Critical Properties of Phi-4-Theories

by Hagen, Kleinert; Schulte-frohlinde, Verena

• ISBN13: 9789810246594
• ISBN10: 9810246595
• Format: Paperback
• Copyright: 2001-08-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

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. Softcover.

Table of Contents

Preface	v
Introduction	1	(31)
Second-Order Phase Transitions	1	(2)
Critical Exponents	3	(6)
Correlation Functions	3	(2)
Other Critical Exponents	5	(2)
Scaling Relations	7	(2)
Models for Critical Behavior	9	(6)
Landau Theory	9	(2)
Classical Heisenberg Model	11	(4)
Fluctuating Fields	15	(8)
Ginzburg-Landau Energy Functional	16	(2)
Ginzburg Criterion	18	(2)
Kleinert Criterion	20	(3)
General Remarks	23	(9)
Appendix 1A Correlations and Structure Factor	27	(1)
Notes and References	27	(5)
Definition of &phis;4-Theory	32	(9)
Partition Function and Generating Functional	32	(2)
Free-Field Theory	34	(3)
Perturbation Expansion	37	(2)
Composite Fields	39	(2)
Notes and References	40	(1)
Feynman Diagrams	41	(11)
Diagrammatic Expansion of Correlation Functions	41	(5)
Diagrammatic Expansion of the Partition Function	46	(1)
Connected and Disconnected Diagrams	47	(2)
Multiplicities of Disconnected Diagrams	47	(1)
Cancellation of Vacuum Diagrams	48	(1)
Connected Diagrams for Two- and Four-Point Functions	49	(1)
Diagrams for Composite Fields	50	(2)
Notes and References	51	(1)
Diagrams in Momentum Space	52	(7)
Fourier Transformation	52	(3)
Free Two-Point Fuction	52	(1)
Connected n-Point Function	53	(2)
One-Particle Irreducible Diagrams and Proper Vertex Functions	55	(2)
Composite Fields	57	(1)
Theory in Continuous Dimension D	58	(1)
Notes and References	58	(1)
Structural Properties of Perturbation Theory	59	(18)
Generating Functionals	59	(1)
Connectedness Structure of Correlation Functions	60	(2)
Decomposition of Correlation Functions into Connected Correlation Functions	62	(2)
Functional Generation of Vacuum Diagrams	64	(4)
Correlation Functions From Vacuum Diagrams	68	(1)
Generating Functional for Vertex Functions	69	(5)
Landau Approximation to Generating Functional	74	(1)
Composite Fields	75	(2)
Notes and References	76	(1)
Diagrams for Multicomponent Fields	77	(14)
Interactions with O(N) Cubic Symmetry	77	(1)
Free Generating Functional for N Fields	78	(1)
Perturbation Expansion for N Fields and Symmetry Factors	79	(1)
Symmetry Factors	80	(11)
Symmetry Factors for O(N) Symmetry	81	(3)
Symmetry Factors for Mixed O(N) and Cubic Symmetry	84	(3)
Other Symmetries	87	(1)
General Symmetry Factors	88	(2)
Notes and References	90	(1)
Scale Transformations of Fields and Correlation Functions	91	(11)
Free Massless Fields	91	(2)
Free Massive Fields	93	(1)
Interacting Fields	94	(3)
Ward Identities for Broken Scale Invariance	95	(2)
Anomaly in the Ward Identities	97	(5)
Notes and References	100	(2)
Regularization of Feynman Integrals	102	(28)
Regularization	102	(2)
Dimensional Regularization	104	(8)
Calculation in Dimensional Regularization	104	(2)
Dimensional Regularization via Proper Time Representation	106	(2)
Tensor Structures	108	(1)
Dimensional Regularization of 't Hooft and Veltman	109	(2)
Subtraction Method	111	(1)
Calculation of One-Particle-Irreducible Diagrams	112	(18)
One-Loop Diagrams	112	(2)
Two-Loop Self-Energy Diagrams	114	(4)
Two-Loop Diagram of Four-Point Function	118	(2)
Two-Loop Vacuum Diagrams	120	(1)
Appendix 8A Polar Coordinates and Surface of a Sphere in D Dimensions	120	(1)
Appendix 8B More on Dimensional Regularization of 't Hooft and Veltman	121	(2)
Appendix 8C Parametric Representation of Feynman Integrals	123	(3)
Appendix 8D Expansion of Gamma Function	126	(2)
Notes and References	128	(2)
Renormalization	130	(26)
Superficial Degree of Divergence	131	(2)
Normalization Conditions	133	(7)
One-Loop Mass Renormalization	137	(1)
