Vi Brazilian Symposium Theoretical Physics

A A) VI BRAZILIAN SYMPOSIUM N THEORETICAL PHYSICS Erasmo Ferreira, Belita Koller, Editors

Nuclear Physics, Plasma Physics and Statistical Mechanics B. Giraud, C.S. Wu, P.H. Sakanaka, C. Tsallis E.J.V. Passos, G. Toulouse, Contributors VOLUME I

CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO VI BRAZILIAN SYMPOSIUM ON THEORETICAL PHYSICS r

VOLUME I VI BRAZIL SYMPOSI ON THEORETI PHYSIC Erasmo Ferreira, Belita Koller, Editors

Nuclear Physics, Plasma Physics and Statistical Mechanics B. Qiraud, C.S. Wu, PH. Sakanaka, C. Tsallis E.J.V. Passos, G. Toulouse, Contributors VOLUME I

CONCHO NACIONAL D£ DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO Coordenação Editorial Brasilia 1981 CNPq Comitê Editorial Lynaldo Cavalcanti de Albuquerque José Duarte de Araújo Itiro lida ----- Simon Schwartzman Crodowaldo Pavan Cfcero Gontijo Mário Guimarães Ferri Francisco Afonso Biato

Brazilian Symposium on Theoretical Physics, 6., Rio de Janeiro, 1980. VI Brazilian Symposium on Theoretical Physics. Bra- sília, CNPq, 1981. 3v. Bibliografia Conteúdo — v. 1. Nuclear physics, plasma physics and statistical mechanics. — v. 2. Solid state physics, biophysics and chemical physics. — v. 3. Relativity and elementary particle physics. 1. Física. 2. Simpósio, l. CNPq. II. Título. CDU 53:061.3(81) FOREWORD

The Sixth Brazilian Symposium on Theoretical Physics was held in Rio de Janeiro from 7 to 18 January 1980, consisting in lectures and seminars on topics of current interest to Theoretical Physics. The program covered several branches of modern research, as can be seen in this publication, which presents most of the material discussed in the two weeks of the meeting. The Symposium was held thanks to the generous sponsorship of Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq (Brazil), Academia Brasileira de Ciências, Fundação do Amparo a Pesquisas do Estado de São Paulo (Brazil), Organization of American States, Sociedade Cultural Brasil- Israel (Brazil), Comissão Nacional de Energia Nuclear (Brazil), and the governments of France and Portugal. The meeting was attended by a large fraction of the theoretical physicists working in Brasil, and by about 30 scientists who come from other countries. The list of participants — about 250 — is given at the end of this book. The Pontifical Catholic University of Rio de Janeiro was, for the sixth consecutive time, the host institution, providing excellent facilities and most of the structural support for the efficient running of the activities. The community of Brazilian Theoretical Physicists is grateful to the sponsors, the host institution, the lecturers and the administrative and technical staff for their contributions to the cooperative effort which made the VI Brazilian Symposium on Theoretical Physics a successful and pleasant event.

Organizing Comittee Guido Beck (CBPF) Carlos Bollini (CBPF) Hélio Coelho (UFPe) Darcy Dillenburg (UFRGS) Erasmo Ferreira (PUC/RJ) Henrique Fleming (USP) Belita Koiller (PUC/RJ) Roberto Lobo (CBPF) Moysés Nussenzveig (USP) Antônio F. Toledo Piza (USP) Ronald Shellard (IFT) Jorge A. Swieca (U.F. São Carlos e PUC/RJ) Zieli D. Thomé (UFRJ) WELCOME ADDRESS VII

Ladies and Gentlemen: It is a great pleasure and honor for me to welcome to our University the participants of the Sixth Brazilian Symposium on Theoretical Physics. It is on such occasions that the true spirit of the university is manifested in the search for new knowledge and in the communication of new discoveries. Renowned physicists in several areas of Physics, and from leading institutions in Brazil and abroad, gather here to share their latest results of research, new hypotheses and unanswered questions. Through the exchange of new insights, suggestions and critical reactions, the academic community emerges, united in its commitment to truth and in its effort to contribute to the understanding of the world in which man lives. The understanding of Nature — which is the original meaning of Physics in Greek - is indeed the goal of this Symposium on Theoretical Physics. Events of recent years have taught us to be cautious in our dealing with Nature, since we do not understand all of its interrelations. The precipitate use of nature's forces, made possible by scientific progress, though intended to make the world more livable, threatens to destroy mankind itself. It is not possible to reduce nature to a single model and it is not possible to reduce history to a human project. The full understanding of nature demands humility and respect in dealing with it. Only under such conditions will the knowledge and use of nature be of serve to man. One of the basic goals of this Symposium is to show the fundamental unity of physical theory. The rediscovery of the deep unity of all the phenomena of nature is undoubtly a step forward in the understanding of nature, because understanding requires unification of many phenomena in one law or theory. But a unified theory, in order to be faithful to its object, must recognize the variety of levels through its use of analogy. It must respect the limits of its understanding of nature through the continual perfection and superseding of its own models. It must admit the existence of other dimensions and other approaches to reality, through openess to interdisciplinary dialogue. It is through the pluralism of viewpoints, consciously accepted as relative and interchangeable, that the global vision of reality progressively emerges, and converges to its final goal, which is man in all his complexity and his self-transcendence. May this attitude - so proper to the spirit of the university - be present in all the work of this Symposium. VIII

In closing I would like to thank ail the Brazilian and foreign foundations and government agencies which are sponsoring this meeting. May the bold initiative of the Department of Physics at PUC in hosting this Symposium, and the generous efforts of its organizers be rewarded by the scientific and human growth of all the participants. Thank you.

Rev. João A. Mac Dowell, S.J. Rector, Pontifícia Universidade Católica do Rio de Janeiro IX

INDEX

Volume I NUCLEAR PHYSICS, PLASMA PHYSICS AND STATISTICAL MECHANICS 1. The Generator Coordinate Method in Nuclear Physics - B Giraud 2. Collective Motion and the Generator Coordinate Method - E.J.V. Passos 3. Some New Radiation Processes in Plasmas - C. S. Wu 4. Magnetohydrodynamics and Stability Theory - P. H. Sakanaka 5. Free Energy Probability Distribution in the SK Spin Class Model - Gerard Toulouse and Bernard Derrida 6. Connectivity and Theoretical Physics: Some Applications to Chemistry -H. E. Stanley, A. Coniglio, W. Klein and J. Ferreira 7. Random Magnetism - C. Tsallis

Volume II SOLID STATE PHYSICS, BIOPHYSICS AND CHEMICAL-PHYSICS 1. Scattering of Polarized Low-Energy Electrons by Ferromagnetic Metals - J. S. Helman 2. Phase Separations in Impure Lipid Bilayer Membranes - F. de Verteuil, D. A. Pink, E. B. Vadas and M. J. Zuckermann 3. RMMtt Progress in Biphysics - G. Bemski 4. Meotrooic Aspects of Enzymatic Catalysis - R. Ferreira and Marcelo A. F.GOMK

Volume III RELATIVITY AND ELEMENTARY PARTICLE PHYSICS 1. Gauge Fields as d Uaifying Ideas - C. G. Bollini 2. On the Personality of Prof. Abdus Salam - J. J. Giambiagi 3. Gauge Unification of Fundamental Forces - A. Salam 4. Geometric Formulation of Supergravity - S. W. MacDowell 5. Weak Interactions - R. Chanda 6. Applications of Quantum Chreatodynamics to Hard Scattering Processes - C. O. Escobar 7. Topological Aspects in Field Theory and Confinement - B. Schroer

AUTHOR INDEX - AT THE END OF THE VOLUME CONTENTS xl

Bertrand Giraud THE GENERATOR COORDINATE METHOD IN NUCLEAR PHYSICS 1. Introduction 1 2. Elementary Technology of Generator Coordinates 3 3. Search for Collective Paths 6 4. Application to Nuclear Reactions 14 5. Other Examples of Application 22 6. Conclusion 25

Emerson J. V.Passos COLLECTIVE MOTION AND THE GENERATOR COORDINATE METHOD Abstract 27 1. The Kinematics of Generator Coordinates 28 2. Dynamics in the GCM Collective Subspace — Collective Hamiltonian, Collective Operators and Collective Wavefunctions 31 3. Special Cases and Examples 33 References 42

Ching S. Wu SOME NEW RADIATION PROCESSES IN PLASMAS Foreword 43 1. Spontaneous Emission near the Electron Plasma Frequency 44 2. Direct Amplification of Radiation near the Electron Cyclotron Frequency by Cyclotron Maser Instability 66 3. Parametric Amplification of Radiation by Stimulated Scattering 89 Closing Remarks 103

Paulo H. Sakanaka MAGNETOHYDRODYNAMICS AND STABILITY THEORY Abstract 105 1. Magnetohydrodynamic Equations 105 2. Energy Principle for the Ideal Magnetohydrodynamics 108 3. Extension to a—Stability 131 4. Energy Principle for Dissipative Systems 134 References 139

Gerard Toulouse FREE ENERGY PROBABILITY DISTRIBUTION IN THE SK SPIN GLASS MODEL 1. Introduction 143 2. Annealed and Quenched Models 144 XII

3. The Moments < Zn > 146 4. High Temperature Phase 151 5. The Behavior at T = J Í 55 6. Low Temperature Phase 157 7. In the Presence of an External Magnetic Field 162 8. The Spherical Model 169 9. Conclusion 170 References 171

H. Eugene Stanley CONNECTIVITY AND THEORETICAL PHYSICS: SOME APPLICATIONS TO CHEMISTRY Outline 173 0. Philosophy 174 1. The Phenomenon 176 2. Model # 1: Randon-Bond Percolation 176 3. Model # 2: Correlated Site-Bond Percolation 178 4. The Solvent Itself: Are Correlated Percolation Concepts Relevant to Water? 182 5. Water at Low Temperatures: The Observed Phenomena 182 6. Why are we Interested? 185 7. Possible Clues 186 8. A "Hunch" 189 9. Tests of the "Hunch" 196 10. Conclusions 197

Constantino Tsallis RANDOM MAGNETISM 1. Introduction 199 2. Well Defined Systems 201 . 3. Random Systems 206 4. Conclusion 216 References 216

SYMPOSIUM LECTURERS \l

THE GENERATOR COORDINATE METHOD IN NUCLEAR PHYSICS

B. G. Giraud Dph-T, CEA-CEN Saclay, B. P. 2 à 91190 Gif sur Yvette, France

ABSTRACT

The generator coordinate method is introduced as a physical description of a N-body system in a subspace of a reduced number of degrees of freedom. Special attention is placed on the identification of these special, "collective" degrees of freedom. It is shown in particular that the method has close links with the Born- Oppenheimerapproximation and also that considerations of differential geometry are useful in the theory. A set of applications is discussed and in particular the case of nuclear collisions is considered.

I. INTRODUCTION. NUCLEAR COLLECTIVITY

A major, and somewhat surprising property of nuclear dynamics is that the low- lying spectrum of a nucleus is often dominated by collective states and transitions. The observation of collectivity results from experimental data and more recently, from time-dependent-Hartree-Fock (TDHF) calculations. Indeed, one often finds bands of nuclearlevels,or a path of nuclear states, strongly connected by specific transitions, and the point is that in such cases the nuclear dynamics is expressed in terms of a small number of simple modes (translations, rotations,quadrupole vibrations, octupolevibrations, and so on).

Such a reduction of the number of degrees of freedom of a nucleus can be described by phenomenological models of course. It would be more satisfactory to understand why it occurs. A first, and qualitative answer, is available. A certain amount of correlation between the single-nucleon coordinate degrees of freedom x....x (or the associate momenta p.•••?») is obviously the result of the binding of nucleus, into a quantal liquid drop, with its surface tension and its surface and volume waves. The

Pauli principle, furthermore, obviously correlate x....xA. It is therefore not surprising that some combinations £(x,g....x.g ) emerge finally as normal, dominant degrees of freedom rather than x....x,.

The exact nature of the new variables Q. as well as an explicit change of variables from x^'s to 1's represent, however, a formidable mathematical and physical problem. The purpose of these lectures is to show that a few steps to solve this problem 2 B.G. Giraud have been undertaken, in particular with the help of the generator coordinate method.

The central concept in these lectures is that of the collective subspace. This subspace must be, up to a good accuracy, an eigensubspace of the many-body Hamiltonian J"" . For a significant amount of decoupling is necessary to isolate a degree of freedom.

It must also likely contain, up to a good accuracy, the orbit of a Lie group, generated by the algebra of the collective degrees of freedom Q, and their associate momenta'?- For a time-dependent description of the dynamics will replace the evolution operator exp(-i^tt) by an approximation exp[-i ^>£ (Q,/?>t)] where 3f , a time-dependent function of the-collective variables, will often be linear like in the case of translations, rotations and small amplitude vibrations.

As a third property of the subspace, its collective states $ have to show a large amount of coherence with respect to Q_, best expressed by the limit case of factorization between Q*s and complementary (spectator) degrees of freedom Ç,

í..j(xr..xA) = \,{0,O = wu(Q) x(C) , (LI)

where the "spectator" wave function \ weakly depends on the collective quantum numbers v in the subspace. This ensures that transition matrix elements inside the subspace < <*v.jel*v> = «PV.|«!V XIX> . (1-2) are large, with = '•

Last but not least, the collective operators 0, y are expected to be essentially one-body operators. For an exponential exp(-i3£ ), with ÍÊ linear in Q., ^p , can then be interpreted as an evolution operator for which all nucleons move alike, since the one-body nature of Q^ for instance, reads

a- I ÍL (i-3) i=l Another advantage of one-body operators is that their commutators remain one-body operators, a useful condition for the closure of a Lie algebra. Furthermore it is known that exponentials of one-body operators convert Slater determinants into Slater determinants, a result of interest for the connection of the theory with the time- dependent Hartee-Fock (TDHF) approximation. One thus expects to span the collective subspace by means of a path of Slater determinants. It may also be noticed that well- known collective operators, such as the centre-of-nass position and momentum, the total angular momentum, the quadrupole, octupole (etc) momenta and various SPA operators are one-body operators. Finally it nay be conjectured that there; is less ergodicity to be expected from the exponential of a one-body operator than from the true evolution operator (-iXt), the latter being known to induce ergodicity in many general cases. Prevention of ergodicity is essential for the isolation of the degrees of freedom under study. GENERATOR COORDINATES 3

The main emphasis in the^e lectures will be laid upon the fulfilment of these four criteria, in an attempt to derive collectiveness from jfi rather than guess it. This is the subject of subsections III A to III D. As a preliminary subject, the formal aspects of the generator coordinate method will be described briefly in section II. Finally a set of applications is displayed in sections IV and V.

II. ELEMENTARY TECHNOLOGY OF GENERATOR COORDINATES

The one-channel, one-generator-coordinate ansatz

¥(x,...xA) = jdq f(q) * (x,...^) , (II. 1)

can be generalized into a multi-channe1, raulti-generator-coordinate form

qi...dqNfn(qr..qN) ^^ (?,...x^), (11.2)

which is of interest in a roupled-wav? theory of nuclear collisions with polarization phenomena. We shall mainly be concerned in the following with the simpler fore, eq.(II.I).

The continuous label (generator coordinate) q acquires a physical meaning if it is the expectation value of the collective coordinate Q in the generator function Í1

It will be assumed in this section that fl(x.p....xp ) is known in advance and that the set {<{• }, usually a path of Slater determinants, is also known.

The physical meaning of q becomes even more transparent if it turns out that the fluctuation

2 2 Aq H |0 | > - q (II.4)

remains small when compared to a typical order of magnitude q of the range of q in the problem under study. It is then clear that $ represents a wave packet narrow with respect to 0,

where the wave packet Y has a width Aq and one recognizes from eq.(I.l) the "spectator" wave function X and spectator degrees of freedom Ç. It must be checked separately whether x depends strongly on q or not. A weak dependence will be assumed in the following.

When x Í3 fixed the ansatz, eq.(II.I), induces

(II-6) 4 B.C. òirav.c with g(a) = jdq " (^j flq) . UI.:>

It must be stressed here, and this is maybe the main property of the generator coordinate formalism, that eq.(II.6) achieves in practice a change of coordinate from tho microscopic coordinates x....x, to the collective and spectator coordinates Q, .r. This change of coordinates was impossible in an explicit, direct way. It is remarqua- ble that the ansatz, eq.(II.l), has solved this problem in an indirect way, while all calculations remain explicitly in the representation x....x , a most practical feature.

It is interesting at this stage to introduce the "sharp basis", where the change of variables becomes even more obvious. The diagonalization of Q in the generator subspace proceeds via the equation

1 jdq «Pq|((HO|*qi> jk(q') = 0 , (II-8)

to generate "sharp states"

|k> = jdq' jk(q') |*ql> . (II.9)

Under suitable conditions of normalization, spectrum density and so on, the sharp states obey the conditions

-k|k"> = i5(k-k') (II. 10)

-k 101k'> = k o(k-k') . (11.11)

The sharp states, as approximations to eigenstates of Q., yield the interpretation

= ô(Q-k) x.(Ç) • (II-12) rC They still make a basis of the generator (collective) subspace, but this basis identifies the label k with the coordinate itself. The change of variables fron x....x to Q.j 5 is thus very explicit. In this sharp representation the collective Haniltonian is now trivial. Its matrix element reads

hcoU(k,k') = , (11.13)

where again all calculations remain in the x....x representation. A collective Schrodinger equation may result from an expansion of the non locality of h .. up to second order GENERATOR COORDINATES

(U 14J oll\ 2 7 - provided of course ra .. remains positive.

The diagonalization of ''* in the sharp basis has thus more physical interpretation than the usual Griffin -Hill-Wheeler derivation of the energy basis

:E-|dq' fE(q') !q. , (11.15)

where fg(q') is given by the integral equation

jdq'

With the usual energy and overlap kernels

H(q.q') = <Í'qrJ'tM';>q,> . (11.17)

Kq,q') = • (11.18)

the matrix notation (H-EI) f = 0 of eq.(II.16) can be reduced into a diagonalization 1/2 form more general than the usual transform via the square root I . Any kernel J, possibly non Hermitian, which fulfills the equation I = J J, may provide a reduced Hamiltonian kernel h under the condition H = J hJ. The final equation to solve is thus J+(h-E) Jf = 0 , (11.19)

where J may be erased and (Jf) is the new unknown.

In particular, if one uses a matrix notation {|k>} = jilq'5"} for eq.(II.9), then eq.(II.I3) reads h = j H j. Although j may be nearer to a rectangular than a square matrix (because of null states in the subspace), it is obvious that the sharp basis corresponds formally to the special case J = j . All these considerations can be made more rigorous, the details being omitted in the present lectures.

There is another technical point with generator coordinates which is not connected with diagonalization, however. It has rather to do with inversion and is of special interest for the theory of nuclear collisions since this theory makes great use of the resolvent (E-^ò)

Consider the functional

F = <¥f \ + <.?' |fi> - <4" \(E-H)\T-> , (11.20)

where Y. f are given, square integrable vectors (usually the product of an optical wave and a residual potential). A variational principle with respect to 4" gives B-G. Giraud

= -':f i " •'i'.CE-Jt' ) = O . (11.21)

With respect to ': ' one obtains

|•;•,-• = (E-JÍ. )!';••• . (11.22)

As a consequence of eqs.(11.21) and (11.22) the stationnary value of F is

1 F = -.7£|(E-Í(i J' !'^-- • (11.23)

There is no reed to stress that such a resolvent matrix element is exactly what is needed in a theory of nuclear collisions.

In the generator coordinate method, with a trial function H' defined by ec. (II.1), the form taken by eq. (11.22) reads

fdq1 <0 |Q(G-E)|$ ,> f(q') t- <>!> \y^ = 0 , (11.24)

which is an inhomogeneous generalization of the Griffin-Hill-Uheeler equation. The same holds, of course, for eq. (11.21), with a source term given by 't' instead of "í. .

III. SEARCH FOR COLLECTIVE PATHS

A. Curvature as a necessary condition

As mentionned ealier the generator coordinate description of an eigenstate (time independent) is provided by the ansatz, eq.(II.I). Collective motion can rather be expressed by time dependent states

H-Ct) = jdq f(q,t) Í , (III.l)

and, if furthermore the generator states are sufficiently coherent to be long-lived, it then occurs that f may be a sharply peaked wave-packet around a classical posi-

tion in phase space, 1QM

f(q,t) = T [q-qo(t)] . (III.2)

The path {í>q (t)J> and actually { } itself if one takes q rather than t as current parameter, is therefore more than an abstract basis of a subspace. It can actually be understood like the physical trajectory of the system in its Hilbert space when the collective motion occurs.

As mentioned also in section I, it is felt that this trajectory is (at least part of) an orbit of a Lie group, generated by an algebra of collective operators 0. All these geometrical intuitions lead to the need for the definition of a metric, for instance tlhe following. GENERATOR COORDINATES

in order to define various characterizations of the trajectory, such as curvatures.

The choice, eq.(III.3), of the metric may not be unique. It appears at first, however, like that which naturally uses the length in Hilbert space and applies it dd$$Q to the tangent —7-1 of the trajectory. The unit tangent vector is then

and the unit normal vector comes with the next derivative, ii curvature C is then defined by

In the special case, of wide interest, where $ is a Slater determinant or a q correlated RPA state and where the time dependent evolution is approximated by

eX l ikt *qo(t) " ? ~ ® *qo(o> '

with 0 a quasi-bason operator, it can be checked that the curvature resulting from 9 eqs.(III.6) and (III.5) reaches a critical value, C = 3, when !; i? d traditional collective; quasi-boson operator. The basic step of the proof of this result is to consider the spreading of the particle-hole matrix elements 0 , . One finds that the curvature reaches a maximum when all Q_, have the same order of magnitude. -ph It is empirically well known that the particle-hole matrix elements of those RPA boson operators which correspond to maximum collectivity are evenly spread. The same magic value C = 3 can also be found in the orbits of various groups such as SU(3), SP(1,R), SP(3,R)... which are known to be of interest in the theory of collective motion. It can be concluded that a curvature close to /3 is likely necessary (but certainly not sufficient) for a path {i> } to be a good candidate for collectivity.

&. Born-Oppenheimer approach

Let us assume that (£ is known in advance. Such a collective degree of freedom must likely have a large inertia, since all nucleons move alike when governed by this mode Q.. The other degrees of freedom Ç must therefore have a smaller inertia.

If these degrees of freedom £ are light enough when compared to Q_, the Born- Oppenheimer approximation becomes valid. Indeed, like in eq.(II.12), a spectator wave function Xi,(Ç) develops with a minimal energy locked to the sharp value k taken by Q. The sharp basis of the generator coordinate formalism thus proves the 8 B.G. Giraud equivalence between this formalism and the Born-Oppenheimer approximation.

To find Xj,on e mav think of a minimization of the total energy (involving both Q_ and Ç) under the constraint 6(Q.-k). Such a 6-function constraint, however, is obviously too sharp for practical purposes, for it restricts all mooents "&|Q, |k> at the same time, whatever the value of the exponent n. A milder constraint, clearly a more practical one, takes care of the first moment <$ |Qj$ > only. This is nothing but the form shown by eq.(I1.5). When $ is a Slater determinant, it is thus obtained by a constrained Hartree-Fock calculation.

There are cases, however, where precautions are necessary in a constrained variational principle. This occurs when nay diverge while <36> remains finite. Let for instance Q be the quadrupole moment operator and consider a fixed wave-function 0 , with a finite energy e = <$ |^£|$ > ancl a finite quadrupole moment q = <$ |Qj$ >. Consider now an auxiliary wave function $, lying around like the "Saturn ring" around the planet Saturn. This function $. is designed to have an enormous quadrupole moment q., through its extension and ellipsoidal deformation in space. Simultaneously, the nuclear matter density shown by §. is designed to be very thin. Because of this thin • spreading, $, carries no kinetic or potential energy. Because of $. lying around $ , and far enough, there is no coupling matrix element between

3>o and $. . Therefore any small admixture of $. into 4> , of the form Y = $ + £§., keeps for ¥ the ^nergy e . It is clear, however, that the quadrupole moment shown by H1 will be q + e q , which can differ arbitrarily from q . This arbitrary value of the constraint for a fixed value of the energy «H1 |^í<|¥> makes the constrained variational principle inefficient. Such a breakdown casts doubt on many theories of fission, for instance, if the fission barrier has been predicted by a constrained calculation. Indeed,the life time is very sensitive to the barrier and it has just been seen that the barrier may be washed away by Saturn-ring-like admixtures.

To solve the paradox one may notice that the admixture of $j also increases enormously the fluctuation Aq as defined by eq.(II.4). Indeed the quadrupole moment is a local operator, whose eigenstates are localized while 0. is thinly,spread If therefore one considers a doubly constrained variational principle including both moments and ,a. reduction of Aq forces $. to shrink. The shrinking produces density (and phase) gradients, thus kinetic energies, and it also induces larger den-

sities, thus potential energies. The energy will then differ from eQ and generate a significant energy surface.

To summarize these considerations, an equivalence between the Born-Oppenheimer and the generator coordinate approximations can be derived, provided one keeps under control the fluctuations of the collective coordinates Q.. GENERATOR COORDINATES 9 C. Fiber bundle and adiábatic time-dependent Hartree-Fock approach It is obviously desirable to have a theory which contains more dynamics than the considerations of subsection A, which are only geometrical,and those of subsection B, where one is prejudiced about the nature of 0. It is known that the random phase approximation provides a dynamical description for which the collective operators £ are by-products of the theory rather than prerequisite. Unfortunately the RPA is restricted to small amplitude nuclear motions. Large amplitude collective paths must therefore be identified by a more general theory. The adiabatic time-dependent Hartree-Fock (ATDHF) approximation has been suggested by Baranger and Veneroni for that purpose. It will now be considered.

Let $ be a Slater determinant and p the corresponding one-body density matrix. Out of p and the many-body Hamiltonian }6 = *6 + V" , one makes the Hartree-Fock Hamiltonian

li) =13 +11 , (III. 7) where the average, one-body potential \L is given by the usual antisymmetrized trace

lL=Tr Vp . (III.7a) The time-dependent Hartree-Fock (TDHF) approximation then specifies which determinant will occur at time dt on the path if one starts with 0 at time t = 0

S(dt) = [1-i dt W] í> + 0(dt2) (III.8) = exp[-i dt W] $ + 0 (dtZ) The two forms of eq.(III.8) are equivalent up to order dt only, of course. It is clear, however, that they suggest that W is the generator of a Lie group, actually the generator of displacement along the TDHF trajectory. One expects a close connection between W and a collective operator Q,.

Since III depends on $, see eq. (III.7a), the connection between W and Q^ will likely provide too many (J's if one follows an arbitrary TDHF trajectory. To reduce the variety of Q_'s we might follow the suggestion by several authors to consider only periodic TDHF solutions. This is clearly a global approach. Alternately, one may try to select Q_'s which locally correspond to several TDKF trajectories. This can be achieved in the framework of the adiabatic (ATDHF) trajectory. It is useful at this point to realize that in the simplest and most frequent case a generator coordinate path {$ } is made of real Slater determinants . More q q precisely the single-particle orbitais which are filled in 0 can then be described by real wave-functions in coordinate representation and the density p is even under time reversal. An even density like that will be denoted p in the following and the 10 B.G. Giraud corresponding Hártree-Fock Hamiltonian by WQ. It is clear that {'1^, given by

W = ^ + Tr ^p , (III.9) o o is also real (even under time reversal).

On the other hand, a TDHF trajectory is complex most of the time. As shown by eq.(III.8), if one starts at time t = 0 with a real , then i'(dt) acquires at once an imaginary part. Also W(dt) is complex

W(dt) =t + Tr

-i dt[M0>P0]) = W - i dt W. , (III.10) o 1 with W, = Tr \r [Wo,Po] (III. 11)

The main feature of the ATDHF r.heory is that 4>(t) always remain almost real. Imaginary components in wave functions and operators are associated to velocities and kept small. The splitting of Ü into a real and imaginary component W and W can thus be retained as of interest despite its validity limited to first order with respect to imaginary parts, see eq.(III.lO). It can be stressed at this point that there are one and only one, well defined W and W. for each real determinant . Indeed 0 defines p , then one finds-U through eq.(III.9) and W. through eq.(III.ll).

One then notices that ll' is a real, Hermitian operator and that i (I is a purely imaginary (odd under time reversal) and Hermitian operator. If one uses i C'p. as a Lie group generator

4>o (d6) = [ I + i dfl iW.1l * o (0)

= (I-d8 Tr \P[ti ,p ]) (0) (III. 12)

it is a clear that $ (dt) is real, like $ (0). This induces a step along a real path and can be considered as the beginning of the making of a generator coordinate path.

This path equation can be written, from eq.(III.12)

w TrVpq, Pq]) 0q , (III. 13)

where p is the real density defined by i> and one writes $ instead of 0 (0) in order to specify that one is running along the path, step by step. It generates a path as soon as one initial determinant 0 is given. The question, of course, is whether the path is acceptable as collective.

One first check is to measure the curvature along the path, as discussed in subsection A. A second check consists first in noticing that i Wj, the displacement GENERATOR COORDINATES 11 operator, behaves like a momentum since it is imaginary and Hermitian. One then notices that W , real and Hermitian, behaves like a position operator. Indeed, consider eq.(III.8) with 0 = $ real. It must here be remembered that real determinants, in the Baranger-Veneroni analysis, are interpreted as motionless. The imaginary component -i dt U' $ which then appears in $(dt) is thus associated to the creation of collective velocity, under the acceleration operator W . The acceleration being usually given by the coordinate operator, one may relate W to the collective coordinate. Because of the high coherence which has been assumed for the collective states in section I, the second check now consists in assuming that ('.' (coordinate) and i ft1, (momentum) have proportional particle-hole matrix elements, like in RPA,

[•g + Tr Vpq - A Tr V"[T+Tr Upq,Pq], pq] = 0 . (III. 14)

It is interesting to point out that this path equation, eq.(III.14), is a constrained Hartree-Fock equation. Indeed Í1 is nothing but the Hartree-Fock Hamiltonian for the real determinant $ . Then the constraint is Aft 1., with A as Lagrange multi- plier and it', as a rsLher unusual, because anti-Hermitian, constraining operator. A consequence of eq.(III.14) is the existence of a fiber bundle of ATDHF trajectories. Indeed, consider at time t = 0 the set of initial determinants

(0) = (1-i dT t' ) on the path. The path {<£ } will be the o q q basis of the bundle. The functions $(0) parametrized by the real and snail number dx will be the first fiber of the bundle. The point is that, at time dt, one obtains

4>(dt) = [1 - i dt(W o - i dTft',)] 1 0(0 ) = (1 - i dx' (V ) * (0) (III. 16) o o with

1 dT = dt + dT - i ÈLJ1 t (III. 17) A because the particle-hole matrix elements of W and (!/, are proportional. Except for the fact that dx' is complex, the form of eq.(III.16) is exactly that of the fiber, eq.(III.15).

Actually nothing prevents to write eq.(III.16) as

4>(dt) = (1 - i dT" WQ) $Q(de) (III. 18) 12

with

- (1 - dx dt Wj) *o(0) (III. 19)

and

dl" = dt + dT, d6 = dx dt. (III.20)

It can be stressed that í>Q(d6) is real, like dx dt, and that dx" is again real. A new fiber, strictly analogous to that described by eq.(III.15), has been described by eq.(III.18),

The physical meaning of this fiber bundle approach is obvious. Under various initial velocities dx as specified by eq.(III.15) the system follows always the same kind of steps on the basis of the bundle , the path generated by eq.(III.19) and acquires always the same kind of accelerations, described by the fiber, eq.(III.18). In other words the system is stable under, changes of velocities. Although this conclusion is based on first order arguments in dt and dx and should be checked up to second order at least, it corresponds to a decoupling of the mode, governed by W and ill., from other modes perpendicular to the fiber bundle. Thus the number of degrees of freedom has been (approximately) reduced.

Last but not least a third check is necessary. For I'.' (and ('.'.) may change strongly along the path, while one expects the collective algebra to be reasonably constant. Despite this last difficulty the two path equations, eqs.(III.13) and (III.14), seem to provide a selfrcontained (dynamical) derivation of a generator coordinate path of some interest for collective motion.

D. Time as a generator coordinate

Let f(t) be the solution of the time-dependent SchrcJdinger equation. Any eigenstate ¥_ can be recovered by a Fourier transform

+iEt fE = [dt e ¥(t) , (III.21)

which is actually nothing but a generator coordinate ansatz with respect to time where fg(t) = exp(iEt). Of course one needs that the initial condition f(0) have a non-vanishing component on 1f_.

One may notice at this stage that any eigenstate can be obtained from eq.(III.21) if ¥(0) is "random" enough, namely, in the present context, if it has components on all eigenstates of J*6. This may be understood as a kind of ergodicity, the trajectory ¥(t) winding around so much on the unit sphere that the ansatz, GENERATOR COORDINATES 13

eq.(III.21) allows one to pick up any unit vector.

In that sense, che opposite case of "non ergodicity" corresponds to ^(t) having components upon the states of a collective band only. Incidentally, if the number of eigenstates in that band is finite, the trajectory {^(t)} is seen to coil around a kind of multidimensional torus.

This geometrical image reminds one of the fiber bundle discussed in the previous subsection. Since an exact time-dependent SchrBdinger trajectory is not available in practice, one may rather consider the ansatz

¥E = Jdt fE(t) $(t) , (III.22)

where (t) is now a TDHF trajectory. Obviously there is now no reason for fp.(t) to be equal to exp(iEt), so that one needs to solve explicitly a Griffin-Hill-Kheeler equation identical to that shown by eq.(II.16). Conversely, if it turns out that the solution f-Xt) looks like exp(iEt) one may conclude that TDHF is an excellent dynamical approximation in that special case.

Of special interest of course would be the case where $(t), instead of showing a large amount of "ergodicity", would remain collective at "all" times (during at -21 least a typical time for collective processes, namely a few 10 s). Although the curvature criterion of subsection 1 is neither necessary nor sufficient, it is clear that one may calculate the curvature as a function of time and that time intervals where the curvature remains of order /3 may provide an interesting first gues". for $(t) in the ansatz, eq.(III.22).

An other advantage of this ansatz relates with the theory of nuclear collisions. It turns out that, as time goes to +°°, a TDHF solution $(t) usually does not reach an asymptotic state. This difficulty prevents one to use in TDHF the standard concept of channels, and TDHF "cross sections" are thus often of a classical and statistical nature. On the contrary it can be shown that a suitable choice of boundary conditions for fg(t) allows to reconstruct a quantal scattering wave ¥ from eq.(III.22). The details of the proof are too technical to be given in these lectures, but they have been published elsewhere. It can be stressed here that the TDHF theory is non linear, hence the difficulties for asymptoticity, while the generator coordinate ansatz is linear. It is the linearity which allows the various channels to evolve independently and provide cross-sections without spurious interferences.

A third advantage of the TDHF ansatz, eq.(III.22), is a stability property of the energy of ^ with respect to a change of the trajectory {(t)}. Indeed, the energy is

1 E = jdt dt f*(t) < fE(t'), (III.23) 14 B.G. Giraud whose variation under a modification 6 $(t) is

6E = f*(t) jdt' <6 4>(t) |*6 | i(t') > fE(t'). (III.24)

It is reasonable to consider as dominant, in this integral, the neighbourhood of t = t'. Since 6 $(t) is then a set of particle-hole elements with respect to $(t), the full Hamiltonian can be replaced by the Hartree-Fock Hamiltonian. One thus obtains < & 4>(t) |3fi | f(t) > = < 6 *(t) |W| *(t) > , (III.25) and, because of the TDHF equation of motion

< 6 $(t) |W| = i < & *(t)l|| > • (III.26)

A significant variation of the generator coordinate trajectory, however, must at d least be orthogonal to the trajectory itself. For both $(t) and 4>(t+dt), thus ^ , are already in the generator subspace. Therefore < 5 (t) | dO/dt > identically vanishes in eq.(III.26), which is the stability property considered earlier for E.

IV. APPLICATION TO NUCLEAR REACTIONS

A. Definitions

Typically one considers here an elastic scattering a + A •+ a + A, with the pro- jectile a and the target A, or a transfer reaction a + A -* b + B, where b is considered as the core of a, and A as the core of B. In other words a = (b+v.) and B = (A+v^), where V. and v- are the initial and final clouds of transferred nucleons.

One defines the creation operator A (r) of nucleus A around the shell model center r AT(r) = I n}(r) ... n^Cr) , (IV.1)

where r\ (r) is a single-nucleon creation operator in a shell model state tp(x-r) cen- tered around r and where the necessary configuration mixing is shown symbolically by h In coordinate representation one demands

+ = I^ÍXj-t xA-r)

Here |o> is the vacuum, the variables x^ are the single nucleon coordinates, 'PA is

the total wave function of the nucleus, gA is the center of mass coordinate and one defines the usual (A-l) Jacobi coordinate (internal degrees of freedom) £. It is

i GENERATOR COORDINATES 15 important that "Í factorize into a wave packet r, for the center of mass and an internal wave-function T. /. Otherwise the formalism would be plagued by center of mass spuriosity. 'For" the sake of simplicity T. will be assumed to be Gaussian in the following. i" "J" *f* •** In the same way one defines creation operators a (r')t B (r"), b (r"1) , v!(r') ! ~ and v, (r") for the other nuclei and clouds of nucleons around their respective shell tttttt model centers. One finds in particular that a = v.b and B = v A .

The generator function for the channel a 5 A + a is then

|ar> = A+(N r/N) af (-N. r/N) |0> , (IV.3) where N , N and N are the mass numbers of a, A and the total mass number, respectively. In coordinate representation this means

where dk. is the antisymmetrization operator.

Let pa = R - RA be the channel degree of freedom and R be the total center of mass coordinate. The Gaussians T and T have a product which is also Gaussian with respect to p and R

ra(?a+NA E/

It can happen that "€L also factorizes

where F may be called the channel wave-packet. In such a case the generator function |ar> is non spurious, because the total center of mass factorizes out. If furthermore a generator function |(3r'> for any other channel B is designed like |cer>, in a way strictly analogous to eq.(IV.3),

+ + |Br'> = B (Nb r'/N) b (-NB r'/N) |0> , (IV.7)

one may demand that the corresponding product T V, factorizes into a product of Gaussians G T. with the same G(R) as that for channel a, eq.(IV.6).

The multichannel ansatz 1 l^ = Jdr £a(r) |ar> + Jdr' fgir ) |Br'>

+ ...(other channels y) (IV.8) 16 B.G. Giraud

reads in such a case, after insertion of eqs.(IV.A) and (IV.6),

Y = G(R) crk flnt: 4j/int \jiln'- g + ... , (IV.9)

where the sine G(R) factorizes out for all channels (and commutes with Jt ) and one defines channel wave-functions

g ( } = d or d ) f ( or r f ( or a 6 6a B | ^ i' a 6 2a R"E '> a 8 i l'^ ' (IV. 10)

It is clear that the factorization, eq.(IV.6), prevents any center-of-mass spuriosity from creeping into the generator coordinate formalism. This property will be assumed until subsection D, where it will be s.^en how it can be relaxed without damage.

