Critical properties of [Greek letter phi]4-theories / Hagen Kleinert, Verena Schulte-Frohlinde.

Includes bibliographical references and index.

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1. Introduction. 1.1. Second-order phase transitions. 1.2. Critical exponents. 1.3. Models for critical behavior. 1.4. Fluctuating fields. 1.5. General remarks -- 2. Definition of [symbol]-theory. 2.1. Partition function and generating functional. 2.2. Free-field theory. 2.3. Perturbation expansion. 2.4. Composite fields -- 3. Feynman diagrams. 3.1. Diagrammatic expansion of correlation functions. 3.2. Diagrammatic expansion of the partition function. 3.3. Connected and disconnected diagrams. 3.4. Connected diagrams for two- and four-point functions. 3.5. Diagrams for composite fields -- 4. Diagrams in momentum space. 4.1. Fourier transformation. 4.2. One-particle irreducible diagrams and proper vertex functions. 4.3. Composite fields. 4.4. Theory in continuous dimension D -- 5. Structural properties of perturbation theory. 5.1. Generating functionals. 5.2. Connectedness structure of correlation functions. 5.3. Decomposition of correlation functions into connected correlation functions. 5.4. Functional generation of vacuum diagrams. 5.5. Correlation functions from vacuum diagrams. 5.6. Generating functional for vertex functions. 5.7. Landau approximation to generating functional. 5.8. Composite fields -- 6. Diagrams for multicomponent fields. 6.1. Interactions with O(N) cubic symmetry. 6.2. Free generating functional for N fields. 6.3. Perturbation expansion for TV fields and symmetry factors. 6.4. Symmetry factors -- 7. Scale transformations of fields and correlation functions. 7.1. Free massless fields. 7.2. Free massive fields. 7.3. Interacting fields. 7.4. Anomaly in the ward identities -- 8. Regularization of Feynman integrals. 8.1. Regularization. 8.2. Dimensional regularization. 8.3. Calculation of one-particle-irreducible diagrams -- 9. Renormalization. 9.1. Superficial degree of divergence. 9.2. Normalization conditions. 9.3. Method of counterterms and minimal subtraction -- 10. Renormalization group. 10.1. Callan-Symanzik equation. 10.2. Renormalization group equation. 10.3. Calculation of coefficient functions from counterterms. 10.4. Solution of the renormalization group equation. 10.5. Fixed point. 10.6. Effective energy and potential. 10.7. Special properties of ground state energy. 10.8. Approach to scaling. 10.9. Further critical exponents. 10.10. Scaling relations below Tc. 10.11. Comparison of scaling relations with experiment. 10.12. Critical values g*, n, v, and w in powers of e. 10.13. Several coupling constants. 10.14. Ultraviolet versus infrared properties -- 11. Recursive subtraction of UV-divergences by R-operation. 11.1. Graph-theoretic notations. 11.2. Definition of R- and R-operation. 11.3. Properties of diagrams with cutvertices. 11.4. Tadpoles in diagrams with superficial logarithmic divergence. 11.5. Nontrivial example for R-operation. 11.6. Counterterms in minimal subtraction. 11.7. Simplifications for Zm2. 11.8. Simplifications for Z[symbol] -- 12. Zero-mass approach to counterterms. 12.1. Infrared power counting. 12.2. Infrared rearrangement. 12.3. Infrared divergences in dimensional regularization. 12.4. Subtraction of UV- and IR-divergences: R*-operation. 12.5. Examples for the R*-operation -- 13. Calculation of momentum space integrals. 13.1. Simple loop integrals. 13.2. Classification of diagrams. 13.3. Five-loop diagrams. 13.4. Reduction algorithm based on partial integration. 13.5. Method of ideal index constellations in configuration space. 13.6. Special treatment of Generic four- and five-loop diagrams. 13.7. Computer-algebraic program -- 14. Generation of diagrams. 14.1. Algebraic representation of diagrams. 14.2. Generation procedure -- 15. Results of the five-loop calculation. 15.1. Renormalization constants for O(N)-symmetric theory. 15.2. Renormalization constants for theory with mixed O(N) and cubic-symmetry. 15.3. Renormalization constant for vacuum energy -- 16. Basic resummation theory. 16.1. Asymptotic series. 16.2. Pade approximants. 16.3. Borel transformation. 16.4. Conformal mappings. 16.5. Janke-Kleinert resummation algorithm. 16.6. Modified reexpansions -- 17. Critical exponents of O(N)-symmetric theory. 17.1. Series expansions for renormalization group functions. 17.2. Fixed point and critical exponents. 17.3. Large-order behavior. 17.4. Resummation -- 18. Cubic anisotropy. 18.1. Basic properties. 18.2. Series expansions for RG functions. 18.3. Fixed points and critical exponents. 18.4. Stability. 18.5. Resummation -- 19. Variational perturbation theory. 19.1. From weak- to strong-coupling expansions. 19.2. Strong-coupling theory. 19.3. Convergence. 19.4. Strong-coupling limit and critical exponents. 19.5. Explicit low-order calculations. 19.6. Three-loop resummation. 19.7. Five-loop resummation. 19.8. Interpolating critical exponents between two and four dimensions -- 20. Critical exponents from other expansions. 20.1. Sixth-order expansion in three dimensions. 20.2. Critical exponents up to six loops. 20.3. Improving the graphical extrapolation of critical exponents. 20.4. Seven-loop results for N = 0, 1, 2, and 3. 20.5. Large-order behavior. 20.6. Influence of large-order information. 20.7. Another variational resummation method. 20.8. High-temperature expansions of lattice models -- 21. New resummation algorithm. 21.1. Hyper-borel transformation. 21.2. Convergence properties. 21.3. Resummation of ground state energy of anharmonic oscillator. 21.4. Resummation for critical exponents.

