Nikolai F. Nelipa

Introduction to Gauge Field Theories

Units and Notation.- I Invariant Lagrangians.- 1. Global Invariance.- 1.1 The Global Lorentz Group. Relativistic Invariance.- 1.1.1 The Lorentz Group.- 1.1.2 The Lie Algebra of the Lorentz Group.- 1.1.3 Representations of the Lorentz Group.- 1.1.4 Reducible and Irreducible Representations.- 1.1.5 Relativistically Invariant Quantities.- 1.1.6 The Lagrangian Formalism.- 1.1.7 The Hamiltonian Formalism.- 1.1.8 The Operator Form of the Quantum Field Theory.- 1.2 Global Groups of Internal Symmetry. Unitary Symmetry.- 1.2.1 Internal Symmetry Properties.- 1.2.2 Condition for the Invariance of the Lagrangian.- 1.2.3 Classification of Groups.- 2. Local (Gauge) Invariance.- 2.1 Locally (Gauge-) Invariant Lagrangians.- 2.1.1 The Group of Local Transformations.- 2.1.2 Gauge Fields.- 2.1.3 Conditions for the Local Invariance of the Lagrangian.- 2.1.4 Connection Between the Globally and the Locally Invariant Lagrangians.- 2.2 Gauge Fields.- 2.2.1 Transformations of the Gauge Fields.- 2.2.2 The Lagrangian for Gauge Fields.- 2.2.3 Conserved Currents.- 2.3 The Abelian Group Ux. The Electromagnetic Field.- 2.4 The Non-Abelian Group SU2. The Yang-Mills Field.- 3. Spontaneous Symmetry-Breaking.- 3.1 Degeneracy of the Vacuum States and Symmetry-Breaking.- 3.2 Spontaneous Breaking of Global Symmetry.- 3.2.1 Exact Symmetry.- 3.2.2 Spontaneous Symmetry-Breaking.- 3.3 Spontaneous Breaking of Local Symmetry.- 3.3.1 Exact Symmetry.- 3.3.2 Spontaneously Broken Symmetry.- 3.4 Residual Symmetry.- II Quantum Theory of Gauge Fields.- 4. Path Integrals and Transition Amplitudes.- 4.1 Unconstrained Fields.- 4.1.1 Systems with One Degree of Freedom.- 4.1.2 Systems with a Finite Number of Degrees of Freedom.- 4.1.3 The Boson Fields.- 4.1.4 The Fermion Fields.- 4.2 Fields with Constraints.- 4.2.1 Systems with a Finite Number of Degrees of Freedom.- 4.2.2 The Electromagnetic Field.- 4.2.3 The Yang-Mills Field.- 5. Covariant Perturbation Theory.- 5.1 Greens Functions. Generating Functionals.- 5.1.1 Path-Integral Formulation.- 5.1.2 Generating Functional W (J).- 5.1.3 Greens Functions in Perturbation Theory.- 5.1.4 Types of Diagrams.- 5.1.5 Generating Functional Z (J).- 5.1.6 Generating Functional ? (?).- 5.1.7 The Tree Approximation for ? (?).- 5.1.8 Another Expression for W (J).- 5.1.9 Expression for the Matrix Elements of the S-Matrix in Terms of Greens Functions.- 5.2 The ?4 Interaction Model.- 5.3 A Model with Non-Abelian Gauge Fields.- 5.4 The 1/TV-Expansion.- III Gauge Theory of Electroweak Interactions.- 6. Lagrangians of the Electroweak Interactions.- 6.1 The Standard Model for the Electroweak Interactions of Leptons.