B. Connection with the resonating group formalism

It is trivial that * r r i G(R) i# r(p -r) M^ ^n = G(R) ds ^(r-s) |<&6(p -s) ^ V™1 . (IV.11)

The antisyraaetrized function between brackets is nothing but the resonating group basis function |as> . Indeed, in the resonating group theory, the channel degree of freedom p is strictly localized by the corresponding 6-function. (The total center- of-mass wave-packet can be integrated out and forgotten.) Alternately, one may interpret [as> „ as the sharp basis of the generator coordinate subspace, as defined by eq.(II.12), with p as the collective degree of freedom, —ot Let us compare a generator coordinate kernel and a resonating group kernel, for instance the following

HaB(r,r') = , (IV.12a)

it' As seen in eq.(IV.ll), a state |ar> is a linear supersposition of states |as> >' ~ ~ RGM and an analogous result holds for any other channel. It is thus obvious that

H ( ) = d d r ( ) h { X) V i > • aB E'i' J ? 5* a I"? ae> ~'~ $ Z'~Z" ' (IV.13) i, j

|.-.' In a short notation this reads HD = r *hD*ro,in order to express the convolu- : < i tion product. In the same way eq.(IV.lO) can be shortened into g o = V D*f „• [."• o-tP H,B a,p " This convolution correspondence between the resonating group and the generator coordinate formalism is then general, and can trivially be extended to any Hamiltonian, overlap or transition kernel such H , I and so on. CXCt Otp GENERATOR COORDINATES 17

C. The high frequency catastrophe. Various methods of solution

Except for unessential coefficients, the Fourier transform of eq.(IV.lO) reads

g(ir) = exp(-TT2) f(Tt) , (IV.H) where IT is the momentum conjugate to p (or r) and f, g are the Fourier transforms of f, g respectively. As it is known from the resonating group theory that g behaves like a wave-function, the singularities of g in the continuum are thus of a mild nature (principal parts, etc). It is then clear, however, that f diverges strongly, like exp(+ir2 ), when ir •+• +=>. Thus a Fourier transform of f(ir) back to a coordinate space f(r) will diverge. This serious difficulty has been analyzed by many authors and brings many numerical technicities in the solution of the Griffin-Hill-Wheller equation. As an example, consider the two-channel equations, which read, with obvious notations

fa (IV.15a) to = 0

rJ(haa"E W ga (IV. 16) 8a

In coordinate space, there is difficulty in defining an inverse operator V „. For, as shown by eq.(IV.14), the matrix element of T „ in momentum representation reads (X, p (unnecessary coefficients omitted)

<3r|r|Tj'> = exp(-ir2) S(H-TT') , (IV. 17)

and thus one obtains ^Ir"1!;^ = exp(+Tr2) 6(J-TT') , (IV. 18)

the Fourier transform of which to coordinate representation is obviously singular. Nothing prevents, however, to consider eqs.(IV.16) in momentum representation and 18 B.G. Giraud multiply the upper (lower) equation by T~ (Fg ) from the left. One then obtains

h E + (h E < aa- W 8a aff V »B " ° (IV ,9)

The transition from eqs.(IV.15) to eqs.(IV.19), which can be made without singularities in the momentum representation, is a first method to obviate the high frequency catastrophe. There is a second method, however, which has the advantage of avoiding the detour through the momentum representation. Indeed, as found by the Takacsy, a special case of eq.(IV.lO) reads, after projection onto a channel partial wave

2 2 sin(kp+n) - [dr expj+k - (p-r) | sin(kr+n) + gs>r.(P) . (IV.20) where T] is the phase shift and g is a square integrable (short range) contribu- s. r • tion to the resonating group wave-function. In other words, the asymptotic parts of g and f are proportional. It is easy to show that the same result is also valid for Coulomb waves 2 2 sin(kp-7 Iog2kp+n) = fdr exp|+k - (p-r) 1 sin(kr-Y Iog2kr+n) + 8S r (pX* (IV.21)

The phase shifts in the generator coordinate theory can thus be deduced from the short range properties of f, like in the usual theory of collisions. Practically the second method may go as follows. One defines the short range part of f (in each channel) by the condition

r fs r _ - g - r fas , (iv. 22)

where £ is known from eqs.(IV.2O) and / or eq.(IV.21), except for the phase shift n, to be determined later. Then the right hand side of eq.(IV.22) is the short range part gs.r - of g is that channel. It can be expanded, with strong convergence properties, in a suitable basis of square integrable functions, {u },

n and thus.one. obtains

£sr = I cn r"1 Un • (IV'24) n

where the unknown coefficients CQI and later the phase shifts, can be obtained by an insertion of eq.(IV.24) into eq.(IV.15). This makes eq.(IV.]5) inhomogeneous, since a source term is brought by the identity f = f + £ . The phase shifts, which S«7« AS GENERATOR COORDINATES 19 ÍJ, are unknown parameters in the source term, must be determined by any usual variatio- ' • nal principle of the theory of collisions or any standard matching procedure of f as and fsr>, or gas and g^. In practice one truncates the expansion, eq.(IV.23), at a finite order of components u . It is easy to show that one may always choose this basis {u } in such n -1 n -1 a way that, although T is singular in coordinate representation, the product Y u is still a regular wave function. Finite order expansions, eq.(IV.24), of the singular function f are thus regular and retain the convergence properties of eq.(IV.23) as regards the fact that the coefficients c make a square integrable se-

D. The biased density matrix

Among the various methods proposed by many authors we now discuss that of the "biased" case, namely that which is still available when eq.(IV.5) does not simplifly into eq.(IV.6). When this factorization does not occur, the generator coordinate ansatz, eq.(IV.8), breaks down completely since the total center of mass and the relative degree of freedom, R and p, are spuriously correlated.

An analogue of eq.(IV.Il) is available, however. Indeed, one obtains from eq.(IV.4) and (IV.5) the trivial result } R ds ^(Ç.f-s)

6(R-R) \ift <5(p -s) vjnt

where again there appears the bracket [as> „„. Here however, the total center of mass is frozen at position R, while it could be integrated out in the case of eq.(IV.ll).

According to eq.(IV.25) and ary identical equation for another channel, a kernel H~ now reads

3 d d «irlKlPr' ' " } 5 S ds' ^a(R,r-s) ©giR.r'-s') RGM«*fl#? I^'>RGM . (IV.26)

where one has taken into account that the total center of mass must be frozen at the same location R in the bra and the ket provided by eq.(IV.25). If now one defines

one obtains from eq.(IV.26) H' f(r2.E'-S'> VW^' 20 E.G. Giraud

In other words, the information carried by the resonating group kernel is still embedded inside the generator coordinate kernel H „, although the ansatz,eq.(IV.8), is

meaningless . If h o can be recovered from H „, one may still solve eqs.(IV.16). This Otp Up is of great physical interest for the frequent case where the generator functions |ctr> are derived from Gaussians of unequal parameters, corresponding to the collision of ions of unequal masses. The derivation of h „ from H „ is simple when H . is a Gaussian. For, as seen earlier, the choice of T. and T as Gaussians makes trivially $ „ also a Gaussian. A a 4(ip It is then easy to check that h „ is also a Gaussian. More precisely, assume that, op except for unessential coefficients,

Hag = exp í-(r,r') í>'(ji)| . (IV.28)

where '!)' is a (2x2) matrix,and denote the Gaussian form of % o as 0Ctp , (IV.29) J

where Sis also a 2 x 2 matrix. Then eq.(IV.27) provides

with

•X>= I©'"1 - € "'] (IV.31) L J except for special cases, where h „ is a local kernel, and which can be handled explicitly. In the more general case of Gaussian wave functions for clusters where H „ (and all the other kernels under consideration) is slightly more complicated, namely contains products of polynomials and Gaussians, the result, eqs.(IV.3O) and

(IV.31)fcan be extended. One also finds polynomials and Gaussians. Incidentally this method induced by eq.(IV.31) can also be applied in the special case where eq.(IV.6) is valid. In this special case Ç is just diagonal.

It is now possible to define the "biased density matrix" which, analogously to eq.(IV.27), takes into account the conservation of the information about a state although some spuriosity is introduced by a spectator degree of freedom (the center of mass). Let d(p,g') be a density matrix inRGM representation. Assume that the RGM kernel k(p,p') and GCM kernel K(r,r') derive from each other through an equation analogous to eq.(IV.27)

K(r.r') - fdp dp'

d(p\p) = [dr dr1 Í (r-P.r'-P1) D • (IV.33)

Then the RGM and GCM expectation values are equal,

Tr k d = [dp dp1 k(p.p') d(p',p)

f,dp dp1 dr dr1 k(p.p') $ (r-p.r'-p1) D(r',r)

[dr dr1 K(r.r') D(r',r)

Tr K D . (IV.34)

E. Distorted wave Born approximation

Rather than the methods described in subsections C and D, which are attempts towards a full solution of the dynamical problem of collision, one may look for a DWBA approximation.

The DWBA form factor for the transition from channel a to channel 0 has then obviously to do with the generator coordinate kernel

Kag(r',r) = <6r'|VT|ar> , (IV.35) Kag( where Vis the two body potential operator present inland the subscript T means that V1 has been truncated into a residual potential such as the post potential for instance.

More precisely, one may define a specific set of Wick's theorem contractions

Kga(r\r) = <0|b(NBr'/N) [AÍN^VN) vf(Nbr'/N)j \TT

A+(N,r/N) [v?(-H.r/N) b+(-N,r/N)l |0> (IV.36) A" [_ 1 A— A~ J

where Avf corresponds to the decomposition of B into core and cloud and, similarly t t + vjj corresponds to the decomposition of a into core and cloud. The rules for contractions are the following,

i) one pair of creation-annihilation operators of "\T contracts from y_ to V.,

ii) the other pair contracts from b to b ,

iii) what is left inside Av, contracts with what is left inside A v., for instan- t X ce A contracts with A and what is left in vf contracts with what is left in V^f 22 B.G. Giraud

iv) what is left inside b contracts with b . The first two rules, i) and ii), correspond to a truncation of Tf into the residual interaction between core b and the cloud of transferred nucleons. The last two rules, iix) and iv), correspond to an inert core overlap approximation relating A to the core of B (and the core of a to b)when connecting the ket to the bra.

Once K R is obtained, nothing prevents to define a resonating group kernel kR (p',P) by a formula analogous to eq.(IV.27) or eq.(IV.32). Let now ga(p) and g (p1) be optical waves for the initial and final channel, respectively. The DWBA amplitude is then |de de' V£f-e> «fe *S • (IV-37)

Alternately one may consider g (p) gj(p') as a density operator d(p,p') and define a a - p ~ ~ - density D(r.r') by a formula analogous to eq.(IV.33). One then obtains

d' KBa(í''í) ^'^ • CIV'38)

V. OTHER EXAMPLES OF APPLICATION

A. Axial rotations

This may be the oldest case of a generator coordinate model and it is well known since the paper of Peierls and Yoccoz. Let $„ be an eigenstate of J , with eigenvalue K, usually a deformed Slater determinant. A rotation band is derived from the angular momentum projected states

¥JK = jd(cos g) d^CB) exp(-ie Jy) \ • (V.I)

The generating function is here, obviously, that obtained from $ by a rotation of . J angle 6 around the y-axis. One then uses the reduced rotation matrix element d as Griffin-Hill-Wheeler amplitude.

B. Triaxial rotations

The deformed (intrinsic) Slater determinant $ is no more an eigenstate of J . The generating function is obtained by the most general rotation, depending upon three Euler angles a (5 -y

= eXp( i J ) ex f 'Wr " ^ z P -i B Jy) exp(-i Y Jz) 4> . (V.2)

The Griffin-Hill-Wheeler equation can be solved in two steps. The first one is a quasi-band and angular momentum projection,

d(cos B)dY?Í(aBT) $a0Y, (V.3) GENERATOR COORDINATES 23 with thee rotationn matrix element !)„O „ as amplitude. The second step is a residual ~ — - MMrK diagonalization between quasi-bands,

K The index K disappears, to be replaced by an index K carried by the eigenvalue E JK and the amplitude vector f . Full degeneracy with respect to M is obviously found.

C. Three-body problem

An exact, and technically available solution to the non-relativistic three-body problem is the set of Faddeev equations. When discrete degrees of freedom (spin, isospin, etc) are involved, however, the number of coupled channels in these equations may become unwieldy. As an example, if three quarks are heavy enough to be treated non relativistically and if the effective quark-quark interaction depends strongly on spin, flavor and color, it may be useful to look for an approximation such as the Hartree-Fock approximation.

For the sake of simplicity let the discrete degrees of freedom be discarded and the 3 quarks be considered as distinguishable. Assume, furthermore, that the quark bag amounts to a harmonic oscillator. The oscillator can be deformed, with three different constants a, B« Y along the three axes x, y, z. With V a Gaussian, a generating function reads, in an obvious notation

r(yj/B) r(y2/B) r(y3/g)

r(z,/Y) r(z2/Y) r(z3/Y) . (v.5)

It is easy to check that it has no center of mass spuriosity, namely that it factorizes

Here x is a deformed Gaussian for the center of mass R and ¥ is also a deformed Gaussian for the'two Jacobi coordinates Ç, Ç1.

Since x depends on a, B, Yi the GCM ansatz

V = [da dg dY f (ctBY) 3^ • (V.7)

introduces center of mass spuriosity. It is clear however that the overlap kernel 24 B.G. Giraud

makes sense. Also, if "J^ depends only on the Jacobi coordinates, the energy kernel

also makes sense. AlthoughllZ depends only on Ç, Ç' it can still be written in terras of r., r», r, and the numerator of the left hand side of eq.(V.9) can easily be calculated in second quantization. As regards denominators <$( ,g, ,|x gy5"» they can be easily derived analytically and/or tabulated.

The equation

(H-EI) f = 0 (V.10) is then equivalent to the ansatz

da dB (V.ll) a theory of bag vibrations. We insist, however, that all calculations can be nade in the single quark coordinate representation.

D. Hard cares

It is usually assumed that a single nucleon orbital is concentrated in a certain domain, but that its tail extends everywhere. Nothing prevents, however, to restrict a single nucleon orbital

Assume now that, for i t j, J>. and J). have no point at a distance smaller than a certain radius r. . If the nucleon-nucleon interaction has a strong short range repulsion (hard core) of radius smaller or equal to r , that repulsion cannot act between J)• and '£).. A Slater determinant built out of such orbitais with disconnected domains is then compatible with the hard core. In other words, the Hartree-Fock theory applies to hard cores as well.

The parameters of such determinants are then the shapes of the domains S>. themselves. The radius r is also a parameter, for one may always take a larger separation than needed. In any case, the ansatz

in [D

"may be understood as a geometrical," or functional integral, generalization of the GCM ansatz. Here J) ... £) is a set of shapes for the domains allowed for N nucleons, £) -r\ is the result of a Hartree calculation for that set of frozen shapes, and the symbol n[D 53.][means superposition of all allowed shapes. In a restricted appro-

i ^ - - , • . ; - ximation the amplitude f may he understood as a function of a more or less finite set of parameters for the shapes of JJ....3J.

This opens the way to a theory of a new possible mode of nuclear vibrations, which can be understood as correlation vibrations.

VI CONCLUSION

There may be three ideas which can be retained from the above considerations. The first one is the amazing flexibility of the generator coordinate formalism. Any macroscopic degree of freedom can be forced into the generator function and the resulting dynamics studied. The second idea is the level of mathematical technology involved. A certain amount of caution is necessary in order to avoid unphysical singularities, but the situation is less complicated than it looks at first sight.

The third idea is probably the most important and corresponds actually to a major, and yet unsolved physical problem. Since the generator coordinate method finally amounts to diagonalization in a subspace and a reduction of the number of degrees of freedom, it is essential to understand uhy such subspace is of interest and why such degrees of freedom emerge from the many body dynamics. It is hoped that section III has brought a slight progress on this fundamental question.

The considerations in these lectures are the result of a many-year collaboration with many coauthors, among whom B. Grammaticos,J. Le Tourneux and D. Rowe should receive special credit for the difficult question of collective motion. It is a pleasure to thank the organizers of the Symposium for the opportunity of giving these lectures and thir warm and stimulating hospitality. A tew reterences

Equivalence between the generator coordinate method and the Born-Oppenheimer approximation : Nuclear Physics A233, 373 (1974).

Curvature : Journal de Physique Lettres f5, 179 (1979).

Fibration : Preprint Dph/T 78-59, Saclay.

Time as a generator coordinate : Nuclear Physics A299, 342 (1978).

Theory of reactions : Annals of Physcis &9_, 359 (1975).

General review : C.W. Wong, Physics Reports J5£ (1975) n°5.

General review : J. Le Tourneux, June-July 1976 Lectures at Institut de Physique Nucléaire de Lyon. 27

COLLECTIVE MOTION AND THE GENERATOR COORDINATE METHOD

E.J.V.de Passos* Instituto de Física, Universidade de São Paulo • 05508 - São Paulo SP, BRASIL

ABSTRACT

The generator coordinate method is used to construct a collective subspace of the many-body Hubert space. The construction is based en the analysis of the properties of the overlaps of the generator states. Some well-known misbehaviours of the generator coordinate weight functions are clearly identified as of kinematical origin. A standard orthonormal representation in the collective subspace is introduced which eliminates them. It is also indicated how appropriate collective dynamical variables can be defined a posteriori. To illustrate the properties of the collective subspaces applications are made to

a) translational invariant overlap kernels b) to one and two-conjugate parameter families of generator states.

•Partially supported by the Conselho Nacional de Desenvolvimento Cientifico e Tecnológico. 28 E. Passos

I - THE KINEMATICS OF GENERATOR COORDINATES

In the generator coordinate method (GCM) there is a clear separation of two stages in the setting up of a phenomenological scheme to describe collective motion of many-body systems. In the first stage we select a subspace of the many-body Hilbert space (HS) which is spanned by states constructed as a linear superposition of the generator states |a> ,

|f> = J^ f (a) |a> da I.I with square integrable weight function f(a). The generator states |a> are specified a priori usually on the basis of phenomenological considerations in a way to reflect the distortion of the many-body system during the collective motion. The parameter a. is the generator coordinate and it establishes a one to one mapping between the generator states and points in a label space. In contrast to other approaches , which use product type wave functions, the GCM does not require an explicit reference to any collective dynamical variable at this stage. Indeed, the specifi- cation of the appropriate collective degrees of freedom is in general very difficult, the adequacy of a given choice being a dynamical question which cannot be settled without explicit reference to the many-body hamiltonian. Furthermore even in the cases when one has a good phenomenological knowledge as to the nature of the collective motion under consideration it is, in general, very difficult to find an explicit expression for the collective degrees of freedom in terms of the microscopic ones . These desadvantages are absent in the GCM. The dynamics is established in the second stage with the determination of the weight function f(o) , the only unknown in eq. I.I. The states given by the ansatz I.I are used as trial wave functions in the variational principle

6 = 0

resulting in the Griffin-Hill-Wheeler (GHW) integral equation for

J{ - E )f(a')da' = 0 1.2 GENERATOR COORDINATES 29

The GCM is probably less a accurate than methods involving the explicit introduction of collective dynamical variables in cases for which this explicit introduction is straightforward, unfortunately this is not the case in general and the GCM provides a scheme which is very easy to apply in practice . However in this lecture I would like to discuss the GCM description of collective dynamics with especial emphasis on questions of interpretation. To do so, the first difficult that one has to face is the well known fact that, in many cases, the weight functions f(a) have undesirable mathematical properties . The consequence of this fact is that the GHW ansatz (I.I) will associate highly singular weight functions to vectors defined in the many-body H.S.. This difficulty can be shown to be simply a consequence of the use of the non-orthogonal and, in general, linearly dependent generator states |a> as a representation in the GCM collective subspace Sand it is eliminated by introducing a standard orthonormal representation in S. This standard orthonormal representation is constructed in terms of the adopted set of generator states and it allow us to define, a posteriori, the collective hamiltonian, collective dynamical variables and collective wave functions. The basic tool of the method is the diagonalization of the overlap kernel (4f5'6)

1 1 f*" uk(o')da = A(k)uk(a) 1.3

The eigenfunctions of the overlap kernel form an orthogonal and carolete set

= 6(k-k')

4" \(«)\ (of)dk = fi(a-a') 1.4

In eq. 1.3, A(k) is always a semi-positive definite function of k (the overlap kernel is a norm) and for simplicity it will be assumed to be a monotonic decreasing function of k vanishing only at |k|-*«> . The general case is discussed in ref. 6. To proceed we need to be specific and we define the GHW space as the space spanned by the many-body vectors constructed as in i.l with square integrable weight functions. The scalar product of two such vectors is equal to E. Passos

1 = f f*(a) f2(a") da da a 00

= 1 ^(kjf,(kj ) dk 1.5 where f(k) is equal to

f(k) = • A(k") ]_„ f(a)u*(a) da

.+0° = i— da u* (a) /AW '_«, *

= 1.6

In eq. 1.6 we introduced the orthogonal states |k>r which are defined as

]k>. = - r |a>u.(a)da 1.7 /TíkT ; — k = O(k-k')

Therefore to each vector defined in the GHW space we can associate a square integrable function f(k), which is equal to the scalar product of |f> with the orthogonal state |k>. However the inverse is not true . To show this we write the weight function as

f(a) = /dk u.(a) f(k) 1.8

Thus f (a) is a regular function only if f (W/vA^kT goes to zero when |k| goes to infinite. Otherwise it is singular. At this point we consider the extension of the GHW space to a subspace of the many-body H.S. in which the vectors have square integrable f(k). This extended space is the GCM collective subspace S and the projection operator in S is given by

S= /dk |k>

From the previous discussion it is clear that not all vectors defined in the GCM collective subspace S belongs to the GHW space. However it can be shown that the GHW space is dense in S and the subspace S is the smallest (closed) subspace which GENERATOR COORDINATES 31

contains the G.H.W space. "Therefore we can say that the singular behaviour of the weight function is related to the fact that strictly speaking, the generator state is a well-behaved representation of the GHW space but not of the GCM collective subspace.

II - DYNAMICS IN THE GCM COLLECTIVE SUBSPACE - COLLECTIVE HAMILTONIAN, COLLECTIVE OPERATORS AND COLLECTIVE WAVE-FUNCTIONS

The description of the dynamics in the GCM is given by the projection of the many-body dynamics onto the collective subspace S

t | , | II.

where s f In eq. II.1 Hg*-" is the GCM collective hamiltonian. It is the projec- j-; tion of the many-body hamiltonian onto the GCM collective subspace S,

= S H S II.2

Using the expression 1.9 of the projection operator S, we can write the dynamical equation II.1 as a wave-equation in the "momentum" representation | k>

/ h(k,k') ítk-.tJdk1 = i 3»(k'fc) II.3 3t

where

h(k,k') =

(k) /A(k')

and <|>(k,t) is the collective wave-function in the "momentum" representation

i))(k,t) = (t)> II.5 32 E. Passos The above discussion was carried in the specific representation that diagonalizes the overlap kernel. Although this representation is very convenient for sorting out the kinematical oddities inhe- rent to the GCM, other representations may be prefereable from a physical point of view. They can be obtained by unitary transformations in S. One which has been considered often in the literature ' ' is the "coordinate" representation obtained by an unitary transformation of the "momentum" representation given in terms of the eigenfunctions of the overlap kernel

L ^ dk II.6

As pointed out before, in the GCM there is no explicit reference to any collective dynamical variable. However once one has a representation in the collective subspace, one can define, ã posteriori, collective dynamical variables in S. These collective dynamical variables would allow us to describe the dynamics in terms of a small (9) number of specialized degrees of freedom. As an example we can associate to the coordinate representation II.6, a pair of canonical operators in S,

Qnh> = n|n>

P |n> = i3/3n|n> ii.7

= is

These canonical collective operators in S can be easily expressed in terms of the microscopic degrees of freedom . We can

also express any operator defined in S in terms of Qn and Pn. For example, the GCM collective hamiltonian is given by* '

HGCM _ - 1^ ; ?m 5(B, (1) ; jiQ m=0 2 n

In eg. II.8 the normal order is defined as

m anti-ocnnutators GENERATOR COORDINATES 33

and

m 5< > (n) = (dÇ-tÜHÜ

m-oannutators

This shows that we can always write the dynamical equation II.1 in the form a Schroedinger type equation in the coordinate representation |TI>. However, in general, this Schroedinger type equation has a "velocity"-dependent potential and a "mass-parameter", which depends on the coordinate.

Ill - SPECIAL CASES AND EXAMPLES

III-l - TRANSLATIONAL INVARIANT OVERLAP KERNELS(6)

Translational invariant overlap kernels depend only on the difference of the generator coordinates

= N(a-a') III.l

and they are diagonalized by a Fourier transformation,

1 /uk{a')da = 2irA(k)uk (a) III.2

where

VL.(a) = -^— eika III. 3 ^ /2?

A(k) = — /N(a) e~ika da III.4 2ir

In this case the "coordinate" representation is the Fourier transform of the "momentum" representation 34 E« Passos

1 J Ir v |x> = -±- /|k> e 1KX dk III.5 and the pair of canonical collective variables defined as

'x> • xix* HI.6

satisfy the eqs.

Q |k> = -13 |k> s * III.7 Ps|k> = k|k>

The gaussian overlap kernel is a special case of a translational invariant overlap kernel where

— e o III.8 /ir

Until now we considered only the continuous "coordinate" or "momentum" representations. We could also have considered discrete representations which diagonal!ze a boson number operator constructed

in terms of Qg and P_. Taking the case of a Gaussian overlap kernel as an example and expanding the reduced energy kernel, h(a,ct')

in a power series in the generator coordinate one has ,

v h _ hi a,a J — n'm nlm! Introducing the boson operators

and using the expression of the GCH collective hamiltonian in the (8) "coordinate" representation (see eg. II.4), one can easily show that GENERATOR COORDINATES

gGCM n,m n'.ml

Therefore, the expansion of the reduced energy kernel in the generator coordinate is equivalent to a boson expansion of the collective hamiltonian, where the boson is constructed in terms of the collective canonical operators in S, Qs and Fg. In order to shed light on the origin of the singular behaviour of the weight function consider again the case of a Gaussian overlap and a quadratic approximation to the reduced energy kernel.

z l2 h(a,a') =h +i f" h- (a -i-a ) +2hI1cta'l + ... III.9 O í L 2O Xi- _[

The collective hamiltonian 5S^ can be written in this case as,

2 -GCM= h 1 - C O 2M «j M S P Z where the zero point energy E„, the mass parameter M and the spring constant K^aie respectively given by

EPZ = bo hll

1 M" = 2(h11-h2o)b; HI. 11

The hamiltonian III.10 is a standard harmonic oscillator hamiltonian and its eigenfunctions are harmonic oscillator wave functions

(x/bc) and its eigenvalues refered to the unperturbed ground state are

En=f«oc(n+l/2)-Ep2

where

h L- 2o K2 11 2o E. Passos . í To proceed, consider the ground state wave function

x2 1 - /2bC l c *o(x/bc) e III. 12

The weight function associated to this state is (see eq. 1.8)

f(a) = JO W

2 which exists as a regular function as long as b *bo • The nature of this singular behaviour is equivalent to the high-momentum divergence considered in ref. 10 and it stems from the fact that when bc

2 2 /b~ -bc k /2 =/_£ e and /b~ -k2b2/2 = e-lka/-°- e which shows the correctness of our statement

III-2 ONE AND TWO-CONJUGATE PARAMETER FAMILIES OF GENERATOR STATES*9*

The one and two-conjugate parameter families of generator states (OPF and TCPF) are defined respectively as

III.13 iBQ |OfB> = e— e |0> where Q and P are canonical operators in the many-body Hubert space

= i

The overlap kernel of the OPF is a translational invariant GENERATOR COORDINATES 37 overlap kernel and so it can be diagonalized by a Fourier transformation

I 1 /uk(a')da =

where ika ^=r /2ir and A(k) =

Ilf is the Peierls-Yoccoz projection operator associated with the operator P

-PY _ 1 (i*a -iaP nk -—— Je ee da III.14 = 6(P-k)

Thus, the standard orthonormal "momentum" representation in the GCM collective subspace S, associated with the OPF of generator states is identical to the normalized Peierls-Yoccoz projection of the reference state |0>, associated with the operator P

III. 15 /

By construction one has

and the pair of canonical collective operators in S^ are given by

Q = S1QS1 III.16

However, in general

so S^ is not an eigenspace of Q and we cannot find a base in S^ which 38 E. Passos diagonalizes Q. On the other hand, the overlap kernel for the TCPF is

and its eigenfunctions and eigenvalues are determined by the equation

I 1 1 /<(>n;k(a ,B )da'dB = 2ITXn(k)n.k (a,B) .

It can be easily shown that $ . (a,g) is given by ' XI7 iC

eika n;k ' /2ir n and the X (k) are independent of k.

The functions (f>n(B) and the X are eigenfunctions and eigenvalues of the semi-positive definite Hilbert-Schmidt overlap kernel 5(6,6')

N(B,6') = <0|e"ieQ6(P)eiB'Q|0> III.17

The Hilbert-Schmidt kernel III. 17 can have zero eigenvalues and when they occur there are two-important consequences. One is that the weight functions defined in the null space of N gives -rise to vectors of zero norm in the many-body H.S. Therefore there is no loss of generality if we restrict the weight function space to the orthogonal complement of the null space of N. The other is that the existence of eigenvectors of N with zero eigenvalues implies that the generator states are not linearly independent. In this case the standard orthonormal representation in the GCM collective subspace S, associated with the TCPF of generator states is given by the Peierls-Thouless projection of the reference state 10") associated with the operator P

0> —. X 9*0 III.18

In eq. III.18 fij:T is the so-called Peierls-Thouless double projection operator

dad* ^ * (B-k) fir = Í n k,n 1 2TT GENERATOR COORDINATES 39

By construction one has

^•k.n"* -^Vn^ IIX'19

and it can be easely shown thatv '

Therefore the pair of canonical collective operators associated with the continuous label k are in this case given by

PS = S2P = PS2 III.21

= S2Q = QS2

Thus in the TCPF case S2 is an eigenspace of both Q and P and by an unitary transformation we can find a basis which diagonalizes

Q. This basis is the Fourier transform of the states |t|/k n>T and it is given by the Peierls-Thouless projection of the reference state |0> associated with the operator Q,

III.22 1 ( i , ^ -ikx ,, = — *,. >T e dk 2ir ) K,n x

To proceed in the discussion about the physical properties of the GCM collective subspaces, one introduce a canonical transformation from the microscopic degrees of freedom to collective, Q and P, and intrinsic degrees of freedom. Together with Q and the remaining

N-l intrinsic operators Ç_,Ç2,...Ç ., which by the canonical nature of the transformation must commute with both Q and P, we can arrive at a coordinate representation of the full many-body H.S. defined by the kets |Q,Ç> chosen as eigenkets of Q and Ç. This representation is a product representation and the states |Q> span a H.S. of one single degree of freedom, the collective space and the \K> are likewise associated with a H.S. of N-l degrees of freedom, the intrinsic space.

However it should always be kept in mind that both S, and S2 carry all the N degrees of freedom of the many-body system under consideration . They are distinguished from the full many-body H.S. in that 40 E. Passos they contain various imposed correlations among the N degrees of freedom. The discussion which follows will be aimed precisely at exhibiting the general nature of these correlations in each of the two cases. We begin by considering the wave function associated to the states \. >„, by the |Q,Ç> representation ,

T = —— x (Ç) III.23 where

/Xn and 0 (Q) is the Fourier transform of $ (B). n n The states X (Ç) are orthonormal and depend only on the

intrinsic variables and so \i>. >T comes out as a product of a collective wave function and an intrinsic wave function, and this holds even when |0> is not itself a product wave function. Indeed, the wave function associated to the reference state |0> by the |Q,Ç> representation can be shown to be given by (91

1> = I .

Thus we see that the reference state |0> is given by a sum of products of collective and intrinsic states, the number of terms in the sum being equal to the number of eigenvectors of N(B,g') with eigenvalue different from zero. In the case of the OPF ,the wave function associated to v the states l^t>v by the |Q,£> representation are '

where

/r;

Therefore the states |i|ik> are given by the product of a collective wave function and an intrinsic wave function. However, the intrinsic wave function depends on the eigenvalue k of the operator P. This property is responsable for the fact that S, is not an eigenspace of Q. GENERATOR COORDINATES 41

On the other hand if the reference state itself is a product wave function

= otQ) XQ(Ç) ,

in which case N has only one eigenvector with eigenvalue different from zero, one has

|t|), ->_ becomes equal to I^v>y an^ t'le subspaces S-^ and S becomes identical. If we identify the canonical operators Q and P with the center of mass coordinate and momentum respectively, the above discussions shows that in general the subspace associated with the OPF of generator states is not galilean invariant. It is galilean invariant only when the reference state itself is a product wave function. This fact is responsable for the incorrect translational mass that one in general obtains in GCM with generator states chosen as in eq. III.13. On the other hand, the súbspace associated with the TCPF is always a galilean invariant subspace and this is true even in the case when the reference state is not itself a product wave function. However when the reference state admits itself a factorization into a product of collective and intrinsic wave-functions both spaces are galilean invariant and identical. To conclude, in general the TCPF which depends on two- generator coordinates describes two-degrees of freedom, one collective and other non-collective (it depends only on the intrinsic degrees of freedom) , and its nature depends only on the correlations imposed on the reference state I0y>. When we make this identification we are supposing that the dynamical variables which are diagonal in the basis obtained by the diagonalization of the overlap kernel are the appropriate ones (if not, they can be found by unitary transformations in S). Furthermore the GCM collective subspace has the property that those two degrees of freedom are kinematically decoupled. They are coupled by the dynamics, in other words by the GCM collective hamiltonian, which can be easily written down in terms of these two degrees of freedom fullowing ref. 9. However we are in general interested in cases when the coupling of the collective and non-collective degrees of freedom is small so that one has almost decoupled bands. In that case we can always restrict the non-collective degree of freedom to be in the lowest energy state and so the dynamics reduces to the collective dynamics only x'. on the other hand when the generator states are so redundant that there is only one eigenvector of N 42 E. Passos with non-zero eigenvalue the TCPF which depends on two parameters describes only one degree of freedom, the collective one. In this case the TCPF is redundant in the sense that the collective subspace associated with the one and two conjugate parameter family of generator states are identical^ . This happens when the reference state itself factorizes into a product of a collective wave function and an intrinsic wave function. However in the case when the reference state does not admit this factorization, the two subspaces are different and the collective and non-collective degrees of freedom are knematically coupled in the GCM collective subspace associated with the OPF of generator states.

ACKNOWLEDGEMENTS

The author would like to thank Dr. C.P.Malta for a critical reading of the manuscript.

KEFEHENCES

1 - D.L.Hill and J.A.Wheeler - Physics Rev. 89, 112 (1953); J.J.Griffin and J.A.Wheeler - Phys.Rev. 108, 311 (1957). 2 - C.W.Wong - Phys. Rep. 15C, 283 (1975). 3 - D.M.Brink and A.Weiguny - Nucl.Phys. A120, 59 (1968); C.W.Wong - Nucl.Phys. A147, 545 (1970). 4 - L.Lathouvers - Ann. of Phys. 102, 347 (1976). 5 - A.F.R. de Toledo Piza, E.J.V. de Passos, D.Galetti, H.C.Nemes and M.M.Watanabe - Phys.Rev. C15, 1477 (1977) . 6 - A.F.R.de Toledo Piza and E.J.V. de Passos - II Nuovo Cimento 45B (1978),1. 7 - B.Giraud and B.Grammaticos - Nucl.Phys. A233, 373 (1974). 8 - N.Onishi and T.Une - Prog, of Theoretical Phys. 53, 504 (1975). 9 - E.J.V. de Passos and A.F.R. de Toledo Piza - Phys.Rev. C21, 425 (1980) . 10 - B.Giraud, J.C.Hocqueghem and A.LumbrosD- Phys.Rev. C7, 2274 (1973) . 11 - K.Goeke and P.G.Beinhand - to be published in Ann. of Physics. \l 43

SOME NEW RADIATION PROCESSES IN PLASMAS *

C.S.Wu Institute for Physical Science and Technology, University of Maryland College Park, Maryland 20742 - U.S.A.

FOREWORD

Since emissions of electromagnetic waves from- high temperature plasmas have profound implications to laboratory experiments and astrophysical problems, the subject has attracted much attention for many years. In the early days, theoretical discussions of plasma radiation processes relied to a large extent on the assumption that the radiation from a many-particle system can be estimated by considering the statistical average of the emission from individual particles. Now it becomes increasingly clear that such an approach is far from satisfactory. Recent studies indicate that new radiation processes can occur in plasmas which are unpredictable by the above-mentioned method. It has been found that collective plasma effects can give rise to enhanced spontaneous emission as well as induced radiation processes. The latter processes are particularly important. The purpose of the present series of lectures is to review some of the new plasma radiation processes. It is not our intention to present a comprehensive and complete survey of the existing works concerning this topic area. Rather, we shall only discuss a few cases which, in our opinion, may represent the high lights of recent theoretical developments. These cases are: (1) emission near the electron plasma frequency, (2) direct amplification of radiation near the electron cyclotron frequency, and (3) parametric amplification of radiation by stimulated scattering. The first case is a spontaneous process whereas the second and the third case are induced processes. These three cases are discussed separately in the following sections.

*A series of invited lectures presented at the 6th Brazilian Symposium on Theoretical Physics, held in Rio de Janeiro, Brazil, during January 7-18, 1980. 44 CS. Wu

I. SPONTANEOUS EMISSION "NEAR THE ELECTRON PLASMA-FREQUENCY

1,1 The General Expression for the Emissivity

The instantaneous power radiated from a plasma, P , can be calculated according to

3 P - - J d r ó^.t) (1) where ÔJ (;£, t) is the microscopic source (or extraneous) current and ôE^r^t) is the microscopic radiation field generated by the source electrons. The source current can be expressed as

N f S[*-%(«:)]v^(t) (2) where ToC^) aB^ Kl^ denote the position and velocity, respectively,

of the &th electron at time t , and, moreover, N is the total number of source electrons in the system. Evidently it is desirable to consider the average value of P which we denote by P . Thus 1 av d3r " J (3)

where the angular brackets denote the operation of an ensemble average. We assume that the plasma forms a quasi-stationary and homogeneous medium and expect

(2TT) RADIATION PROCESSES IN PLASMAS 45

Here <6E • 6,J >. represents the Fourier transform of <

3 .6J> -fd af dT

with R 5 £ - £* and T = t - t1 .

be It is instructive to point out that <<5j£ 'ils>k m can further expressed in terms of the Fourier-Laplace transforms of the source

current and radiation fiel*1

f d3r exp(iu)t - At-iJ^)«£(;£, t) o

dt | d3r

where w is the real frequency and A is a small, positive number

which insures the convergence of the time integral. It can be shown

that2

(4)

where the asterisk indicates the complex conjugate. As a result

4»

2A <í£qS,d>HA).ç£qç,»HA) . (5) (2ir)

A self-consistent expression relating to the radiation field and the source current can be obtained by the solution of the linearized Klimontovich-Maxwell equadons,^ 46 CS. Wu

where

and e_Qç,w + IA) denotes the plasma dielectric tensor. From Eqs. (5) and (6), we obtain

x p i f 3 r ii av * "% d k dü) Í where A Qc,(o) and \ (^c,u) are the determinant and the adjoint of Aj. (^u + iA) , respectively, in the limit as A approaches zero from above, and

5 £5 2A<ÔJaiqc,ü*iA)6J*;1Cfc,o>fiA>> . (9)

Hereafter we write

(10)

For the case of interest to us A,(£,ÜJ) •+• 0 , Eq. (8) can be rewritten as

X ( >ü>) P 1 f d 3k T du ss £ 6[A (k ü av ' ~22 J j -^ITIT- r -' _ ao

(ID RADIATION PROCESSES IN PLASMAS 47 where we have made use of the relation

which holds for propagating modes [1. e. , A Qs.iu) = 0] and where

X ss denotes the trace of X.i j.(k,io *v ) and â is the unit rpolarization vector of the propagating mode, i. e.,

At this point, it is apparent that the calculation of the quantity

T..(Jj,(i)) , the spectral function of the source current fluctuations,

is a crucial task. The details are given in an earlier publication by

Freund and Wu. Here we simply present the result. It is found

3 + » f d u F (u) / j j A ô(u- mü /y — k|>vii) , (13) 11 e m——°° ^lu&m 6 ii where

, Vm(b)]

b - kji^/n,, Y - (1 + u /c ) , «e = |eBQ/mec| , ^ = ^/me .