One-Loop Coupling Constant Renormalization	137	(1)
Two-Loop Mass and Field Renormalization	138	(2)
Method of Counterterms and Minimal Subtraction	140	(16)
Minimal Subtraction Scheme	142	(3)
Renormalization in MS-Scheme	145	(5)
Recursive Diagrammatic Subtraction	150	(3)
Appendix 9A Overlapping Divergences	153	(1)
Notes and References	154	(2)
Renormalization Group	156	(30)
Gallan-Symanzik Equation	156	(2)
Renormalization Group Equation	158	(2)
Calculation of Coefficient Functions from Counterterms	160	(3)
Solution of the Renormalization Group Equation	163	(4)
Fixed Point	167	(2)
Effective Energy and Potential	169	(3)
Special Properties of Ground State Energy	172	(2)
Approach to Scaling	174	(1)
Further Critical Exponents	175	(2)
Specific Heat	176	(1)
Susceptibility	176	(1)
Critical Magnetization	177	(1)
Scaling Relations Below Tc	177	(4)
Spontaneous Magnetization	177	(1)
Correlation Length	178	(1)
Specific Heat	178	(1)
Susceptibility	178	(1)
Transverse Susceptibility and Bending Stiffness	179	(1)
Widom's Relation	180	(1)
Comparison of Scaling Relations with Experiment	181	(1)
Critical Values g*, η, ≠, and ω in Powers of &epsis;	181	(2)
Several Coupling Constants	183	(1)
Ultraviolet versus Infrared Properties	184	(2)
Notes and References	185	(1)
Recursive Subtraction of UV-Divergences by R-Operation	186	(11)
Graph-Theoretic Notations	186	(1)
Definition of R- and R-Operation	187	(3)
Properties of Diagrams with Cutvertices	190	(1)
Tadpoles in Diagrams with Superficial Logarithmic Divergence	191	(1)
Nontrivial Example for R-Operation	192	(1)
Counterterms in Minimal Subtraction	192	(2)
Simplifications for Zm2	194	(1)
Simplifications for Z&phis;	195	(2)
Notes and References	196	(1)
Zero-Mass Approach to Counterterms	197	(29)
Infrared Power Counting	198	(1)
Infrared Rearrangement	199	(6)
The R-Operation for Massless Diagrams	202	(2)
Zero-Mass Simplifications for Z&phis;	204	(1)
Infrared Divergences in Dimensional Regularization	205	(2)
Nonexceptional External Momenta	206	(1)
Exceptional External Momenta	206	(1)
Massless Tadpole Diagrams	207	(1)
Subtraction of UV- and IR-Divergences: R*-Operation	207	(14)
Example for Subtraction of IR- and UV-Divergences	207	(2)
Graph-Theoretic Notations	209	(3)
Definition of R*- and R*-Operation	212	(1)
Construction of Infrared Subtraction Terms of Subdiagrams	213	(4)
IR-Counterterms	217	(4)
Examples for the R*-Operation	221	(5)
Appendix 12A Proof of Interchangeability of Differentiation and R-Operation	223	(2)
Notes and References	225	(1)
Calculation of Momentum Space Integrals	226	(28)
Simple Loop Integrals	226	(5)
Expansion of Loop Function	228	(2)
Modified MS-Scheme and Various Redefinitions of Mass Scale	230	(1)
Further Subtraction Schemes	231	(1)
Classification of Diagrams	231	(1)
Five-Loop Diagrams	232	(1)
Reduction Algorithm based on Partial Integration	233	(8)
Triangle Diagram	234	(2)
General Triangle Rule	236	(2)
Reduction Algorithms	238	(3)
Method of Ideal Index Constellations in Configuration Space	241	(7)
Dual Diagrams	241	(1)
Star-Triangle Rule for an Ideal Vertex	242	(2)
One Step from Ideal Index Constellation	244	(1)
Transformation of Indices	245	(1)
Construction of Tables	246	(2)
Special Treatment of Generic Four- and Five-Loop Diagrams	248	(1)
N-Shaped Diagram	248	(1)
Computer-Algebraic Program	249	(5)
Appendix 13A Fourier Transformation of Simple Powers in D Dimensions	250	(1)
Appendix 13B Further Expansions of Gamma Function	251	(2)
Notes and References	253	(1)
Generation of Diagrams	254	(19)
Algebraic Representation of Diagrams	254	(2)
Generation Procedure	256	(17)
Vacuum Diagrams	257	(1)
Two-Point Diagrams	257	(4)
Four-Point Diagrams	261	(1)
Four-Point Diagrams for Mass Renormalization	262	(1)
Check for Number of Connected Diagrams	262	(10)
Notes and References	272	(1)
Results of the Five-Loop Calculation	273	(13)
