John B. Kogut

The renormalization group and the ε expansion

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ϵ = 4− d is explained [ d is the dimension of space (statistical mechanics) or space-time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1–8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6–8 include the approximate renormalization group recursion formula and the Feyman graph method for calculating exponents. Sections 10–13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11–13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.

The Renormalization group and the epsilon expansion

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ϵ = 4− d is explained [ d is the dimension of space (statistical mechanics) or space-time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1–8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6–8 include the approximate renormalization group recursion formula and the Feyman graph method for calculating exponents. Sections 10–13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11–13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.

Phase Transitions in Abelian Lattice Gauge Theories

We study the Euclidean partition function of Abelian lattice (gauge) theories in various dimensions. Using generalizations of mathematical methods developed recently to study the XY model in two dimensions, we obtain useful expressions for the partition functions and physical pictures of the phases of these more complicated theories. Approximate duality relations and dilute gas approximations yield estimates of critical coupling constants which separate confining and non-confining phases for the rotor model in three dimensions and Abelian lattice gauge theory in four dimensions. Generalizations of this work to non-Abelian continuum theories are discussed.

The renormalization group and the ϵ expansion

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ϵ = 4− d is explained [ d is the dimension of space (statistical mechanics) or space-time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1–8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6–8 include the approximate renormalization group recursion formula and the Feyman graph method for calculating exponents. Sections 10–13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11–13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.

The parton picture of elementary particles

Abstract The parton theory is developed along several lines. We begin by developing quantum mechanics in the infinite momentum frame. The Galilean analogy is worked out and the use of non-relativistic reasoning in relativistic contexts is illustrated. Applications of infinite momentum quantum mechanics include the computation of the radius of a relativistic bound state, space time visualization of the multiperipheral model, and the eikonal approach to high energy scattering. The classic phenomenological applications of the parton model are reviewed and explained. These include deep inelastic electroproduction, the shrinking photon effect and heavy lepton pair production. The string-model of hadrons is formulated in the infinite momentum frame as a parton model. We consider currents and the distribution of spins among the partons of the hadronic string. We suggest that it is profitable to view a hadron as a one-dimensional lattice of spins and isospins and show that many of the properties of the lattice can be related to the meson spectrum. Applications of the spin lattice idea are made to deep inelastic electron and neutrino scattering. Predictions are made for the behavior of the structure functions in these processes. Multiparticle production is examined in the string model. We derive the distribution of secondaries in longitudinal and transverse momentum, the charge per secondary as a function of rapidity and the correlations among secondaries at different rapidities. Speculations are made about a class of phenomena which go beyond the string model. These phenomena we call hard parton effects. They include the production of large transverse momenta among secondaries, logarithmically increasing total cross sections and power behaviors of wide angle exlusive cross sections. We conclude with some speculations about the breakdown of the parton model.

A quantitative approach to low-energy quantum chromodynamics

Abstract A general method for solving the low-energy spectrum of an infrared unstable field theory is presented. The method involves a strong coupling expansion of the lattice approximation to the theory. Ultimately the results must be continued to zero-coupling constant in accord with the asymptotic freedom of such theories. The method is applied to the pure gauge field (glueball) part of quantum chromodynamics. The spectrum of lowest-lying states consists of a scalar and tensor which are almost degenerate and an axial vector with mass ≈1.6 times the scalar mass. The same procedure applied to the Abelian gauge theory yields unstable results which may indicate the presence of a phase transition.

Massive μ-pair production in hadron-hadron collisions and the asymptotically free parton model☆

Abstract Recursive (renormalization group) equations for the momentum space distribution functions of asymptotically free constituents of hadrons are obtained. The transverse momentum distributions are discussed qualitatively in a simple model field theory. The physical picture is applied to μ-pair production and we conclude that the μ-pairs transverse momentum distribution will spread at moderate values of √Q2, the mass of the μ-pair, but its rate of growth will become negligible at high Q2 as the theorys invariant charge vanishes.

Estimates for W±, Z0, and μ-pair production in the asymptotically-free parton model☆

Curves are presented for massive μ-pair, W± and Z0 production from proton-proton collisions at center-of-mass energies between 100 and 600 GeV assuming mW ≈ 60 GeV and mZ ≈ 77.5 GeV. The calculations are done using a generalization of the Drell-Yan formula to asymptotically free quantum chromodynamics. The cross sections are large and interesting, and in many cases the asymptotic freedom estimates exceed the naive quark parton model calculations.

On the Absence of an Exponential Bound in Four Dimensional Simplicial Gravity

We have studied a model which has been proposed as a regularisation for four dimensional quantum gravity. The partition function is constructed by performing a weighted sum over all triangulations of the four sphere. Using numerical simulation we find that the number of such triangulations containing V simplices grows faster than exponentially with V . This property ensures that the model has no thermodynamic limit.

HARD AND SOFT PROCESSES IN HADRODYNAMICS.

Abstract We consider the consequences of a parton-parton interaction which is long range in rapidity and capable of transferring large momentum. The results of our analysis for hadron-hadron inclusive processes are presented.