J and J' are respectively the usual Bessel function and its m is first derivative of integer order. Let us now return to Eq. (11) and introduce the definition of the spectral emisslvity n 48 CS. Wu

d2P

where 9 is the angle between ,£„ and k and S2, represents an element of the solid angle subtended by the wave vector \ . Thus

S [A.

T(kyo)

2 . e N 2 , . r , nX e ti ,3 . , , r , 2 ss —= =3• I d u F, (u) L I dn -= 2 c3 ' i I, A A ( ' I 3n~ r x ô(n"-n.'2 ) PXi '3* • j Ii 2 5f i 'j in

2 7 2 2

In Eq. (15), n = c~k /a) denotes the square of the index of refrac-

tion and 2 represents a sum over the roots of the dispersion equation i A (n,(u,S) = 0 . Clearly in c[A (k,.v) 1 we have imposed the approximation that the thermal effects may be neglected. This implies that

the frequency u) should not be too close to the cyclotron frequency 2 2 2 2 2

H , and, moreover, ni/c « SI /u is also true. The cold electron approximation leads to the usual Apple ton-Har tree dispersion

6 2 2 relation 2 = . 2a (l - a ) n+ = 1 = = Í 2(1-^) - S (sin 9 + p)

where a - u> Im , B = Í2 /tu , UJ = 4irn e /m , and e e e e e

2 _ 2 2 2 P = 4(Ü)W)(1 - a ) cos 0 . . •! -_-• ' ,.KI „•/.-:-

RADIATION PROCESSES IN PLASMAS 49

For a given mode, the unit polarization vector à(k, u) may be determined. It may be written as

here

X12 " " £12(£33 "" a±sin 8)

+ 2 2A 2 X22 = elle33 " n±(eiisin 9 + £33COS e)

42 2 n± - n±[(2 + sin 9

with Che definitions

Ell E22 1 2 Ul - < 2, oi a E12 " ~ £21 = e e 2 i 0)(ü)2 - n )

£33 1 2 01

E23 = £31 ' £13 = E32

The plus (minus) root in Eq. (16) describes the ordinary (extraor-

dinary) mode [or simply the 0(X) mode]. 50 CS. Wu

Ic is straightforward to show

(20) n • n+ + n_

where

2 e N p+ sin 6

-n+8|, cos9) -mfie/Y][V+Jm(b+)+ S^(b+)] • (21)

In Eq. (21) v,,/c , B± = vjc

2 2 u r (l-a )(l - n+B cos 9) - n^sin 9 "I (22) 4 % L n.sin 8(sin28 + p) -I

mn+B,sin 1 - (23)

Furthermore we have used

0) ÍÍ p e e (24) 2 2 2 2, 2 O2. n =n+ u (ID - Si )

Equation (21) is the general expression for the spectral emissivity

of ordinary and extraordinary modes which includes the essence of col-

lective plasma effects. This result has been obtained by several

authors. ' ' RADIATION PROCESSES IN PLASMAS 51

1,2 Spontaneous Emission from a Plasma with a Population of

Energetic Electrons

In many laboratory experiments or-space environments one often finds the situation that in addition to a background plasma there exists a population of supra thermal electrons. One example is that in the "tokamãk" device used for controlled thermonuclear fusion research, during the process of ohmic heating, a population of so-called runaway electrons are often produced. It is important to understand

the spectrum of spontaneous radiation resulted from these runaway electrons.

Although the following discussion is mainly concerned with the emission produced in tokanak plasmas when a fractional component of runaway electrons is present, the analysis is applicable to any physical

system which displays similar situation.

We shall mainly pay our attention to the runaway electrons,

particularly the emission near the electron plasma frequency. The

theoretical study was motivated by the recent observations of enhanced 9—11 radiation flux in the frequency range from several tokamak experiments.

The analysis presented1 in the following is taken from Refs.13 and 14.

The physical model under consideration is that the plasma consists

of two components, the thermal background and the energetic runaways

which have energies typically in the range of several hundred KeV

along the ambient magnetic field and several KeV in the transverse

direction. We shall direct our attention mainly to the emission from

the runaway electrons.

Under the combined influence of an external electric field and

Coulomb scattering, the runaway electron distribution function is 52 CS. Wu generally expected to possess a long flat tail in the direction of the ambient magnetic field. Since we find that the expression for

the emissivity does not depend sensitively on the detailed structure of the model distribution function, we shall adopt a distribution which facilitates the computation but without loss of generality.

The chosen distribution function, F , in the laboratory frame,

is as follows:

22 / 2 uu. (1 P 1\ "

for U|| > 0 (25)

u for u,| < 0

Again in Eq. (25) JJ = jj/m is the relativistic velocity; u

characterizes the perpendicular momentum spread; u and -s, charac-

terize the parallel momentum and momentum spread. It is assumed that

u « u^ and u_ << u. . Distribution (25) displays the essential

features of runaway electrons; in particular, it has a long tail when

u, « u, and vanishes for small and negative u|j .

Before we proceed with the discussion, some consideration of the

relativistic gyroresonance is appropriate. In wiew of the small mo-

mentum spread in the transverse direction, the relativistic resonance

condition may be written as

(26)

2 2 where the term of order u /c has been neglected. Since we require P RADIATION PROCESSES IN PLASMAS 53 that the resonant U|| be positive, the Cerenkov (m = 0) resonance yields the minimum resonant u(| where n+cos 6 > 1 and therefore dominates in the computation of the emissivity. Because we are interested in the emission near the plasma frequency and also because the index of refraction exceeds unity only for the extraordinary mode when ui *\< tu , we shall ignore the ordinary mode in our discussion.

To discuss the resonant velocity U|| based on Eq. (26) we ovtain the following analytic expression

u* (m|£i n.cos 6 + [mV + u2(n2cos26 - 1)] ™ e - e = (27) w|n2cos26 -

Furthermore making use of Eq. (21) we obtain after some algebraic manipulation that the emissivity associated with the runaways is

2 2 e n oiu n+ °° + _n

2TT cp (u., - u-) m=

±2 - ü)2(l - n2cos26)]|ex| p (" "f3" )

±2

(28)

where n is the density of runaway electrons 54 CS. Wu 2 2 r - 1 tu % 2.2fl K = 1 -J 2 n±9111 9 > e 5 1+ 2 *£ li f V2 2) ¥2 ta 2LV (l-a2)cosV

2 +2 2

2 2 Xr m = (1 - a ) - Yrm -7Tmii n?sin± 9 .

As mentioned earlier, the Cerenkov resonance is expected to be most Important. Thus we shall only consider the case m = 0 . For 2 2 to ^ u , 9 < TT/2 , u « c and m = 0 , the emissivity associated

with the runaway electrons for the extraordinary mode is

n sin en imc Y n 9(P - sin 9)

H(n2cos 9-1) 2 2 - exp(-ur0/u2)]

(29)

where n is the density of the runaway electrons and Y Q RADIATION PROCESSES IN PLASMAS 55

1.3 Numerical Study of the Emissivity near the Electron Plasma

Frequency

Computational results of the emissivity near the electron plasma frequency are presented in Figs. 1 to 4. It is found that in this frequency range, the emissivity is highly enhanced. The emission is due to a relativistic, Cerenkov resonanca which is possible only when the index of refraction exceeds unity. Furthermore, we find that this enhanced emission can only occur to the energetic runaway electrons which are assumed to have energies greater than 100 KeV. Within the context of our model, the thermal electrons with energies of several

KeV cannot give rise to the said emission.

The numerical study is based on parameters of interest to the -3 2 2 tokamak experiments. We choose n /n =10 , oi /fi = 0.6 , v, /c =• 3.9 x 10~3, u£/c2 = 3.95 x 10~Z , u2/c2 = 1.98 and u^/c2 = 2.05 x 10"1.

This corresponds to a mean perpendicular and parallel energy of 10 and

200 KeV for the runaways, respectively. The plasma frequency, 2 h 0) = (te^e /m ) , is calculated based on the background thermal

electron density, n, . The thermal electrons are assumed to have a Maxwellian distribution with a thermal speed v, .

In Fig. 1 we plot the emissivity n of extraordinary mode associated with the runaways which is normalized by the peak emissivity of the thermal electrons occurring at the second gyroharmonic in the limit of perpendicular propagation. (Since the ordinary mode in the frequency range of interes has an index of refraction less than unity, the Cerenkov emission cannot happen.) It is shown that n is sharply peaked for frequencies to >^ U) and angles of propagation 9 .< 5° . Its magnitude can be much higher than the peak thermal emissivity. I.! o , = 2.5 3 nr/nb=!0"

2 2 II LO Up/C =3.95xio" CD u2/c2=l.92 u|/c2=2.05x|0H

I O 0,5

n 01 0.600 0.605 0.610 0.615 c Fig. 1. Plot of the normalized etnissivity due to the runaway electrons as a function of a) for various values of 8. RADIATION PROCESSES IN PLASMAS 57

It is important to recognize that the computation is based on a conserv- ative estimate of i^/n^ .

In Fig. 2 we illustrate the overall dependence of the emission on both frequency and angle of propagation. We consider the same parameters used in Fig. 1 and plot contours of constant n wersus u) and 3 ,

The shaded region in the upper portion of the figure denotes the region

in which n cos 8 S 1 and no Cerenkov resonance is possible. In order

to include the effects of cyclotron absorption, we exclude the regime

in which the resonance velocity for cyclotron damping |vn| = |w - fi \

/|k.,| { 3v, . The sharpness of the peak in the emissivity for frequ-

encies (u "V a) and small 6 is evident from the figure, where every

other contour represents an order of magnitude interval.

In order to determine the range of frequencies of the enhanced

emission, we numerically integrate the emissivity over dR, ( = 2 us in 3 d8).

The results of the integration are shown in Fig. 3 in which we plot

dP/dü)(= J dilri) versus m for (a) B. = 40 kG , w = 0.6 !i ,

n^i^ = 10"3 , u2/c2 = 0.205, and u2/c2 = 1.92, 4.36, and 7.74 (which

corresponds to average parallel runaway energies of 200, 400, and 600

3 KeV respectively), and (b) BQ = 60 kG , (ofi = 0.4 Çl& , nr/nb = 10" , u_/c = 0.205, and u /c » 3.03, 7.74, and 13.25 (which correspond

to average parallel energies of 400, 600, and 900 KeV respectively.)

Note that, for given U) , the chosen bounds on the integration are

those shown in Fig. 2. Examination of Eq. (29) shows that rf increases

sharply with decreasin frequency until n_cos 9 S 1 , at which point

the Cerenkov resonance is no longer possible. This is the source of

the sharp peak in the intensity shown in Fig. 3 for w > ID . There

is no emission at the plasma frequency (for which n cos Q < 1) , and

the sharp maximum in the figure does not imply the presence of a in

n 0J 0.3 CO üi/fla

3 2 2 Fig. 2. Graph of contours of constant n versus U) and 9 for u) =0.6 Í2 , BQ = 40 kG, nr = 10" , Uj/c = 1.92, and Uj/c = 0.205. for oi > ü) ana small 6 is clearly shown, Every other contour represents an order of magnitude, and the sharp peak e RADIATION PROCESSES IN PLASMAS 59 0.22 ü)e/íie=0.6

B0=40kG 3 nr/nb=!0" u|/c2=0.205

e Fig. 3a. Graph of the spontaneously emitted power per unit frequency and volume versus (i). 60 CS. Wu

0.14

cüe/ÍIe=0.4

0.12 B0=60 kG 3 nr/nb=!0"

— 0.10 IO 'E 0.08

i 0.06 3 1 -a 0.04

0.02

0.4 0.5 0.6 0.7 0.8

Fig. 3b. Graph of the spontaneously emitted power per unit frequency and volume versus ID. RADIATION PROCESSES IN PLASMAS 61 singularity in the intensity. The initial decrease in dP/dü) as the frequency increases relative to the plasma frequency is indicative of the rapid decline of r£ with u( >, ü>e) . However, for the parameters chosen in the calculation, the slope of dP/dcu is reversed for 0.68 fi 4- (D < 0.79 Q because in this frequency regime the range of 8 over which radiation can occur increases faster than the magnitude of nr decreases. Finally, the precipitous decline of dP/du for ül $, 0.8 ft is the result of the inclusion of the effects of cyclotron absorption. Thus, it is evident that substantial emission is possible over a relatively broad range of frequencies. The range of 6 over which substantial emission can occur is illustrated in Fig. 4, for the same parameters chosen previously. In this figure we plot dP/dfi. (« / dun) versus 6 , and note that, as before, the bounds on the integration are those indicated in Fig. 2. It is clear from the figure that while ry_ reaches a maximum for small 8 , dP/díí. is, initially, an increasing function of 6 . For the parameters chosen, the peak in the emission occurs for 8 i 27° . At higher 6 , dP/dil decreases sharply due to the effects of cyclotron absorption, and emission is expected primarily for 8 .£ 35s. We remark again, that no emission occurs for 8 - 0". To sumarize the preceding discussion we conclude that the presence of plasma effects can result in the emission near the electron plasma frequeny. In the classical approach which only accounts the emission of radiation from individual particles cannot predict this radiation process. The plasma effects which affect the emissivity may be briefly described as follows: 62 CS. Wu

0.16 r

Fig. 4a. Graph of the spontaneously radiated power per unit volume and solid angle subtended by Jc versus 6. RADIATION PROCESSES IN PLASMAS 63

Fig. 4b. Graph of the spontaneously radiated power per unit volume and solid angle subtended by Jj versus 9. 54 CS. Wu

(1) First the index of refraction, n , is no longer of unity. In the Appleton-Hartree approximation, n is a function of frequency u> and propagation angle 8 . Because n may be greater than unity, Cerenkov emission becomes possible. (2) The finite electron density gives rise to the well known dielectric polarization. Consequently the emitting electrons are effectively "dressed". This process in turn can affect the radiation field, as can be seen in Eq. (6). In short, it is not difficult to see that the consideration of various plasma effects can give rise to significant modification of the theory of the radiation processes associated with energetic electrons. The emission near the plasma frequency illustrated in this section merely represents one of the newly discovered radiation processes which are peculiar to high temperature plasmas produced in some laboratory experiments. RADIATION PROCESSES IN PLASMAS 65

REFERENCES

1. G. Bekefl. Radiation Processes in Plasmas, (Wiley, New York, 1966).

2. Yu. L. Klimontovich, The Statistical Theory of Non-Equilibrium

Processes in a Plasma, (MIT Press, Cambridge, Mass. 1967), Sec. 14.

3. Ref. 2. Chapter 2.

4. A-G. Sitenko, Electromagnetic Fluctuations in Plasmas, (Academic

Press, New York, 1967). Chapter 2.

5. H.P. Freund and C.S. Wu, Phys. Fluids 20, 963 (1977).

6. N.A. Krall, and A.W. Trivelpiece, Principles of Plasma Physics,

(McGraw-Hill, New York, 1973). Chapter 4.

7. D.B. Melrose, Astrophysics.Space Science TL, 171 (1968).

8. K. Audenaerde-, Plasma Phys. 19, 299 (1977).

9. A.E. Costley, and T.F.R. Group, Phys. Rev. Lett. 3f±, 1477 (1977).

10. I.H. Hutchlnson and D.S. Kenan, Nucl. Fusion J7> 1077 (1977).

11. P. Brossier and G. Ramponi, in International Topical Conference

on Synchrotron Radiation and Runaway Electrons in Tokamaks.

(University of Maryland, College Park, Md. 1977), paper B4.

12. H.P. Freund, L.C. Lee, and C.S. Wu, Phys. Rev. Lett. ^0, 1563(1978).

13. H.P. Freund, C.S. Hi, L.C. Lee, and D. Dillenburg, Phys. Fluids ^i.

1502 (1978).

14. H.P. Freund, L.C. Lee, and C.S. Wu, Phys. Fluids (to appear in

1980). 66 c-s- Wu

II, DIRECT AMPLIFICATION OF RADIATION NEAR ELECTRON CYCLOTRON

FREQUENCY BY CYCLOTRON MASER INSTABILITY

II.1 Introduction

It is «ell known that radiation processes nay be divided into

two categories; namely spontaneous and induced radiations. The

former is «ell discussed in classical electromagnetic theory. How-

ever, most of the early discussions are concerned with spontaneous

radiation from single particles. Until recently collective effects

have been ignored in the theory. An approach which takes account

of the plasma effects is already presented in the preceding section.

Hereafter let us turn our attention to some induced radiation pro-

cesses .

First of all, one may ask what is an induced radiation process.

To understand this, let us consider the situation In which initially

a low level of radiation is present in a plasma (or other medium).

Due to the interaction of the radiation field and the medium, the

radiation field can get amplified under certain conditions. Consequently

enhanced radiation can be produced. This is what we call induced

radiation processes, although the specific amplification mechanism

may be different from caae to case. In the present series of talks

we discuss two different induced radiation processes. First, let us

consider the cyclotron maser instability. Before going further, we

think that a few introductory remarks may be appropriate.

In general induced radiation processes are much more intense

than radiation generated by spontaneous processes. Strong evidence

acquired from astrophysical observations indicates convincingly that

induced emissions occur in various regions in space. The auroral RADIATION PROCESSES IN PLASMAS 67

1 2 kilometric radiation of the earth, the Jovian decaraetric radiation, and a variety of solar radio emissions represent a few of the well- known examples. On the other hand, important applications of induced radiation processes to laboratory experiments have also been devised. 4 5 The gyrotxon (or cyclotron maser) and free electron laser experiments are highly representative.

Basic concept of Induced emission is first discussed in the 6 7 8 early works by Twlss, Schneider, and Gapanov. Review of some 9 of the early works is given by Be kef i. Since then there have been many recent developments. Among the theoretical discussions, several processes of considerable Interest are the cyclotron maser instability and the stimulated scattering. The former gives rise to the direct amplification of electromagnetic waves, whereas the latter results in parametric amplification of radiation. In this section we first discuss the cyclotron maser instability. The theories of cyclotron maser instability are presented in many publications. References ' ~ are some of the recent publications

from which the readers may easily find other relevant works. However, it is appropriate to remark at this point that most of these theories are concerned with the experiments of gyrotron. The beaming electrons are considered to have very small momentum spread so that the unperturbed electron distribution function may be approximated by the following form.

Of course, the approximation of zero momentun spread is not always

justified. For example, if we consider the caw of parallel propagation, 68 CS. Wu

It Is already seen that the approximation requires

» kAv![-= w^Av /c , (2)

where on is the growth rate, Av^ denotes the velocity spread along

BQ , to is the real frequency, and c is the speed of light. Evidently,

if u./u ^ 10~ , we musmust require

Av,i f « 10 L (3)

This condition can be easily violated in many astrophysical cases when the electrons of interest have a temperature greater than several KeV. Furthermore, these previous theories are often developed for the particular geometric configuration and specific modes in the waveguide of interest In the gyrotron experiment. Consequently these discussions are usually not applicable to astrophysical problems. For this reason in the following we present an analysis which may be more interesting to those readers who are not particularly concerned with the gyrotron experiment but are interested in the general physical process of in-

duced emission by direct amplification.

11.2 Physical Model and Dispersion Equation

The physical situation under consideration in this paper is that the plasma consists of two tenuous components: a cold background plasma and a population of energetic electrons which are weakly relativistic, with energies ranging from 5 KeV to 100 KeV. The background electrons are Important in regard to the determination of the dispersion relation and the wave propagation characteristics, whereas the energetic electrons dictate the damping or excitation of the radiation. RADIATION PROCESSES IN PLASMAS 69

Before going further, we first consider a coordinate system such

that the ambient magnetic field JJ_ is parallel to the z axis,

Furthermore, the wave vector £ of the radiation is designated as

k • It ê + k ê where e and e are unit vectors. If we denote " X Z Z X Z

the plasma frequency of the background electrons by u g and the rest mass gyrofrequency by ti (>0) , the dispersion equation can be

readily obtained from the linearized Vlasov equation as follows

2,2 to _ c k» _ P 2 at -SI -Q

2.2 DC&w) - 0 2 yy ÜJ -i •Ç.

2,2 2 c k, w l - ^zx 2 2 (i> 01 (4)

Here, j^ represents the contribution from the suprathermal electrons

2 +=> <= ÜH. f dr, f dv, ^(v, 3Í- - v,( ^-

2 -K» 00 !k „, f iL JL ^!!( a . a dVt + v 3v,/ e U2 J II J iy[ 3v, Y V ~ 3v,| " 3v_,

(s) •K» I (5) 70 CS. Wu

v 2 2 -)í 2 2 where Y = (1 - v /c ) , u " 4irn e /m , F is the unperturbed pss s s 6 distribution function of the superthermal electrons, and the tensor £^ is defined as

«i 2T,2 lv«Wi (6)

2T2

where J^ = J^Cb) and b

The density of the supra thermal electrons, n , is considered to be smaller than that of the background electrons, n . However, it is assumed that the gyrofrequency Si is larger than the plasma frequencies u and m , t. e., Í1 > u > ui .In the pe ps e pe ps following we shall use the approximation that in calculating u , the contribution of the supra thermal electrons to the dispersion equation may be neglected. Consequently, if we define the index of refraction, N » kc/d) , the governing equation of N is well known and may be written as

AH4 - BN2 + C - 0

2 2 where A » E sin 8 + T\ cos 9 B = - g2sln28 + en + e(e sin29 + n cos29)

2 2 c - n(e - g ) RADIATION PROCESSES IN PLASMAS 71

2 2 g»u n/w(« -n) pe e r r e 2 2 pe' r and 6 denotes the angle between £ and ^. . From Eq. (7), It is seen that

, B + (B2 - h l 2Ã

Solutions with subscripts "+" or "-" correspond to the ordinary node or extraordinary mode respectively. For the extraordinary mode, 2 2 ^ the upper branch has a cutoff frequency u »i2(l+2w/fi)". x e pe e In the following we assume that the distribution function F of the suprathermal electrons can be represented by F_(y) " F (v,)

Fe||(v|() where

Fexa TTBTO!

The index a in Eq. (9) characterizes the width of F or velocity

spread in v_, space. For a given temperature Tj_ , it can be

readily shown that

2T, B2? -

and the distribution function F peaks at BL(1 + «) v Q . As

a +« we see 72 C.S. Wu

6(v, - v )

However, in the "ollowlng discussion, we shall limit the index

II.3 Growth Rate and Numerical Results Since the population of suprathermal electrons is tenuous, tht; dispersion equation D0j,u) » 0 may be approximated by neglecting terms proportional to or smaller than n . Thus, if we postulate that the growth rate u os small in comparison with the frequency u , the growth rate may be approximately expressed as

Im US. » 1- (11)

After some algebraic manipulation, one obtains from Eq. (4),

Im D - ^Im <4 + *yy Im

where

^ =» N4 sin29 - N2(e sin26 + n) + en

ij/ » - N2(e sin2e + n cos28) + en

*zz - N4 cos28 - N2 e(l + co£26) + e2 - g2 K, ' 2N2(N2 - e)sin 8 cos 8) AS 2 2 *TO - 2g(n - N sin 8)

2 * 2 * 2N 8 sin 0 cos 9 CS. Wu 73

S In the calculation of Ira , Im Q , etc., we can use the ap- yy proxlmatlon

where P denotes the principal value. For weakly relativistic electrons, we may approximate y by

Moreover, in Re D we nay ignore the contribution from the supra-

thermal electrons. Itsis found that

9 2 2 2 2 •Vãèr-^MP-^l-^r^UMOsin 9 - u cos 8

in2(u2 - SI2) u2n2sin28 \ e (13) (% Q2)2 - *- *Jt fl2)2)

+ 2 r'\ re' rre

To evaluate the growth rate by numerical method we first define several basic quantities

i 2y\*K*

r Í d3v 74 CS. Wu

where v is a streaming velocity as shown in the distribution function S given by Eq. (10).

In the following, we consider the situation »,»<». This

choice is mainly for simplicity. In order to investigate the effect due

to the variation of F on the instability, we also choose 6

so that as a. •*•">, F approaches that described by Eq. (1). How- ever, we remark that for very large a , the method which we have employed in calculating w, becomes inaccurate because it implicitly

assumes

wi j

Obviously, when B)t is sufficiently small, condition (16) may be violated. We shall adopt a more general approach to this problem later. In the present discussion, let us restrict our investigation to the case in which a is not too large, say liai 100. In the following, we discuss the numerical results. We first consider the fast extraordinary mode (or the x-mode) and the ordinary mode (or the 0-mode). These two modes are normally escapable from a high field region to a low field region and therefore are of interest in the study of radiation processes in astrophysical problems. Next, we will consider the fast extraordinary mode (or the z-mode). Although this mode in general cannot escape from a source region, the trapped radiation may play significant roles in the plasma heating and other transport processes. Since the argument of the Bessel function RADIATION PROCESSES IN PLASMAS 75

J.(b) In Eq. (6) is on the order of o(v/c), which is small, the I » 1 tens is moat important in Eq. (5). In the following calculation, only the

Jl - 1 term will be retained.

(1) The Fast Extraordinary Mode and the Ordinary Mode

First let us study emissions near the fundamental of the electron

cyclotron harmonics. The first set of parameters which we choose is

that Gj. • 20 KeV, £g - 20 KeV, and %/^& - 0.2. The growth rate

is computed for various values of a . The results are presented in

Fig. 1 in which we plot the maximum growth rate ai of tha x-mode x f ni3X or (a + 1) . It is seen that as ct increases the

growth rate u. also increases. The corresponding values of k||/k

are also given in the figure.

In Fig. 2, we plot (i). of the x-mode versus w /ft . It is i,max e e interesting to notice that a peaks at to /ft ^ 0.2. For large XjUlSX 6 IS u /ft , the cutoff frequency becomes significantly high above the gyro-

frequency ft the resonant parallel velocity v() must be large. The

population of the resonant electrons diminishes and so does the growth

rate. On the other hand, since the maximum growth rate is normalized

n n ic such that (u m]t/Üe) ( e/ a) . diminishes as ^e/\ * 0 .

We now choose to examine in detail, the case in which (J /Í2 =0.2

and a » 9 (the corresponding

(o for both the x and 0-mode are plotted as functions of u /Q . i,max r re The results are shown in Figure 3. The x-mode growth rate vanishes as

the frequency to approaches the cutoff frequency because the index

of refraction diminishes. This implies again that the resonant elec-

trons must have exceedingly large v(. and consequently their popu-

lation vanishes. As expected, the growth rate of the ordinary mode

is much smaller than that of the extraordinary mode. Thus the 0-mode

is less interesting. 76 CS. Wu 10r\

£ =20keV ± 0.9 f =20 keV '

10 - Growth rate 0.8 c (X-mode) X 0.7

2 -3 2 10 h kn/k 0.6

0.5

10 0.01 0.025 0.05 0.1 0.25 0.5

Fig. I. The maximum growth rate u. of the X-mode is plotted as a function _i i,max of ou Ic or (a +1) , which is related to the velocity spread in the energetic electron distribution. The corresponding values of k, /k are also plotted. The maximum growth rate occurs at the real frequency cor 1.06 Í2 . The plasma parameters chosen are:

«e/fie = 0.2, - <5g 20 keV. HADIATION PROCESSES IN PLASMAS 77

= 20 keV v-2 10 = 20 keV 6,,= 2 keV •a- 9 05 C

C IO

.E 4 10

5 1<5 0.3 0.2 0.3 0.4 0.5 cue/ne

Fig. 2. The maximum growth rate u^ of the X-mode and the corresponding value of k||/k are plotted as a function of the frequency ratio ue'^e* The Plasma Parameters used are: S> = 20 keV, <§ = 20 keV, ay =2 keV and a = 9. The maximum growth rate peaks at ID in - o.2. 78 CS. Wu

cue/ae=o.2 £±=20keV 1.0

0.9 tf) Growth 3 rate 1, IO 0.8 c (X-mode)

03 0.7

a -4 £10

Growth rate (0-mode) 105 0.4

I 4.0 1.05 1.10 1.15 1-20 1.25 1.30

Fig. 3. The maximum growth rates fc). and the corresponding k./k for both the X- and 0-modes are plotted as functions of u /ft for u /í2 = 0-2, & lê - 20 keV, &.. -2 keV, and a = 9. The growth rate of the X-mode peaks at in = 1.06 íí . re RADIATION PROCESSES IN PLASMAS 79

In Fig. 4, we fix G>e/8 - 0.2 , a - 9 , and - Q.I and

plot oij max for the x-mode as a function of . The growth

rate peaks at about 20 KeV. This feature is Insensitive to the index a chosen in the calculation. To show this, a similar plot is shown in Fig. 5 in which a » 3 is chosen. The growth rate again peaks at abour 20 KeV, although the maximum growth rate is slightly smaller than that shown in the preceding figure. For small 6, , say Ç 1 KeV, the growth rate is insignificant because the population of the resonant electrons, which can contribute to positive w. is small. Nevertheless, as », increases, the gradient of F in v, space decreases. This may explain why the maximum growth rate decreases at sufficiently high £ 2 2 values of », . Of course, approximation y - 1 + v /2c Imposed in

our analysis is inaccurate when (£>, :>100 KeV.

a The second set of parameters chosen for numerical computation is ? fir * 6 KeV and a - 3 . In Figs. 6 and 7, we plot to —' S 1 m /ft (for

fe, »20 K , except that the maximum growth rate for the x-mode de-

creases rapidly as to /ft exceeds 0.2.

(2) The Slow Extraordinary Mode

For the slow extraordinary mode, or the z-mode, wave frequencies

are below the upper hybrid frequency. We consider the case in which

i • Ga 6 KeV , £|| - 3, a - 1 , and 0.5. The maximum growth rate u and the corresponding frequency u are plotted as functions of k../k in Fig. 8. It is seen that US. peaks at k.i/ k - - 0.8. ii x t H13X

In Fig. 9, we plot u. versus oi /Í2 for kfJV. » - 0.8 and find that

the maximum growth rate occurs at ai /ÍÍ » 0.8.

Clearly, the growth rate d> for the z-oode is greater than i,inax 80 C.S. Wu

1 t i

0JA /ílp =0.2 W - W £II/6S = 0.1

-2 \0 -1.0 c / ^^ 0.9 c / Growth rate / "* -3 / (X-mode) / 0.8 ^1Ç3 0 / / ^< \ / O \ v / 0.7 3" -0.6

i i 0.5 73 10 30 100 £, ;6 (keV)

Fig. 4. The maximum growth rate us. and the corresponding k./k.for the Í )in3x II X-mode are plotted as a function of the particle streaming energy &_(.& = ê ). The plasma parameters chosen are: u /Si = 0.2, S S J. 6 6 én/é • 0.1 and a • 9, The growth rate peaks at & c 20 keV. it S X RADIATION PROCESSES IN PLASMAS 81

to c

10 30 100 &i;£s(keV)

Fig. 5. The maximum growtg h ratt e ÜJ.. as a function of the particle l iHRx energy &s(é>s - &±) for é,,/é » 0.25 and a = 3. 82 CS. Wu

1.0 1.05 LIO 1.15 1.20 1.25

Fig. 6. The maximum growth rates ü). and the corresponding k,/k for both i,max II the X- and 0-modes are plotted as functions of u /B . The plasma parameters chosen are: ia /ti » 0.2, ê = & = õ keV, ».. =1.5 keV CG X S || and a » 3. RADIATION PROCESSES IN PLASMAS 83

:i £ Growth Rate 1=6keV 8 10-2- s=6keV a=3

10^ c

10-5

0 0. 0-5

Fig. 7. The maximum growth rates u. and the corresponding V../V. for both ivmax II the X- and 0-modes are plotted as a function of the frequency ratio We^e* The plasma Parameters used are: è' - é =6 keV, en -1.5 keV and a - 3. 84 CS. Wu

-10 1.1 £s =6keV Eu =3keV

Growth rate 0.7

-4 l 1 10 -1.0 -0.6 -0.2 0.2 0.6 1.0 k,,/k

Fig. 8. The maximum growth rate u. and the corresponding wave frequency OJ for the Z-mode are plotted as a function of ^i/k- The plasma parameters chosen are: w Iti = 0.5, é = & = 6 keV, c e e ' l s "% &.. =3 keV and o » 1. The growth rate peaks at k,,/k >v -0.8. RADIATION PROCESSES IN PLASMAS

1 w I I I Ci-/p / A Z/ Ci — vJ 0 6, =6keV en 10 -• f\ _L ~" c \ £s=6keV \ £,, = 3keV C 1 MM* \ d - \ ~~ \ k,,/k = -0.8 10H \ —

102

3 1Õ —

I I I I \ 0.75 0.8 0.85 0.90.95 1.0

Fig. 9. The growth rate to. or the Z-mode is plotted as a function of the wave frequency u^/ti^ for k,/k - - 0.8. The plasma parameters chosen are: u /fi = 0.5, §, = é • 6 keV, é,, =3 keV and a = i. 86 C.S. Wu that of che x-mode. Here ic is necessary to remark that in calculating

_j. we have used a method which assumes u. « ?. . Thus, for the peak

value (x./f. ) (n /ns) = 10, the method is justifiable only if n /n « 1. s e The z-mode in general has a large group velocity (comparable

to the speed of light), whereas the x-mode may have a much smaller group velocity when the frequency u approaches the cutoff frequency.

The group velocity of the x-mode is shown in Fig. 10 for various values of ID /ÍJ . Thus the effective amplification distance of the x-mode

can be much smaller than that of the z-mode, if the growth rates are

the same for the two modes.

The main results can be summarized as follows:

(1) Both the ordinary and extraordinary modes can be excited by the

suprathermal electrons with a loss-cone distribution. The growth

rate for the extraordinary mode is significantly higher than that

for the ordinary mode.

(2) The velocity spread in the distribution function of the suprathermal

electrons may reduce the growth rate of the x-mode by a factor

of "u 10 from that in the monoenergetic beam case.

(3) The growth rate of the x-mode peaks at electron energies ^ 20 KeV

and at u IS. *\J 0.2. The maximum growth rate fo the X-mode occurs 2 2 k near the cutoff frequency u « (Í2 + 2 u ) .

(4) The maximum growth rate for the z-mode occurs when waves propagating

in a direction antiparallel to the streaming velocity of the

energetic electrons (kjj/k. ^ -0.8) ; the maximum growth rate

for tha x-mode occurs when waves propagating in a direction paral-

lel to the electron streaming velocity (k||/k ^ 0.6).

The discussion presented in Sec. II.2 and II.3 are based on a recent

work by lee and Wu which reoresents a continuation of an earlier work

by Freund and Wu. RADIATION PROCESSES IN PLASMAS °*7

Fig. 10. The growth velocity v of the X-mode is plotted as a function of the wave frequency U) /fi for k./k = 0.5 and u /ft = 0.1, 0.2, 0.3, 0.4, or 0.5. T!

1 j A/1 88 CS. Wu

REFERENCES

1. D.A. Gurnett,-J. Geophys. Res, _79. 4227 (1974).

2. T.D. Carr and S. Gulkis, Ann. Rev. Astro. 7_, 577 (1969).

3. M.R. Kundu, Solar Radio Astronomy, Interscience, New York (1965).

4. Relevant references may be found in Ref. 10 of chis section.

5. Relevant references may be found in Section III.

6. R.Q. Twiss, Austr. J. Phys. _11, 564 (1958).

7. J. Schneider, Phys. Rev. Lett. _2, 505 (1959).

8. A.V. Gapanov, Izv. VÜZ, Radiofizika 2., 450 (1959).

9. G. Bekefi, Radiation Processes in Plasma. (Wiley, New York, 1966).

10. P. Sprangle, V. L. Granatstein, and A. Drobot, J. de Physique,

Supplement ^8, c6-135 (1977).

11. This is to be discussed in Section III.

12. H.S. Uhm, R.C. Davidson and K.R. Chu, Phys. Fluids _21, 1866 (1978).

13. H.S Uhn, R.C. Davidson, and K.R. Chu, Phys. Fluids 21, 1877 (1978).

14. H.S. Uhm and R.C. Davidson, J. Appl. Physics 50_, 696 (1979).

15. D.E. Baldwin, I.B. Bernstein, and M.P.H. Weenink, Advances in Plasma

Physics, Vol. 3 (ed. A. Simon and W.B. Thompson) (Wileym New York)

(1969) p. 32.

16." L.C. Lee and C.S. Wu (to be published).

17. H.P. Freund and C.S. Wu, Phys. Fluids 2Q, 619 (1977).

18. C.S. Wu and L.C. Lee, Astrophys. J. .230, 621 (1979). RADIATION PROCESSES IN PLASMAS 89

III. PARAMETRIC AMPLIFICATION OF RADIATION BY STIMULATED SCATTERING

III.l Basic Concept of Parametric Excitation

Let us first review the basic concept of parametric excitation which is relevant to the theory of stimulated scattering. We consider

two oscillators X(t) and Y(t) which may be described by the

following model equations

2 d X r dX ., 2 2 n, L1X =~T+ 2F1 di+ ("i+'i^ " ° {1> at

L_Y - ^-^ + 2ro ^r- + (u, + vl)Y = 0 (2) "2 dt2 2 dt 2 2 where tu and V denote frequency and damping decrement associated vith

each oscillator, respectively. Evidently they are independent to each other.

Now let us consider the situation in which an external field Z(t) is

present. Z(t) takes the form

Z(t) =• 2ZQ cos wQt (3)

where Z. is a constant. If this external field is coupled to the

oscillators X(t) and Y(t) , Eqs. (1) and (2) are modified to the

following form

L2Y » X2Z(t)X(t) , (5)

where X and X_ are two coupling coefficients.

Let us further assume 90 CS. Wu

r.(t) - X(u) exp(-iü)t) (6) Y(t) » TC(iD)exp(-iut)

From Eqs. (•>), (5) and (6) we obtain

di0) + Y(w - uiQ) j (7)

X((Ü + 2w0) ] (8)

where D (u) - - t»2 - 2iT u + u)2 + I"2 (n - 1,2) . n n n n The term X(u + 2 cu_) in Eq. (8) can be ignored because it is nonresonant. From Eqs. (7) and (8), it is found

1] D2(ui - u>0) J

his is the dispersion equation from which we can discuss the process

of parametric excitation.

