Mikko Lainea and Aleksi Vuorinenb aAEC, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland bDepartment of Physics, University of Helsinki, P.O. Box 64, FI-00014 University of Helsinki, Finland Abstract These lecture notes, suitable for a two-semester introductory course or self-study, offer an elemen- tary and self-contained exposition of the basic tools and concepts that are encountered in practical computations in perturbative thermal field theory. Selected applications to heavy ion collision physics and cosmology are outlined in the last chapter. 1An earlier version of these notes is available as an ebook (Springer Lecture Notes in Physics 925) at dx.doi.org/10.1007/978-3-319-31933-9; a citable eprint can be found at arxiv.org/abs/1701.01554; the very latest version is kept up to date at www.laine.itp.unibe.ch/basics.pdf. Contents Foreword ........................................... i Notation............................................ ii Generaloutline....................................... iii 1 Quantummechanics ................................... 1 1.1 Path integral representation of the partition function . ....... 1 1.2 Evaluation of the path integral for the harmonic oscillator . ..... 6 2 Freescalarfields ..................................... 13 2.1 Path integral for the partition function . ... 13 2.2 Evaluation of thermal sums and their low-temperature limit . .... 16 2.3 High-temperatureexpansion. 23 3 Interactingscalarfields............................... ... 30 3.1 Principles of the weak-coupling expansion . 30 3.2 Problems of the naive weak-coupling expansion . .. 38 3.3 Proper free energy density to (λ): ultraviolet renormalization . 40 O 3 3.4 Proper free energy density to (λ 2 ): infraredresummation . 44 O 4 Fermions.......................................... 49 4.1 Path integral for the partition function of a fermionic oscillator . ...... 49 4.2 TheDiracfieldatfinitetemperature . 53 5 Gaugefields........................................ 61 5.1 Path integral for the partition function . ... 61 5.2 Weak-couplingexpansion ............................ 67 5.3 Thermalgluonmass ............................... 73 2 5.4 Free energy density to (g3) .......................... 80 O 6 Low-energyeffectivefieldtheories. ..... 85 6.1 Theinfraredproblemofthermalfieldtheory . .. 85 6.2 Dimensionally reduced effective field theory for hot QCD . .. 91 7 Finitedensity....................................... 99 7.1 Complex scalar field and effective potential . 99 7.2 Dirac fermion with a finite chemical potential . 105 8 Real-timeobservables ................................. 112 8.1 DifferentGreen’sfunctions ........................... 112 8.2 From a Euclidean correlator to a spectral function . .... 124 8.3 Real-timeformalism ............................... 130 8.4 HardThermalLoops............................... 135 9 Applications........................................ 152 9.1 Thermalphasetransitions............................ 152 9.2 Bubblenucleationrate.............................. 161 9.3 Particleproductionrate ............................. 171 9.4 Embeddingratesincosmology ......................... 180 9.5 Evolution of a long-wavelength field in a thermal environment . .... 189 9.6 Linearresponse theory and transport coefficients . ...... 194 9.7 Equilibration rates / damping coefficients . 202 9.8 Resonancesinmedium.............................. 210 Appendix: Extended Standard Model in Euclidean spacetime . ......... 220 Index .............................................. 223 3 Foreword These notes are based on lectures delivered at the Universities of Bielefeld and Helsinki, between 2004 and 2015, as well as at a number of summer and winter schools, between 1996 and 2018. The early sections were strongly influenced by lectures by Keijo Kajantie at the University of Helsinki, in the early 1990s. Obviously, the lectures additionally owe an enormous gratitude to existing text books and literature, particularly the classic monograph by Joseph Kapusta. There are several good text books on finite-temperature field theory, and no attempt is made here to join that group. Rather, the goal is to offer an elementary exposition of the basics of perturbative thermal field theory, in an explicit “hands-on” style which can hopefully more or less directly be transported to the classroom. The presentation is meant to be self-contained and display also intermediate steps. The idea is, roughly, that each numbered section could constitute a single lecture. Referencing is sparse; on more advanced topics, as well as on historically accurate references, the reader is advised to consult the text books and review articles in refs. [0.1]–[0.16]. These notes could not have been put