This work explains in detail how to perform perturbation expansions in quantum field theory to high orders, and how to extract the critical properties of the theory from the resulting divergent power series. This work explains in detail how to perform perturbation expansions in quantum field theory to high orders, and how to extract the critical properties of the theory from the resulting divergent power series. These properties are calculated for various second-order phase transitions of three-dimensional systems with high accuracy, in particular the critical exponents observable in experiments close to the phase transition.;Beginning with an introduction to critical phenomena, this book develops the functional-integral description of quantum field theories, their perturbation expansions, and a method for finding recursively all Feynman diagrams to any order in the coupling strength. Algebraic computer programs are supplied on accompanying World Wide Web pages. The diagrams correspond to integrals in momentum space. They are evaluated in 4-epsilon dimensions, where they possess pole terms in 1/epsilon. The pole terms are collected into renormalization constants.;The theory of the renormalization group is used to find the critical scaling laws. They contain critical exponents which are obtained from the renormalization constants in the form of power series. These are divergent, due to factorially growing expansion coefficients. The evaluation requires resummation procedures, which are performed in two ways: (1) using traditional methods based on Pade and Borel transformations, combined with analytic mappings; (2) using modern variational perturbation theory, where the results follow from a simple strong-coupling formula. As a crucial test of the accuracy of the methods, the critical exponent alpha governing the divergence of the specific heat of superfluid helium is shown to agree very well with the extremely precise experimental number found in the space shuttle orbiting the earth (whose data are displayed on the cover of the book).;The phi4-theories investigated in this book contain any number N of fields in an O(N)-symmetric interaction, or in an interaction in which O(N)-symmetry is broken by a term of a cubic symmetry. The crossover behavior between the different symmetries is investigated. In addition, alternative ways of obtaining critical exponents of phi4-theories are sketched, such as variational perturbation expansions in three rather than 4-epsilon dimensions, and improved ratio tests in high-temperature expansions of lattice models.

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