Fran Eq. (9) we see that Instability can arrise due to two types

of resonance; (1) D-Cu) » 0 and D2((o - IU ) [or D2(u + u )1 • 0,

and (2) D2((D+ U-) - 0 and D2(ü) - UQ) » 0 . To proceed with the

discussion, we treat D- with the so-called resonant approximation.

Assuming u_ = u2 and iu «-UL. , we obtain

- (u

m + RADIATION PROCESSES IN PLASMAS 91

6 = uQ - w2 . (10)

Thus Eq. (9) can be rewritten as

If we consider F, • ?, • 0 , Eq. (11) reduces to

2 9 (!) - Ü) W2 LW m J

One can now show that Eq. (12) supports complex solutions to » oi + iA. Here we consider the two cases mentioned earlier.

(1) Dx(u) =0 and D2(tu - iüQ) = 0 . In this case we assume w » u, = a)_ - u . Thus from Eq. (12) we find 2

(2) D2(ÜJ - u0) =0 and Djdu + u>0) - 0 but D^^di)) i< 0 In this case we postulate u » iA . It is readily seen that Eq. (12) may be reduced to

2 2 2 2 ^1^2Zfl^ - (5 + A ) (u^ + A ) -

which implies that the assumed form of solution is indeed compatible, 92 CS. Wu

if XjXjjiS < 0 . The above discussion illustrates the process of parametric excitation by considering a set of simple model equations for two harmonic -oscillators coupled to an external field. In the absence damping (F - T, - 0) excitations can occur regardless the strength of the external field. If T^ and I"2 f 0 , excitation occurs only when the amplitude of the external field exceeds a certain threshold value. A more comprehensive analysis of the mathematical nature of of parametric instabilities may be found in Ref. 1.

III. 2. Coupling Mechanism and Ponderomotive Force In the preceding section we have described the process of parametric excitation based on a simple mathematical model. Evidently the key point is that there must exist a physical mechanism which can result in the coupling of the external field Z(t) and the two oscillators X(t) and Y(t). In the previous discussion we have postulated that such a coupling process occurs without giving any justification. The purpose of this section is to discuss the physical aspect of the coupling process. To proceed with the discussion, we consider the situation that

a high-frequency electromagnetic wave (u0»JO incident on a plasma. At the same time we assume that the electron density of the plasma can fluctuate at a certain natural frequency, say the electron plasma frequency o> , and wave vector £ . Then a scattered wave (oi ,Jc ) can occur, if the matching conditions

U m 0) + (i) s pe — 0 RADIATION PROCESSES IN PLASMAS 93 are satisfied. As a result of the interaction of the scattered wave and the 2 incident wave, a ponderomotlve force is created. This force is characterized by the low frequency oo and can enhance the electron pe density oscillations. Consequently, it leads to the development of parametric excitation. Through this ponderomotive force the interacting waves become coupled. To illustrate the creation of the ponderomotive force we first define £. and E to be the electric field and %~ and £ to be the magnetic fields of the incident and scattered waves, respectively. The electric fields £Q and J^ can result in oscillatory motions with corresponding velocities v. and v ,

Mjs imüjs

Here implicitly we have assumed that ,vQ and ^ are sufficiently

small so that the relativistic corrections to vQ and v can be ignored. These motions csn give rise to a low frequency force

provided Ug = oi - uiQ and k = ]f. - J^ . Since the incident and scattered waves are electromagnetic, we find 94

Evidently this force is parallel to the low-frequency electrostatic field which can reinforce the original electrostatic oscillations at (w ,k) , Indeed the discussion is so far very qualitative. How- ever, it enables us to understand the physical origin of the instability.

III. 3. Stimulated Raman Scattering The process just mentioned is called the stimulated Raman scattering. In this section a brief theory is presented. Let us consider that the electric field associated with the incident electromagnetic wave can be written as

1 So = 2io cos %'Z - V5 • (1) i •?j where in_ and Jj» satisfy the linear dispersion relation

Í mZ - c2k2 + u2 . (2) o pe "! i In the presence of this field electrons execute oscillatory motion. A characteristic velocity

eE 0 , (3)

where m denotes the electron mass. We now further consider that the situation is perturbed by a low frequency electrostatic wave (w,£1 with w = u) . Since electrostatic waves can give rise to RADIATION PROCESSES IN PLASMAS 95 density perturbation and this perturbation under the influence of the incident wave can further result in currents with characteristic frequency and wave-vector, in + S.IDQ and Jc + HJcQ where the harmonic number SL is an integer. Eventually these currents can generate the so-called side-band modes. If we shall only consider i = 1 , and Raman scattering, the side-band mode is the back-scattered electromagnetic waves with frequency and wave-vector

ui = a) - ;u

*- " * " *0 •

As mentioned in the preceding section, the backscattered wave can, in turn, interact with the incident pump wavu field so that the ponderomotive force is produced to modulate the low frequency density oscillation (id ,^.J . This process consequently leads to parametric amplification. To discuss the instability problem associated with the stimulated Raman scattering, we adopted the following dispersion equation

where

(s = e.i) (5)

2 Z 2 Z 2 2 D± - c kj - iuf + io e = c k + 2%-% c + 2oxo0 - io . (6)

In Eq. (6) we have made use of the dispersion relation [Eq. (2)]. The dispersion equation is readily obtainable from a general form- 96 CS. Wu alism due to Drake et ai. Readers who are interested in the derivation of Eq. (4) may refer to this work.

In the case of backscattering the anti-Stokes component D+ is nonresonanC and therefore can be neglected. On the other hand, the Stokes component D may be written as

D_ = 2u)Q[w - Jj-y - 6] = 0 (7)

2 2 2 where v » JcQc /uQ and 6 = k c /2iu0 For electron plasma oscillations, the quantities x and X, may be approximated as follows

Xt = 0 (8) 2 ID Xe T (ÜI+ iYL) where y denotes a damping decrement (for example v, the landau L> L damping decrement if collisional damping is negligible). Indeed, a more satisfactory expression of x should include the term pro- Z 2 portional to k X^ (XD is the electron Debye length). However,for the present discussion which is presented only for a qualitative illustration, the crude form given by Eq. (9) is sufficient. Making use of the above mentioned approximations we can simplify Eq. (4) to

2 2 (w - upe + ivL) (OJ - Jj-j^ + &) " - sin * k VgW e/4u)Q , (10) RADIATION PROCESSES IN PLASMAS 97 where iji denotes the angle between Jc_ and ^_ . From Eq. (10)

one can show that If he sets w = ui + iy ,

, 2 ,. 2 2 . 2. Y - IY + k v sin $ LT TL o y - > 0 (11)

meaning instability always exists. However, if we assume that the incident wave can experience a collisional damping, 9ay y , then Eq. (XX) is modified to

YL + V ~ [(YL ~ V)2 + k2v Y *-* ^ (12)

In that case, y is positive only when

2 2 2 uo

III. 4. Free Electron Laser The stimulated Raman scattering process discussed in the pre- eding section has a very important practical application. That is, the development of the so-called free electron laser. The possibility of developing lasers in which the active medium is a beam of free electrons has been demonstrated both theoretically ~ and experimentally. The continuous tunability and the high power operation are two of the several main advantages. In short, free electron lasers have profound implication to the technology of high-power microwave generators. The basic notion of the free electron laser is essentially the stimulated back scattering process associated with a re lati v- iatlc electron beam passing through a periodic external magnetic 98 C.S. Wu

field which behaves as a pump wave in Che beam frame. In this case the pump wave is circularly polarized, whereas in. the preceding section the pump wave is considered to be plãne-polarized. In this section we present a brief analysis to illustrate the parametric amplification of coherent electromagnetic waves via the aforementioned system. The physical model is described as follows. First, there is an externally Imposed static magnetic field which has the form

cos V + % sin ko2)

Of course, it only represents an approximate form since it is not

really curl free. If we consider that a coherent relativistic elec-

tron beam passing through the rippled magnetic field, the zeroth

order electron momentum is

V 4+ Sl n k0Z

2 2 2 2 So far we have Implicitly assumed k c » u /y [where ID » un p nu p 2 2 2 2 !<

4itnQe /m and YQ • (1 + PQ/ni c ) ] so that the self-magnetic field is negligible. The pondercDotive force can be calculated as

B0 3in k0z " Vo C0S RADIATION PROCESSES IN PLASMAS 99

k0z " PyBO cos V

Hereafter we shall consider a one-dimensional problem and express the perturbations of magnetic field <5B and 6B in terns of the vector * y potential A such that

and

From Eqs. (3) and (4) we can write the z-component of the equation

of motion as

where

eB / 3A 3A O ^ V08 k°z "^ " Sin k°- ^ )J *

The electrostatic field E is associated with the density perturb-

ation 6n and is described by the Poisson's equation

3Ez •^5- - - 4m5ne . (7)

Furthermore, 6n and p are related by the continuity equation

(8)

It is also evident that 100 C'S- Wu

p -5. A (9)

p rnS- y c

The vector potentials A and A and the momenta p^ ard p are

in turn determined by the field equatiore

4™ne r /*„ POzP2 \ ~!

mY°C (11)

It Is apparent that if we consider Sn = 0 and p =0 Eqs. (9) - (11) describe primarily the propagation of an electromagnetic perturbation, whereas Eqs. (3) - (8) determine the dispersion equation of an electrostatic perturbation, if the ponderomotive force is ignored. In the presence of F and the rippled magnetic field the two perturbations become coupled and thereby eventually parametric excitation can occur under certain conditions.

to proceed with our discussion hereafter let us look for solutions of the following form:

5G =• I $Gn exp [i(k+ Me )a - lot] . (12) J, p L

However, we shall only consider Í. » + 1 . From Eqs. (9) - (11) we find

(13) 2 2 where UJ =» eB./mc and m is the electron mass, and OJ * 4im_e /m. D U D U RADIATION PROCESSES IN PLASMAS 101

In obtaining Eq. (13) we have made use of Eq. (8). We now go back to Eqs. (5) and (6) and consider backscattering. After some algebraic manipulation it is obtained that

{'-

Equation (14) describes the coupling of the electrostatic mode with the electromagnetic wave due to the presence of the ripple magnetic

field, ^ . If we assume that it is possible to find a real frequency u such that

then we may look for a solution of Eq. (14) by writing ID » u + 6w. Assuming in » fiu we obtain

6co - + 4 -x -i 5 p- (16)

Of course, in general the beaming electrons may have finite velocity spread which can result in Landau damping and thermal correction to the dispersion relation. The electrimagnetic wave may subject to collisional damping. Furthermore, if condition (15) is not satisfied, there is a frequency mismatch. All these factors can affect the growth rate and instability can occur only if the amplitude B. exceeds a certain threshold value which depends upon the real situation. The preceding analysis follows from a more general theory given in Ref. 12 in which a uniform guide magnetic field is included. 102 C"S

Kwan and Dawson furthermore consider some other effects which are omitted here. Results from computer simulation are also presented in. Réf. 12. The above discussion illustrates the basic principle of free electron laser. We see how a relativistic beam passing through a rippled magnetic field can result in parametric amplification of radiation. This induced radiation represents another new and important emission process which has eixeedingly valuable applications to the development of high-power microwave generator.

REFERENCES

1. K. Nishikawa and C.S. Liu; Advances in Plasma Physics, Vol. 6,

(ed. by A. Simon and W.B. Thompson) Wiley, New York) p. 3 (1976)

2. Â simple physical description of the ponderomotive force may be

found in Introduction to Plasma Physics by F.F. Chen, Plemm Press,

(New York, 1974) p. 256

3. D.F. DuBois and M.V. Goldman, Phys. Fluids ÍS, 1404 (1965)

4. J. Drake, P.K. Kaw, Y.C. Lee, G. Schmidt, C.S. Liu, and M.N.

Rosenbluth, Phys. Fluids .17, 778 (1974)

5. W. Manheimer and E. Ott, Phys. Fluids 17_, 1413 (1974

6. P. Sprangle and V.L. Granatstein, Appl. Phys. Lett. 25_, 377 (1974)

7. P. Sprangle, V.L. Granatstein, and L. Baker, Phys. Rev. A12, 1697

(1975)

8. V.I. Miroshnichenko, Soviet Tech. Fis. Lett. 1, 453 (1975)

9. N.A. Krall and W.A. Mcttullin, Phys. Rev. A17_, 300 (1978)

10. T. Kwan, J.M. Dawson, and A.T. Un, Phys. Fluids 2P_. 581 (1977)

11. P. Sprangle, R.A. Smith and V.L. Granatstein, NRL Memo Report 3888

(1978) RADIATION PROCESSES IN PLASMAS 103

12. T. Kwan and J.M. Dawson, Phys. Fluid 22, 1089 (1979) 13. P.C. Efthimion and S.P. Schlesinger, Phys. Rev. A15, 633 (1977) 14.~ T.C. Marshall, S. Talmadge, and P.C. Efthimion, App. Phys. Lett. J31, ~ 320 (1977) 15. L.R. Elias et ai. Phys. Rev. Lett. j£, 717 (1976) 16. D.A.G. Deacon et al. Phys. Rev. Lett. .38., 897 (1977) 17. Y. Camel and J.A. Nation, J. Appl. Phys. 44, 5268 (1973)

CLOSING REMARKS

We have discussed three cases to illustrate some of the new

radiation processes which can occur in plasmas. It is necessary to

reiterate that the present review is by no means of exhaustive.

Furthermore, there exist many other emission processes which are not

even mentioned in the present series of talks. For example, interactions of large amplitude electrostatic waves can result in emissions of electromagnetic waves via nonlinear processes. This subject has been discussed extensively by V.N. Tsytovich. However, most of the radiation processes discussed in that article are not as efficient as the ones discussed in the present talks.

V.N. Tsytovich, Plasma Radiation Mechanisms in Astrophysics, Soviet Physics - Uspekhi, 12_, 42 (1969). 105

MAGNETOHYDRODYNAMICS AND STABILITY THEORY

P.H.Sakanaka Instituto de Física, Universidade Estadual de Campinas 13,100 Campinas, SP - BRASIL

ABSTRACT

The Energy Principle far the stability analysis of magnetized plasmas

is presented,in U'lia lecLui-e. IIQUJ. Firstly, a complete developnent of the energy principle far the ideal magnetohydrodynanics is given, with an example for sharp boundary plasmas. Secondly o-stability concept is introduced, and lastly, the energy principle far p. dissipative fluids and plasmas is given.

I. MaGNEKraDRODflEMIC EQUATIONS

An assanbly of ionized atcms is called plaana. The statistical description of plasma is given in terms of many-body distribution function which obeys the Liouville equation. This description may be reduced to one-body distribution function which obeys the Boltzmann equation if the so called plaana parameter, g, is small [13. The plasma parameter, g, is

defined as: g = nk- , where n is the density of electrons, X_ = vQ/u> is the Debye length, v. is the electron thermal velocity and w is the plasma frequency, therefore g may be interpreted as the inverse of number of particles in a Debye sphere.

The kinetic description of plasma may be reduced to multiple fluid description by taking moments of the Boltanann equation and calculating the higher moments of the distribution function and collision terms by Chapman-Enskog approximation [2,3].

This may further be reduoed to one fluid description with the appro- xtaations: (a) quasi-neutrality, | p — p. | « p , where p . are electron or ion 6 1 6 Qf X P. Sakanaka charge density; (b) anall mass ratio, m « m.; (c) kw-frequency, IÜJ «c, wherecis the speed of light [4,5], L the characteristic length, and u angular frequency. lhe resulting set of equations is know as Resistive MHD Equations:

|| + V-pjj = 0 , (continuity) (1-1)

p it t + p)i'V)i + vP + 7*;Ç+P9-§JxJ-l = 0 , faction) (1-2)

||| + | jj-Vp + | pV-jj = nj2 , (heat balance) (1-3)

i-^£ + VxjÇ=O , (Faraday) (1-4)

"c" í= V x 8 ' (Ãnplre) (1-5)

E= j--uxB. (Ohm) (1-6)

where, p(jj,t) is the mass density, ]^(}j,t) the fluid velocity p(jj,t) the total pressure, jr the traceless pressure tensor, g the acceleration of gravity, i(x,t) the current density, the magnetic field, Jg(;ç,t) the electric field and n the plasma resistivity. This is given by n = 3.80 x 103 ZSLnA/T2^2 (Ohm-cm), where UnA ranges between 7 to 22 for the widest possible existing plasma conditions [13 and T is the electron temperature given in degrees Kelvin [6], and Z the degree of the ionization. The Gaussian system of unit is used throughout this text. Initially the following condition need to be satisfied

• jg = 0 (1-7) MAGNETOHYDRODYNAMICS AND STABILITY 107

This set of aquations is generally valid for low temperatures. For intermediate temperatures the so called Magnetic Reynolds Number, 1^ 5 4jrLu/nc , is much larger than one, and, as aconsequence, the resitive terms in equations (1-3) and (1-6) become negligible. Equation (1-3) becomes simply the adiabatic equation of state. The resulting set is known as Ideal MHD Equations:

- + V . pu = O , (1-8)

pp~Y = O , (I-1O)

V • B = O . (1-14)

5 where y =

ih' v 108 P. Sak?"?. a

II. ENERGY PRINCIPLE FOR THE IDEAL MAGNETOHYDRODYNAMICS A. LINEARIZED IDEAL MHD EQUATIONS FORCE-OPERATOR EQUATIONS

Consider a magnetized plasma which is in a metal chamber separated from the wall by a magnetic field. It is assumed that this plasma is described by the ideal MHD equations. The question posed here is: how can one study the stability of such magnetically confined plasmas?

There are three basic methos used to determine the stability of a plasma: The intuitive method, the energy principle and the normal mode analysis.

In the intuitive method one considers the forces which arise when a perturbation is made on an equilibrium state. If the forces enhance the perturbation, then, the perturbation grows and there is an instability. In the energy principle one calculates the variation in the potential energy of the plasma, SW, due to the perturbation. If 6W is negative it means that the plasma can attain a lower energy state, and as a result, the perturbation grows. In the normal mode analysis the time dependence of the perturbation is Fourier analysed in terms of e lutl The perturbation equation is written as a boundary value problem and one solves for the eigenvalue

un" If un has a ne9ative imaginary part, the systems is unstable.

This note treat only the energy principle [7,8,9, MAGNETOHYDRODYNAMICS AND STABILITY 109

A stability problem is separated in three parts: the equilibrium, the perturbation equation, and the solution of the problem. In any stability analysis one assumes the existence of an equilibrium, the proof of which may be, in some cases, very difficult. Consider the ideal MHD equations (1-8) to (1-14), rewritten here

f| + Vp^u = 0 , (II-l)

PA U + pU'Vu +'P-|jxB=0 , (II-2)

^ pp"Y = 0 , (II-3)

±H i = V x B (II-5)

and v-B = 0 . (II-7)

To obtain the equilibrium equations one assumes that the solutions are time independent, -^ ( ) = 0 and, E = u =0. Here the subscript ( ) represents the equilibrium solutions. From the above set of equations one gets the equilibrium equations:

I 3o x So = V*o ' t11"8* T- ío = V x So i11-9»

and v-j=$o = 0 (11-10) P. Sakanaka

Combining (H-8) and (II-9) one has the equilibrium equation,

p = 1 (7 x BQ) x BQ (11-11) where B is subjected to equation (11-10) . Next, one considers the perturbation of the equilibrium quantities

p(r,t) = pf(r) + Px(r,t) , (11-12)

j(r,t) = ;jo(r) + Jx(r,t) , (11-16) and

E(.r,t) = EQ(r) + E^r^t) , EQ(r) = 0 (11-17) where the subscript ( ,) represents the perturbed quantities which are considered much smaller than the corresponding equilibrium quantities. Inserting expressions (11-12) to (11-17) into equations (II-l) to (II-7) and subtracting (II-8) from (11-10) results in the perturbation equations

•k pi

-o" 3O

it pl + «1 • 'Po + rPoV-Ui = 0, (11-20)

•5" A Si + ' x Si = °' (II"21) MAGNETOHYDRODYNAMICS AND STABILITY ill

*!• JL = V X Bx , (11-22)

and 7 • B^ = 0 . (11-24)

This set of fourteen first order partial differential equation with variables p, , u, ( p. , B,, j and E, can be reduced to one second order vectorial equation, with one variable, namely, u^..

è (v x «o>x tv x (ui

The other variables are obtained through time integration of functions of u.

£ Sl = V x tux x Bo) , (11-26)

À Pl = -Ul'Vpo ~ TPoV'«l ' (II"27)

and p1# ^ and Ej^ are solved from (11-18), (11-22) and (11-23). With a given set of initial conditions u, (r,0) and ^(^,0), and appropriate boundary conditions, equation (11-25) can be solved. The above equations are written in Eulerian formalism. The variable u^trft) and the equilibrium quantities, such as,

pQ(r) are expressed in terms of the Eulerian position £, that is 112 P. Sakanaka

to say, for a fixed~position £ the fluid is passing by with a velocity u. (:ç,t) , pressure p(r,t) etc. It is more convenient, however, to write equation (11-25) in terms of the Lagrangian displacement E; as in Fig. II-l. This is the displacement of any elejaent from is equilibrium position and is expressed in terms of the equilibrium

position ç and time t as |(£oft) [7, 12, 13J.

path of fluid element

equilibriuti position of the fluid element.

Fig. II-l Lagrangian Perturbation of a fluid equilibrium

The time derivatives of £j(r ,t) at fixed r describes the velocity of the same fluid element as it varies in time.

(11-28) •MAGNETOHYDRODYNAMICS AND STABILITY 113

The Eulerian velocity at r, u(£,t), coincides with £(rQ,t) within the same order of magnitude for

by expanding it for small £ around r . The second RHS term

is a second order term, for £; and u are both first order,

therefore

JüíiEo'*5 -í^'*» (II'30) (For a complete description of this subject refer to Goedbloed [7J).

One may write equations (11-25) to (11-27) in terms

of the Lagrangian displacement |. Integration of equation

(11-26) results in

For later convenience the quantity B. is renamed 0..

S = v x % x ?o)" (11-32)

From (11-27) results

V T (II 33) Pi = -|- PO " VS ' " and from (11-25) one obtains the equation of motion

where one defines the force-operator

V + TO = <|"PO YPOV-|) + ^ [(V X jg) X Bo + (V x Bo) X g ]

(11-35)

The equation of motion (11-35) may be solved when initial conditions £(£ ,0) and £(£ ,0), and boundary conditions are given. 114 P. Sakanaka

B. BOUNDARY CONDITIONS

Assuming that the plasma considered here is confined by the magnetic field, there could be a vacuum region between plasma and the wall. Therefore, it is needed to set up plasma/ vacuum interface boundary conditions, and, vacuum/conducting wall boundary conditions.

Plasma/Vacuum Interface 2 T + _BJ (II 36) IT n - o " ««me[I " l> I[B 1 ?j -[TB 1 == 0 (11-38) Vacuum/Pe rfe c tly Conducting Wall Interface

1 x £ = 0 (11-39) JJ • jç B = 0 (11-40)

The symbol j[q ]] means the jump of the value of q across the interface, that is, {[q1]] = q - q1 where, q is the value just inside and q1 is the value just outside the plasma. The vector n is a unit vector normal to the interface pointing into the vacuum. Its counterpart is n1 which points into the plasma. Equation (.11-35) is written in terms of the equilibrium position r . When there is a displacemnet e, the interface

; changes position. This has to be taken into account in these :!•" boundary conditions, but still the quantities have ti be written q in terms of çQ. This complicates the equations a great deal. Equation (11-36) may be written explicitlys MAGNETOHYDRODYNAMICS AND STABILITY 115

(I-I-41)

Since r = £ + £, one may expand each equilibrium term around

ç with displacement £.

Collecting terms and using (111-33) one obtains the

first order boundary conditions:

-yp V-£ + 7T- B -Q + i-E-VB2 = -p- B' -B' + •—E-VB'2 (11-44) O * 4TT "^o ^5 Srr* 'v-O 4ir "^O "v

Now, from (11-37) and Ohm's Law (II-6) the following relation is obtained

n x E' = i (n .u)B' (11-45)

Since E and u are first order quantities, and considering the vector potential £ with Coulomb Gauge (v»A = o), defined as

E = - — •— A , B,1 = V X A (11-46) >\< c at >v ' •vi <\i one has, after integrating in time

" So x A = (So * *'«i (II"47) Condition (11-39) at the interface vacuum/wall becomes

no x A = 0 (11-48)

-Finally, collecting the equations of motion and the boundary conditions, and droppidroppinr g the subscript ( ), for there is no more need for it, one has P. Rakanaka

h k W (II"49) where

C(^) = V(j|-Vp + YPV|) + i [(V x 0.) x B + (V x B) x 0.] (11-50) with the boundary conditions

2 2 -YP V| + ^B.q + i$-VB =ÍB'-V x A + Í£-VB' (11-51)

and QoX ft= ° (II-52)

This completes the specifications and the problem is now well defined.

C. SELF-ADJOINTNESS OF THE FORCE OPERATOR

It may be proved that the force operator is a self- adjoint operator. In order to prove it one has to, show first that an energy integral exists, and proceed to demonstrate the self-adjointness. The system of equations (II-l) to (II-7) may be written in the conservation forms. In particular, the equation of state may be transformed into the energy conservation equation

(See, for instance, Goedbloed [7]. MAGNETOHYDRODYNAMICS AND STABILITY 117

Integrating (11-53) over all space, and assuming that the outgoing energy flux is zero at the wall one obtains the energy integral

U = Í (4- pu2 + -^rr + "I—"I) dt = constant (11-54) •p i. y—X. BIT where dx is the volume element, P represents the total plasma space (including the vacuum region) , and U is the total energy. One may separate U into kinetic K and potential W energy:

U = K + W (11-55) with K = ( -i- u2dt (11-56) and W = I (~5j- + -§—)dx (11-57) P

Since uQ = 0 the variation of the kinetic energy <5K due to the perturbation is exactly

n 2 f l " ' K = ( i pu (ix = ( i PÊ-Èdx (II-58) yP -T» The variation of ;:he potential energy due to the perturbation, represented by 6W, depends on p and J3, and since these variables and their perturbations do not depend on £ but only on £, one can assume that 6W is a functional of | as SW[£,£]. From the conservation of energy K = - «Ã (11-59) and from J(11-58) with (11-49) P 118 P. Sakanaka

But, -sw = -

,and\inserting_(11-60) ^and Jll-61) in (ll-59K;one has Jk-Vk)dz- - 6WUti -

Since the boundary conditions (11-51) and (11-52) are satisfied for I as-well as £, | may be considered as another pertur-

bation, TJ - 4i» and itL satisfies the equation. Therefore one may cons truct X r£! = -6W h»Ú -sw U'fll' (n-63) r ^-s^)^ = - 6WU'.Q] -sw U'.|] (n-64)

Since the RHS of equations (11-63) and (11-64) are equal one obtains the proof of the self-adjoinbnass of the force operator, that is,

^p and also

The self-adjointness of the force operator may be proved directly from the definition of F (see Goedbloed [7J, but this is a very long and tedious work.

Now, substituting (11-49) into (11-66) one obtains the variation of the potential:

> (11-67) MAGNETOHYDRODYNAMICS AND STABILITY 119

The integration is performed o_ver the space occupied by the plasma only/ excluding the vaccum regions. Using the following vector identities

.) + (V • |) [ <£'7 P + ^P <7 " & 1

(11-68)

B) • (?XOJ

= Q2 + £.[(VxQ)xB] , • (11-69) and |-[(V x B)xig] = - (v x B) • (£ x $g) , (11-70)

in (11-66) and applying the divergence theorem results in two surface terms. Using (11-51) and (11-52) and rearranging the terms, one obtains a more interesting form of the variation of the potential energy

6W(£) = 6Wp(|) + 6Ws(^) + 6Wv(|) (11-71)

where

!-7P> +"TiT + Ad T '

\ ( W2 n • IV(P+ TirB2 ) IId T '

2 6Wv(£) = J^-l- (7 x^) dx . . (11-74)

The subscript p represents the plasma volume, V the vacuum volume and S the plasma/vacuum interface. The boundary conditions

n x A = " (JJ

for the plasma/vacuum interface and n x $ = 0 (11-76)

for the vacuum/perfectly conducting wall still have to be satisfied. Equation (11-66) means that the variation of the potential energy is the work done agaisnt the force F(|). This work leads to an increased potential energy of the plasma, W ,

of the surface W and of the vacuum, Wv# It is interesting to notice that from (11-67) it seems that the variation in the potential energy does not depend on the vacuun region. This is only apparent Lecause its dependence is hidden under the form of boundary contritions (11-51) and (11-52), for (11-71) shows that 6W has contributions from the plasma, the interface and the vacuum. Consider, now, a normal mode solution with an exponential time-dependence exp(-iwt). The time derivative of the equation of motion (11-34) may be eliminated so that

P~V|> = - u2! . (11-77)

where £ = |(r) function of r only, and with the boundary conditions it becomes an eigenvalue problem. The natural definition for the inner or scalar product of the two vector fields f and TJ for the study of this problem is

Then, one may talk about the spectrum of the operator P F. Its ? spectrum consists of the collection of eigenvalues { - u 2} n i I corresponding to the eigenfunctions £; (r) . ' Since the force operator is self-adjoint (or Hermitian) its eigenvalue should necessarily be real, positive or negative, MAGNETOHYDRODYNAMICS AND STABILITY 121

and therefore m is real or purely imaginary. The eigenvalue mn being real means that the mode is a stable oscillation, and a . fj imaginary means an exponentially growing mode, that is, an unstable mode. The Rayleigh-Ritz variational principle may be applied to this eigenvalue problem: The eigenfunctions of the operator p F are obtained for functions £ for which the functional

6W(|) *— (H-79)

becomes stationary [7], where the virial

and SW(ç) is given in (11-66). The virial is written I(t) when it is necessary that the parameter t be explicit. If one assumes that the eigenfunctions of the operator p~ £ are a subset of the class of square-intagrable func tions, then one may say that the eigenfunction ^ is a linear

combination of a base functions { Q\ tt\2,...} of the square- integrable functions. Starting with a trial function

2. n=l one may find, using the variational method, a set of coefficients

2 { an } which extremizes the functional S2 (|). Then, the eigenfunction is given by £. = S a n and its corresponding eigen- 1 h^l n * value ai.2 = flz{£.). In particular, the ground state is that P- Sakanaka which makes SI2 (£) minimum. Thus one may find a set of eigenvalues [ ui } and their corresponding eigenfunctions. However the representative spectrum of a typical MHD plasma contains both discrete and continuous regions, such as, in the models studied by Goedbloed 77,14,15']. Grad ^16] and Kruskal and Oberman [17] have also shown earlier that the operator p F does not have a complete set of eigenfunctions in the space of square integrable functions. This introduces complica- tions for the analysis of next subject: the energy principle.

D. ENERGY PRINCIPLE

The energy principle states that an equilibrium is stable if and only if SW(|) > 0 for all displacements £,(r) that are bound in norm and satisfy the boundary conditions. This is the Raylelgh-Ritz variational principle applied to marginal stability analysis, i.e., ui2 = 0. To apply this principle one uses a finite class of trial functions covering a subspace of the Hilbert space of the system to calculate 6W given in (11-71). If any one of 6W is negative, that is enough to say that the equilibrium considered is unstable. If one has a good physical intuition and constructs a trial function which renders 6W < 0, the problem of instability is solved.

If the modes are all normal and have exponential dependence in time exp(-w t) the demonstration of the principle is straight forward. In this case one considers a complete basis MAGNETOHYDRODYNAMICS AND STABILITY 123

' (ri"82)

and the orthonormality: Um»5n> = «^ • (11-83) An arbitrary trial function is expaneded in the eigenfunctions:

Therefore, applied to SW results

6W = - (í.p-^íí)) = Jx an*«na (H-85)

Thus 6W can be made negative if and only if there

2 exists at least one fjn(Ç) with eigenvalue u < 0. This completes the proof.

A correct proof of necessity and sufficiency is given by Laval, Mercier, and Pellat [9][, in which no assumption of a complete basis of discrete eigenfunctions is made. Besides the proof, they provide the lower and upper bounds for the growth rate of the unstable perturbation.

They proceed to use the virial and the energy conservation equations to demonstrate that the kinetic energy of the possible perturbation remains bound for ill values of time if and only if SW > 0 for all . They use the concepts of unstable equilibrium according to Lyapunoff's definition. Virial Equation:

T7 *W = ^r (£'&> = 2(t4> + 2^'k) = 2K " 26W-

(11-86)

Energy conservation equations H=K+6W,H=0 - (11-87) 124 P. Sakanaka

Sufficiency; If 6W > 0 for all 5, then the kinetic energy k is bound for all values of time. Proof: If 6W > 0 for all £, whatever the evolution of £j(t) , then from (11-87) k < H, for all t. Since H is a constant of motion, k is bound. Now if £ grows linearly in time, £ *> t, then £ = constant, therefore k is a constant and bound. Hence, fiW > 0 does not exclude the linearly growing instabilities. These instabilities arise from the continuous spectrxim which reaches the origin

oi = 0.

Necessity; If or a displacement Q, íW(ri) < 0, the system will present an unbound kinetic energy. Assume 6W(r.) = - Y2KT;)« Y real. (11-88) One chooses e(r,0) = n(r) and l(r,0) = ynU)- (11-89) Inserting (11-89) in (11-87) one obtains

K[|(r,t)] + 6W[|(r,t)] =H[|(r,0)j =

and from (11-86) and (11-87)

I(t) = 2K - 2fiW = 4K - 2H = 4K > 0 (11-91)

The Schwartz inequality gives

I2(t) = 4(|,|)2£ 4(£,|)(|.,!) = 4I(t)K = í(t)I(t) (II-92)

Then

i(t) ii(t) , (II-93) I(t) 7^

(11-94) 1(0) MAGNETOHYDHODYNAMICS AND STABILITY 125

From (11-87)

ii§j- = 2Y (11-96)

From (11-95) and (11-96)

I(t) ^1(0) exp (2Yt) (11-97)

consequently, £ grows at least as fast as exp(yt). q.e.d. The upper bound for the growth rate is demonnstrated by: Theorem: If one can find a positive number r such that for all functions F

-r2 1 - Y2(|) = 6W(£)/IC£) (11-98)

then <|(t) cannot grow faster than exp(rt). Proof; i' = 2K(t) - 26W(t) = 2H(t) - 4fiW(t) <_ 2H(t) + 4r2I(t).

(11-99) Hence, I(t) - 4r2I(t) <_ 2H(t) = 2H(0). (11-100) Therefore, I(t) cannot grow faster than exp(2rt) and |(t) cannot grow faster than exp(rt). a.e.d.

In conclusion, the energy principle is a very practical stability analysis device, for one does not have to solve the time dependent equation nor the eigenvalues equation for the normal analysis. From the analysis of all possible displacement from equilibrium one can check its stability, and not only that, one can obtain the upper and lower bound for its instabilities . if

126 P • Sakr.naka

The energy principle is a very powarfull tool to study /i the stability of MHD plasmas. The extension of this principle to the double-adiabatic equations is made by Bernstein et. al [8].

E. APPLICATION OF THE ENERGY PRINCIPLE STABILITY OF SHARP BOUNDARY PLASMAS

Consider a non magnetized plasma with a constant pressure, confined by a vacuum magnetic field. Assume that the plasma is representable by the ideal MHD equations and that there is an equilibrium. . .

What is the stability of this equilibrium? {:

One may apply the energy principle to answer this ques- ! k tion. With no magnetic field, B = 0, and constant pressure, Vp = 0 the equation for 6W (11-71) becomes ';

. , 2 1 I 2 VB'^ 6W = i | YP(V-|)" dt + j \ <.§.'%)* n —^ dS

Observing that V-Ej contributes with a positive definite term in W, the mos unstable cases should be those with £ which are divergence free, that is, incompressible perturbations of the fluid,

V-| = 0 (11-102) Then 5W becomes SW = B~-¥ f\

The vacuum term is a stabilizing contribution since it is also a positive definite term. One cannot set V x £ = 0 arbitrarily here because j^ must satisfy the boundary conditions with (11-75) and (11-76) and the Maxwell equations v x v x £ = 0,

with = 0. However one minimize 6Wv by chosing the trial func-

tions £; and £ such that iW is negligible compared to 6Wg. 2 The surface term is a destabilizing term, for n-VB1 is negative when the field lines are curved out from the plasma, and so its magnitude decreases away from plasma. If £ and A^ may be chosen such that SW is negligible compared to 6W and 2 which n-VB* < 0 on some surface region, then there is a makes 6W < 0, that is, the equilibrium is unstable.

Stable Curvature

If n-VB' > 0 everywhere on interface the equilibrium is stable, for 6W > 0.

into page

Field line curved into plasma

B = radius of curvature (a) (b) Fig. II-2 (a) Stable Magnetic Configuration. Cusp Field (b) Hadius of Curvature of Magnetic Field in Vacuum 128 P- Sakanaka

If the confined plasma in Fig. II-2 (a) has an axial symmetry, such as the topology obtained by rotation around the xx-axis, or a toroidal symmetry, such as the figure obtained by rotation around the zz-axis the plasma is surrounded everywhere by a stable magnetic configuration. It is presumed that the equilibrium is stable. However for plasma confinement purpose this is not enough because of the loss of particles through the magnetic point or line cusps. These cusps have windows of the size of the ion Larmor radius. In Fig. II-2 (b) one defines the radius of curvature R of the vacuum magnetic field lines. In vacuum V x B1 =0 for j1 =0. Therefore, from Stokes theorem the integral around the closed curve ABCD gives S.B ' = S B '. The arc S. is proportional to R^, so vV =BO'V- (I Therefore

The stability of the confining field is then related to the sign of n'§-

Unstable Curvature

If n-VB1 < 0 for some region of the interface the equilibrium is unstable since 6W < 0 for some e. This statement has to be proved. Consider a coordinate system (x,y,z) such that the plasma/vacuum interface is tangent to the plane x = 0 touching at the P°int So = <0'°'°>- L«* Ç'(íEo> "B(o 'o) Sz' and B^o» = «xb e the mormal to the equilibrium interface. Choose the trial dis- MAGNETOHYDRODYNAMICS AND STABILITY 129 placement | and the vector potential A so that

£x(0,y,2) = ço f(y,z) sin ky , (11-106)

oy) _kx and A(x,y,z) = f(y,z) V (— cos ky e KX ) , (11-107) k where f is a function of order unity which falls to zero in a characteristic distance a such that ka >> - >> 1 (11-108)

The vector potential A satisfies the boundary conditions (11-75), and satisfies V x V x fy = 0 with V«j|V = 0, up to the order of 0(l/ka). The function f(y,z) is determined from an equation of the order of 0(l/ka). The other components of E, may be chosen BO that it satisfies V-| = 0. The vacuum contribution is written

ço -kx 1 2 E x V(-g- B1 (x,y) cos ky e )J dt

-ikx j o AS ,__ ._-. —=— e dt = 5— ^r;- (11-109) a^ 8 a'5 2k while the surface contribution is

6W = ( )2 n 7B 2 s T?7 J 4'« ' ' dS = °16ilR *S (11-110)

v R/a Therefore 5^ = ^ < < 1 (11-111)

Then, if n-VB1 < 0 a trial function £ may be chosen so that 6W < 0. Therefore the equilibrium is unstable. q.e.d. One may proceed to calculate the lower bound value of the growth rate from (11-97). 130 P. Sakanaka

Expression (11-106) is the value of £ for x = 0. One may choose an x-dependence of ç within the validity of the X above analysis , so that one defines

Cz = 0,

kx çy = ço f e cos ky. (11-112) „ kx sen f e ky. • The virial results in 2 •fiedz = (11-113) Therefore <5W 2 B-2 k

In conclusion, the growth rate is at last 12, that is to say

Y > £2= V —- (I]

2 2 where vft = B' /4 irp is the Alfvin velocity in a plasma of density p in the magnetic field jg'. The great advantage of this principle is that the stability of plasma is decidded without having to solve the equations with time dependence. It is enough to find a negative variation of potential energy, 6W, to know that the system is unstable. '•I

MAGNETOHYDRODYNAMICS AND STABILITY 131 tA

III. EXTENSION TO o-STABILITY

The purpose of studying MHD stability is to learn the mechanism and features of instabilities which shorten the confinement time of thermonuclear fusion plasmas, and from that knowledge improve the configuration to eliminate or at least to damp those instabilities. If the instabilities grow so slowly that for the characteristic time T needed for fusion its growth is negligible then the configuration may be considered stable for all practical purposes. One could also, take T to be another time-scale, such ,. "<• i as the time-scale for which the ideal MHD model is a valid : ' [•• . description, or one may choose T to be the time-scale for the j decay of external currents used for the magnetic confinement >; experiments [7]. For all these purposes one may allow perturbations that grow at most like exp(at), where o = 1/T. We shall ( call o-stable equilibria, those equilibria which do not have instabilities growing faster than exp(at) £18] . From theoretical point of view the o-stability concept serves also a purpose. The MHD spectrum has a continua which reaches the origin u =0 from positive side and from negative

side there is a set of point eigenvalues with an accumulation at u =0, Fig. III-l [IS"].

a - unstable -a2

, . jun- • i — _j ° ~ stable

-ff2 0

Fig. III-1 - MHD spectrum 132 p- Sakanaka

The marginal point u. = 0 is then, a singular point with a continuum from one side and an accumulation point from other and defies any numerical analysis. Besides that, its practical value i= small. The o-stability analysis simplifies this analysis enormously. In o-stability analysis,only point eigenvalues are considered. Therefeore one may define equilibrium to be a-stable if no point eigenvelue w < - o exist, and a-unstable if such eigenvalues do exist.