together without the helpful influence of many people, vary- ing from students with persistent requests for clarification; colleagues who have used parts of an early version of these notes in their own lectures and shared their experiences with us; colleagues whose interest in specific topics has inspired us to add corresponding material to these notes; alert readers who have informed us about typographic errors and suggested improvements; and collaborators from whom we have learned parts of the material presented here. Let us gratefully acknowledge in particular Gert Aarts, Chris Korthals Altes, Dietrich B¨odeker, Yannis Burnier, Ste- fano Capitani, Simon Caron-Huot, Jacopo Ghiglieri, Ioan Ghisoiu, Keijo Kajantie, Aleksi Kurkela, Harvey Meyer, Guy Moore, Paul Romatschke, Kari Rummukainen, York Schr¨oder, Mikhail Sha- poshnikov, Markus Thoma, Tanmay Vachaspati, and Mikko Veps¨al¨ainen. Mikko Laine and Aleksi Vuorinen i Notation In thermal field theory, both Euclidean and Minkowskian spacetimes play a role. In the Euclidean case, we write X (τ, xi) , x x , S = L , (0.1) ≡ ≡ | | E E ZX where i =1, ..., d, β 1 dτ , ddx , β , (0.2) ≡ ≡ ≡ T ZX Z0 Zx Zx Z and d is the space dimensionality. Fourier analysis is carried out in the Matsubara formalism via iK X K (k , k ) , k k , φ(X)= φ˜(K) e · , (0.3) ≡ n i ≡ | | ZK P where ddk T , d . (0.4) ≡ k k ≡ (2π) ZK kn Z Z Z P X Here, kn stands for discrete Matsubara frequencies, which at times are also denoted by ωn. In the case of antiperiodic functions, the summation is written as T k . The squares of four-vectors { n} read K2 = k2 + k2 and X2 = τ 2 + x2, but the Euclidean scalar product between K and X is n P defined as d K X = k τ + k xi = k τ k x , (0.5) · n i n − · i=1 X where the vector notation is reserved for contravariant Minkowskian vectors: x = (xi), k = (ki). If a chemical potential is also present, we denote k˜ k + iµ. n ≡ n In the Minkowskian case, we have (t, x) , x x , = , (0.6) X ≡ ≡ | | SM LM ZX 0 where dx x. Fourier analysis proceeds via X ≡ R R R 0 i (k , k) , k k , φ( )= φ˜( ) e K·X , (0.7) K≡ ≡ | | X K ZK dk0 where = 2π k, and the metric is chosen to be of the “mostly minus” form, K R R R = k0x0 k x . (0.8) K · X − · No special notation is introduced for the case where a Minkowskian four-vector is on-shell, i.e. when = (E , k); this is to be understood from the context. K k The argument of a field φ is taken to indicate whether the configuration space is Euclidean or Minkowskian. If not specified otherwise, momentum integrations are regulated by defining the spatial measure in d = 3 2ǫ dimensions, whereas the spacetime dimensionality is denoted by − D =4 2ǫ. A Greek index takes values in the set 0, ..., d , and a Latin one in 1, ..., d . − { } { } Finally, we note that we work consistently in units where the speed of light c and the Boltzmann ~ constant kB have been set to unity. The reduced Planck constant also equals unity in most places, excluding the first chapter (on quantum mechanics) as well as some later discussions where we want to emphasize the distinction between quantum and classical descriptions. ii General outline Physics context From the physics point of view, there are two important contexts in which relativistic thermal field theory is being widely applied: cosmology and the theoretical description of heavy ion collision experiments. In cosmology, the temperatures considered vary hugely, ranging from T 1015 GeV to T 3 ≃ ≃ 10− eV. Contemporary challenges in the field include figuring out explanations for the existence of dark matter, the observed antisymmetry in the amounts of matter and antimatter, and the formation of large-scale structures from small initial density perturbations. (The origin of initial density perturbations itself is generally considered to be a non-thermal problem, associated with an early period of inflation.) An important further issue is that of equilibration, i.e. details of the processes through which the inflationary state turned into a thermal plasma, and in particular what the highest temperature reached during this epoch was. It is notable that most of these topics are assumed to be associated with weak or even superweak interactions, whereas strong interactions (QCD) only play a background role. A notable exception to this is light element nucleosynthesis, but this well-studied topic is not in the center of our current focus. In heavy ion collisions, in contrast, strong interactions do play a major role. The lifetime of the thermal fireball created in such a collision is 10 fm/c and the maximal temperature reached is in ∼ the range of a few hundred MeV. Weak interactions are too slow to take place within the lifetime of the system. Prominent observables are the yields