- 6.1.1 The Standard Model.- 6.2 Quark Models of Hadrons.- 6.2.1 Hadrons.- 6.2.2 SU3-Symmetry. Three Quarks.- 6.2.3 SU4-Symmetry. Charm.- 6.2.4 Coloured Quarks.- 6.2.5 Quark-Lepton Symmetry.- 6.2.6 Two Types of Models with Coloured Quarks.- 6.2.7 Heavy Quarks.- 6.3 The Standard Model of Electroweak Interactions of Quarks.- 6.4 Non-Standard Models.- 6.4.1 Ambiguity in the Choice of the Model.- 6.4.2 The SU3 x U1-Model.- 7. Quantum Electrodynamics.- 7.1 Covariant Perturbation Theory for Quantum Electrodynamics.- 7.2 Differential Cross-Sections.- 7.2.1 Formula for the Differential Cross-Section.- 7.2.2 Cross-Section of the Compton Scattering.- 8. Weak Interactions.- 8.1 Processes Caused by Neutral Weak Currents.- 8.1.1 Diagonal Terms.- 8.1.2 Non-Diagonal Terms.- 8.1.3 Comparison with Experiment.- 8.1.4 Non-Standard Models.- 8.2 Processes Caused by Charged Weak Currents.- 8.2.1 Charged Weak Quark Current.- 8.2.2 Leptonic Processes.- 8.2.3 Semi-Leptonic Decays.- 8.2.4 Non-Leptonic Hadronic Decays.- 8.3 CP Violation.- 8.3.1 NeutralAT-MesonDecay.- 8.3.2 CP Violation and Gauge Models.- 9. Higher Orders in Perturbation Theory.- 9.1 Divergences of Matrix Elements.- 9.2 Renormalization.- 9.2.1The Renormalization Procedure.- 9.2.2 Dimensional Regularization.- 9.2.3 The R-Operation.- 9.2.4 Generalized Ward Identities.- 9.2.5 Renormalization of Gauge Fields.- 9.2.6 Unitarity of the Amplitude.- 9.2.7 Anomalies.- 9.2.8 Renormalization of the Standard Model.- IV Gauge Theory of Strong Interactions.- 10. Asymptotically Free Theories.- 10.1 Renormalization Group Equations and Their Solutions.- 10.1.1 Multiplicative Renormalizations and Their Groups.- 10.1.2 Dependence of the Factors Zi on the Dimensionless Parameters.- 10.1.3 The Renormalization Group Equations.- 10.1.4 Effective Charge and Asymptotic Freedom.- 10.1.5 Method of Investigation.- 10.2 The Models.- 10.2.1 Models Without Non-Abelian Gauge Fields.- 10.2.2 Models with Non-Abelian Gauge Fields.- 11. Dynamical Structure of Hadrons.- 11.1 Experimental Basis for Scaling.- 11.2 Exact Scaling and the Parton Structure of Hadrons.- 11.2.1 The Quark-Parton Model.- 11.2.2 Sum Rules.- 11.2.3 Relations Between the Structure Functions.- 11.2.4 Comparison with Experiment.- 11.2.5 Quark and Gluon Distribution Functions.- 12. Quantum Chromodynamics. Perturbation Theory.- 12.1 Covariant Perturbation Theory for Quantum Chromodynamics.- 12.1.1 The Lagrangian for Quantum Chromodynamics.- 12.1.2 Covariant Perturbation Theory.- 12.1.3 Renormalizability of Quantum Chromodynamics.- 12.2 Examples of Perturbation-Theory Calculations.- 12.2.1 Basic Processes.