To developed a-stability theory, one defines the mod- fied force operator;

2 This force is reduced by pa | in relation to the force f to drive instabilities under the usual definition. Next, one introduces the modified potential energy 5W°,

SWa = 6W + a2l , (III-2)

where a part of kinetic energy is abosrved by the potential energy. If one assumes that the plasma is surrounded by a conducting wall,the modified potential energy becomes:

(III-3) MAGNETOHYDRODYNAMICS AND STABILITY 133

Furthermore the modified force operator is self-adjoint and the energy principle applies here too. Modified Energy Principle: An equilibrium is a-stable if and only if 5Wa(|)> 0 for all possible trial functions |. Its demonstration is quite analogous to the demonstration of the ordinary energy principle. Sufficiency: If SWa > 0 for all £, then 6Wa = H - (K-52I)> 0 for all I. Since H is a constant (K-6"I) cannot grow without bound for then the energy conservation would be violated. Necessity: If for some rj SW0^) < 0, one defines

m tf 2 2 2 n -a =Y -a >0.

Then

Y2 = _ _^£ = V2 + a2 > ^2

so that £(t) grows at least as exp(yt) = exp p2 +az t and the equilibrium is, therefore, a-unstable.

The advantage of o-stability analysis over the normal mode analysis is that while in the normal mode analysis u is determined as an eigenvalue, in the former analysis o is a fixed' parameter and one needs only to find the sign of 6W° . Aplications; This new stability concept has been applied to cylindrical plasma with circular cross sections by. Goedbloed and Sakanaka fl4, 20], where stability of a wide range of equilibrium profile has been studied. More recently, effects of toroidal correction has been studied under a-stability analysis [21] . P. Sakanaka 134

IV. ENERGY PRINCIPLE FOR DISSIPATIVE SYSTEMS

A. ENERGY PRINCIPLE

The energy principle described in Chapters II and III are applied to conservatine systems, in which the Lagrangian displacement, £(x,t), from equilibrium satisfies the equation of the form:

p'( (x,t) + Q £(x,t) = 0 (IV-1) where Q is a symmetric operator with no time derivatives. For a dissipative fluid, such as, resistive and viscous plasma, occupying a volume U with surface S, one may find a convenient representation to describe perturbations around an equilibrium in the form:

NY(x,t) + (P + M)Y (x,t) + QY(x,t) = 0 (IV-2) where Y(x,t) is a vector representing perturbations, N positive and M positive non-zero symmetric operators, Q symmetric operator, P antisymmetric operator, and x and t space and time variables. The variable Y(x,t) satisfies appropriate boundary conditions on S, dictated by physics, and certain properties imposed on Y(x,t) in U, such as, incompressibility, V . £; = 0. The symmetry of above operators are defined according to the inner product. (Z(x,t), Y(x,t)) =5 JI o Z* . Y dx (IV-3) where Z* de notes the complex conjugate of Z, dx is the volume element, and Z(x,t), Z(x,t), Y(x,t), Y(x,t) are in D, where D is defined to be the set of all functions which satisfy the boundary conditions on S, the properties imposed ond Y(x,t) in Ü, and n-times continuously differentiable on U. The differ- entiability n is 2,3 or greater, depending on the physical system. Barston [22, 23 "j and Tasso [24] have demonstrated that any system described by equation (IV-2) is stable if, and only MAGNETOHYDRODYNAMICS AND STABILITY 135 iff (Z, QZ) 2. °- Tne stability depends only on Q. The opera- tores, N, P and M affect the values of growth rate, but do not change the stability. It follows then:

ENERGY PRINCIPLE: A necessary and sufficient condition for the stability of the system described by equation (IV-2) is (Y,QY) _> ° f°r all Y in D.

(I) Sufficient Condition:

If Q is a non-negative operator, that is (Y,QY) >_ 0 for all Y in D, then the system is stable. Proof: If one considers the inner product between Y(x,t) and the terms of equation (IV-2) , it results in:

(Y, NY) + (Y, PY) + (Y, MY) + (Y, QY) = 0 . (IV-4)

Since N and Q are symmetric and P antisymmetric operators ,

(Y, NY) = \ -g| (Y,NY) , (IV-5)

(Y, QY) = \ -jç- (Y,QY), (IV-6)

and (Y, PY) = 0, (IV-7)

hence equation (IV-4) becomes: I -^- [("/,NY) + (Y,QY)] =-(Y,MY). (IV-8)

The time integration of (IV-8) gives:

(Y,NY) +

(IV-9) X36 P. Sakanaka

Since M is a non-negative operator, (Y,NY) + (Y,QY): is a monotonically decreasing funtion of time, and since N is apositive operator, if (Y,QY) _> °/ one concludes that Y cannot grow in time, that is, the system is stable. q.e.d.

(II) Necessary Condition: If (Y,QY) < 0 for some Y in D, then the system is exponentially unstable. Proof:

Consider the following definitions: , _ inf [ (Y,QY) "I , „ , , , . A = D [ (Y,Y) J ' ( " A '0) 'n

D H < Z / Z in D, (Z,QZ) < oS , (IV-11) \ kjjff (Z'MZ)2 . (Z,QZ,QZ) 111/ 1/2 _ (Z,MZ) 2\L (ZNZ)2 (Z''NZ)NZ)JJ lZ'NZ) for Z in D (IV-12)

r = sup Y (IV-13) D M = 2yN + M (IV-14)

2 Q^ = Y N + Y(M + P) + Q (IV-15)

Since (Z,QZ) < 0 in 5, YZ > 0, so that r > 0. Let e such that 0 < e < r. Consider a particular Z, in 5 such that r - E < y < T Zl Now, by definition,

(IV-16)

Since N and M are positive and (Z.,,PZ,) = 0, (Z]rQ Z,) is a mono- MAGNETOHYDRODYNAMICS AND STABILITY 137 tonically increasing function of y which vanishes for y = y z

(from definition (IV-12) , therefore for y = V - F. < vn , its zl value is negative, ^I'^V-E^X^ K "' Now let Y(t) be a solution of equation (IV-2) satisfying the initial conditions Y = Y(0) = Z, and Y

(r e)t = Y(0) = (a -£)ZX. Then the function X(t) = Y(t)e" " satisfies the equation

NX + (P+Mr )X + Q X = 0 (IV-17)

One can verify that inserting definitions (IV-14) and (IV-15) equation (IV-17) becomes equation (IV-2) for Y(t). Carrying out the inner product with X, equation (IV-17) becomes

-~ [(X,NX) + (X,Qr_EX)] = - 2(X,Mr_EX) <_ 0 . (IV-18)

From definition (IV-10), positivity of N and M, and equation (IV-18) one obtains

A(X,X) <_ (X,QX) _< (X,NX) + (X,Qr_gX)

1 (Xo,NXo) + (Xo,Qr_eXo) = (Zl,Qr_eZl) < 0,

(IV-19)

where (XQ,NX ) = 0 by construction.

Inequality (IV-19) implies that

1/2 1/2 [[Xj = [(X,X)] > [cz1flr_eZ1y/A] (IV-20)

Hence [[Y(t)U = [Tx(t)Ie(r-E)t > [(Z^Qj,,^^]1/2 e(r-e)t ,

(IV-21) 138 P. Sa^anaka and this holds for sufficciently small positive E, i.e. the exponential growth rats r may be approached arbitrarily closely by solutions of the equation. q.e.d.

In conclusion, if (Z,QZ) >0 for all Z in D, then the system is exponentially stable, while if (Z,QZ) <0 for some Z in D, then the system is exponentially unstable whith the maximal growth rat r given by

(IV"22) where D is the subspace of D in whicn (Z,QZ) < 0, Z in D .

B. APPLICATIONS

Barston in £ 22 ^applied this principle for the stability of one dimensional resistive sheet pinch and has shown that: (a) there is no exponentially growing "tearing" mode for a constant resistivity equilibrium with no gravity; (b) a non constant resistivity equilibrium with insulating boundaries cannot be gravitationaly stabilized; and (c) a non constant resistivity equilibrium with gravity but constant mass density and the magnetic field non-unidirectional is exponentially unstable to all perturbations. Barston in ^23} has applied this principle to the viscous fluids and plasmas, and has shown that the necessary and sufficient conditions for exponential stability is

VpQ . V*o <_ 0 where p and * are, respectivily, the unperturbed mass density and gravitational potential. Tasso has applied the principle to two and three dimensional resistive instabilities [ 24,25,26] with effects of viscosity and finite Larmor radius effects, FLK, were included. Basically he has^shown that: (a) the viscosity and MAGNETOHYDRODYNAMICS AND STABILITY 139

FLR effects do not change the stability of a system but the growth rates; (b) for: the resistive kink or ideal internal kink the growth rate reduces by a factor of 100 by inclusion of the viscosity parallel to the magnetic field; (c) helical perturbations can always be unstable. Caldas and Tasso in [27,28,29} have applied the dissipative energy principle for the stability of two-fluid plasma model taking into account viscosity, gyroviscosity, resistivity, compressibility, the Hall term, and electron inertia. Most previous results on MHD, resistive and universal-type instabilities have been recovered in a simple way that demonstrates the power of this formalism. This allows 3-dimensional tearing mades for general cross-sections to be investigated. In summary, the energy principle is a very powerful tool to study stability of fluids and plasmas, in which inclusion of complicated geometries and physical effects are possible without turning the problem insoluble. One cannot obtain the growth rates but one may estimate its value.

REFERENCES;

1. D.C. Montgomery and D.A. Tiaimsn, Plasma Kinetic Theory, Chap. 4, "The BBGKY Kinetic Theory", McGraw-Hill, 1964.

2. S.I. Braginskii, "Transport Processes in a Plasma", in Reviews of Plasma Physics, Vol. 1, Ed. M.A. Leontovich, 1965.

3. S. Chapman and T.G. Cowling, Mathematical Theory of Nonuniform Gases, Cambridge University Press, 1953. 4. N.A. Krall and A.W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, 1973.

5. P.H. Sakanaka, "Macroscopic Plasma Properties. Magneto- hydrodynamics and Stability Theory. Thermodynamic Approach for Stability of Plasmas", in Proceedings of Autumn College on Plasma Physics, ICTP, Trieste, 1979. •^4Q P. Sakanaka

6. L. Spitzer, Physics of Fully Ionized Gases, Interscience, 1962. 7. j,p. Goedbloed, "Lecture Notes on Ideal Magnetohydrodynamics", Universidade Estadual de Campinas, Campinas, 1979. 8. I.B. Bernstein, E.A. Frieman, M.D. Kruskal, and R.M. Kulsrud, Proc. Roy. Soc. (London), A244, 17 (1958). 9. G. Laval, C. Mercier, and R. Pellat, Nuclear Fusion, 5_, 156 (1965).

10. W.A. Newcomb, Ann. Phys. 1£, 232 (1960). 11. G. Schmidt, "Physics of High Temperature Plasmas", Academic Press, New York, 1966. 12. W.A. Newcomb, Nuclear Fusion, 1962 Suppl. Pt. 2, 451 (1962) . 13. W.A. Newcomb, "Lecture Notes on Magnetohydrodynamics" unpublished.

14. J.P. Goedbloed and P.H. Sakanaka, Phys. Fluids L7.- 908 U974) . 15. J.P. Goedbloed, Phys. Fluids 1J3, 1258 (1975). 16. H. Grad, Plasma Physics Theory Meeting, Gatlinburg, Tennessee, 1954. 17. M.D. Kruskal, C.R. Oberman, Phys. Fluids .L, 275 (1958). 18. J.P. Goedbloed and P.H. Sakanaka, "A New Approach to Magnetohydrodynamic Stability. I. A Practical Stability Concept", Phys. Fluids 1_7, 908 (1974). 19. J.P. Goedbloed, "Spectrum of Ideal Magnetohydrodynamics of Axisymmetric Toroidal Systems", Phys. Fluids ^0, 1258 (1975). 20. P.H. Sakanaka and J.P. Goedbloed, "New Approach to Magneto- hidrodynamic Stability. II. Sigma-Stable Diffuse Pinch Configurations", Phys. Fluids, 17, 919 (1974). MAGNETOHYDRODYMAMICS AND STABILITY 141

21. R.M.O. Galvão, H. Shigueoka, and P.H. Sakanaka, "Influence of Toroidal Effects on the Stability of the Internal Kink Mode", Phys. Rev. Lett., 4L, 870 (1978).

22. E.M. Baraton, "Stability of the Resistive Sheet Pinch", Phys. Fluids 12:, 2162 (1969) . 23. E.M. Barston, "An energy principle for dissipative fluids" J. Fluid Mech. 4_2, 97 (1970) . 24. H. Tasso, "Energy Principles for non-ideal MHD", Plasma Physics and Controlled Nuclear Fusion Research 19 76, Vol. Ill, 371 (Sixth Conf. Proceed., Berchtesgaden). 25. H. Tasso, "Energy Principle for Two-Dimensional Resistive Instabilities", Plasma Physics 1_7, 1131 (1975).

26. H. Tasso, "Energy Principle for Resistive Perturbations in Tokamaks", Plasma Physics 19_, 177 (1977). 27. I.L. Caldas and H. Tasso, "Application of the Two-Fluid Energy Principle to Large Aspect Ratio Tokamaks", Plasma Physics 2Q_, 1299 (1978) . 28. I.L. Caldas and H. Tasso, "Stability of Large Aspect Ratio Tokamaks in the Two-Fluid Dissipative Theory", Plasma Physics and Controlled Nuclear Fusion Research 1978, Vol. I, 637 (Seventh Conf. Proceed-, Innsbruck). 29. I.L. Caldas, Instabilidades Dissipativas em Magnetohi- drodinâmica, Dissertation for Doctoral Degree, Univer- sidade de São Paulo, Slo Paulo 1979. J 143

FREE ENERGY PROBABILITY DISTRIBUTION IN THE SK SPIN GLASS MODEL

Gerard Toulouse (') Bernard Derrick (2) (') Laboratoire de Physique de 1'Ecole Nomale Supérieure 24 rue Lhomond, 75231 Paris Cedex 05, France and (2) CEN Saclay, B. P. n°2, 91190 Gif-sur-Yvette, France

1. Introduction

The elaboration of a thorough and correct mean field theory of spin glasses appears now desirable. Indeed, the lack of such appears as the main barrier to any major progress of understanding in this dcmain of physics. A mean field theory might be defined as a theory valid in the limit of large space dimensionality. Without further ado, it will be defined here as the theory which solves the famous Sherrington- Kirkpatrick (SK) infinite-ranged model ^ . In this direction, during the last five years, two main paths have been opened and vigorously pursued. The replica method has revealed the depth of the problem, where an apparently innocent procedure leads to the paradox of a negative entropy . Subsequently, replica-symmetry breaking schemes of increasing complexity have been contrived, at an ever longer stretch fran physical intuition . The second method, initiated by Thouless, Anderson and Palmer , involves consideration of mean field equations for an arbitrary sample followed by a delicate averaging over disorder. The results presented here all stem from a simple central idea, vrtiich may be seen as defining a third path. The idea is to reconstitute the probability distribution p(Z) from the knowledge of its moments , where Z is a partition function and < > an average over disorder. This approach is a priori more ambitious than the replica method because it aims at a knowledge of the full p(Z) instead of just and the average free energy. However, it may prove to establish ]_44 G. Toulouse the larger settr within which the problem is to be understood.

One is-reminded-that-the SK model is characterized by the

Hamiltonian of N interacting (Ising) spins :

(ij) ij i i i whore the J. . are distributed randanly according to the Gaussian law :

2 o(J. .) = . exp ^— , with o2 = TT- , (2) 1 1D /ãTa 2o2 '' and H is an external magnetic field (taken as zero unless otherwise specified).

2. Annealed and quenched models

In the quenched (physical) model, one looks for the free energy, obtained from an average of the logarithm of the partition function :

F = - T < log Z > , k_ = 1 .

However, one can consider an annealed version where the free energy would be defined as

F = - T Log < Z >

The average of the partition function is easily performed and gives : 2 N N4~ - J < Z > = 2[ e , Y = BJ = ± , (3) where N, the-nur.iber of spins, is assumed to be large. Hence,

F a J2 if" = " 4T ~ T L°9 2 • (4)

Since this is precisely the hiqh tanperature (T > J) expression that is

found for the quenched model, it is instructive to plot the free energy

F and the entropy S.= - -rpr- of the annealed version as a function of ct a dx temperature (Fig.1). SPIN GLASS MODEL 145

SQ/N Log2 ._

:J J/2 VLÕfli

Fa/N

i -I

Figure 1

On the same figure, have been plotted in dashed lines the numerical simulation estimates for the entropy and free energy of the quenched model at low tenperature (T < J) . It appears clearly that the quenched model, which behaves like the annealed model at high temperature, must depart from it, at some temperature, because the latter suffers an entro:// catastrophe (entropy becoming negative) below T = T = 2.f/5g2" G. Toulouse

An approximation, to be called later the zeroth order approximation .(or :the random energy model), suggests itself immediately, with the quenched free energy following Fg down to Ta, and staying put at the value - J /Log 2 below Ta. This behaviour, with minimal avoidance of the entropy catastrophe, may be derived from a physical

hypothesis : the assumption that all the energy levels are independent N random variables (this means 2 independent variables instead of the N (N—l) correct number of —•*=—- , the number of bonds). It predicts a IP ground state energy — = - /Log 2 « - 0,83 , which is not that bad MJ an approximation to the number predicted by numerical simulations : - 0,765. SomSfihat reminiscent of the ideal Bose-Einstein condensation

(a minimal avoidance of a chemical potential catastrophe), the random-

energy model can be worked out in many details and thus provides many suggestive hints (4)

3. The moments < Sn >

Clearly, the quenched and annealed free energies will be iden-

tical if < Zn > is of the form e8" for 0 f n í 1.

A knowledge of < Zn > can be obtained, for n integer, by the

replica method . Traditionally, this knowledge nas been used in the

restrictive context of the "replica trick", where a limit n •* 0 is being

taken to reach < Log Z >. Here, we want to use all the information con-

tained in the moments : actually, we simply recognize in < Zn > = < en ^

the characteristic function of the random variable s = Log Z.

As an illustration of this point of view, consider an expansion

in powers of n for the logarithm of < zn > :

n T Z > = e™""" *" (5) SPIN GLASS MODEL 147

The coefficients a, b, ... are related to the cumulants of the distrib-

ution of Log Z. Thus, for the first-cumulant Kj ,

a = Ki = < Log Z >

this is just the content of the "replica trick" ; besides, for the

second cumulant K2 ,

b = &• = \ [> (Log Z)z > - < Log Z >2 ] = lp , (6)

where 0" is the standard deviation of the distribution of Log Z.

Quite generally, < Zn > is obtained, after standard procedures ,

n 1/2 = exp( £ n(N-n))/ [n ( -^ ) d yag]exp{- |

a B a + N Log Tr exp(y V y DS S +6 7 S ) } (7) a (aB) aS a

where the indices a, 3 run from 1 to n, (ag) denote distinct pairs with

a^i, and Tr is the trace over the 2n values of the Sa = ± 1. This a formula includes the effect of an external magnetic field H, with 6 = = . Of course, for n = 1 and n = 2, simpler expressions may be given : Y2 < Z > = (2ch 6)Ne T

o N "T *N~2' , IW ~1 yZ rw -w-iN < Z2 > = 2Ne 4 . / dy/|- e * [eTy ch 2S+e Yy] (9)

For n > 2, a saddle-point integration is sought after in a space of

^-5—- variables (the integration variables y .). For n > 2, this is

not a trivial task ; in their initial calculation, SK restricted the

search to the first diagonal (all y equal, and moreover positive), 148 G. Toulouse

"since the replicas are indistinguishable". Within this hypothesis (no replica-synmetry breaking), < Z > is obtained as

< zn> = expN{^n + nLog2- R(y)} (10)

With R(y) = I^L y2 + SL y - Log T^y) , (11)

t% n In(u) = -i- J^dt e" [ch(t^u + 6)] , (12)

the position of the saddle point being determined by the stationarity condition :

mâ. = o (13)

1t> be consistent with the saddle point integration, only terms of larger order in N have been kept in the expression of < Zn >. The saddle-point condition (13) can be rewritten as :

% = C (u) , u = Ty , (14)

Yn-2(u) with CCiui ) = i--ii-= , T Tn(u)

It is quite instructive to solve graphically Eq.(14), firstly in the absence of an external field. In this case (6 = 0), the function C and the integral I will be written without a bar. For u small, one has the following expansions : nu nf3n-'>1 V "5 l"l I (u) = 1 + 2p + nun -' u2 + n -r| n(n-2)+ -± u3+0(u") , (16) n 2 8 |_16 3J

2 3 1 Cn(u) =u+ (n-2)u + n(n-5) + -=-^ u + Oiu *) . (17) SPIN GLASS MODEL 149

Whereas, for u large and positive, one has the expansion :

un2 u(n-2)2 (n even) -^ CT^ 1-n e +C e , (18) In(u) = 2 n" n-1 u 2 2 (n odd ) CR .e

and thus, 2(n 1)u Cn(u) % 1 - 4 e- + ... , n > 2 (19)

Besides, some further useful bits of information are listed below u 1 Io (u) = 1 , Ij (u) = e , I2 (u) = y^

and for u large and positive :

- iflk t2 The expression of Ji (u) = —— / dt e Log ch(tÂi) will also

be useful, with Jj (u) = lim^Q - • For u large and positive, one has the expansion :

1 2p2u jj (U) = /^H - Log 2 + /§ . £ t^* e /" e" "^ dt (20) " p " 2ptiT

so that ^ - Log 2 + ç (2) +

This knowledge enables to draw the function C (u), for u positive (Fig.2), and to derive expansions of < Zn >, around T = J and T = 0. 150 G. Toulouse

Figure 2

For a given value of n, one can distinguish two regimes, a high

temperature regime and a low temperature regime.

In the high temperature regime, the dominant saddle point is at

the origin and

< Zn > x 2Nn e 4 , and thus < Zn > -v. < Z >n , (21)

asymptotically in N.

In the low temperature regime, the dominant saddle point is away

from the origin

u = y2 + 0(Y) 3<-KI

Hn2y2 Zn > o. 2N. e" (22)

asymptotically in N and y. However, the stability of this saddle point

is proved only for n > 2, and actually, for n = 0, this formula makes

no sense. For n > 2, the position of the dominant saddle point changes

discontinuously (first order behaviour) at a temparature T > j, whereas

it varies continuously (second order behaviour) for n = 2. At the temper-

ature T = J, the saddle-point at the origin becanes locally unstable for

all values of n In >, 2). 151 SPIN GLASS MODEL

4, High temperature phase It is instructive to plot a diagram (n,T) showing the dorains o^existence of the t*o characteristic regies (Fig.3). The dividing

line is n = nc(T), and its equation around the point (n = 2, T = J) is 2 r— ^C , whereas for T large, nc **• T . n f- 2 ^ /3r , with T rn— C Ac

JFigure 3

The shaded region in Fig.3 is the dcmain of validity of the high taiçerature regime, with the daninant saddle point at the origin (at least for n > 2), and where, asynptotically in N, < Zn > = < Z >n As a consequence, the free energies of the annealed and quenched models are equal (justifying the statements of Section 2) and the probability distribution of Log Z is quite sharp, since its second cumulant vanishes. Actually, it is possible to go to next order in ^ : X52 <->. Toulouse

£. n(N_n) _ nlpl c Zn > -. 2Nn e 4 (1 - Y?) , n < n (T* , (23) c and therefore

< Log Z > = N [Log 2 + ^-] + •£ Log(l-Y2) (24)

2 i 2 2 2 K2 = < (Log Z) > - < Log Z > = - ^- - -2 Log(l-Y ) • (25)

The expression (24), which agrees with the results of TAP (not with

those of KS ), shows a divergence of the subdominant term. The ex-

pression (25), which to our knowledge is new, shows a similar divergence

in the second cumulant. This behaviour suggests that the spin glass

transition is marked by a change in the probability distribution of

the free energy, most notably in its width.

The expression (23) for < Zn > can be written

2 n _ an+bn ,_c. < Z > = e (26)

This is the analytical form of the moments of a log-nonral distribution

(provided it holds for ail values of n, which is not the case here, ex-

cept asymptotically in T). The log-normal distribution has special

properties : for _.:tance, it is not fully determined by its moments .

It is found for quantities which are products of many independent random N variables. In the 3K model, the partition function is a sum of 2 terms N (N—1) which are themselves products of „ ' random variables. This is the

reason why the log-normal distribution .will be recurrently found as a

limiting behaviour, when one term of the sum dominates.

Thus, in a situation where all the moments < Zn > would follow the law (26),

the normalized probability distribution q(s) of s = Log Z can be written,

modulo the undeterminacy mentioned above, as : 7-PIN GLASS MODEL 153

(s-a) 2 q(s) = _i— e 4b , and (27)

< Zn > = J+" qisKe"3. ds

If —a + 0, when N is made large, the parabola in the exponent of (27) becomes very narrow and the distribution q(s) tends toward a singular S-function behaviour. Returning to the high temperature phase of the SK model, and keeping only dominant terms in N, it is found convenient to make the change of variable : s = Nx , o and to write : Nf(x ) q(s) = e" , so that

Zn > = N rHV"^VA|-luy dx . (28)

The integral (28) is then evaluated by saddle-point Integration

f'(xn) = n , (29)

< Z > ^ e

The function f(x), depicted in Fig.4, which is a combination of a v2 point (at xQ = Log 2 + -^- , with f (x ) = 0) and of an arc of curve 2 (given approximately by f (x) = —X - - Log 2, with x > x ), reproduces yd O well the dominant contributions in IJ for the moments. 154 G. Toulouse f(x)

Figure 4

Indeed, for n snail enough, the saddle-point condition (29) is nowhere satisfied on the arc of curve and the integral is dominated by the point at x , yielding o •* Nnx < Zr > -v e °

which is the correct high temperature regime. When n increases, the condition (29) can be satisfied for a point

xn o> x but it gives a subdominant contribution until xn reaches the value x for which the tangent to the curve passes through the point at x . One has,approximately, x -v 2 leg 2 , so that the critical value n is given as a function of the temperature by

2 n Y = 4 Log 2 or 2 nc -^ T SPIN GLASS MODEL 155

For n > n , the dominant contribution cones from the saddle-point now at x •> x, and one gets : ,2,,2 < Zn" > -v. 2N" e 4 Zn > -v. 2N which is the correct asymptotic expression in N and n.

Note that, in the framework of the zeroth order approximation, the arc of curve in f (x) is rigorously an arc of parabola, and the phase transition is obtained, at T = — , when the intersection point of the arc with the abscissa axis, at x = /Log 2, reaches x . It should also be noted that the manents < Zn > are blind to the

portion of the curve in the interval x < x < x, or at the left of xo- One may think that this curious curtoiguity might be lifted by considering the subdominant contributions to < Z > • however, there is no reason why the expression (28) should be valid at all orders. Actually, examination of formula (23) shows that the scaling expression (28) does not hold at next (after leading) order in N and reveals how the singular limit for f (x) is reached.

5. The behaviour at T = J At T = J, the function R(y), defined by Eg. (11), can be written, up to fourth order in y, as

R(y) = - n(n-l)(n-2) y3 _ nfrpH r(n_2) (n_3)_ h yM + Q[S) (3(J) 6 o J'

As a oonsequence,with the proviso of no replica-symnetry breaking, the moment < Z > is obtained as tin < Zn > i. ^e4 , forni2 , (31)

this is the high tanperature behaviour (21) at y = 1 ; whereas, for n > 2, 156 G. Toulouse

(32) where the beginning of an expansion in powers of (n-2) is written in the exponent. In terms of the function f (x), defined by Eq. (28), this implies that V3 f(x) T> 2(x-x J +| (33) for x & x = log 2 + -r . This behaviour is depicted in Fig.5.

f(x)

tongant with slops * 2

Xo X Figure 5

Indeed, it is easily checked by saddle-point integration, that for n Í 2 the dominant contribution canes from the DOint at x and o gives (31), whereas for n >, 2 the expansion (33) reproduces correctly the term in (n-2)" in the exponent of (32). Carparison of Figs. (4) and (5) illustrates the modification of f(x), when the transition temperature T = 5 is approached from above. SPIN GLASS MODEL 15 7

A The appearance of the critical exponent -^ in the probability distribution of Log Z is interesting : as a matter of fact, it can be written as -^r— , where the exoonent 6 assumes its classical o value, 6 = 3, for a homogeneous ferromagnetic transition.

Finally, note that this "critical" behaviour differs from the

one of the random-energy model, not only in the value of Tc and xQ,

but also because, in the latter model, the slope of f (x) at xQ is equal to 2/Log 2 and there appears no non integer exponent.

6. Low tatperature phase

This section contains

i) a new and direct proof that the SK solution is incorrect below T = J,

ii) speculations about the shape of f (x) (i.e. the probability distrib-

ution of Log Z), and its physical significance,

iii) an approximation for the ground state energy and the low-tenperature

behaviour of the thernodynamic functions.

Firstly, the proof

If one calculates < Zn >, within the hypothesis of no replica sym-

metry breaking (SK solution), not only up to first order in n, but up to

order nz, one gets < zn > = e (34)

for T = J (1+T), T < 0, up to fourth order in T, and

< Zn > = e

at very low temperatures, for y asymptotically large.

This shows that, both below T and at low temperatures, the SK

X '• 158 G. Toulouse

solution (saddle-point integration on the positive first diagonal in y-space) leads to an unacceptable result : a negative second cumulant for the distribution of Log Z. Note that this negative number appears to increase in absolute value from T down to T = 0, and that it starts c initially being fourth order in T, whereas the deviation of the free energy (from the continuation of its high temperature expression) is initially of order T3, leaving a possibility that this first term is correct. Secondly, the speculations. An exact calculation of the low tenperature phase properties for the SK model appears difficult. Some attempts, through various replica synmetry breaking schemes, have been very interesting but, so far, un- controlled. Within the present analysis, and in the absence of the final solution, two typical possibilities (among others) emerge naturally and are worth examining. They are best discussed on drawings for the function f(x), see Fig.6. f(x) f(x)

Xe Xe

(a) (b) Figure 6 t:

SPIN GLASS MODEL 159

In the drawing of Fig.6a, the curve f (x) intersects transversally f ! the abscissa axis, as it is the case at Tc (Fig.5). The slope of f (x) at x_j n (T), might be a decreasing function of taiperature ; for n < n (T), one would get < Zn > = < Z >n iitplying that (reminder : at dominant order in N) the probability distribution of Log 7. has no width in the low tanperature phase, like in the high temperature phase, except possibly at T = 0. This is actually what occurs in the random energy ncdel. In the drawing of Fig. 6b, the curve f (x) touches tangentially the abscissa axis. One assumes that, in the vicinity of x , f (x) can be expanded as

2 3 f(x) ^ A(x-XQ) + 0[(x-xQ) ] , (35) which yields, for the logarithm of < Zn >, the following expansion in power:, of n : it 4. X < Zn > * e " ~ . (36)

Therefore, in this hypothesis, the second cumulant of the distribution of Log Z would be of order N in the low temperature phase : otherwise stated, the ratio of the standard deviation .of (see Eq. (6) ) to the average x of Log Z would be, in the low temperature phase,

o M whereas, in the high temperature phase, it is

x~ ^ N (38) 160 G. Toulouse

Strangely enough, within this hypothesis and as far as the function f (x) is concerned, the high temperature phase would appear more singular than the low tenperãturê phase. Thirdly, an approximation for the very low temperature properties. In the framework of the second hypothesis (corresponding to Fig. 6b), one can make an Rnsatz for the function f (x), requiring that it satisfies some known constraints, and derive frcm it the ground state energy and an expansion in T for the thernodynamic quantities. As an example, consider the following Ansatz :

fW = ~T- X(Y>.x + |§4--Log 2 , (39) with x = Y Ijog 2. The first and last terms have been chosen because they yield the correct low temperature asymptotic regime for < Zn >, the middle terms have been chosen on grounds of convenience and simplicity. It is required now that f (x) is tangent to the abscissa axis :

f(xQ) =0 , f(xo) = 0 , (40)

in order to satisfy the unitarity requirement :

< Zn > = 0 for n = 0

Besides, it is required that the first mcrant assumes its value (3), implying that f (x,) -xi=-Log2-^- , f' (xj) = 1 . (41)

It is easily checked that these requirements are satisfied, asymptotically

in y, i.e. at low temperatures, with X »(Y) = ^T+0 (TT) , b = boY SPIN GLASS MODEL 161

The ground state energy is then obtained, independently of \ and b, as

The agreement with the number obtained from numerical simulations should not be over-emphasized because the choice made for X was partly made ad hoc. At higher order in Tr one finds for the free energy

with X Co = 4 I 1 - , , I * 0,26

As a consequence, one finds for the specific heat an initial contribution linear in T, with a coefficient of value snail enough to be conpatible with numerical simulation estiirates :

§ = 0,016. Z + O (T2)

Finally, the Ansatz function f (x) can be expanded in the vicinity ot its minimum : 2 3 f(x) = A (x-xQ) +o[(x-xo) ] ,

and the coefficient A is then found to be of order Tz, at low temperatures,

A = 23 (4-) +0 (T3) J

As a consequence, from formulae (6) and (36), the second cumulant K2 of the 162 G- Toulouse distribution of Log Z is found to behave at low temperatures, like

This somewhat primitive calculation has been presented as an illustration. One could contemplate a larger programe, incorporating more constraints (behaviour of < Z2 >, systematic corrections for high n) and encompassing the whole temperature range from T = 0 to T = Tc. 7, In presence of an external magnetic field This section examines the modification of the results of the previous sections, in presence of a magnetic field H. The annealed model of Section 2 is easily solved since Ny2 < 2 > = (2 ch 6)N e 4 , 5 = pH (8) at dominant order in N. Therefore the annealed free energy F is obtained as

•f =-g-TLog(2Ch|) . (42)

In the phase diagram (T,H), there is a line T (H) on which the entropy

S= vanishes. Since, for H small,

•f = LOg2-g2--^r +

and, for H large, S - Si

one gets for T (H) the drawing of Fig.7. Ü?IN GLASS MC. Ei 163

Figure 7

Por H sjrall, T. (H) depends quadratically on H

(43) Ta(H) 2/tog 2

the external field H producing simply an effective vádth for the probability 'distribution of the bond values. An important fact mist now be noted : the quenched and annealed models are no longer identical in the presence of an external field. Indeed, the saddle-point in y-snace moves away from the origin, as is shown on Fig.8 (vrtiich replaces Fig.2).

th'S

Figure 8 - Toulouse

For u small, Eqs. (16) and (17) are replaced by

n n 2 2 In(u) = (ch 6) + ^ (ch5) Qn-l)th 6 + Ü + 0(u ) (44)

2 2 H 2 Cn(u) = th 5 + uQ + 2(n-2)th 6 - (2n-3)th 6] + O(u ) . (45)

As a consequence, at daninant order in N, < Z > written as N{n£_L(y)) < Z > = e with

y - Log In fty) - n

u = yy and ^ = Cn(u)

can be expanded at high tetrperature (in powers of y and 6) and for n not too large, n < n (T,H),

Nn N{Sli + ^ T2th*fi + ... } < Zn >

It is recognized that, in contradistinction to (21), one has now, even at high temperature, < Zn > f < Z >n

abolishing the equivalence between quenched and annealed models. Indeed, the expansion above (46) for < Z > has the log-normal form (26), and one can extract from it a parabolic expression for the corresponding function f(x), defined in (28), - •

z f(x) = A(x-xQ)

yielding for the average of Log Z

\ < Log Z > = xQ = Log 2 ch f + ^ [l - th" |] , (47) 165 SPIN GLASS MODEL and fear the second cumulant

2 2 1 (48) K2 = < (Log Z) > - < Log Z > = ^r- zF** T

One is reminded that these expressions are not closed analytic expressions but beginnings of expansions at high T and low H.

This widening of the probability distribution of Log Z is interesting because it suggests how an external field might allow a snooth

transition from the high temperature phase, in a way which would be

superficially analogous but deeply different from what occurs for a homo-

geneous fermiagnet. The physical idea would be that the external field

has the effect of breaking the system into a mosaic of disconnected parts

leading thus to a smearing of a transition, when the temperature is varied

under fixed applied field. Of course, this interpretation would be in

best agreement with the second hypothesis presented in Section 6.

