• Frontiers in Physics A Lecture Note and Reprint Series DAVID PINES, Editor P. W. Anderson CONCEPTS IN SOLIDS: Lectures on the Theory of Solids N. Bloembergen NUCLEAR MAGNETIC RELAXATION: A Reprint Volume · · . Geoffrey F. Chew S-MATRIX THEORY OF STRONG INTERACTIONS: · . · • . A Lecture Note and Reprint Volume . · ·• • .. · . .. . · · . · . · . R. P. Feynman QUANTUM ELECTRODYNAMICS: A Lecture Note · • . . · . . · ·. . . . and Reprint Volume · · . · ·. . · .. . · .• · • .• . · R. P. Feynman THE THEORY OF FUNDAMENTAL PROCESSES: · . . · . . • • A Lecture Note Volume · • . · . · · . • • · . · ·. . Hans Frauenfelder THE MOSSBAUER EFFECT: A Collection of Reprints . · with an Introduction . • • • Robert Hofstadter NUCLEAR AND NUCLEON STRUCTURE: A Reprint • • • · • • Volume . • • • Leo P. Kadanoff and Gordon Baym • • • QUANTUM STATISTICAL MECHANICS: A Lecture • • · . • . Note Volume · . . • - .- • • • - - - • • • • • • A. M. Lane RECENT DEVELOPMENTS IN NUCLEAR THEORY: • • • .• • • ·- . .- - • • - A Lecture Note and Reprint Volume • • • • • • • • • • • • • • • • • • • • R. Omnes and M. Froissart • • • • • • • • • • • • • • • • • • MANDELSTAM THEORY AND REGGE POLES: • • • • • • • • .• • . • - - An Introduction for Experimentalists • • • • • • • .• . • • • - - • • • • • • • • • • • George E. Pake PARAMAGNETIC RESONANCE: An Introductory • • .• · . · • . Monograph · . • · . · • . • · . David Pines THE MANY-BODY PROBLEM; A Lecture Note and · . Reprint Volume J. E. SqUires COMPLEX ANGULAR MOMENTA AND PARTICLE PHYSICS: A Lecture Note and Reprint Volume L. VanHove, N. M. Hugenholtz, and L. P. Howland PROBLEMS IN THE QUANTUM THEORY OF MANY PARTICLE SYSTEMS: A Reprint Volume . . . . . . . , , , , , , , , " , " " " " " " " " "'"'' ""'"'' ""'"'' .,,"'"'' """'" "..".." '" ., . .." ." "" .,.,."., ".,"" ',"',, ....,.. ,. ... , .,,. . , ,, ", ,' ,, ,' , , ..........,. ,...... ",." """", , . , ........." ".." ,, , ........ .,.. "",, , .........., .. ,',", , ,,".,,'"" ,"',, MANDELSTAM THEORY AND REGGE POLES AN INTRODUCTION FOR EXPERIMENTALISTS * , R.OMNES , Institut des Huutes Etudes Scientiflques M. FROISSART Centre d'Etudes NucIeaires de Saclay W. A. BENJAMIN, INC. 1963 New York Amsterdam MANDELSTAM THEORY AND REGGE POLES An Introduction for Experimentalists Copyright © 1963 by W. A. Benjamin, Inc. All rights reserved Library of Congress Catalog Card Number: 63- 22795 Manufactured in the United States of America The final manuscript was received on July 1, 1963; this volume was published December 15,1963 The publisher is pleased to acknowledge the assistance of Zeb Delaire, who composed the volume; Cecilia Duray-Bito, who produced the illustrations; and William Prokos, who designed the Idust iacket • • W. A. BENJAMIN, INC. New York, New York EDITOR'S FOREWORD The problem of c:ommunicating in a coherent fashion the recent devel opments in the most exciting and active fields of physics seems particularly pressing today. The enormous growth in the number of physicists has tended to make the familiar channels of communication considerably less effective. It has become increasingly difficult for experts in a glven field to keep up with the curn:nt literature; the novice can only be confused. What is needed is both a consistent account of a field and the presentntion of a definite "point of view" concerning it. Formal monographs cannot. meet such a need in a rapidly developing 6eJd, and, perhaps more im portant, the review article seems to have falIen into disfavor. Indeed, it would seem that the people most actively engaged in developing a given field ace the people least likely to writc at length about it. "'Frontiers in Physics~ has been conceived in an effort to improve the sihlation in several ways. Fir~t, to take advantage of the faet that the leading physici~t:s today frequently give a series of lectures, a graduate seminar, or a graduate course in their special fields of intere5t. Such lec tures serve to summarize the present status of a rapidly developing field and may well constitute the only coherent account available at the time. Often, notes on lectures exist (prepared by the lecturer himseH, by gradu ate students, or by postdoctoral fellows) and have becn distributed in mirneogr.l.phed fOITn on a limited basis. One of the principal purposes of the "Frontiers in Physics" series is to make sllch notes available to a wider audience of physicists. It should he emphasized that lecture note.~ are necessarily rough and informal, both in style and content, and those in the series will prove DO exception. This is as it should he. The point of the series is to oHer new, v • EDITOR'S FOREWORD VI - . rapid, more informal, and, it is hoped, mOre effective ways for physicists to teach one another. The point is lost if only elegant notes qualify. A second way to improve communication in very active fields of physics is by the publication of collections of reprints of recent articles. Such collections are themselves useful to people working in the field. The value' of the reprints would, however, seem much enhanced if the collection would be accompanied by an introduction of moderate length, which would serve to tie the collection together and, necessarily, constitute a brief survey of the present status of the field. Again, it is appropriate'that such an introduction be informal, in keeping with the active character of the field. A third possibility for the series might be called an informal monograph, to connote the fact that it represents an intermediate step between lecture notes and fOllnal monographs. It would offer the author an