Foundations of Quantum Chromodynamics

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• GENERAL SURVEY

• QUARKS AND COLOR
  • Low-lying baryon states
  • Quark confinement
  • Indirect experimental evidences

• NEED FOR ASYMPTOTIC FREEDOM

• NOTATION AND CONVENTIONS

Quantum chromodynamics is a quantized non-Abelian gauge field theory. We start with an elementary introduction to gauge field theories based on the gauge principle and then describe the method of quantizing gauge fields. We briefly review perturbation theory and present the Feynman rules for QCD. Methods of regularizing divergent integrals needed for practical higher-order calculations are explained with emphasis on dimensional regularization. Renormalization theory is discussed in some detail in consideration of its later use in the operator product expansion …

The finite renormalization of Green functions constitutes a group called the renormalization group. The physical predictions are invariant under renormalization group transformations. An analytic expression of this property is given by the renormalization group equation. We derive the renormalization group equation in the minimal subtraction scheme and compare it with the other renormalization group equations. We present its general solution, and through it we characterize the short-distance behavior of the underlying theory. The method of the renormalization group equation is applied to quantum chromodynamics leading to an understanding of asymptotic freedom. The meaning of the quantities appearing in the renormalization group equation is clarified in terms of the scale transformation …

The product of two electromagnetic (or weak) currents relevant to deep inelastic lepton-hadron scatterings is written as a series expansion called the operator-product expansion which enables us to extract a short distance piece in the scattering cross sections. This piece is under our control in the sense that it is calculable through the QCD Lagrangian by the use of the renormalization group method. We prove the operator-product expansion to all orders of perturbation theory by employing the BPHZ renormalization procedure. The renormalization group equation for the expansion coefficients is derived for later use …

Equipped with the operator-product expansion and renormalization group method we are ready to apply perturbative quantum chromodynamics (QCD) to physical processes dominated by short-distance effects. As typical examples we deal with the total cross section of e⁺e⁻ annihilations, deep inelastic structure functions and hadron jet distributions in e⁺ e⁻ annihilations. The dependence of the physical predictions on the renormalization scheme chosen in perturbative calculations is explained and its phenomenological relevance is emphasized. The factorization theorem which is a generalization of the operator product expansion is then briefly discussed and applied to the Drell-Yan process. In the course of our discussions of these physical applications there appear infrared (soft and collinear) divergences due to the presence of the massless gluons and light quarks. These infrared divergences are disposed of by a standard method which will be explained in Chapter 6 …

Although infrared divergences are of long-distance nature, they often play an essential role in the verification of the validity of the perturbative treatment of short-distance phenomena. Relying on a one-loop example we explain how the infrared (soft as well as collinear) divergence emerges and verify its cancellation by the real emission process. A general proof of the cancellation of soft divergences is given within the framework of quantum electrodynamics. The absence of infrared divergences in off-shell Green functions is shown in massless renormalizable field theories (the Kinoshita-Poggio-Quinn theorem). It is also shown that infrared divergences cancel out in the physical cross section if the degeneracy of the states is introduced (the Kinoshita-Lee-Nauenberg theorem)…

The following sections are included:

• APPENDIX A ONE-LOOP CONTRIBUTION TO SUPERFICIALLY DIVERGENT FEYNMAN AMPLITUDES IN QCD

• APPENDIX B USEFUL FORMULAS INVOLVING THE SU(N) GENERATORS

• APPENDIX C RENORMALIZATION CONSTANTS IN QCD

• BIBLIOGRAPHY

• INDEX