The previous results can be illustrated by a drawing of the function

f (x), as sketched in Fig.9. It shows, in reverse, hew the singular limit

of Fig.4 is reached in zero field.

f(x)

Figure 9 166 G. Toulouse

The calculation above has been done assurrting no replica-symnetry breaking, i.e. searching for the domijiant saddle-point on the positive first diagonal in y-space. Staying within this hypothesis, as the temperature is lowered, the following expansion in y will be useful. Defining, as in Section 3,

< Zn > = exp tl { ^- + n log 2 - R(y)} ,

R(y) is obtained, up to fourth order in y, and for 6 snail, as

n 2 íi 2 2 2 t 2 R(y) = - n log chi - -Ç^-.th 6.Yy+ ^-Dl-v )-2(n-2)th 6+(2n-3)Y th' 6]y

|jn-2) + (n-3)

2 5 )- 3 + O(th 6)]YV + 0(y ) , (49)

and the saddle-point condition remains (13)

dR(y) _ n

This expansion allows to calculate < Zn >, up to second order in n, and up to fourth order in 6, at T = J ,

_ N{nL4 + ^ 2 + T " 3^ + 96 ^J+ 47T L1" ~r -6-J}' (50)

u where 6 = — . The second cunulant is found to be third order in 6 , in- J stead of second order at high temperatures, and still positive, though with a dangerous negative fourth order contribution in 6. Actually, if one considers the same quantity, but up to third order in 6 and up to fourth T-T order in T, with T = c , t < 0, one will get a positive contribution _ c in 63 coming from (50) and a negative contribution in T1* caning from (34). SPIN GLASS MODEL

The second cumulant vanishes on a line given, in the vicinity of the

point (H = 0, T = Tc), by the fontula

!t>st probably, the calculation has became incorrect at temperatures higher than the ones given by the above condition, that is before the vanishing of the second cumulant is reached. At low tençeratures, and for the same purposes, one writes

2 3 R(y) = n Ri(y) + n R2(y) + O(n ) with

Ri(y) =-Ç + ^

H2iy) = i + [J where Jl(u) = -4=- f° dt e't?i Log —CO

J2(u) = -^ f^dte"^ (Log

In presence of an applied field, formula (20) becomes

/ e dt+e^6 / e dt } (52)

The position of the saddle-point is found as : 16S S. Toulouse

y - .-/I.e-^+ and finally, one obtains

Zn > = e

-i- n2[-Y2( - e"6-^ - i )+O(Y)]+O(n3)} (53)

From this expression, one extracts the following results of the SK hypothesis for the low temperature behaviours of the free energy and

the second cumulant of log Z :

2 *L = _ II . e-^H _ ^ ^ e"t>2 dt+ J^ e-^ + 0(T2) (54) NJ -6 2TTJ

f-eo(H)]- | /I .e" ^ with eo(H)= -ò/l^- ^ /"*! e^ dt H

Tne last formula shows that the second cumuiant becomes unacceptably negative for

e£(H) > H2 +Í- (55)

It is then possible to sketch in the (H,T) phase diagram the line where

the second cumuiant (55), calculated à la SK, vanishes (Fig.10). SPTM CLASS MODEL 169

H \ • \ '. \ • \ \ . \ . \

T J Ficiure 10

On the same Fiq.10, the dotted line sketches the locus where the entropy, calculated ã la SK, vanishes and the dashed line the locus where the instability of the SK saddle-point, as calculated by de Almeida- Thouless (8), occurs. Fig. 10 is a summary of the bewilderment which springs from the innocent-looking hypothesis of no replica-symmetry breaking.

8. The spherical model The spherical model is obtained by relaxing the condition that each spin has unit norm and replacing it by the weaker constraint that 2 I S± = H

It is also obtained as the limiting behaviour, for m -*•«•, of the exchange Hamiltonian (1) with m-component spins. Indeed, following the calculation of Ref.9, with the normalisation choice |s| = iin, one finds at low temper- 170 G. Toulouse atures T < J :

< Log Z > = Urn [_G + ^J where G is a function of y, and

2 2 2 (Y 1)2 K2 = < (log Z) > - < Log z > = Nm [m3 - ^ ] .

This shows how, in the limit m •+ •», one recovers a oositive K2. Moreover, this kind of behaviour is in accordance with the second hypothesis of Section 6, corresponding to Fig.6b, that has been formulated for the Ising case.

9. Conclusion

This study, which stresses the importance of the probability distribution of Log Z, calls for a net/ generation of numerical simulation studies. More detailed knowledge of the variation of this distribution, with temperature-and applied field, would provide much useful guidance for solving the analytical mysteries, .^steries which constitute one of the main challenges in the history of statistical mechanics.

Acknowledgements Daily discussions With Jean Vannimenus are acknowledged with high gratitude. SPIN GLASS MODEL 17

References

(1) D. Sherrington, S. Kirkpatrick, Phys. Rev. Letters, 35, 1792 (1975), S. Kirkpatrick, D. Sherrington, Phys. Rev. B _T7, 4384 (1978).

(2) A.J. Bray, H.A. Moore, Phys. Rev. Letters, 4l_r 1068 (1978), A. Blandin, J. Physique, C-6, 39_, 1568 (1978), G. Parisi, Phys. Rev. Letters, 43, 1754 (1979). (3) D.J. Thouless, P.W. Anderson, R.G. Palmer, Phil. Mag. _35# 593 (1977). (4) B. Derrida, to be published (Phys. Reports). (5) S.F. Edwards, P.W. Anderson, J. Phys. F 5, 965 (1975). (6) D. Sherrington, J. Phys. Ajl, L 185 (1978) . (7) W. Feller, An Introduction to Probability Theory and its Applications, (Wiley, 1966). (8) J.R.L. de Almeida, D.J. Thouless, J. Phys. A JU, 983 (1978).

(9) J.R.IJ. de Alneida, R.C. Jones, J.M. Kosterlitz, D.J. Thouless, J. Phys. C U_, L 871 (1978). 173

CONNECTIVITY AND THEORETICAL PHYSICS: SOME APPLICATIONS TO CHEMISTRY *

H. E. Stanley ('), A. Conigiio, W. Klein, and J. Teixeira (2) (') Center for Polymer Studies and (2) Department of Physics, Boston University, Boston, Massachusetts 02215 - U.S.A.

When Belita asked me to give the opening talk of this symposium, I accepted: I can never resist the urge to proselytize. In my opinion, the essential connectivity problems posed by polymer gels and low-temperature r^O and D2O present some of the greatest challenges to theoretical physics today. Although there are but a handful of theoreticians nowadays concerned with these problems, striking progress has recently occurred—in part through the close interplay between experimentalist and theorist. The gelation experts in the audience have already heard what I have to say on gel theory in tauch more detail than I could possibly present in an hour. Therefore I shall deliver a very elementary and non-mathematical introduction to the subject, suitable—at best—for "bedside reading." A brief outline of this talk is presented in Fig. 0. After a word of philosophy, we will start from the very beginning with a description of the basic phenomenon of gelation. Then we will describe "Model #1", a very elegant and simple model of gelation due originally to Paul Flory and elaborated extensively by Walter Stockmayer. We shall see that this model is equivalent to the random—bond percolation problem. Flory solved his model exactly for the case that the lattice is a Cayley tree—i.e., a pseudolattice in which cycles or "loops" do not occur. Flory's solution allows one to obtain properties exactly for systems with dimensionality d greater than six! The inclusion of loops (d < 6) makes the problem much harder (as yet no exact solution has been found), though great progress has been made in recent years using techniques of modern statistical mechanics such as renormalization group, Monte Carlo computer simulation, and series expansions.

OUTLINE 0. Philosophy 1. The Basic Phenomena of Polyfunctiona! Condensation 2. Model #1: "Random Bond Percolation" a) Classical theory, Flory 1940: "no loops" (d > 6) b) "Loops" (d < 6) 3. Model #2: Site-Bond Percolation with Correlations a) Solvent effects b) "Queer" points (£p -+ °°. ÇT •*• °°) A. The Solvent Itself a) Is liquid water a "gel", at least at low temperature? b) Is amorphous silicon hydride a "gel"?

FIG. 0

*Talk delivered by HES. **John Simon Guggenheim Memorial Fellow, 1979-1980. •Permanent address: Laboratoire de Physique Thermique, École de Physique et Chimie, Paris. ++Supported by ONR, ARO, and AFOSR. 174 H.E. Stanley

We shall see that Model #1 neglects the solvent. To see that solvent Is Important, 1 want you to examine this specimen of a common gel—polyacrylamlde—which I am passing around the audience. By the end of the conference, I predict that this gel will occupy a great deal less volume, as it is prepared with a recipe that calls for 19 parts water of 1 part of acrylamlde and the water solowly evaporates from the gel network. Consideration of solvent effects by us was motivated entirely by our interactions with the experimental group of Toyo Tanaka, which has undertaken a detailed study of some unusual solvent effects. We shall see that a simple and straightforward elaboration—correlated site-bond percolation—of the original Model III is sufficient to obtain good qualitative agreement with the Tanaka experiments.

Finally, we will consider the gel solvent itself. In the sample you are examining, the solvent is water—~and we shall see that water itself is a hydrogen-bonded liquid whose highly unusual properties may be usefully Interpreted in terms of simple percolation and gelation considerations. We shall see that the same correlated-site percolation model used to explain the unusual properties of liquid water has also been very recently applied to elucidate the unusual behavior of amorphous silicon hydride, an amorphous semiconductor.

Before beginning, It is appropriate to state my owa degree of indebtedness to a number of individuals. In addition to those of my colleagues who join my as co-authors of this chapter, I must als acknowledge the fact that my own interest in connectivity arose from interactions with Bob Birgeneau, in connection with trying to understand novel features he had discovered in random magnets near their percolation thresholds. The theoretical models to be presented here were motivated strongly by discussions of experimental phenomena with C. A. Angell, R. Bansil, and T. Tanaka. I must also acknowledge fruitful Interactions with R. Blumberg, M. Daoud, A. Geiger, A. Gonzales, J. Leblond, S. Muto, P. Papon, S. Kedner, P. J.Reynolds, P. Ruiz, and G. Shlifer. Finally, I should state my indebtedness to the "grand masters" of polymer theory, the physical chemists P. J. Flory and W. H. Stockmayer. It is still very much a debatable point whether or not we physicists will ever significantly augment the marvelous theoretical framework built 30-40 years ago by the chemists!

0. PHILOSOPHY

The overall theme of this talk is that exhaustive si-udy of simple models—suitably generalized—can lead to Insight and eventual understanding of comparatively complex systems.

Much of the progress of the last decade in statistical mechanics can be summarized in the simple process illustrated in Fig. 1. One first seeks to abstract the essenti.il physics of a given system, and then one constructs a model that reflects this essential physics. Two familiar examples are

(1) a fluid near its critical point, for which the essential physics is an interparticle interaction potential characterized by a hard-core repulsion and short-range attraction, and

(ii) a dilute polymer solution, for which the essential physics is the hard-core repulsion (or "excluded volume") alone. CONNECTIVITY 175

SYSTEM ESSENTIAL PHYSICS • • MODEL

FLUID H.C. Repulsion I SR Attraction | • {_ n=l) (near C.P.) J DILUTE POLYMER Excluded Volume •—• {_ n=0 ) GEL Connectivity •• [s°l| (near gel pt.)

METRO MAPS

S-state discrete spins (Potts'52)

n-component continuum spins

FIG. 1

OH OH OH X ÇH, -CHjOH. HCHrO-CH-i

CH2 OH OH OH OH - H lal FIG. 2

•W- X H A

(bl (cl 176 H.E. Stanley

Useful progress has resulted on system (1) and system (11) using, respectively, two cases of the n-vector model—n = 1 and n ~ 0.

The third example shown In Fig. 1 is that of polymer gelation. In my opinion, the essential physical feature of a gel is connectivity, and to this end one expects the s = 1 case of the s-state Potts hierarchy to be relevant. On the other hand, temperature-dependent effects such as those due to the presence of solvent are excluded from this simple "pure percolation" case—and we shall see below that some straightforward extensions are necessary.

1. THE PHENOMENON

The simplest example that illustrates the basic phenomenon of gelation is polyfunctional condensation of monomers. The simplest example of polyfunctional condensation Is the situation shown in Fig. 2.

Suppose all our monomers are identical, and that each has f functional groups that can raact with one of the f groups of another monomer. The simplest case— f • 0 —produces no reactions at all; an example might be argon! The next simplest case, f • 1, results in dimers only. If f • 2, we can have unbranched linear polymers. For f ^ 3, we form branched polymers, as illustrated in Fig. 2 for particular f " 3 example, trimethoyl benzene. Each benzene ring has three reactive methyl groups. Methyl groups from two different monomers can react to form an ester linkage. This process is characterized by a parameter ot, termed the conversion, which is the fraction of reacted methyl groups. Clearly if a " 0 only monomers are present. If a > 0, there exist finite polymers of all sizes. However the probability of an infinite polymer is zero for all a less than a critical value a • For a > a , however, there is non-zero probability of occurrence of a single poSymer that is Infinite in spatial extent. The probability of this infinite molecule to occur jumps from zero (for a < a ) to unity (for a > a ). Thu3 the connectivity of the system changes drastically at a » ac, and this "phase transition" is termed the gelation threshold.

2. MODEL #1: RANDOM-BOND PERCOLATION

The first successful model to capture the essential physics of the gelation threshold was proposed 40 years ago by P. J. Flory, and developed in series of classic papers by both Flory and W. S. Stockmayer. The Flory-Stockmayer (FS) model not only predicts the gelation threshold, a - a , but also provides the "critical exponents" characterizing the behaviour 8f various quantities in the immediate vicinity of a .

The FS model is most easily explained after first introducing the problem of random bond percolation. Random percolation is to "connectivity" phase transitions what the Ising model is to "thermal" phase transitions: an appealingly simply model that embodies the essential physics yet is sufficiently subtle that it eludes exact solution. In percolation, however, we are still waiting for an "Onsager" to provide us with an exact solution in two dimensions!

Random bond percolation can be easily illustrated with the use of Fig. 3. Suppose we have an Infinitely high and infinitely long wire fence. Imagine also CONNECTIVITY 177

(c) P - 0.6 (d) p - 0.8

FIG. 3

(a) infiniCe cluster finite clusters plus t~mi.ce ciusLers

(bl gel phase ("infinite" molecule) sol phase plus (finite molecules) sol phase

FIG. 4 that a randomly chosen fraction p of the links of this fence are conducting while the remaining fraction (1 - p_) are insulating. Computer simulations of a finite (16 x 16) section of this fence are shown in Fig. 3 for p - 0.2, 0.4, 0.6, and 0.8. Clearly for pB small, as in part (a), the system consists of small clusters of conducting bonds. 178 H.E. Stanley

In (b) the conducting fraction p has doubled, yet the system still consists of only finite clusters . . . the "scale" has increased, but not the essential macroscopic conductivity. In (c), p - 0.6, and the system is macroscopically different: in addition to the finite clusters, there is a single cluster that is infinite in spatial extent (of course, the fence must be infinite if the cluster is to be infinite!). For some value of p in c between part (b) and (c), there is a threshold pfl ; below p c, the fence cannot conduct, while above p it can. Thus its macroscopic properties change suddenly as a microscopic parameter p c c increases infinitesimally from p - & to pn + 6 (Fig. 4a). o o

What is the effect of allowing for loops? Clearly the threshold is expected to increase, since extra bonds will be formed that will merely create a loop rather than contributing to the formation of an infinite branched network. Moreover, we expect that the behaviour of the system in the immediate vicinity of the gel point to be characterized by different critical exponents. In fact, if we are to believe the utility of lattice models, then it turns out that exponents are shifted from their Cayley tree values quite considerably (Fig. 5). Of course, one could well question the appropriateness of a lattice model to represent a continuum system. Hence much-needed are calculations for "continuum percolation" that are sufficiently accurate to make meaningful predictions concerning critical-point exponents.

Is there experimental evidence for departures for "classical" critical point exponents? A literature search focussed on answering this question was recently carried out by Brauner, who concludes that no clear-cut answer emerges despite rather extensive analysis of existing data. Much of the problem has to do with the paucity of data extremely close to the gel point—since only there are exponents expected to depart from their classical values. The very recent Ph.D. thesis of Manfred Schmidt, working in Walther Burchard's laboratory in Freiburg, may answer this question more definitively.

3. MODEL #21 CORRELATED SITE-BOND PERCOLATION

In Model #1, solvent effects are not included. Nor are temperature effects included in any statistical mechanical fashion: all 2 states ol a CONNECTIVITY 179

(a) THERMAL (b) PERCOLATION (c) GEL

coupling constant bond probability "conversion"

v • tinh (J/kT.) PB • percolation pt. °c » gel point

droplet bond cluster polymer

M P(p) Wg = wt. fraction XT S(p) D.P. = deg.of poly

C(H,T) < N . > nolfcules clusters

Husimi-Tsmperley Fisher-Essam Flory

d+ - (2B + y)N -4 d+ = 6 d+-6

FIG. 5

system consisting of N bonds are equally probable. This simplifying feature has great merit in that Model #1 Is extremely tractable. However we know that solvent effects are quite profound in gelating systems.

In FS theory, all sites are occupied by monomers. Two simplifying assumptions of the FS theory are

(i) the absence of solvent molecules, and (ii) the absence of correlations between the molecules.

Recently both solvent effects and correlations have been taken into effect via "Model #2".

(a) Solvent Effects

Suppose we allow the slteB to be of two sorts, M and S. M sites are occupied by monomers and S sites by solvent. The original "random-bond" percolation problem is now a "random-site-bond" problem, and the phase diagram is given in Fig. 6, and the FS critical point in the simple phase diagram of Fig. 4b is now an entire "line" of critical points.* If p is the density of M sites, then FS theory corresponds to the special case p - 1 (heavy solid line). A typical experiment corresponds to moving to the right along the horizontal dashed line. 180 H.E. Stanely

FIG. 6

-U

FIG. 7 (b) Correlations Among Molecules

We shall assume that the monomers and solvent molecules are not randomly distributed among the sites. Rather, we shall assume a correlation of the standard lattice-gas model sort (Fig. 7). In specifying the interactions, we must consider that the monomers can interact with each other in two ways. One is the usual van der Waals interaction, and the other is a directional interaction that leads to chemical bonds. The particle-particle interaction of this system is reasonably approximated by the following nearest-neighbor interaction.

— W " solvent- solvent

W ~ c-U * solvent-monomer

i- W van der Waals (with weight P ) J HW U

*•— E bonding energy (with weight 1 - Pu).

The energy of a given configuration is then c M ^ H - - - W I (1)

where the summations are over nearest-neighbor parts of sites and

*There is RG evidence for "universality" along this entire line. CONNECTIVITY 181

TT S (IT ) = 1 if site i is occupied by a solvent molecule (monomer), and zero otherwise; obviously ir.s + i, "1 since each cell must be occupied by either a solvent molecule or monomer.

Finally, we take into account the fact that the bond probability is temperature dependent (Fig. S).

The resulting model has been solved for the.Cayley tree—i.e., in the intramolecular interactions. The resulting phase diagram is shown in Fig. 9. He see that in addition to (i) a line of connectivity critical points separating the high-temperature sol phase from the low-temperature gel phase, there exists also (ii) a phase separation curve characterized by a consolute temperature Tc below which the system splits into two seperate phases. We should emphasize that curve(i) corresponds to looking at the system through the eyeglasses of a "Percolation freak", whose only concern is connectivity, curve(ii) corresponds to looking at the same system wearing now the spectacles of an Onsager whose concern is with ordinary thermal phase Trans itions'and critical phenomenon.

The calculations are in qualitative agreement with recent experiments of Tanaka and co-workers. Quantitative agreement is not to be expected due to the complete neglect of intramolecular interactions.

It is possible to adjust the experimental parameters in such a fashion that the sol-gel phase boundary intersects the consolute point. At this point, labeled Q in fig. 10, we expect that both connectivity fluctuations and concentration fluctuations are simultaneously critical. Percolation is relevant to the former, while the lattice-gas (Ising) model is relevant to the latter. A particularly intriguing open question concerns the sort of phenomena that one might expect in the immediate neighborhood of such a "Queer" point. There is some evidence that the Q point has a much richer structure than the ordinary percolation point.

PJT)

•mall

FIG. 8 182 H.E. Stanley

SOL

T..

FIG. 9

SOL »M«B Get PM»SS

if,--.

(a) t (b)

FIG. 10

4. THE SOLVENT ITSELF: ARE CORRELATED PERCOLATION CONCEPTS RELEVANT TO WATER?

In Model t2, we neglected the possibility of bonding between solvent molecules. This assumption may be appropriate for many gel solvents, but almost certainly is not OK for water—which is a strongly hydrogen-bonded solvent. The hydrogen bonds are thought to be directly responsible for the highly unusual properties of water at low temperatures. Three of these are described in the following section.

5. WATER AT LOW TEMERATURES: THE OBSERVED PHENOMENA

(i) Mass Density

Fig. lla shows schematically the temperature dependence of the mass density p~ found for many liquids, while Fig. lib shows the situation for water. At high temperatures, water is qualitatively not unlike a normal liquid, but at low temperatures p~ actually'decreases as T is lowered. This unusual behaviour is accentuated when one examines the Bupercooled domain (Fig. lie). In fact, at the lowest temperatures on which reliable density data exist (about -32°C>, P has decreased almost 3Z from its maximum value (at +4 C). CONNECTIVITY 163

DENSITY

a) Normal Liquid

b)H20

4°C

c)Supercooled H20

4°C

FIG. 11

(11) Isothermal Compressibility

The isothermal compressibility K Is a measure of the density of the system to a small pressure change, it. is also a measure of the spatial density fluctuations. Fig. 12a shows the situation for water. Again, at high temperatures water is qualitatively not unlike other liquids, but at low temperatures K actually increases as T is lowered. This unusual behaviour is exaggerated as T is decreased below 0°C (Fig. 12c). At the lowest temperatures, K has increased by roughly 100Z from its minimum value at 46 C. 184 H.E. Stanley

COKPRESSIBILITY

a) Normal Liquid

KT1

b)H2O

O°C 46°C T

KT1

c)Sup«rcool«d HO

46°C T

FIG. 12

(ill) Transport Quantities; Self-Dlffualon and Shear Viscosity

For many liquids, a semi-log plot of the self-diffusion coefficient D vs. 1/T is roughly linear (Fig. 13a). At high temperatures, water is roufhly the sane; however, at low temperatures D drops off.much faster (Fig. 13b). Finally, foe supercooled water, this anomalous decrease becomes even more pronounced (Fig. 13c).

There are many other anomalies in transport properties. For example, at high temperatures pressure acts to decrease D , just as for most fluids. However at low temperatures water moleculis diffuse "faster" under pressure! The above remarks about D apply equally to [ n J"1 , where n is the shear viscosity. CONNECTIVITY 185

SELF-DIFFUSION SHEAR VISCOSITY log 08 logl

a) Normal Liquid

I/TV I/T

log D, logl

b) H20

I/T, I/T M h logD, logl. c) Supercooled H,0

I/T, I/T M FIG. 13

6. WHY ARE WE INTERESTED?

One reason we are Interested in the behaviour of supercooled water is that the anomalous properties of normal water are greatly exaggerated. Fcr example, the thermal expansivity of normal water is negative only for 0 < X < 4 C, while if we include supercooled temperatures this temperature range is increased by almost a factor of ten.

A second reason is that Angell and his collaborators have discovered that many properties of water, when plotted on log-log paper against the variable t= (T - T )/T are found to lie on straight line. Here T is regarded as § frie variable and is adjusted to make the data linear: for most functions a TB -45°C, which lies on the vicinity of TH (the homogeneous nucleation temperature)', a temperature 10-20 C lower than the lowest temperature at which reliable data are generally obtained. The obvious caveats are that A

186 H.E. Stanley

(I) The smallest values of e are perforce somewhat large; hence instead of data over, say, several decades (as in many critical phenomena experiments), one has data over less than one decade.

(II) The exponent values are somewhat unusual (e.g., y - 0.3) unless one takes pains to try to extract the singular part of the function in question, and of course there is considerable flexibility in this procedure. Angell and collaborators have displayed considerable ingenuity and physical insight in adapting this procedure to the data in question—with the result that, e.g., the effective compressibility exponent increases to 0.9 or so. Interestingly, most of the exponents for other singular functions are rather close to unity, when suitably corrected.

7. "POSSIBLE CLUES" I'd like to present this section somewhat like a mystery story. First one identifies clues, then one make a "hunch" (of course, in theoretical physics a hunch is termeti a conjecture!). Finally one must test the hunch by the extent to which it is consistent with all the known facts of the case.

(1) The first clue is provided by examining the solid phase. Fig. 14 shows a schematic illustration of the local environment of an oxygen atom in the phase "ice I.", indicating the fact that there are four nearest-neighbor oxygens, C linked by hydrogen bonds. The ice I lattice formed by the oxygen atoms * has coordination number z - 4, and the angle made by any two of the four J hydrogen bonds is very close to the tetrahedral angle, cos~l(-1/3) - 109°. J, (ii) Â second clue is provided by observations of Linus Pauling on the - isoelectronic sequence H2Te, H^Se, ^S, and H.O. As we see in Fig. 15, .-. the melting and boiling temperatures are elevated by roughly 100 C from what one might have anticipated from the values for the non-hydrogen bonded '•• molecules. Also the latent heat of vaporization is increased by 100%, indicating that the additional "interactions" (hydrogen bonds) present in H.O have a rather profound effect in stabilizing the more ordered liquid (or solid) phase, relative to the less ordered gas (liquid) phase.

(Ill) A third clue is provided by examination of the latent heats of fusion, vaporization, and sublimation (Fig. 16): clearly most of the hydrogen bonds do not break on melting ice!

(iv) A fourth clue concerns the fraction of intact hydrogen bonds as T is lowered below 0 c. The IR spectra reproduced in Fig. 17 suggest that the fraction of broken hydrogen bonds is a smooth function of T, even when T passes through 0°C.

FIG. 14 From D. Eisenberg and W. Kauzmann, The Structure and Properties of Water (Oxford University Press, New York, 1969). 187 CONNECTIVITY

i- L

x x.

1 o o I- O o o o o pi"-

H.E. Stanley 188

0.7

0.6

0.5 UJ o 0.4 m S 0L3 tn m < 0.2

0.1 0 1.30 1.40 1.50

FIG 17 Part (a) is from CA. àngell. Cháp. 1 of F. Franks, Hater: A Compre: hensive Treatise, Vol. 7 (Plenum, NY, 1980). CONNECTIVITY ' 189

8. A "HUNCH"

I have always believed that an essential feature of theoretical physics is willingness to "play"—as soon as we take ourselves too seriously, we are less apt to play, creativity is hampered, and we become as "pompous" as most of academic society. So let us put the "clues" in the back of our mind for a minute, forget about real water, and simply play the following well-defined game.

Begin with a huge ice 1^ lattice, and randomly place bonds between a fraction pB of the nearest-neighbor pairs of sites. Now color a site black if it has four intact bonds, and white otherwise. This is all there is to the game.

Next we ask two questions:

Question (i): The "Census-Taker"

Since the bonds are placed randomly, the fraction of 4-bonded "black" sites is simply

f4 - *B4- <2> Thus we conclude that (a) f, is quite small unless p is rather large, and (b) there is an "amplification mechanism" implied by the form of equation (2): a 1% increase in pg implies a 4% increase in f,. For example, when pgis 0.8, f. - 0.4096 (Fig. 3d), but when p = 0.9, f. has increased by 60% to 0.6561.

Question (li): The "Sociologist"

A more sophisticated question concerns the "connectivity" aspects of the black sites. At small values of f^, tt.e black sites mostly form isolated one-site clusters, but as f, continues to increase the mean cluster size S(f^) increases rather dramatically. Moreover, the clusters of black sites are more "compact" (less ramified) then the corresponding clusters in random-site percolation. This is because the problem we have defined in our game is a correlated-site percolation problem. In random-site percolation, each of thiê2^ sites of an N-site system is equally probable, while in the present corre.tated-site problem, many of these 2 states cannot occur at all (e.g., fi£. 18).

We can visualize the "compact" nature of these black-site clusters easily by means ofcomputer simulations. Since it is easier to show you- 2-dimensional rather than 3-dimensional lattices, let us consider the square lattice. Like the ice lattice, it too has coordination number z - 4. The sites of each of the four random-bond -simulations of Fig. 3 have been colored according to the rules of our game. Fig. 19 is still another simulation, with p increased still further, and the number of sites increased from Z56 to 2500. Since it is hard to visualize the clusters of black sites when p_ is so large, we have removed all bonds except those joining nearest-neighbor pairs of black sites.

Thus far, we have accomplished nothing—other than defining a rather novel sort of correlated-site percolation problem. It is "novel" because the correlation arises from a purely random process, that of random bond occupancy, rather than 190 H.E. Stanley from an interaction such as the lattice-gas interaction that served to partition the sites of "Model #2" into two colors.

Host games that we theoretical physicists play are not relevant to anything. However despite the fact that;there are almost as many different models of - water as there are authors on the subject, it seems that no model has systematically applied the concepts of conectivity (or "percolation theory") to the hydrogen-bond network present in water—at least at low temperatures. At first glance, it might be tempting to regard our game as the construction of topological maps that correspond to the connectivity patterns present in water at low temperatures. Thus we could call the sites "oxygen atoms", and the bonds would then be "hydrogen bonds". Computer simulations, such as those of Figs. 3 and 19, correspond to photographs with the following two very peculiar properties: :- ..-._-

(a) The shutter speed is much shorter than the bond lifetime (which is probably less than a picosecond), and

(b) the film is sensitive only to the connectivity properties of the

o—®-—o o— I*) to

FIG. 18 CONNECTIVITY 191

ia) pB = 0-875 (6) Pt = 0 90

(c) pB = 0-925 (d) Pa = 0-88

FIG. 19 From H. E. Stanley, J. Phys. A12, L329 (1979). 192 H.E. Stanley hydrogen-bonded network—two atoms can be close to each other but if their net potential interaction energy is not beiow a threshold V , they are not considered bonded and no line is drawn. The subtleties Involved in the precise choice of the "threshold parameter" V has been thoroughly elucidated by Stillinger, Rahman, and their co-woriers in seeking to interpret results from molecular dynamics. Since the Interaction potential between tuo isolated water molecules is strongly dependent on their relative orientation, it is to be expected that a given water molecule can be unbonded to molecules that are closer to it then those to which it is bonded. Indeed, x-ray evidence suggests that more Chan four water molecules are within a "nearest-neighbor" sphere about a central water molecule, yet for realistic choices of VHB essentially no water molecules have more than four intact hydrogen bonds.

In short, our "game" results In the construction of an infinite hydrogen-bonded network (i.e., a "gel"), characterized by bonds of extremely short lifetime. Since this system cannot "act like a gel" unless we were to interact with it using some probe of characteristic time much shorter than the bond lifetime, one might at first wonder if this game has any physical implications at all!

To answer this legitimate concern, let us focus on the simulation of the hydrogen bond networks of Figs. 3d and 19. We have a gel, well above the gelation threshold, and characterized by the existence of patches the mean size of which is rather small. It is important to stress that these patches are local regions of the infinite hydrogen bonded network characterized by four-bonded molecules; in no way do they resemble "clusters" of ice in an unassociated "normal liquid".

Suppose p = 0.9, and let us return to our metaphore of a Maxwell demon sitting on a given oxygen atom. Oxygen atom A is in the center of such a patch, and our demon measures the distances to his nearest and next-nearest bonded partners. Oxygen atom B is not in a patch, but does belong to the gel molecule.* We can reasonably assume that the distances to the nearest and next-nearest bonded partners of atom B will be modestly larger than for atom A—from the very fashion In which bonding is defined. To the extent that this assumption is plausible as a working hypothesis, we can conclude that the local density well inside the patch is just slightly smaller than the global density of the entire hydrogen-bonded network (gel).

It is possible that one might confuse our proposed picture of low-density patches of an infinite hydrogen bonded network with a class of "cluster" models that date back to Roentgen (1892) and have been developed by many workers, including a series of papers by Scheraga and his collaborators in the 1960's and early 197O's. The cluster models by definition Imply that the majority of oxygen atoms are unbonded and four-bonded—i.e., they lead to a "bimodal" histogram for the density of each species f . of oxygen atoms with j intact hydrogen bonds. By contrast, the present theory leads to unimodal distribution, namely the simple binomial distribution V

•Almost all the atoms belong to the infinite conected network, since the fraction f of unbonded oxygens is (1 - p ) - 10 PB) CONNECTIVITY 193 where z * 4.

A second possible confusion concerns the relations betweeen topologic distinction between real space configurations of H,0 molecules, and the sort of connectivity (or "topological") diagrams such as those of Figs. 3 and 20. The connectivity diagrams are cnly representations of the specific connectivity patterns among oxygen atoms, and are definitely not representations of the real-space configurations of these atoms. Indeed, many different possible molecular distributions in real space can correspond to a single unique connectivity pattern: to each oxygen atom configuration in real space there corresponds a unique configuration in our "connectivity" space, but the converse is of course false.

A third possible confusion is the apparent absence of any "energy term" that can give rise to the apparent "clumping" of the four-bonded oxygen atoms (I am grateful to Frank Stillinger for emphasizing this concern). The black sites clump because they are correlated, yet no energy is present!

Previously-studied correlated connectivity problems involve "clumping" than specifically arises from an energy—e.g., the lattice-gas or Ising interaction. Here, the clumping arises from a purely random process, that of random bond occupancy. This is a subtle and extremely surprising point, at first glance, and certainly it took its instigator (HES) a long time to believe that the black sites could clump without being "driven" by an energy term. Indeed, the magnitude of this "énergy-less correlation" can vary from an infinitely strong interaction (e.g., a site whose four neighbors are black must itself also be black) to very weak (e.g., a site only one of whose neighbors is black has but a slightly increased probability of being black). A useful example is shown in Fig. 18 for which p - 0.8, so that f, - 0.4096. In (a) we consider an arbitrary site A of an ice lattice, all four of whose neighbors are black. The conditional probability pA that site A is black is of course

PA-l»f4. (4a)

In (b), three of the neighbors are black and nothing is known about the fourth neighbor, so that

\ - PB > V . W>) In (c), two neighbors are black, and PA 'P B2 " °'64 > V (4c) In (d) one neighbor is known to be black end nothing is known about the other three neighbors, so that

P = p 3 A B ° 0-512 > f4. (4d)

Finally, in (e) nothing is known about the four neighbors, and hence

4 PA - pB - 0.4096 - f4. (4e)

The above example concerns the tendency of blaclc sites to clump, despite the 194 H.E. Stanley

2 3 4 j O I 2 J3 4 j (b) (C)

FIG. 20

IO9O, (cm*S-!) 10

2 I 5

.2 .1 .05 -50 50 T/#C

FIG. 21 CONNECTIVITY 195 clear absence of any interaction. The converse effect—the tendency of black sites to appear Infrequently in regions of white sites—is also readily exemplified. For example, we can calculate the probabilty W^ that a randomly chosen site has four white neighbors; H^ is, equivalently, the weight fraction of the gel consisting of 1-site patches.

In the uncorrelated problem, we have

since the probability that each of the four neighbors is white is simply one minus the probability that it is black. On the other hand, for the correlated problem,

w C 4<1 3)4 (5b) i " PB ~ PB • by the same reasoning. Clearly

VJ^BJ", (6)

with the equality sign holding only in the limit p. —> 0. In fact, for our example with p • 0.8, a C W1 - 0.023 ; Wj" - 0.050. (7)

The effect of correlations is thus to reduce by more than half the probability of finding an isolated black sitel Consistent with intuition, the ratio H. /W. decreases as pjj increases, in a very non-linear fashion. The effect of the black-site correlation becomes stronger and stronger and as pg approaches unity, this ratio approaches (3/4) - 0.32.

In the time that it takes to enjoy a good feijoada, one can calculate on one's napkin the probabilities W for s-site patches of black sites in a gel for s up to 3 or 4; beyonS that, it is a job best left for the computer. Clearly the sum on all s of these weight fractions W must equal p , the fraction f, of all the sites that are black. . . and tnis "sum rule" will make you more confident in the accuracy of your calculation.

The interesting fact is that in all cases, the effect of black-site correlation increases as p_ increases. Since p increases when the temperature T decreases, we have a clumping effect which increases as T decreases, just as might have been expected were there an explicit "energy term".

Before concluding, we must make clear that the present "hunch" is not in contradiction with any of the current "random network models" and molecular dynamics pictures of water—in fact, we shall see in the next section that the results of molecular dynamics provide one of the more striking tests of the quantitative accuracy.of this approach. However it is important to qualify our enthusiasm by emphasizing that this picture is very much a "zeroth order approach"—and many systematic improvements immediately Buggest themselves. In the next section we shall see that even in its zeroth order form, the present picture provides a physical mechanism for understanding—at least qualitatively—the unusual behavior of water at low temperatures. 196 H.E. Stanley

9. TESTS OF THE "HUNCH"

A. EXPERIMENTAL DATA

It is one thing to construct models, and It is quite another to put these models to qualitative and quantitative tests. A detailed comparison with existing experimental data—as well as suggestions for further experiments designed to test the "hunch"—is under development. The results are quite satisfactory in the sense that all measured properties are in qualitative agreement with the predictons. Here space permits us to summarize only three of the findings, corresponding to the three anomalies discussed above in Section 5. A complete discussion has recently been submitted to the Journal of Chemical Physics.

(i) Mass Density.

In part (i) of section 5, we discussed the fact that the density falls off increasingly rapidly as T is lowered below 4°C, the TMD ("temperature of maximum density"). This fall-off is clearly rationalized by the picture, as is the decrease of the TMD with pressure and with the addition of most impurities, both of which serve to "weaken" (or "strain") the hydrogen bonds.

(ii) Isothermal Compressibillty.

The isothermal compressibility is directly proportional to the density fluctuations. For most liquids, these decrease as T decreases. However in our picture, there is a second competing effect: as we lower the temperature, the low-density patches (and hence the density fluctuations) increase. The two physical phenomena—normal vs. bond-dominated phenomena—balance at 46°C, while below 46°C, the net compressibility actually increases 1

(iii) Transport Quantities; Self-diffusion and .Shear Viscosity.

Thus far our discussion has concerned the static properties of water—each computer simulation corresponds to a snapshot of water. Suppcse we take

iigg snapshots, each separated by a time interval TB (the hydrogen bona lifetime). This is a very crude picture of dynamics—but it is amusing to note that it provides a qualitative explanation of the transport anomalies of low-temperature water. As T is lowered, the weight in the unimodal distribution of Fig. 20 shifts from low values of IL,. to larger values of njjB, and there is a corresponding decrease in the "mobility". Suppose Fj. is the fraction of "mobile" molecules; e.g., F.. might be some linear combination of the unbonded species [occurring with density f » (l-p)4] and the singly-bonded species [occurring with density f, " 4p(l-p)3], The fraction of immobile atoms is then Fj. - 1 - F... If we neglect the spatial correlations, then the probability that oxygen atom A Is immobile during the time interval 0 < t < TB is given by F_ . The picture of dynamics mentioned in the previous paragraph implies that after time TB, we create a new random-bond realization, so that, the probability that oxygen atom A remains immobile during the time interval 0 < t < 2TB 1B FJ . let us now define the parameter n through i, • nig, where xR is the rotational relaxation time. Then we have

TR/T FI«=(FI) B =1 (8) CONNECTIVITY 197

TR - <-ln2)TB/ln Fj. . (9)

Equation (9) relates the rotational relaxation time to the mean hydrogen bond lifetime and the density of "immobile" molecules. As T decreases, F decreases and T_ certainly does not decrease; hence (9) shows that the effect, of the hydrogen bonds Is to increase i . Since n~Tp and D ~ n the experimental anomalies mentioned in Section 5(iil) are explained qualitatively. Preliminary calculations suggest that Eq. (9) is also in quantitative agreement with experimental data (Fig.21).