opportunity to present his views of a field that has developed to the point at which a summation might prove extraordinarily fruitful, but for which a formal monograph !night not be feasible or desirable. Fourth, there are the contemporary classics....,.papers or lectures which constitute a particularly valuable approach to the teaching and learning of physics today. Here one thinks of fields that lie at the heart of much of present-day research, but whose essentials are by now well understood, such as quantum electrodynamics or magnetic resonance. In such fields some of the best pedagogical material is not readily available, either be cause it consists of papers long out of print or lectures that have never been published. "Frontiers in Physics" is designed to be flexible in editorial format. Authors are encouraged to use as many of the foregoing approaches as seem desirable for the project at hand. The publishing format for the series is in keeping with its intentions. Photo-offset printing is used throughout, and the books are paperbound, in order to speed publication and reduce costs. It is hoped that the books will thereby be within the financial reach of graduate students in this country and abroad. Finally, because the series represents something of an experiment on the part of the editor and the publisher, suggestions from. interested readers as to format, contributors, and contributions will be most welcome. DAVID PINES Urbana, Illinois August 1961 PREFACE The present book stems from separate lecture courses given by the authors at Professor Leprioce-Ringuet's laboratory in the Ecole Poly" technique and at Saclay. As the audience was essentially made up of experimentalists, these courses were of a descriptive character. The book is addressed to students who arc not acquainted with the most recent work in the theory of strongly interacting elementary particles. Our aim has been to extract the most important physical ideas, with proofs frequently omitted or replaced by heuristic arguments. The level has been kept as elementary as possible; the reader is assumed to know only elementary qu,.'mtum mechanics. We have made no attempts at mathematical rigor or completeness, since further developments may be found in the book by G. F. Chew, S-Matrir Tlzeory of Strong Interactions, published in the same series. As is customary in elementary textbooks, we have not explicitly referred to the original authors of the work under description. We feel ju.sti.£ied in not doing so because our heuristic presentation would hardly give them due credit. In addition, we have given only a short bibliography, intended to provide a reading program for those who may want to acquaint them selves further with the subject. t One of tlle authors (R. O. thanks Mr. L. Motchane for his hospitality at the Institut des Hautes Etudes Scientifiques. We both thank Mile Helene Noir for her invaluable aid in preparing the manuscript. R. DMN", M. FROISSAnT Pam, France September 1963 vii I • • • I CONTENTS Editor's Foreword v Preface vii Chapter 1 Nonrelativistic Scattering Theory 1 1-1 Formulation of the Problem 1 1-2 Integral Form of the Schrodinger Equation 4 1-3 The Born Series Expansion 10 1-4 Calculation of the Scattering Amplitude 11 Chapter 2 Properties of Partial Waves 15 2-1 Separation of the Schrodinger Equation in Partial Waves 15 2-2 Study of the Radial Equation 20 2-3 Analytic Properties of the Partial-Wave Amplitude as a Function of k2 21 2-4 The Left-Hand Cut 22 2-5 The Right-Hand Cut 23 2-6 Analytic Properties of the Partial-Wave Amplitude 25 2-7 Physical Aspects of the Partial-Wave Formalism 26 Chapter 3 Analytic Properties in Energy and Momentum Transfer 29 3·1 Kinematics 29 3-2 Analytic Properties in Energy 30 3·3 Analytic Properties in Momentum Transfer 33 3-4 Treatment of Exchange Forces 36 3-5 Bound States 37 ix , • x CONTENTS Chapter 4 Introduction of Complex Angular Momenta 41 4-1 Interpolation between Physical Values 41 of Angular Momentum 43 4-2 The Sommerfeld-Watson Fonnula 4-3 Physical Interpretation of Regge Poles 46 4-4 Behavior of Regge Trajectories 48 4-5 Exchange Forces-Signature 50 52 4-6 Asymptotic Behavior in Momentum Transfer Chapter 5 Kinematics of Relativistic Scattering, Crossing 54 5·1 Kinematics 54 5-2 Crossing 58 5-3 Crossed Reactions 62 5-4 Kinematics of Crossed Reactions 64 5-5 First Example: p-n Scattering 66 5-6 Second Example: Pion-Nucleon Scattering 66· 5-7 Third Example: Decay and Dalitz Plot 66 K/L~ 5·8 A General Case: Associated Production 70 "a Chapter 6 Relativistic Scattering Theory la Mandelstam" 72 6·1 FomlUlation of the Problem 72 6·2 The Invariant Scattering Amplitude 74 6·3 75 The Nonrelativistic Region 6·4 Another Energy Cut 77 6-5 An Example: Proton·Antineutron Scattering 78 6-6 The Mandelstam Hypothesis 80 6-7 Application to the Properties of Nucleon- Nucleon Potential 80 6-8 Exchange of Other Particles 82 6·9 Application to Pion-Nucleon Scattering , 86 6-10 The High-Energy Domain 88 6·11 Asymptotic Properties of A (8, t, u) 92 94 6-12 Dispersion Relations Chapter 7 Relativistic Regge Poles 100 7-1 100 The Paradoxes 7·2 Comparison with Nonrelativistic Theory 102 7·3 Chew's Hypothesis 104 7·4 Exchange Forces 105 7·5 Trajectories 106 7-6 Generalizations 107 Chapter 8 Comparison with Experiment 110 8-1 Experimental Tests in the Direct Channel 110 CONTENTS xi 8-2 Experimental Tests in the Crossed Channel U2 8-3 Total Cross Sections. The Pomeranchuk Pole U3 84 Total Cross Sections. The Satellite Poles U5 8·5 The Diffraction Peak U7 8·6 The Inelastic Experiments 120 Bibliography 122 Index 123