Renormalization Constants for O(N)-Symmetric Theory	274	(5)
Renormalization Constants up to Two Loops	274	(1)
Renormalization Constants up to Five Loops	275	(1)
Ratios between Bare and Renormalized Quantities up to Five Loops	276	(3)
Renormalization Constants for Theory with Mixed O(N) and Cubic-Symmetry	279	(5)
Renormalization Constants up to Two Loops	279	(3)
Renormalization Constants up to Three Loops	282	(1)
Ratios between Bare and Renormalized Quantities up to Three Loops	283	(1)
Renormalization Constant for Vacuum Energy	284	(2)
Notes and References	285	(1)
Basic Resummation Theory	286	(18)
Asymptotic Series	286	(4)
Pade Approximants	290	(1)
Borel Transformation	291	(3)
Conformal Mappings	294	(2)
Janke-Kleinert Resummation Algorithm	296	(5)
Reexpansion Functions	297	(3)
Convergent Strong-Coupling Expansion	300	(1)
Relation with Conformal Mapping Technique	300	(1)
Modified Reexpansions	301	(3)
Choosing the Strong-Coupling Growth Parameter s	302	(1)
Notes and References	302	(2)
Critical Exponents of O(N)-Symmetric Theory	304	(10)
Series Expansions for Renormalization Group Functions	304	(1)
Fixed Point and Critical Exponents	305	(2)
Large-Order Behavior	307	(2)
Resummation	309	(5)
Notes and References	313	(1)
Cubic Anisotropy	314	(14)
Basic Properties	314	(2)
Series Expansions for RG Functions	316	(3)
Fixed Points and Critical Exponents	319	(1)
Stability	320	(2)
Resummation	322	(6)
Pade Approximations for Critical Exponents	322	(1)
Resummations for Cubic Fixed Point	323	(3)
Notes and References	326	(2)
Variational Perturbation Theory	328	(36)
From Weak- to Strong-Coupling Expansions	328	(1)
Strong-Coupling Theory	328	(3)
Convergence	331	(4)
Strong-Coupling Limit and Critical Exponents	335	(2)
Explicit Low-Order Calculations	337	(5)
General Formulas	338	(1)
Perturbation Series	338	(1)
Critical Exponent ω	339	(2)
Critical Exponent ≠	341	(1)
Critical Exponent γ	341	(1)
Three-Loop Resummation	342	(4)
Five-Loop Resummation	346	(8)
Critical Exponent ω	346	(4)
Critical Exponent ≠	350	(1)
Critical Exponents η and γ	351	(3)
Interpolating Critical Exponents between Two and Four Dimensions	354	(10)
Critical Exponents ≠	356	(4)
Critical Exponents η	360	(1)
Notes and References	360	(4)
Critical Exponents from Other Expansions	364	(46)
Sixth-Order Expansion in Three Dimensions	364	(3)
Critical Exponents up to Six Loops	367	(13)
Improving the Graphical Extrapolation of Critical Exponents	380	(2)
Seven-Loop Results for N = 0, 1, 2, and 3	382	(4)
Large-Order Behavior	386	(5)
Influence of Large-Order Information	391	(5)
Another Variational Resummation Method	396	(5)
High-Temperature Expansions of Lattice Models	401	(9)
Notes and References	406	(4)
New Resummation Algorithm	410	(18)
Hyper-Borel Transformation	410	(5)
Convergence Properties	415	(3)
Parameters s and ω	416	(1)
Parameter σ	416	(1)
Parameter ρ	416	(1)
Parameter δ	417	(1)
Parameter β0	417	(1)
Resummation of Ground State Energy of Anharmonic Oscillator	418	(2)
Resummation for Critical Exponents	420	(8)
Notes and References	426	(2)
Conclusion	428	(3)
A Diagrammatic R-Operation up to Five Loops	431	(25)
A.1 Diagrams for Zg and Zm2	431	(19)
A.2 Diagrams for Z&phis;	450	(2)
A.3 Calculation of IR-Counterterms	452	(4)
A.3.1 Determination of IR-Counterterms by R*-Operation	452	(3)
A.3.2 Pole Terms of IR-Counterterms	455	(1)
B Contributions to Renormalization-Constants	456	(19)
B.1 Contributions to Zg and Zm2	456	(8)
B.2 Contribution to Z&phis;	464	(1)
B.3 Symmetry Factors in O(N)-Symmetric Theory	465	(2)
B.3.1 Symmetry Factors S2O(N) of Two-Point Diagrams	465	(1)
B.3.2 Symmetry Factors S4O(N) of Four-Point Diagrams	466	(1)
B.4 The Symmetry Factors for Cubic Symmetry	467	(8)
B.4.1 Symmetry Factors S2;(L-k,k)cub of Two-Point Diagrams	467	(2)
B.4.2 Symmetry Factors S4;(L+1-k,k)cub of Four-Point Diagrams	469	(6)
Index	475