B. MOLECULAR DYNAMICS COMPUTER SIMULATIONS

At the present time, molecular dynamics computer simulations provide us with detailed information against which to test the microscopic predictions of our "hunch". Very recently, Alfons Geiger of Karlsruhe has analysed the famous Rahman-Stlllinger molecular dynamics data from the viewpoint of both the "census taker" and the "sociologist". He finds complete accord in both " categories of analysis. The fractions f of oxygen atoms belonging to each of the 5 species (j=O,l,. . . ,4) confirm the predictions of equation (3), while the connectivity properties of the black sites are consistent with eqn. (5). Gaiger has also calculated the weight fraction Wa of black sites belonging to clusters with 8*2,3,4,. . . sites and thi agreement with the predictions is quite encouraging. Finally, he has calculated the "total number of black-site clusters", and this too agrees. While Geiger's analysis is still unpublished, the degree to which the predictions are borne out of his microscopic calculations is quite encouraging.

10. CONCLUSIONS

Let me summarize In the brief time remaining. The subject of thermal phase transitions is an ancient subject. Recently, there has been a sustained interplay among both exerimentalists and theorists, and considerable progress has resulted. The relatively simple conceptual topic of connectivity is at least as difficult to treat formally as the subject of thermal phase transitions. It is hoped that a sustained interplay between workers from many disciplines will result in comparable progress in this subject over the next decade or so.

In this talk we have tried to illustrate the potential for this hope being fulfilled. Specifically, we have treated two classical problems—polymer gelation and liquid water. In the first problem, the concept of connectivity has been used since the classic work of Flory and Stockmayer in the 1940's. In the second problem, the concept of connectivity has only very recently been proposed to be of possible relevance—and It remains to be seen if this proposal Is In fact correct.

In both problems, generalizations of the original 1956 percolation problem are necessary to describe the system under consideration. In the first problem, one requires a correlated-site model where the correlation is introduced through a lattice-gas or Ising interaction, while in the second problem one also requires a correlated-site percolation, but the correlation arises 198 H.E. Stanley from the definition of the site species—the fundamental physics is that of random bond occupancy. Ihe first model haa been studied for several years and is almost certainly relevant for a number of physical systems. The second model, on the other hand, is completely new and may not•turn out to be relevant for water.

To conclude on a more optimistic note, we should note that even if the new correlated-site percolation model should turn out to be totally irrelevant to the physical mechanisms operative in low-temperature water, it is still likely that it will be relevant to some physical system. For example, very recently Broksky has successfully applied this model to a solid-state system, aSiHx (amorphous silicon hydride). Specifically, Broksky proposes that pure Si corresponds to our model with pg =1. Hydrogenation corresponds to breaking covalent Si-Si bonds, and therefore correspond to decreasing pjj from unity. By doing band theory calculations on the resulting system of disconnected Si patches, Brodsky succeeds in predictine the results of several hitherto unexplainable experiments. \J 199

RANDOM MAGNETISM

C.TsaUB Centro Brasileiro de Pesquisas Físicas/CNPq Av. V.'snceslau Braz, 71 - Rio de Janeiro - Brasil

ABSTRACT The "ingredients" which control a phase transition in veil defined systems as well as in randan ones (e.g. randan magnetic systans) are listed and discussed within a somehow unifying perspective. Among these "ingredients" W»-£ÍMB> the couplings and elements responsible for the cooperative phenomenon, the topological connectivity as well as possible topological incompatibilities, the influence of new degrees of freedom, the order parameter dimensionality, the ground state degeneracy and finally the &ti(&ti Jo ix\\A_jx\\A "quanticity" of the systemJTH^rgeneral trends, though illustrated in magnetic systems, essentially hold for all phase transitions, and give a basis far connection of this area with field theory. Theory of dynamical systems, etc.

7. — Since the early 70's a very rapidly increasing (theoretical as well as experimental) effort has been dedicated to study partially dpifinpfl systems (glasses, polymers, random magnets, etc) in the sense that the definition of the system itself involves random aspects (e.g. the nature of the elements of the systems, their interactions, their mean positions in condensed matter, etc). In particular, in random magnetism (spin- glasses, band- or site- mixed or dilute magnetic systans, etc) converge several pro- blematics, related to the kind of long range magnetic order (oonmensurate or rot), to the cross—over between different universality classes, to the underlying geometrical structure in what concerns its topology (regular lattices, Bethe's trees, cacti, partially removed loop structures, amorphous systems, etc) or its occupancy (independent or correlated site and/or bond percolation), to the method of preparation (annealed or quenched) of the system, etc. By new only a few of these points are clearly understood, some of them are only partially enlightened by preliminar theoretical and/or experimental work (therefore quite a number of controversies exist an these grounds), and finally a great number of aspects are still to be attacked. Within the theoretical approach a new non trivial difficulty must be handled: configurational averages besides the traditional thermal ones.

In the present work we have no intention of exhaustive review of the area of random magnetism, not even of all of its outstanding featuresiwe shall indicate here and there the appropriate References): we shall restrict ourselves to list down the main "ingredients" which control phase transitions in well defined systems (Section II) and hew they emerge in random systems (Section III), Though the general trends we intend to 200 C. Tsallis point out pratioaliy hold for any phase transition, we have voluntarily chosen mast of our illustrations related to magnetic systems. Let us also stress that the list (certainly inccnplete) has been constructed an simplicity or familiarity (for search- ers in the area) grounds rather than within a strict logical classification; therefore seme of the "ingredients" are closely related among than, and could of course have been i i sted together. tor clearness reasons let us introduce sane precision in the current words we stiall use. A critical phenomenon is characterized by the fact that small variations of the causes lead to big variations of the effects (for a mathematical formulation of this concept see R. Them's Catastrophe Theory; for example Jtef. (1)). "Hie critical phenomenon will be called cooperative (or rum cooperative) according to whether it is (or it is not) driven by the mutual enhancement of a particular tendency of the elements of che system (the fall of a ruler at the edge of a desk can clearly be considered as a nan cooperative critical phenomenon). Each element might "know" about the others through energetic couplings (in the Hamiltonian or Lagrangian), through quantum effects such as syinoetrization-antisyniBBtrization of the wave functions, through particular bounds, etc. Among the cooperative critical phenomena we find the phase transitions and the regime changes, The farmer connect two or wore phases (collective states* of thennodynanical equilibrium) of a macroscopic** system (e.g. ferromagnetism or gravitational condensation of a gas of stars to form a galaxy }; the latter connect two or more regimes (collective states of non equilibrium) of a (not necessarily macroscopic) system (e.g., the trigger of the laser effect, the laminar or turb- ulent water flow through a tap). The theoretical study of a phase transition typically (but not exclusively) leads to the analysis of the thenaodynamic free energy F=CF15 (U,T and S being respectively the internal energy, the teoqperature and the entropy). In the limit T+0, D dominates and the cooperation (hence the order) is favoured; in the limit T», S dominates and the cooperation (hence the order) is depressed. Anything which "helps" D enhances

the phase transition, i.e. the critical temperature Tc increases (Bare generally the less disordered phase expands in the intensive parameter space where appears the phase diagram) and/or the eventual long range arder becomes stronger and/or several correlation functions decay more slowly with distance. We shall next list the various "ingredients" which enter an these grounds.

* Two phases might be different even if there is no breakdown of the symmetry (i.e. both are invariant with respect to the sane group of symmetry), in other words if both have the same type of macroscopic order or disorder (see for example itefs.(2-5)). ** Only in the thernDdynandcal limit (It*») appear the various singularities which characterize a phase transition. •tf

RANDOM HANETISM 201 II - MEU. DEFINED SYggtC ' "' II.1 - Cooperative couplings Obviously the phase transition is enhanced if the coefficients of the couplings responsible for the cooperation become stronger, Ferromagiwtisni exhibits this fact as typically T °tf, where J is the exchange integral which couples neighboring spins (see for example Bef^fi.a)).Another example might be superconductivity where, in the limit up •+ 0 of the BCS approximation, it is (see, for example, Ref. (7)) T^l-l^a" ^PF (T_, u and p_ are respectively the Debye temperature, the electron-lattice interaction

coefficient and the density of electronic states at the Fermi level), hence Tc monotonically increases with u. Finally in gauge theories on lattices where a gauge invariant interaction appears characterized by a coupling constant X (see, for example, Ref. (8)), the correlation function < exp i « $> is assymptotically proportional to e pp if XX is bib g enough and to e~e~~AAA iif X smalsmalll enough (<- - •> denotes the canonical mean value, . an angle associated to the j-th site, 0 a sum over a closed path (Nilson loop), and P and A respectively the perimeter and the area of the Wilson loop). In other words, the less disordered phase corresponds to big values of X.

XI.2 - Cooperative elements Hie phase transition will be depressed if the concentration of cooperating elements decreases. Bus is self-evident and clearly related to the previous Subsection, as if the cooperation comes from let us say a two-body interaction, anything which weakens either the two elements or their interaction will obviously depress the critical phenomenon. To illustrate this let us imagina a regular IÍT«»T- chain Where exist first- anâ second- neighbour ferromagnetic interactions. If we call M(N) a certain magnetic (non magnetic) atom, we will clearly have lower magnetic disorder in a crystal ... MM ... than in a crystal ...HM«...

II.3 - Ttapological connectivity If the connectivity between the cooperating elements of the system is increased, the stabilizing "messages" which a given element receives from the others will also increase, therefore the phase transition will be enhanced. This fact explains why for any particular model (e.g. the Ising model) T itonotonically increases with the space c dimensionality d. By the way, this is the reason why the Mean Field Approximation (HEft) , which underestimates the fluctuations, systematically averestiirales T ; further- c mare it is not surprising that at sufficiently high dimensionality the ME& becomes exact, as there the high connectivity makes the fluctuations: to become neglectable in-

deed. Fran Ref.(9.a)we can obtain an illustration of the preceding statements, namely for the simplest ferromagnetic Ising model (n=l) we have that T/T 1*A (where T and T are respectively the actual and the MEA critical temperatures) equals 0 for d=l (linear chain), 0.57 for d=2 (square lattice), 0.75 for d=3 (simple cubic lattice)and 202 C. Tsallis 1 for d -*••*> (simple hypercubic lattice).. A second illustration (still for n=l) of the influence of the connectivity might be taken from Bef. (10) where, for different two-dimensional lattices, we have that kjr/qJ (where q and J are respectively the coordination number and the exchange integral) equals 0.51 for the honeycomb (q=3), 0.57 for the square lattice (q=4) and 0.61 for the triangular lattice (q=€). A last, illustration can be taken from the range of the cooperative interactions. If we have a ferromagnetic exchange integral J.. which decays with the distance R..

(between the sites i and j) as e'^ij^ (or as VR?.) then Tc must be a nonotonie- ally increasing (or decreasing) function of A (or of x); furthermore the MFA becomes exact in the limit A -»• •» (or x + 0) (see, for example, Kef. (11)).

II.4 - Topological incompatibility There exist situations where topological constraints prevent, even at 'vanishing temperature, the full and simultaneous "satisfaction" of every single few-body microscopic interaction. In such cases, on one hand the total internal energy is not as low as it should be without those topological constraints, on the other hand the entropy tends to be favoured by a high degeneracy of the ground state. As a consequence of these facts, the eventual long range order that might appear is not a simple one dike ferromagnetism or antiferromagnetism), it tends to be inocnnensurate* with the underlying (crystalline) structure (as it is frequently the case of helimagnetism; see.for example, Bef. (6.b)), and its stability is quite vulnerable. Such situation appears in the triangular lattice with first-neighbour antiiierrcmagn- etic Ising interactions. This case is a fully frustrated one in the sense that if we mreriffer an elementary triangular plaquette, there is no up-down configuration of the three spins (at the vertex of the triangle) which simultaneously satisfies the three bonds. Another illustration of the influence of a topological incompatibility is given (14i by the R-S model . ibis magnetic model essentially consists in a set of parallel layers: within a layer the spins are strongly coupled through a ferromagnetic interaction, and between first- and second- neighbouring layers exist exchange inter-

actions respectively characterized by J. > 0 and J2 > 0. At low temperatures the system presents, for sufficiently high (low) values of a = JVJi' a ferronBgnetic (helical) long range order. At high temperatures the system is paramagnetic for any value of a. the three phases coexist on a very special point, the Lifshitz point (noted L on the plane p=l of Fig. 1); another very interesting point (noted M on the same figure) is the multiphase point , which separates at vanishing temperature the two ordered phases. How, all the interesting features of this model cane * through tMs fact appear analogies with the Theory of fractals (see, for example, Bef. (12)}. RANDOM MAGNETISM 203 from the topological inoançatibility which appears far a < 0, namely if we consider three consecutive layers of the system (each of them being ferrcnBgnstically orcter- ed), there is no up-down configuration of the spins which can simultaneously satisfy both first- and second- neighbour interactions. lor such a system T should nonotõn- ically increase with o.

Big. 1 - Possible critical surface of a site dilute R-S model: its "outside" corresponds to the paramagnetic phase (P); the "right" Cleft") part (with respect to the first order transition surface IMJ) of its "inside" corresponds to the f errcuagnetic phase (F) (helimagn- etic phases (H)); UN ,\s a iAfshitz line and Mi a multiphase line; the T=O line connecting (P) with (F) and (H) is oversimplified in the figure as it overlooks the consequences of the fact that the a = 0 percolation cannot be via 2™ neighbouring layers.

H.5 - New degrees of freedom A new degree of freedom, besides those responsible for the phase transition, presents two faces: on one side it "helps" O because it introduces mare possibilities for ex- ploiting the cooperating couplings, but on the other side it "helps" S because the raafeer of fluctuating variables is increased, to a new degree of freedom might be "useful" to the phase transition if it "helps" more D than S, or "destructive" in the opposite case. Let us illustrate the latter situation. An asserbly of localized non interacting (electronic) spins leads, in the presence of an external magnetic field, to the Brilloitin paramagnetism, characterized by a divergence, at vanishing tajper- ature, of the isothermal susceptibility Xj (see Fig. 2). If we release the electronic translational degree of freedan, the overlap of the wave functions will increase and through Pauli principle, the spins will be prevented to be parallel, therefore the function. Xj,(T) will be depressed and the singularity will desappear [Pauli paramagnetism). Of course there is no phase transition in the present case, but if there was the depressing effect would be the same. 204 C. Tsallis Fig. 2 - lhe isothermal magnetic susceptibility for vanishing external field for an ideal assembly of localized (Brillouin paramagnetism) or free (Pauli para- nagnetism) electrons as a function of temperature. Brillouin let us now give an example of an "useful" degree of freedom. In KHJO. (KEPJ-type ferroeleetriclty the protons nove in daubJ.e-wells and their left-right" ordering (isoraorphic to the up-down configurations of Ising pseudo-spins ) leads to tha phase transition, the tunneling through the central harrier of each double-well creates a dynamics which can be seen as T pseudo-magnons (see ilg. 3), When the critical temperature T is approached by above a soft mode appears (in a quasi harmonic framework) for vanishing wave vector: when its frequency vanishes, the crystal beccmes structurally unstable and the ferro- electric phase appears. Now it happens in the real system that, through dipolar interactions, the pseudo-roagnons are bilinearly coupled to a new degree of freedon, namely a particular type of collective oscillations (transverse optic phonons) of the heavy ions (see Fig. 3). lhe diagonalization of the Hamiltonian which describes the «hole system leads to two new types of elementary excitations, which might be raiw» quasi-phonons and quasi-pseudo-magiums. lhe interesting feature is that the instability is now related to a new soft mode whose frequency is slightly Zower than before, hence it will vanish "earlier" (at higher temperature) than before. "Hiere- firc_e we see that the essential effect of the new degree of freedom is to increase T .

gig. 3 - lhe frequencies (as functions of the wave vector q) of the elementary excitations before (pseudo-magnons (m) and phonons (pi) and after (quasi-pseudo-magnons (gm) and quasi-phonons (qp)) taking into account the bilinear interaction between protons and heavy ions, lhe empty (full) dot marks the original (final) soft mode; q^ denotes the frontier of the first Brillouin

H.6 - Ground state degeneracy Die entropy effects at low temperatures are clearly determined by the degeneracy of the collective fundamental level of the system, and a strong degeneracy will provoke a weak phase transition, lhe dimensionality n of the order parameter (or of the field in Field theory) provides a direct manner for modifying the ground state degeneracy. RANDOM MAGNETISM 205

T is a raonotonically decreasing function of n, as n gives the number of fluctuating variables per element of the system; in other words, if n increases, the probability for a given element to be in the "right" direction (the one that has been favoured by the eventual breakdown of the symmetry) decreases. See Table I for d=3 and Fig. 4 for d=2. Within thisccntext it is worth while to recall the:: Onsager's Reaction Field Approximation (KE2V) always underestimates T because it overestimates fluctua- G tions: this is why REA becomes exact in the limit n •*• « (spherical model).

order Tc(n) «•del «W7J n parameter

Ising 1 scalar 1 £~ Si Sj irj

ocnçJfix 2 0.99 XT number irj

Heisenberg 3 vector 0.97

Spherical CD tensor 0.91 i,j OE=1

Table I - Various three-dimensional models («Hand J being respectively the Hamilton- lan and the first-neighbour ferromagnetic exchange integral) and theix approximative critical temperatures (Hef. (9.b)) for the foe lattice.

Besides n there are many other ways of increasing the ground state degeneracy. it«r example the q-state Potts model ' attributes to each sibe of a gLvm lattice a va-

riable 0± (for the i-th site) which might take the values a = i,2,...,q, the Hamilt- onian being, for example.

where J is a first-neighbour positive coupling constant and 6CT.CT . is the Kronecker delta (q=l leads to bond percolation and q=2 to the -|- - spin ising model). The elementary two-body interaction of the present model has a ground state whose degeneracy

clearly equals q. We verify'""* '" that Tc mcnotonioally decreases with q. 1.

206 C. Tsallis II.7 - "Quanticity" The present "ingredient" is directly re - lated to the previous one: more a system is ("•1) classical, the bigger is the number of states within a given « T/Tc energy interval, there M(T)/MIO) fore we can say that in a loose sense, its degeneracy increases (v».= (remember that in Classical Statistical 1 T/Tc Mechanics the Third M(T)/M(O) ft Principle of Thermo- dynamics is violated t t because lim S(T)=-">), hence the phase transition is depressed. < T/Tc A trivial illustration of this factcanbe Fig. 4 - Spontaneous magnetization and isothermal susceptibili- .. _. . . , ty for too-dimensionalferromagneti c Ising (n=l), X5f (n=2) and the Einstein conden ^£3^^^ (n=3) rodeis. For n>Xthe susceptibility diverges for sation of a three- T <_ T and the correlation function decays for T < T (T > T ) ,. , .. , as a Sewer (exponential) law with distance (see Bats. (2-5)). dimensional ideal ^^ gas of bosons of mass m and concentration p : the critical temperature is given (see, for example, Ref. (20)) by T = 0.084 h2 pv 3/m kg

hence the.Planck constant h •*• 0 leads to T_-*• 0. c Another illustration can be given by the MEA results for simple ferromagnetism associated to spins with length S: the spontaneous magnetization (as it is presented in Fig. 5; see, for example, Ref. (6.c)) monotonically decreases with S.

IH - RMCXM SYSTBE

m.l - Introduction Until now we have discussed well defined systems, particularly pure magnetiaD, where the space dimensionality d, the order parameter dimensionality n, the spin size S,

among others, are perfectly defined, and where the site concentration ps and bond concentration p. equal unity. The statistical treatment of such systems involves only RANDOM MAGNETISM 207 thermal averages. We intend now to discuss randan systems, and more precisely randan magnetism, whe re one or more of the "ingredients" mentioned in Section II become randan variables, so that the

system is partially undefined (typically pg < 1 and/or p. < 1). Its statistical treatment obliges vis to simultaneously deal with thermal and configurational averages: a further difficulty appears, There is however a particular case, the pure percolation, where the situation is greatly simpli- fied because it corresponds to the limit of vanishing temperature,* therefore there is no ener - getic uwtefinition, and the statistics reduoes to Fig. 5 - MEA spontaneous magnetiza- configurational averages only. tion as a function of temperature. The spin length S + <= corresponds The systems of interest can be classified into to the classical limit. two main categories: the insulating (where no electronic translation is allowed) arei the conducting (where the electronic translation is released, appearing therefore phenomena like hopping, valence fluctuation , etc) magnetic systems. We shall be concerned here with the former, where the critical aspects appear in a more pure manner. The regions of interest in the external parameters spaoe of dimensionality D (e.g. , T-p space, where p denotes a concentration) can be classified as follows (see Fig.6): a) the critical frontier (or phase diagram) of dimensionality (D-l) (e.g., aline in the T-p space), i.e. where the various singularities appear; in the T-p space the limits p -»• 1 (pure magnetism) and T -*• 0 (pure percolation) are typically less difficult to discuss; the point p=l (or T=0) itself might be approached by let us say high and low temperature (or concentration) series; b) the neighbourhood of the critical frontier, in other words how and what singularities appear; this is where the critical exponents come in, and also eventual crossover from one critical set of exponents to another; c) the rest, which from a certain standpoint is less interesting, as there are no singularities (with possible exceptions at T=0 and/or p=l); this is a very hard region in what concerns exact results, excepting the point (p,T) = (1,0) (where there are no averages at all to be done) and the limits T •*• •» and p •*• 0 which are usually trivial. Finally the physical quantities of interest can be classified into tiro (related)

* There are situations(21) where the pure unoorrelated percolation appears in the limit T -*•«>. 208 C. Tsallis groups: macroscopic and microscopic quantities, Among the former we find the equilibrium thermoaynamical quantities (extensjyertype functions of intensive-type variables, e.g., equations of states, specific heat, isothermal susceptibility, compressibility, piezcmagnetic andmagnetostrictrve coefficients, etc) and the non-equi librium thermodynamical quantities (transport properties, such as viscosity, heat and electric conductivities, etc). Among the microscopic quantities we find static ones (e.g., spin correlation functions such as < S? S? > which might be proportional to exp - -Ü— , where R. . is the distance between the i-th and j-th sites and Ç the correlation length which typically diverges at the critical frontier; or eventually the space Fourier transform S(Sc) of the correlation function, where k is a wave vector) as well as dynamic quantities, which might generalize the static ones (e.g., the correlation function < S?(0) S^(t) > or its space and time Fourier transform S(íc,u) where tiisa frequence ; in this function appear the central peak related to solitons , etc). Let us next see how the various "ingredients" we mentioned in Section II can becane random variables, and their consequences.

III. 2 - Cooperative couplings On these grounds, the typical discussion concerns the so called bond problem. Let us consider, as an example, the following first-neighbour Ising Hamiltonian:

«•- = '««

where J. . is an independent random variable whose distribution law is given by

= (1-p) 6 (J, J2)

As the law is the same for each bond, the present problem is a quenched one. The particular case J,=0 is called the bond-dilute problem? the general case J. f J, is called the bond-mixed problem. If J. and J_ are both of the same sign the problem is normally less complicate than when they have different siqns (as here spin-glass orders might appear). fTc i Die simple dilute problem has received attention fron 1C several authors (Refs. (22-28) among others ). The critic (P) V i al frontier for the case J_ > 0 is indicated in Fig. 6. / i In the pure bond percolation point (A in the Fig.) the / I i slope is infinite because the Ising model has no dynamics, / / (F) • i therefore in the magnons spectrum appears a gap, hence in A| the limit of low temperatures one should assymptotically 0 pc * 'p expect e~ 'c « p - p . In the pure Ising limit p •+ 1 Fig.6 - Critical temperature as a function of bond (point C of of the Fig.) one should expect a linear be- concentration of the ferro- haviour because it is essentially a one hole (or absent magnetic Ising model (d>U; P(F) denotes the paramagnet (29) ic (ferrcmagnetic) phase. bond) problem; however there is no reason for expect- A and C respectively corres pond to the pure peroolat-""" ing the frontier to be analytic in the point p=l, because Ion and pure Ising fixed points; B corresponds to the next assymptotic correction concerns a two-hole the possible random fixed problem; and the two holes "see" each other critically point; the arrows denote the possible flow senses. through the correlation length Ç , i.e. we expect RANDOM MAGNETISM 209 in the limit p •+ 1,

T -T (D-vcted-pJ-tcte1 (l-p)2+x (d i> 1) c c

with' x eventually different from zero. Another interesting aspect concerns the set of crttical exponents,

and now we shall restrict ourselves to (3=2. For p=pc there is a percolation critical set which is different from the pure Ising critical set (p=l). What Tc(0 T about p < p < 1 ? It is commonly believed that the eventual random critical set is different from the percolation one (in Renormalization Group (RG) terms, the pure percolation fixed point is expected to be unstable, with a crossover to another fixed point). Is it the same as the pure Ising one? (in RG terms, is the pure Ising fixed point a stable one?). The Harris criteriun/22* says "yes" if the specific heat

critical exponent a < 0, says "no" if afc > 0, and Tc(P) gives no answer in the marginal case a. = 0, (which is precisely the d=2 case I). Nevertheless the specific heat logarithmic divergence of the pure Ising

model can be represented by afc = + 0, therefore (pc

because it is rather intuitive that the weak loga- Tt(p)

rithmic singularity characterized by afc = + 0 should be smeared under dilution. More precisely, a possible answer could be(30) that presented in Fig. 7. Vfe see that the divergence might become a cusp, which

should of course disappear at p=Pc. Furthermore a Schottky-type anomaly, due to the ensemble of finite clusters which coexist with the infinite one for p > p , could be present. The total area is expected to decrease with decreasing p. Let us finally remark that the smearing effect makes experiences as well t'jg. 7 - Possible specific heat as Monte Carlo (or other computer) treatments quite vs. temperature for two-dimensional bond-dilute Ising models difficult in the neighbourhood of p . (p denotes the bond concentration). 2io C. Tsallis

31 32 In what concerns the mixed problem^ ' ' for the case 0 < ^ < J2 (with a = JJ/JJ) , typical results are presented in Fig. 8 (the different sign mixed problem will be mentioned in Subsection III.5). An experimental realization of the bond-mixed model (33 can be obtained in systems such as ' Co(S Se^ )2 where the Co atoms interact between them essentially via superexchange through the S atoms or the Se atoms (and this of course simulates J, and J2).

III.3 - Cooperative elements The site problem is very analogous to the bond one, the dilution concerning the cooperative elements of the system instead of their interactions. A typical problem is the first-neighbour ferromagnetic site-dilute Ising model; the Hamiltonian and the independent occupancy probability law are respectively given by j*f = - J y~ c. c. S?S? (J > 0) *— 13 13

andp(cjL)=(l-p)6(ci)+pô(ci-l) Vi Because of the independent occupancy Fig. 8 - Critical lineVJ ' separating, for the problem is said to be quenched. . different values of a H J./J-, the ferromagnetic (F) from the paramagnetic (P) chase in In ELg. 9 we compare site and bond the square lattice first-neighbour -L - spin critical frontiers for the same latti- ferromagnetic bond-mixed Ising mode]?. ce and interactions. We verify that T (site) <_ T (bond) for all concentrations. It is easy to understand why; let us think on the square lattice: in order to make a perfect covering we must associate two bonds to each site. If one bond is absent we loose one cooperative interaction, if one site is absent we loose four cooperative interactions, i.e. twice the topological proportion, hence site dilution deteriorates the cooperative phenomenon more than bond dilution. The theoretical treatment of the site problem is more complicate than that of the bond problem: the absence of a bond does not perturbate the contributions of the neighbouring bonds, whereas the impact of a site absence depends on the absence or presence of the neighbouring sites. On the other hand, the site problem is easier than the bond one in what concerns the experimental realizations, and various sys- (36) (37,38) (39) terns have been studied: Mn ZnL_ F 1 F4 Fe .-PC12 (43) (40) p ^ . Mn Te 44 , among others. As an illustration we present in Fig. lo the EPR re- Cdl •~P P * * RANDOM MAGNETISM 211 (Ai) suits obtained with a single crystal of Cd Mn Te (antif erronagnet with the zinc blende struo- - Tc tore). Furthermore, in the same work , the expo- i-x nent x defined by AH=|T-Tc(p) (where AH is the EPR line width) has been measured; the results, for IP) (p,x), are (0.25, 1.00), (0.40, 0.96), (0.50, 0.91) // and (0.60, 0.93), clearly exhibiting universality ' 1 IF) along the critical line T vs. p.

The site-dilute problem might be gereralized into o p• Pt iI P the site-mixed one, where atoms A (B) might indepen- Fig. 9 - Typical critical dently occupy a given site with probability p (1-p). frontiers for site-and bond- Therefore the exchange integral J,AA, will dilute ferromagnetic Ising 2 models. It is always (see,for appear with probability p% j_ with (1-p) and DO example, Kefs. (34,35)) B s J^ with 2p(l-p). Wa see that this problem implies PC < p (the equality holds for structures with no loops, 3 different coupling constants, instead of only 2 like Bethe trees). for its bond analogous. Another possible generalization might be the inclusion of further-neighbour interactions ' : T ('K) if we respectively call J. > 0 and J_ > 0 the C 60 first-and second-neighbour interactions in a given lattice, we have represented in Fig. 11 a SO typical phase diagram parametrized by a s Jy^i • It is easy to understand, through topological connectivity arguments, why T (p) mDnotonically 20 c ; i. increases when a varies from 0 to 1. 10 Let us finally mention another generalization : 0 * • * - > i m the uncorrelated quenched site-bond-dilute pro- 0 0.1 0,1! D.< p blem. The Hamiltonian and probability laws are respectively given by Fig. 10 - Critical temperature vs. concentration obtaii«d(44) through EPR measurement on Cd. _ Mn Te (the arrow points the percolation threshold) .

+pB (J > 0),

s s = (l-p )fi(Ci«) +p Sicr-1) Vi.

The expected critical surface is indicated in Fig. 12. If M (N) is a magnetic (ron magnetic) atom, and V (U) is (is not) an atom which allows for superexchange interaction of the M atoms between them (which otherwise do not interact), an experimental realization of the above model can be imagined with a substance of the type 212 C. Tsallis

«ps Nx_^ V

III.4 - Topological connectivity In order to illustrate the effects of random ness on the connectivity of the system, let us imagine a layered Ising ferromagnet (J > 0 and I > 0 respectively being the intra-layer and inter-layer exchange integrals). If we call a = I/J, we have that the dimensionality d of the problem is respectively 2,3 and 1 for a = 0, a = 1 and a •+ <*>. Furthermore, if we remember that the site occupancy critical probability p (d) monotonically increases Fig. 11 - Perro-para critical line of with decreasing d (and that pc(l) = 1), it a site-dilute first-and second-neighbour is natural to expect for the critical front interacts» (^ and J respectively) t^ - Ising model (ct = J-/JÍ); p (1) and p iers the intrincated tailing behaviour in- (i+2) respectively denote the first and dicated in Fig. 13. The problem of the and first-and-second neighbour critical a ^ probability for site percolation, complex crossovers between different di- T./J mensionalities critical sets of exponents has not yet been attacked, to the best of our knowledge. III.5 - Tbpological incompatibility In Fig. 1 we have represented a possible critical surface for a dilute R-S model1 '. Pract- ically nothing is known about such a model where no doubt very interesting features must exist: one of the most fascinating certainly v is the point N of the Fig., as it is simulta - neously a Lifshitz and a multiphase point. Fig. 12 - Critical surface for the Another interesting illustration of random to- uncorrelated quenched site-bond-di- pological inocmpatibilities certainly is the lute first-neighbour Ising model (the T=0 critical line has recently different sign bond-mixed problem, where a been studied'47'). spin-glass phase might appear (Refs. (13, 48- 50) among others). To fix ideas let us imagine a random distribution of first-neigh-

bour Ising interactions ^ < 0 (with probability (1-p)) and J2 > 0 (with probabili - ty p)on a square lattice of _L - spins. If we consider an elementary square plaquette, we see that, for any bond distribution which contains an even number of each exchange constant, two spin configurations exist which simultaneously satisfy all the bonds, whereas for any bond distribution which contains an odd nuntoer of each exchange constant, RANDOM MAGNETISM 213 there is no configuration which simultaneous ly satisfy all the bonds: the plaquette is Tc (P)/T said to be frustrated J; if this phenome *• 1 non occurs frequently in the system, a phase \ (spin-glass) which is neither ferro- or antL ferromagnetic might be the most stable (the (P) higher the dimensionality of the system, the easier for the spin glass phase to appear); / see Fig. 14.

III.6 - New degrees of freedom _ Let us illustrate this "ingredient" by comp- r) PcW c(i) PcU)-l~p aring the quenched models (where migration P IF) of the elements or of the bonds is forbidden) Fig. 13 - Ferro-para critical lines for and the annealed ones (where that migration a layered site-dilute Ising ferranagnet (a = I/J where I and J are respect- is allowed). Though an annealed crystal is ively the inter-and intra-layer exchan- frequently easier to prepare, its theoretic- ge constants) . al treatment presents usually greater difficulty (with respect to its quenched version), |

precisely because of the additional degree j k.lc l of freedom.

let us fix our ideas on site dilution on a / 1 square lattice ferromagnetic model. If we / 1 have only one absent site, both quenched and / 1 annealed problems are equivalent to a one \ IP) / 1 hole problem, hence their slopes in the li- / i \ / i mit p+1 should be identical. The situation / | is quite different if we have two absent CSG) IF) I

sites. In the quenched problem the two ho- i les will probably be far away one from the 0 1 *p other, therefore 8 cooperative interactions will be lost; in the annealed version, the Fig. 14 - A possible phase diagram for two holes will probably migrate in order to a bond-mixed Ising model with J,<0_ T (quenched) V (n,d). An example(50 ) is presented in Fig. 15 214 C. Tsallis III.7 - Ground state degeneracy Until now we have basically restricted ourselves to the Ising model (n = 1). Not very much is known about randan magnets with n > 1. If we call p*(n) the probability where the critical temperature of a dilute n- model vanishes, and p the c corresponding percolation critical probability , it is obvious that p (n) ^ p V (n,d), because the existanee of an infinite cluster is necessary for the development of a long range order. The equality will hold if it is also sufficient. That seems to be the case for the Ising model, Fig. 151 - Approximate phase diagram^U) of a planar quenched (full i.e. p (1) = p Vd, but the question is oontro- lines) and annealed (dashed lines) c G (51) bond-mixed Ising model (with -J,=J2); P, F and AF respectively versial for n > 1 (for example, Barma et al denote the para-, ferro-, and anti- found p*(3) > p for a particular model; see ferromagnetic phases. cannot decrease with increasing n, as the phase Fig. 16). In any case it is clear that p*(n) transition weakens in this situation, lhe critical frontier is expected, for the same reasons as for n = 1, to be linear in the limit p •* 1. In the simple n- models, T v c is expected to follow a power^law (T <* (p-p*) , with x > 0) in the limit p •*• p*, because of the absence of gap in the spectrum of elementary magnetic excitations (magnons); the absence of a gap is of course due to the fact that, for n > 1, the Hamil- tonian of the system is invariant with respect to a continuous group of symmetry. A very rich (and, to the best of our knowledge, complete ly unexplored) discussion is certainly related to a let us say three-dimensional first-neighbour model like the following one: sM) (J > 0)

We see that a. = 0, 1, °° respectively correspond to n = 1, 16 - Critical line 3, 2, therefore quite a number of unknown possibilities or site-dilute ferromagnetic n- model (in what concerns the critical lines and sets of expo- (d j> 2 and n > 1); P and nents) must be involved for arbitrary values of a. In F respectively denote the para- and ferromagnetic Fig. 17 we have indicated two possible results (and have phases (it is expected omitted the eventual complicate tailing effects for arbi p*(n) 2. P = percolation 0 trary values of o). threshold . Another illustration of randomness related to the ground state degeneracy might be RANDOM MAGNETISM 215 the quenched bond-dilute q-state ferromagnetic first-neighbour Potts model, namely

«•' j

(J > 0)

In Fig. 18 we have reproduced the results obtain (52) ed for this model by Southern and Thorpe

Fig. 18 - Ferro (F) -para (P) critical lines for the quenched bond- dilute q-state ferromagnetic first Fig. 17 - Possible ferro (F) -para neighbour Potts model (52) in square (PÍ critical lines for let vis say lattice. three-diinensional site-dilute mixed model (OFOJI,» respectively correspond ton=l,3,2). III.8 - "Quanticity" (a) p*(3)=p*(2)=p*(l)=p ; (b) p*(3)>p*(2)>p*(l)=i>^. To illustrate this "ingredient" we shall ,(53) restrict ourselves to a single simple example . Let us have a quenched site-mixed first-neighbour ferromagnetic Ising model in a z-coordinated lattice. The family A is made of spins with size 1 (and occupancy probability p) and the family B is made of spins with size ,L (and occupancy probability( 1-p)). For simplicity let it 2 (53) be J.. = J J > o. Then the critical temperature is given within the 3B AB= M MEA, by (2-p)

Therefore T decreases when the weigh of the larger spin increases (i.e. when it becomes Less quantical). 216 C. Tsallis IV - CONCLUSICM Random magnetism is a good "bridge" between geometrical and thermal physics. Though we have been, in the present quick survey, extremely restrictive (we have not gone very much beyond the phase diagrams), an enormous number of almost completely unexplored possibilities has become evident. This fact (as well as the various non uniform convergences that have appeared along this work) is typical of this area where very much is still to be done, on theoretical as well as experimental grounds. Through the general trends, it has been exhibited the relevance of the topological concepts in random magnetism, and through than (as well as for other reasons) the Theory of Critical Phenomena beoones related to the mathematical Catastrophe Theory and Theory of Fractals as well as to other branches of Physics such as Field Theory or Cosmology. We might be a little bit closer to the "fundamental unity" of Physics. My collaborators A.C.N. Magalhães, D.M.H. da Silva, E.M.F. Curado, G. Schwachheim , I.F.L. Dias, L.R. da Silva, R.A.T. de lima, S.V.F. Ijevy and S.F. Machado have, in a way or another, very much contributed in the ideas presented in this work: it is with pleasure that I acknowledge this fact. Several useful related remarks from R.B. Stinchoombe, M.E. Fisher, A. Aharony and M. Thorpe are also acknowledged.