- 12.2.2 Cross-Sections for the Sub-Processes.- 12.2.3 Radiative Corrections.- 12.2.4 The Effective (or Running) Coupling Constant.- 12.2.5 Asymptotic Freedom of Quantum Chromodynamics.- 12.2.6 Scaling Violation.- 12.3 The Method of Operator Product Expansion.- 12.4 Evolution Equations.- 12.5 The Summation Method of Feynman Diagrams.- 12.6 Quark and Gluon Fragmentation into Hadrons.- 13. Lattice Gauge Theories. Quantum Chromodynamics on a Lattice.- 13.1 Classical and Quantum Chromodynamics on a Lattice.- 13.1.1 The Lattice and Its Elements.- 13.1.2 Gauge Fields on a Lattice.- 13.1.3 The Spinor (Quark) Field on a Lattice.- 13.1.4 Classical Chromodynamics on a Lattice.- 13.1.5 Quantum Chromodynamics on a Lattice.- 13.1.6 The Continuum Limit.- 13.2 Strong Coupling Expansion for the Gauge Fields.- 13.2.1 The Loop Average.- 13.2.2 The Area Law. The Quark Confinement.- 13.3 Non-Perturbative Calculations in Quantum Chromodynamics by Means of Monte-Carlo Simulations.- 13.3.1 Monte-Carlo Simulations.- 13.3.2 Results of Calculations.- 14. Grand Unification.- 14.1 The SU5-Model.- 14.1.1 The Lagrangian for the Model.- 14.1.2 Energy Dependence of the Coupling Constants.- 14.1.3 Proton Decay.- 14.2 Structure of the Fermion Multiplets in the SUn-Model.- 14.2.1 Structure of SUn-Multiplets with Respect to the Subgroup SU3.- 14.2.2 Structure of SUn-Multiplets with Respect to the Electric Charge.- 14.2.3 Structure of SUn-Multiplets with Respect to the Subgroup SU2 x U1.- 14.3 General Requirements and the Choice of Model.- 14.4 The SU8-Model.- 14.4.1 Composition of the Model.- 14.4.2 The Yukawa Terms and the Fermion Masses.- 14.4.3 Spontaneous Breaking of SU8-Symmetry.- 14.5 The Pati-Salam Model.- 15. Topological Solitons and Instantons.- 15.1 One-Dimensional and Two-Dimensional Models.- 15.1.1 One-Dimensional Solitons.- 15.1.2 Vortices.- 15.1.3 The Idea of Topological Analysis.- 15.2 Homotopy Groups.- 15.2.1 Homotopy Classes.- 15.2.2 Homotopy Group.- 15.2.3 Determining the Homotopy Groups.- 15.2.4 Topological Charge.- 15.2.5 Dimension of the Space and Compositio of the Model.- 15.3 Monopole.- 15.3.1 The DiracMonopole.- 15.3.2 Gauge Theory of the Monopole.- 15.4 Instantons.- 15.5 Quantum Theory of Solitons.- 15.5.1 The Generating Functional for the Greens Functions..- 15.5.2 Perturbation Theory.- 16. Conclusion.- 16.1 Quark and Gluon Confinement.- 16.2 Potential Approach.- 16.3 QCD Sum Rules.- 16.4 Theory of Loop Functional.- 16.5 Unified Gauge Models.- 16.6 Unified Supersymmetric Models.- 16.7 Gauge Theory of Gravitation.- 16.8 Superunification.- 16.9 Composite Models.- 16.10 The Elementary Particles and the Cosmology.- List of Symbols.

Spontaneous Symmetry-Breaking

As we have seen in Chap. 2, the gauge-invariant theories include massless gauge fields. From the viewpoint of physical applications both massless and massive gauge fields are relevant. Just adding a mass term to the Lagrangian for the gauge field is not allowed since it would lead to the violation of the gauge-invariance of the Lagrangian. Therefore, a different approach has been proposed in which gauge fields acquire a mass by breaking of the gauge-invariance of the vacuum, while the Lagrangian of the gauge field remains gauge invariant (spontaneous symmetry-breaking). Symmetry-breaking of the vacuum may be incomplete. A part of the gauge fields then remains massless. This makes it possible to build theories including both massive and massless gauge fields, a circumstance used in unifying the short-range and the long-range interactions (e.g., the weak and the electromagnetic interactions), whose mediators are massive intermediate bosons and massless photons, respectively.

Covariant Perturbation Theory

In the preceding chapter, the expressions for the vacuum-to-vacuum transition amplitude have been obtained. Of practical interest, however, are transitions of a system of initially free particles into a final system of free particles. As shown in Sect. 1.1, such transitions are described by the matrix elements of the S-matrix. Therefore, we shall find in this chapter an expression for the matrix element of the S-matrix, or for the transition amplitude between two states, in terms of the path integral. Unfortunately, effective general methods of analytic calculations of these integrals are lacking as yet and one has to resort to approximate ones. Most developed are the methods of perturbation theory representing the transition amplitude as a series in the coupling constant.

A new expression for the high‐energy upper bound on the scattering amplitude

The two Froissart–Martin high‐energy upper bounds for forward and nonforward scattering are combined into one formula under the additional assumption that the scattering amplitude is polynomially bounded in energy for all scattering angles inside the Lehmann–Martin ellipse. The method used presents a modification of that of Kinoshita, Loeffel, and Martin. The analogous bound for the scattering of particles with spin is obtained as well. Using the same method, a bound for the case of complex scattering angles is also derived and ways leading to its improvement by using the solution of the Dirichlet problem are suggested.

Lagrangians of the Electroweak Interactions

We proceed with the models of electromagnetic and weak interactions. To construct the Lagrangian for a unified model, the following steps are required. 1) Choosing the gauge group which determines the interaction-mediating fields; the number of the gauge fields is equal to the dimension of the adj oint representation of this group; 2) Choosing primary fermions to underlie the model; 3) Choosing the representations of the gauge group in which the fermions are placed; the lowest representations are usually chosen. 4) Introducing an appropriate number of multiplets of scalar mesons as well as of interaction terms of these multiplets with the fermions (the Yukawa terms) to obtain massive particles; 5) Specifying the final composition of the model; 6) Writing the globally invariant Lagrangian for the model; 7) Writing the corresponding locally invariant Lagrangian; 8) Using the spontaneous symmetry-breaking mechanism to obtain the expression for the Lagrangian and then to diagonalize its free part.

Path Integrals and Transition Amplitudes

There are various formulations of quantum field theory, differing in the form of a basic quantity, viz. the transition amplitude. In the most commonly used operator approach, the transition amplitude is expressed as the vacuum expectation value of the product of particle creation and annihilation operators. These operators obey certain commutation relations (cf. Sect. 1.1). Another formulation is based on expressing the transition amplitude in terms of path integrals over the fields. In studying the gauge fields, the path-integral formalism has proven to be the most convenient.

Dynamical Structure of Hadrons

Let us consider the inelastic processes \({e^ - }N \to {e^ - }X,\;{\mu ^ - }N \to {\mu ^ - }X,\;vN \to {\mu ^ - }X,\;\bar vN \to {\mu ^ + }X,\) where N denotes a nucleon and X stands for all other particles. These processes exhibit in the deep-inelastic region an interesting and important property referred to as scaling. It is this property whose discovery gave a strong impact on the development of contemporary views on the hadron structure. The above processes will be given here a detailed analysis.

Lattice Gauge Theories. Quantum Chromodynamics on a Lattice

In this chapter we shall present one of the versions of quantum chromo-dynamics which provides a means of calculating physical quantities without using perturbation theory. This version is based on replacing the infinite four-dimensional space-time by a discrete space-time in form of a lattice of finite dimensions. Introducing a finite-size lattice makes it possible to carry out computer simulations not involving perturbation theory. It turned out that the Monte Carlo method is most suited for this purpose.

Higher Orders in Perturbation Theory

In the previous chapters, our consideration of the processes caused by the electromagnetic and the weak interactions was confined to the first non-vanishing order of perturbation theory. The present chapter deals with higher orders of perturbation theory.

Local (Gauge) Invariance

In the preceding chapter we have considered the groups of global transformations and the Lagrangian invariant under these groups. A globally invariant Lagrangian can, however, be non-invariant under a certain group of local transformations. To obtain a locally invariant Lagrangian, new fields have to be introduced. These are called gauge fields.