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18) R.B.Pottsf Proc. Camb. Phil, Soc. 48, 1Q6 (1952), 19) D.Rün and R.I.Joseph, J,Phys. C T_, L167 (1974). 20) K.Huang, "Statistical Mechanics" (1963, J.Wiley), page 264, 21) W.Klein, H.E.Stanley, P.J.Reynolds and A.Cbniglio, Phys. Rev. Lett. 41, 1145 (1978). 22) A.B.Harris, J.Phys . C 2, 1671 (.19 74). 23) H. Au-Yang, M.E.Fisher and A.E.Ferdinand, Phys. Rev. B 13_, 1238 (1976). 24) T.K.Bergstresser, J. Phys. C 10, 3831 (1977). 25) M.F.Thcrpe, J. Phys. A 11, 955 (1978). 26) E.Dcnany, J. Phys. C 11, L337 (1978). 27) J.M.Yeomans and R.B.StinctCTiibe, J. Phys. C 1,2, 347 (1979). 28) C.Tsallis and S.V.F.Levy, J. Phys. C 13 , 465 (1980). 29) M.E.Fisher, private camui' nation. 30) M.F.Thorpe, private oannuniection. 31) J.M.Yecmans and R.B.SUnchcarabe, J. Phys. C 12, L169 (1979). 32) S.V.F.Levy, C.Tsallis and E.M.F.Curado, Phys. Rev. B 21, 2991 (1980). 33) K.Adachi, K.Sato, M.Matsuura and M.Ohashi, J. Phys. Soc. Japan 29, 323 (1970). 34) V.K.S. Shante and S.Kirkpatrick, Adv. Phys. 20,.325 (1971). 35) J.W.Essam, "Percolation and Cluster Size" in "Phase Transitions and Critical Phencmena", Vol. 2 (1972, edited by C.Doub and M.S.Green, Academe Press), pages 197-270. 36) J.M.Baker, J.A.J.Iourens and R.W.H.Stevenson, Proc. Phys. Soc. 77, 1038 (1961). 37) R.J.Birgeneau, R.A.Cowley, G.Shirane and H.J.Guggenheim, Phys. Rev. Lett. 37, 940 (1976). 38) R.A.Cowley, G.Shirane, R.J.Birgeneau and H.J.Guggenheim, Phys. Rev. B 15, 4292 (1977). 39) T.E.Vtood and P.Day, J. Phys. C 10, L333 (1977). 4W M.Suzuki and H.Ikeda, J. Phys. C 11, 3679 (1978). 41) L,Bevaart, E.iU:±fckee, J.V.Lebesque and L.J. de Jongh, Phys. Rev. B 18, 3376 (1978). 42) K.Katsumata, M.Kbbayashi, T.Sato and Y.Miyako, phys. Rev. B 19, 2700 (1978). 43) P.ttong and P.«.Horn,Bull .Am. Phys. Soc. 24, 363 (1979). 44) S.Oseroff, R.Calvo and W.Giriat, to be published in J. Appl. Phys. 45) T.Idogaki and M.üryu, J. Phys. Soc. Japan 43, 845 (1977). 46) A.R.Mc Gum and M.F.Thorpe, J. Phys. C 11, 3667 (1978). Í7) H.Nakanishi and P.J.Reynolds, Physics Lett. 71 A, 252 (1979). 48) A.P.Young and R.B.Stinchcoite, J.Phys C 2, 4419 (1976). 49) A.Aharony, J. Phys. C 11, L457 (3.978). 50) J.Jakubczak, Z.Mrcziüska and A.Pekalski, J. Phys. C 12, 2341 (1979). 51) M.Bazma, D.Rmar and R.B.Fandey, J. Phys. C 12, L283 (1979). 52) B.W.Southern and M.F.Thorpe, J. Phys. C 12, 5351 (1979). 53) S.Katsura, J. Phys. A 12, 2087 (1979). SYMPOSIUM LECTURERS

GEORGES BEMSKI Departamento de Física Pontifícia Universidade Católica 22451 - Rio de Janeiro - RJ, Brasil CARLOS G. BOLLINI Centro Brasileiro de Pesquisas Físicas Av. Venceslau Bras, 71 22290 - Rio de Janeiro - RJ, Brasil RAJAT CHANDA Instituto de Física Universidade Federal do Rio de Janeiro 22910 - Rio de Janeiro - RJ, Brasil SIDNEY COLEMAN* Harvard University and Stanford Linear Accelerator Center P.O. Box 4349 Stanford, California 94305, USA CARLOS O. ESCOBAR Instituto de Física Teórica Rua Pamplona, 145 01405 - Sâo Paulo - SP, Brasil RICARDO FERREIRA Departamento de Física Universidade Federal de Pernambuco Recife, Pernambuco, Brasil J.J. GIAMBIABI Centro Brasileiro de Pesquisas Físicas Av. Venceslau Bras, 71 22290 - Rio de Janeiro • RJ, Brasil BERTRAND GIRAUD Centre d'Etudes Nucleaires de Saclay Service de Physique Theorique Boite Postale n9 2 91190 - Gif-sur-Yvette, France HAIM HARARP Department of Nuclear Physics Weizmann Institute of Science Rehovot, Israel JORGE S. HELMAN Centro de Investigacion y de Estúdios Avanzados Instituto Politécnico Nacional Apartado Postal 14740, México 14, D.F., México OSCAR HIPOLITO* Departamento de Física Universidade Federal de São Carlos 13560 - São Carlos - SP, Brasil SAMUEL W. MACDOWELL Department of Physics, Yale University New Haven - Connecticut 06520, USA EMERSON J. V. PASSOS Instituto de Física Universidade de São Paulo 05508 - São Paulo - SP, Brasil PAULO H. SAKANAKA Instituto de Física Universidade Estadual de Campinas 13100 - Campinas - SP, Brasil ABDUS SALAM International Center foi Theoretical Physics P.O. Box 586 34100-Trieste, Italy BERTSCHROER Instituto de Física e Química de São Carlos Universidade de São Paulo 13560 - São Carlos - SP, Brasil H. EUGENE STANLEY Department of Physics, Boston University Boston, Massachussetts 02215, USA JAYME TIOMNO* Departamento de Física Pontifícia Universidade Católica 22451 - Rio de Janeiro - RJ, Brasil GÉRARD TOULOUSE Groupe de Physique des Solides de PEcole Normaie Supérieure 24 rue Lhomond, 75231 Paris Cedex 05, France CONSTANTINO TSALLIS Centro Brasileiro de Pesquisas Físicas Av. Venceslau Bras, 71 22290 - Rio de Janeiro - RJ, Brasil JOSÉ URBANO* Centro de Física Teórica Faculdade de Ciências e Tecnologia Universidade de Coimbra Coimbra, Portugal RUI VILELA MENDES* Centro de Física da Matéria Condensada Av. Gama Pinto, 2 1699 Lisboa Cedex, Portugal CHING S. WU Institute for Physical Science and Technology University of Maryland College Park, Maryland 20742, USA MARTIN J. ZUCKERMANN Department of Physics, McGill University Montreal, P.Q, H3A 2T8, Quebec, Canada

* Not included in this publication. Arte Serviço de Editoração do CNPq Impressão Gráfica de Senado VI BRAZILIAN SYMPOSIUM ON THEORETICAL PHYSICS Erasmo Ferreira, Belita Koiller, Editors

Solide State, Biophysics and Chemical Physics J.C. Helman, M.J. Zuckermann, H.E. Stanley, G. Bemski, R. Ferreira, Contributors VOLUME II

CONSEflHO NACIONAL DE DESENVOLVIMENTO CIENTIFICO E TECNOLÓGICO Erasmo Ferreira, Belita Koiller, Editors

Solide State, Biophysics and Chemical Physics J.C. Helman, M.J. Zuckermann, H.E. Stanley, G. Bemski, R. Ferreira, Contributors VOLUME II

CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTIFICO E TECNOLÓGICO

Coordenação Editorial Brasília 1981 CNPq Comitê Editorial Lynaldo Cavalcanti de Albuquerque José Duarte de Araújo Itiro lida Simon Schwartzman Crodowaldo Pavan Cícero Gontijo Mário Guimarães Ferri Francisco Afonso Biato

Brazilian Symposium on Theoretical Physics, 6., Rio de Janeiro, 1980. VI Brazilian Symposium on Theoretical Physics. Bra- sília, CNPq, 1981. 3 v...... Bibliografia - Conteúdo — v. 1. Nuclear physics, plasma physics and statistical mechanics. — v. 2. Solid state physics, biophysics and chemical physics. — v. 3. Relativity and elementary particle physics. 1. Fi'sica. 2. Simpósio. I. CNPq. II. Título. CDU 53:061.3(81) VI BRAZILIAN SYMPOSIUM ON THEORETICAL PHYSICS

VOLUME II FOREWORD

The Sixth Brazilian Symposium on Theoretical Physics was held in Rio de Janeiro from 7 to 18 January 1980, consisting in lectures and seminars on topics of current interest to Theoretical Physics. The program covered several branches of modern research, as can be seen in this publication, which presents most of the material discussed in the two weeks of the meeting. The Symposium was held thanks to the generous sponsorship of Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq (Brazil), Academia Brasileira de Ciências, Fundação do Amparo a Pesquisas do Estado de São Paulo (Brazil), Organization of American States, Sociedade Cultural Brasil- Israel (Brazil), Comissão Nacional de Energia Nuclear (Brazil), and the governments of France and Portugal. The meeting was attended by a large fraction of the theoretical physicists working in Brasil, and by about 30 scientists who come from other countries. The list of participants — about 250 - is given at the end of this book. The Pontifical Catholic University of Rio de Janeiro was, for the sixth consecutive time, the host institution, providing excellent facilities and most of the structural support for the efficient running of the activities. The community of Brazilian Theoretical Physicists is grateful to the sponsors, the host institution, the lecturers and the administrative and technical staff for their contributions to the cooperative effort which made the VI Brazilian Symposium on Theoretical Physics a successful and pleasant event.

WELCOME ADDRESS

Ladies and Gentlemen: It is a great pleasure and honor for me to welcome to our University the participants of the Sixth Brazilian Symposium on Theoretical Physics. It is on such occasions that the true spirit of the university is manifested in the search for new knowledge and in the communication of new discoveries. Renowned physicists in several areas of Physics, and from leading institutions in Brazil and abroad, gather here to share their latest results of research, new hypotheses and unanswered questions. Through the exchange of new insights, suggestions and critical reactions, the academic community emerges, united in its commitment to truth and in its effort to contribute to the understanding of the world in which man lives. The understanding of Nature — which is the original meaning of Physics in 'j Greek — is indeed the goal of this Symposium on Theoretical Physics. Events of recent years have taught us to be cautious in our dealing with Nature, since we ; do not understand all of its interrelations. The precipitate use of nature's forces, í made possible by scientific progress, though intended to make the world more livable, threatens to destroy mankind itself. It is not possible to reduce nature to a single model and it is not possible to reduce history to a human project. The full understanding of nature demands humility and respect in dealing with it. Only under such conditions will the knowledge and use of nature be of serve to man. One of the basic goals of this Symposium is to show the fundamental unity of physical theory. The rediscovery of the deep unity of all the phenomena of nature is undoubtly a step forward in the understanding of nature, because understanding requires unification of many phenomena in one law or theory. But a unified theory, in order to be faithful to its object, must recognize ? the variety of levels through its use of analogy. It must respect the limits of its j understanding of nature through the continual perfection and superseding of its own models. It must admit the existence of other dimensions and other approaches to reality, through openess to interdisciplinary dialogue. It is through the pluralism of viewpoints, consciously accepted as relative and interchangeable, that the global vision of reality progressively emerges, and \ converges to its final goal, which is man in all his complexity and his self-transcendence. May this attitude — so proper to the spirit of the university - be present in all the work of this Symposium. VIU

In closing I would like to thank all the Brazilian and foreign foundations and government agencies which are sponsoring this meeting. May the bold initiative of the Department of Physics at PUC in hosting this Symposium, and the generous efforts of its organizers be rewarded by the scientific and human growth of all the participants. Thank you.

Rev. João A. Mac Dowell, S.J. Rector, Pontifícia Universidade Católica do Rio de Janeiro IX

INDEX

Volumel ^ :-^ -; - — v - NUCLEAR PHYSICS, PLASMA PHYSICS AND STATISTICAL MECHANICS 1. The Generator Coordinate Method in Nuclear, Physics - B Giraud : 2. Collective Motion and the Generator Coordinate Method - E.J.V, Passos 3. Some New Radiation Processes in Plasmas - C. S. Wu 4. Magnetohydrodynamics and Stability Theory - P. H. Sakanaka 5. Free Energy Probability Distribution in the SK Spin Class Model - Gerard Toulouse and Bernard Derrida 6. Connectivity and Theoretical Physics: Some Applications to Chemistry -H. E. Stanley, A. Coniglio, W. Klein and J. Ferreira 7. Random Magnetism - C. Tsallis

Volume II SOLID STATE PHYSICS, BIOPHYSICS AND CHEMICAL-PHYSICS 1. Scattering of Polarized Low-Energy Electrons by Ferromagnetic Metals -J. S. Helman 2. Phase Separations in Impure Lipid Bilayer Membranes - F. de Verteuil, D. A. Pink, E. B. Vadas and M. J. Zuckermann 3. Recent Progress in Biphysics - G. Bemski 4. Electronic Aspects of Enzymatic Catalysis - R. Ferreira and Marcelo A. F. Gomes

Volume III RELATIVITY AND ELEMENTARY PARTICLE PHYSICS 1. Gauge Fields and Unifying Ideas - C. G. Bollini 2. On the Personality of Prof. Abdus Salam - J. J. Giambiagi 3. Gauge Unification of Fundamental Forces - A. Salam 4. Geometric Formulation of Supergravity - S. W. MacDowell 5. Weak Interactions - R. Chanda 6. Applications of Quantum Chromodynamics to Hard Scattering Processes - C. O. Escobar 7. Topologjcal Aspects in Field Theory and Confinement - B. Schroer

AUTHOR INDEX - AT THE END OF THE VOLUME XI

CONTENTS

Jorge S.Helman SCATTERING OF POLARIZED LOW-ENERGY ELECTRONS BY FERROMAGNETIC METALS 1. Introduction 219 2. The GaAs Source 220 3. Theory 223 4. Elastic Scattering of Polarized Low Energy Electrons by Ni 229 5. Conclusion 232 References 233

M. J. Zuckermann PHASE SEPARATIONS IN IMPURE LIPID BILAYER MEMBRANES Summary 235 1. Introduction 236 2. Theory 238 3. Results and Discussion 245 4. Conclusions 260 References 261

Georges Bemski RECENT PROGRESS IN BIOPHYSICS Abstract 263 1. Introduction 263 2. Hemoglobin 264 3. Photosynthesis 273 4. Spin Density Imaging by Nuclear Magnetic Resonance Technique 276 References 280

Ricardo Ferreira ELETRONIC ASPECTS OF ENZYMATIC CATALYSIS Summary 281 1. Introduction 281 2. Enzymatic Catalysis and Transition-State Theory 283 3. The Orbital Perturbation Theory of Catalysis 283 4. Enzymatic Catalysis 288 5. Oxireductases and Semiconductors 289 References 291

SYMPOSIUM LECTURERS 219

SCATTERING OF POLARIZED LOW-ENERGY ELECTRONS BY FERROMAGNETIC METALS

J. S. Helman Departamento de Física - Centro de Investigation del 1, P. N. Apartado Postal 14-740, México 14, D. F., México

ABSTRACT

A source of spin polarized electrons with remarkable characteristics based on negative electron affinity (NEA) GaAs has recently been developed. It constitutes a unique tool to investigate spin dependent interactions in electron scattering processes. The characteristics and working principles of the source are briefly described. Some theoretical aspects of the scattering of polarized low-energy electrons by ferromagnetic metals are discussed. Finally, the results of the first polarized low-energy electron diffraction experiment using the NEA GaAs source are reviewed; they give information about the surface magnetization of ferromagnetic Hi (110).

1. INTRODUCTION

A source of spin polarized electrons based on photoemission from GaAs, proposed by Siegmann et al. at ETH-Ziirich in 1974, has now become operational2. The availability of spin polarized electron beams opens new fields of research in almost every area of fundamental phys- ics3; its importance can be appreciated by considering the optical analogous, namely the advantages that may represent in many situations the use of polarized over non-polarized light.

The remarkable characteristics of the GaAs source allow for the first time to perform scattering experiments which give direct information on electron spin dependent interactions. Besides providing this basic knowledge the source had already an immediate application to solid state physics as a unique tool to determine the surface magnetic structure of ferromagnetic materials'*.

In this paper I describe some recent experimental results which constitute the first application of the source to the determination of the surface magnetic structure of Ni. They are completed by some theoretical considerations regarding the scattering of polarized low 220 J. Helman enery electrons by ferromagnetic metals. I will start, however, with a brief description of the characteristics and working principles of the GaAs source.

2. THE GaAs SOURCE

Under suitable conditions, electrons photoemitted from GaAs are spin polarized. GaAs is a direct gap semiconductor (Eg = 1.52 eV at 110 K) whose valence band is spin orbit split by 0.34 eV. A scheme of the band structure and the energy levels at the r point are shown in Fig. 1. The relative transition probabilities from the valence to the conduction band levels induced by circularly polarized light of helicity a and cr are also shown in Fig. 1. For instance, if

mj=+l/2 sl/2

Eg* 1.52 eV

mj •+3/2

mj—1/2

Fig. 1.- a) Band structure of GaAs in the neighborhood of the T point (center of the Brillouin zone) showing the spin- orbit split valence band and the direct energy gap. b) Energy levels corresponding to the r point. The electronic transitions induced by light of helicity a+ (a ) are indicated by full (dotted) arrows. The encircled numbers on the arrows denote the relative intensity of the transitions. (Taken from Ref. 2). ELECTRON SCATTERING BY METALS 221 a light of energy just above the threshold for excitation from the : upper valence band to the conduction band is used, electrons are promoted from the level j 3/2; -3/2> of the valence band to the level 11/2; -l/2> (spin down) of the conduction band with relative intensity 3, and from the level |3/2; -l/2> of the valence band to the level |1/2; l/2> (spin up) of the conduction band with relative intensity Í 1. Thus, the theoretical negative (spin down) polarization of the electrons promoted to the bottom of the conduction band is P= -50%. The spin quantization axis is taken parallel to the light beam which is at normal incidence. Analogously, if a" light is used, the electron spin polarization is reversed, that is, P = 50%.

Notice that if the light has enough energy to promote electrons from both the upper and lower spin orbit split bands the net polarization of the electrons in the conduction band is zero5. Thus, in order to obtain polarized electrons in the conduction band circularly polarized light of energy just above threshold has to be used. .} Light from a GaAlAs laser has the adequate wavelength (T-800 nm) for this purpose. In the device the laser light is circularly polarized by means of a quarter wavelength plate.

Under normal conditions, GaAs has a large work function (^ 4 eV) so that the electrons at the bottom of the conduction band cannot be extracted, and hot electrons have a short mean free path (% 10 Â) leading to a very small quantum yield for photoemission. However, when p-type GaAs single crystals (doping ^10 19 cm —3) are cleaved along the (100) direction and the surface is activated by successive depositions

of Cs and O2 the work function is lowered until negative electron affinity (NEA) is reached (Fig. 2). In this situation a large fraction of the electrons photoexcited to the conduction band may leave the crystal. The photoemission is only limited by the diffusion length of the thermalized electrons, which is very large (•%> 1 ym) , and the transmission probability through the surface (^ 0.1). Quantum yield curves for NEA GaAs are shown in Fig. 3. ,

I At present, the NEA GaAs source, although not fully optimized | yet, produces a 20 uA beam of 42% spin polarized electrons per mW of exciting radiation of 1.6 eV .

Spin flip mechanisms, mainly exchange scattering with neutral }' Cs atoms at the surface, reduce the 50% theoretical polarization. 222 J. Helman

GoAs Vacuum

Fig. 2.- Energy band diagram at the surface of NEA GaAs. (Taken from Ref. 2) .

Furthermore, this source has the remarkable characteristic that the electron spin polarization can be modulated (up and down) at a given frequency u without introducing any changes in the intensity of the beam. This is accomplished by rotating mechanically the quarter wavelength polarizer at a frequency co/2. This is the essential feature which allows the use of sensitive lock-in techniques to measure directly the effect of spin dependent interactions in scattering processes.

The polarization of the photoemitted electrons is normal to the crystal surface. For low-energy electron scattering experiments the beam is deflected by a 90° angle so that the outcoming beam has transverse polarization (spin perpendicular to the direction of propagation) .

It is noteworthy that this source does not involve any magnetic field which might hamper some scattering experiments.

Full details about the design, construction, operation and per- formance of the HEA-GaAs source are given in Ref. 2. ELECTRON SCATTERING BY METALS 223

PHOTON ENERGY (eV) Fig. 3.- Quantum yield vs photon energy for a NEA GaAs sample at 110 and 300 K. (Taken from Ref. 2).

3. THEORY

When low energy primary electrons (100-1000 eV) strike a metal, several scattering processes happen. There is elastic scattering and also inelastic scattering due to the excitation of single electrons (inter-and intraband transitions) and of collective modes (plasmons, spin waves). The inelastic scattering gives rise to a characteristic energy-loss spectrum whose peaks correspond to the characteristic e- lectronic excitation energies of the metal.

The interaction potential between the primary electron and the electrons of the target contains a spin dependent exchange interaction which is small compared with the total potential. In nonmagne- tic metals it is impossible to observa the exchange interaction separately since there is no distinction between electrons with spin up and down. However, in magnetic metals it is possible to observe this 224 J. Helman effect by using a beam of spin-polarized primary electrons. In this case the exchange interaction leads to different cross sections depending on the relative orientation of the primary beam polarization with respect to the magnetization of the metal.

This type of experiments can now be performed due to the availability of the intense NEA GaAs source of polarized electrons whose spin orientation can be reversed at a high rate without perturbing the beam intensity. Thus, the difference between the cross sections in any scattering process for an alternately spin-up and spin-down polarized primary beam can be measured by a lock-in technique. At present this technique can reach a sensitivity of 10 . These experiments provide direct evidence of the role played by the exchange interaction on the elastic and inelastic scattering processes and allow the determination of exchange coupling constants as well as the behavior of the surface magnetization.

To fix- the ideas let us consider the following experimental setup: The target consists of a ring shaped Ni crystal whose domains are oriented by means of a small coil (Fig. 4). This special shape is

Ni ring

Magnetization

Einzel Lens

Source •'I ) Lock-in of GaAs LIJ 'Detector

Fig. 4.- Schematic setup for a polarized low-energy electron scattering experiment.

chosen to avoid stray magnetic fields. A beam of low-energy electrons, whose polarization is modulated at about 30 Hz, is directed against ELECTRON SCATTERING BY METALS 225 the border of the ring. The outcoming primary electrons, selected in a given energy interval, are detected through a lock-in technique that picks up only the 30 Hz component of the current. The amplitude of the current is a measure of the differential effect due to the spin dependent part of the interaction between the primary electrons and the Ni crystal.

A. Elastic and magnon-inelastic differential cross sections of spin- polarized low-energy electrons in magnetic Ni and Pe7.

The interaction Hamiltonian between the primary electron and the target is

H± = Hm + V. (1)

Here V = I v(r-R.) describes the spin independent part of the prima- j 3 ry electron interactiointeracti n with each atom in the target. R. is the position of the atom j.

Hm = I Jir-RjJs-Sj (2)

describes the exchange interaction between the spin s of the primary electron and the spin S. of the magnetic metal ion at R.. J(r) is the exchange coupling constant. A localized spin model for the magnetic metal is used, although its validity may be doubtful for the transition metals under consideration. In second quantized formalism

and in the spin-wave region (T << Tc = Curie temperature), tt± can be written in the form

k+ q

-*• •*• • cr cr' lrl+lc+ V • 4» lex 226 J. Kelinan where

v = |d3r v(r) e'1^'1 (4) Q and

= I d3r JCrJe"^'1 (5)

a+ and a+ are the creation and destruction electron operators, k,a k,a + respectively, b_». and b+ are the creation and destruction magnon operators. Q = k"-k' and N is the number of magnetic ions in the normalization volume ft.

The differential cross sections for electrons with spin up (dot/du)) and down (da+/dd)) are calculated in the Born approximation. If the magnon energy loss (^0.1 eV) cannot be resolved in the scattered electron spectrum, the processes involving elastic scattering and magnon creation and destruction can be incorporated together in the differential cross sections. In this case the result is7

(6) Q 5 q

and f

+i I (n+ + 1) S(Q-q) (7) Q + q q

where M denotes the mean thermodynamical value $f S at temperature T, m is the electron mass, S(3) =

The relative current IQ measured with the experimental set up described above is given by

IQ - Pig (8)

with

o = fdoj, _ doij /rda£ da£| {9) Q |_ do) dwj / |_da> dwj e Here Q = IQI = 23; sin-r*where k is the momentum of the primary electron, e is the scattering angle and P is the polarization of the incident electron beam.

The nain contribution to 1° comes from processes in which the spin of the electron is conserved. The processes involving the emi- Z ssion or absorption of magnons are of the order |JQ| . Neglecting z terms of order lJQ| we are left with v J* + v*J 12 - - -2-2 S-S M. (10)

In this approximation the structure factor drops out and the result depends only on the interaction between the electron and a single atom.

The self-consistent Hartree-Fock-Slater potentials for fully polarized Ni (0.6 y_ per atom) and Fe (1.72 u_ per atom) have been calculated by Wakoh8. The results for the self consistent poten-

tial for spin-up electrons V+(r) and for spin-down electrons V^(r) are tabulated. We can write v(r) and J(r) in terms of Wakoh1s potentials as follows

v(r) =| Jv+(r) + V+(r)l (11)

a"hd

J(r) = V+(r) - V+(r). (12)

The Fourier transforms (Eqs. (4) and (5)) were calculated numerical 228 J. Helman ly. Since v_ and JQ are real Eq. (10) reduces to

1° = - (J^/vJM, (13) Q Q' Q' where M is the reduced magnetization. 1° as a function of Q for M=l is shown in Pig. 5 for Hi and Fe.

3 • io rQ

8 9 Q(a.u.)

Fig. 5.- Theoretical IQ VS momentum transfer Q (in atomic units) for fully magnetized Ni and Fe. The arrows indicate momentum transfers at which experiments were performed in Ni.

Wakoh's self consistent potentials are calculated for fully

polarized Ni and Fe (T = 0 K). The evaluation of IQ at a finite temperature T would require the calculation of the self consistent potentials corresponding to a reduced magnetization M. This information is not available at present. At low temperature (T << T ) the behavior of I_ can be estimated using the spin wave approximation together with Wakoh's potentials (Eq. 13).

In the experimental setup described above the primary electrons ELECTRON SCATTERING BY METALS 229 may penetrate only a few angstroms in the metal, therefore in Eq. (13) one should use the surface rather than the bulk magnetization. On the other hand, the dependence of the electron penetration on the primary energy could be used to measure the magnetization profile near the surface.

It is convenient to stress that Wakoh's selfconsistent potentials were derived from a band calculation, while in this work they are assumed to hold for an additional electron in the range 100-1000 eV which is rather doubtful. The idea, however, was to obtain only an order of magnitude of the effect to estimate if it was within the sensitivity of the lock-in technique.

B. Inelastic scattering of polarized electons by plasmons in ferromagnetic metals9.

In a spin polarized electron gas a plasmon involves a charge polarization together with a spin density modulation. Therefore it can be excited by an external electron through both the coulomb and exchange coupling. Using for the exchange energy the form Ge2n'3 with G = 0.84, where n^ is the electron density for spin a10, the Born approximation leads to a reciprocal mean free path for plasmon excitation by an electron of energy E and spin a = ± 1

2/' Gk N" -

Here u = (N, - N,)/N is the relative spin polarization of the elec-

tron gas kc is the plasmon cut-off wave vector, u is the plasmon 3 frequency and aQ the Bohr radius. Then, Gk^N""/ / 12*2 ^ =0.084rg 3 1 with (4TT/3) (rgao) = N"

The first term of Eq. (14) was given by Ferrell11 and Pines12. The electron scattering techniques using spin modulated beams should allow a measurement of the last term.

4. ELASTIC SCATTERING OF POLARIZED LOW ENERGY ELECTRONS BY Ni

I will describe recent experimental results by Celotta, Pierce, 230 J. Helman Wang, Bader and Felcher performed at National Bureau of Standards'*.

The experiments were made on a thin Ni crystal (0.3x5x10 mm ) with the large surface oriented along the (110) direction. The crystal was suspended by Ta rods so that it closed a magnetic cir- cuit of a small c-shaped electromagnet as shown in Pig. 6. The direction (111) of the surface, which is an easy magnetization direction, was chosen to be the magnetization axis and it was oriented to

Fig. 6.- The electron beam at an angle of incidence 6 is diffracted into the detector. The incident electron spin polarization and the crystal magnetization lie in the scattering plane. (Taken from Hef. 4).

lie in the scattering plane determined by the electron beam. The spin polarization of the incoming beam was also in the scattering plane to avoid spin-orbit effects on the electron scattering.

The experimental details regarding the preparation of the sample and the setup are given in the original paper by Celotta et al .

Measurements were made at an intensity maximum of the specular- ly reflected (00) beam at an angle of incidence e = 12° end energy E = 125 eV. This corresponds to a momentum transfer Q = 5.9 a.u.

Fig 7. shows IQ at about 400Kas a function of the magnetic field. IQ follows a known hysteresis curve indicating that it is indeed related to the local magnetization. ELECTRON SCATTERING BY METALS 231

Fig. 7.- A hysteresis curve, I_, at E = 125 eV and 6 12c (Taken from Ref. 4). Q'

Fig. 8 shows i" as a function of temperature in the range 0.5 < T/T < 0.8. The upper half of the graph corresponds to one orientation of the sample magnetization while the lower part was obtained after reversing the magnetization by means of the small coil. Both the upper and lower curves extrapolate to zero at the bulk. Curie temperature. However, if 1° is indeed proportional to the magnetization as an extrapolation of the validity of Eq. (13) to finite temperatures would suggest, then this curve indicates that the surface magnetization differs from the bulk magnetization. The latter has the usual shape predicted by molecular field theory instead of the observed straight line. This effect has already been observed in other magnetic materials, where the surface magnetization is smaller 1 3 than that of the bulk at the same temperature

An experiment was also made at 6 = 12° and incident energy E=20 eV, which corresponds to a momentum transfer Q = 2.3 a.u.. A value IQ = -0.04 was obtained which is larger in absolute value and negative. Comparing the experimental values of l2 with those predicted in Fig. 5 (indicated by arrows) it can be seen that although the simple theory using Wakoh's potential predicts correctly the change of 232 J. Helman

«o o

-.01 -

-.02 .6 .7 .8 .9 1.0 TEMPERATURE (T/Tc) Fig. 8.~ IQ vs temperature, at E = 125 eV and 6 = 12° in the range 0.5 < T/Tc< 0.8. The upper half of the graph corresponds to one orientation of the sample magnetization while the lower part is obtained by reversing the magnetization. The full lines indicate that a linear extrapolation to zero at T is not inconsistent with the data. (Taken from Ref. 4)7

sign of I? it underestimates its absolute value by a factor of about 20.

5. CONCLUSION

The purpose of this paper was essentially to call the attention towards a new tool which to my view is going to play a very important role in the study of electron spin dependent interactions in the various fields of physics. In particular, in solid state physics, the NEA GaAs source together with lock-in electronics provides a unique technique to determine the surface magnetic structure of ferromagnetic materials under a variety of controlled experimental conditions including the effect of contaminants. This is not only interesting from the point of view of understanding basic interactions, it may also be of great practical interest. ELECTRON SCATTERING BY METALS 233 Nothing has been said here about the application of the technique- to the study of the spin-orbit effect in magnetic and non-magnetic materials. Although measurements of a similar kind were already performed in W and contain a wealth of information they have not been fully interpreted yet.

ACKNOWLEDGMENTS

I had many helpful discussions with colleagues at ETH, (Switzer- land) particularly with Dr. H.C. Siegmann and Dr. W. Baltensperger, I want also to thank Dr. D.T. Pierce at NBS (USA) for providing me with information prior to publication, and to Prof. H. Panepucci at USP (Brazil) for a critical reading of the manuscript.

Finally, my gratitude to the Organizing Committee of the VI Brazilian Symposium on Theoretical Physics and in particular to Prof. E. Ferreira for their kind hospitality.

REFERENCES

1 E. Garwin, D.T. Pierce and H.C. Siegmann, Helv. Phys. Acta 47, 393 (1974). 2 D.T. Pierce, R.J. Celotta, G.C. Wang, W.N. Unertl, A. Galejs, C.E. Kuyatt and S.R. Mielczarek, Rev. Sci. Instruments, April 1980. 3 M.S. Lubell, Atomic Physics 5, edited by R. Marrus, M. Prior and H. Shugart (Plenum Press, New York, 1977), pp. 325-373. 14 R.J. Celotta, D.T. pierce, G.C. Wang, S.D. Bader and G.P. Felcher Phys. Rev. Letters 4_3_, 728 (1979) . s A more detailed treatment of the photoproduction of polarized electrons which includes band structure effects around the r point as well as photoexcitation in the neighborhood of the L point in the Brillouin zone is given by D.T. Pierce and F. Meier, Phys. Rev. B 13^, 5484 (1976) . 6 The characteristics of sources using p-GaAs with a small positive electron affinity (PEA) were discussed by B. Reihl, M. Erbudak and D.M. Campbell, Phys. Rev. B 19_, 6358 (1979) and also in Ref. 5. 7 X.I. Saldaria and J.S. Helman, Phys. Rev. B 16_, 4978 (1977). 8 S. Wakoh, J. Phys. Soc. Jpn.2£, 1894 (1965). 9 J.S. Helman and W. Baltensperger, Bull. Am, Phys. Soc. March 1980. An extended version will be published elsewhere. 10P. Hohenberg-and W. Kohn, Phys. Rev. 136, B864 (1964). UR.A. Ferrell, Phys. Rev. 101, 554 (1956). 12D. Pines, Rev. Mod. Phys.~2J[, 184 (1956) . 13S.F. Alvarado and W. Eib, Proc. ICMAO, Aug. 15-18, 1977, Haifa, Israel. 235

PHASE SEPARATIONS IN IMPURE LIPID B1LAYER MEMBRANES *

F. de VeiccuO ('), D. A. Pink (3), E. B. Vadas (3) and M. J. Zuckeimann (') (') Department of Physics, McGill University, Montreal, Que. Canada (2) Department of Physics, St. Francis Xavier University, Antigonish, N. S. Canada and (3) Department of Biochemistry, McGill University, Montreal, Que. Canada

SUMMARY

A physical model is presented to describe theoretically the temperature dependent interactions of lipid bilayers with anaesthetic molecules. Based on an earlier model, a triangular lattice in which each site is occupied by a single lipid chain is constructed and the site of interaction of the anaesthetic molecules is then considered. Small anaesthetics and local anaesthetics at low concentration are assumed to occupy interstitial sites in the centre of each lattice triangle. The phase characteristics of such lipid-anaesthetic mixtures are described in terms of the interaction parameters between lipid-lipid, lipid-anaesthetic and anaesthetic-anaesthetic molecules. Depending on the chemical nature of the interacting species the following three models are formulated: Mçdel I. An interstitial model in which the only perturbation is in the head group region of the bilayer and direct interactions between neighbouring anaesthetic molecules are taken into account. Model II. Here only hydrophobic interactions between anaesthetics and lipids are considered. Model III. Both van der Waals and Coulombic interactions are taken into account. Phase diagrams for the three models are obtained by numerical calculation over a wide range of interaction parameters. It is shown that in all three models lateral phase separation takes place due to the presence of anaesthetics. The heat of transition, however, is found to be virtually independent of the anaesthetic concentration.

*based on lecture given by M. J. Zuckennann 236 M. Zuckermann

1. INTRODUCTION

Many theories of anaesthetic action are based on the change of fluidity of membrane phospholipids by solute anaesthetic molecules [1,2,3] . In consequence there is a considerable range of in-vitro experiments which examine the fluidity of pure and mixed phospholipids as a function of anaesthetic concentration. The experimental methods include differential scanning calorimetry (DSC) [4,5] , turbidity [6], dilatometry

[71, electron spin resonance (ESR) with spin labels such as TEMPO [ 8,9 ] fluorescence spectroscopy [5,10 ] and nuclear magnetic resonance (NÍIR) [ 11,

12,13 ] . The anaesthetic molecules used are both local anaesthetics such as synthetic cocaine derivatives (e.g. procaine, dibucaine, tetracaine), benzyl alcohol, long chain alcohols as well as general anaesthetics (e.g. halothane, enflurane and other inhalation anaesthetics).

The main conclusion of the studies listed above is that most anaes-

thetics increase the fluidity of phospholipids in the liquid crystalline

or disordered state. This increase is related to a decrease in the gel-

liquid crystal transition temperature, T , with increasing anaesthetic con-

centration. Exceptions to this result are the trans saturated and unsat-

urated isomers of tetradecanol and hexadecanol which produce an increase

in T of dimyristoyl phosphatidylcholine (DMPC) and distearoyl phospha-

tidylcholine (DSPC) bilayers with increasing anaesthetic concentration [14] .

However, all anaesthetics appear to broaden the phase transition region

in lipid bilayers.

The purpose of this paper is to present a physical model which pro-

vides a framework for a systematic analysis of these experimental results.

This requires, however, a more detailed description of the pertinent experi-

mental data: BILAY.ER MEMBRANES 237

The DSC experiments of Mountcastle et al.[4] are of particular interest in this context. Here the effects of the volatile general anaesthetics, halothane and enflurane, on dipalmitoylphospnatidylcholine (DPPC) bilayers were examined. Both anaesthetics decreased T and broadened the phase transition region with increasing drug concentration. However, the heat uf. transition was found to be nearly independent of the anaesthetic concentration. Similar results were obtained by Papahadjopoulos et al. [5] in their extensive analysis on the effects of dibucaine dissolved in acidic phospholipid membranes using both DSC and fluorescent probes. It is also interesting

to note that Cater et al. [15] in examining the effect of drugs on the phase

transition temperature of DMPC and DPPC show that the heats of transition are

effectively independent of drug concentration up to 50 mole % for both the

morphine derivative MIV and the tricyclic antidepressant desipramine.

The analysis of DSC data using our model requires some knowledge of

the location of the anaesthetic molecules in the bilayer. Shieh et al. [11]

have shown by high resolution proton magnetic resonance (PMR) that volatile

general anaesthetics act on the hydrophilic groups of the phospholipid mole-

cules at physiological anaesthetic concentrations,and then diffuse into the

hydrophobic region of the bilayer at high concentrations. They conclude that

the general anaesthetics increase the motion of the positively charged head

groups by weakening Coulombic interactions.

Recently Turner and 01dfield[l2] used deuteron magnetic resonance (DMR)

to study the interactions of benzyl alcohol with DMPC bilayers. They find

that the average orientational order parameter of the hydrocarbon chain de-

creases with increasing benzyl alcohol concentration, and the thickness of

the bilayer is unchanged by the anaesthetic at physiological concentrations. 238 M. Zuckermann

They also show that benzyl alcohol causes a decrease in the quadrupole splitting

and hence in the ordar parameter along the hydrocarbon chains. This is in con-

trast to the action of cholesterol in phospholipid bilayers [16] since choles-

terol was found to cause an increase in bilayer thickness at comparable con-

centrations and a large increase in the order parameter along the hydrocarbon

chain. Finally, some preliminary results using DMR on egg phosphatidyl choline

(PC) and PC-phosphatidyl serine (PS) mixtures containing procaine and tetra-

caine have been reported by Boulanger et al.[13]. In this case both the

hydrocarbon ctialtvs of the phospholipid molecules and the aromatic rings of the

Cetracaine were deuterated. 1'he experimental data indicated that charged

tetracaine molecules ware attached to the bilayer at two different sites. At

the first site, the tetracaine molecules are in the aqueous medium and seem

to interact with the polar heads of the lipids through their charged H (CH3)2

group and fast exchange with the aqueous medium takes place. At the second

site, the tetracaine molecules are oriented parallel to the lipid hydrocarbon

chains and hence their aromatic groups lie in the hydrophobic region of the

bilayer. Only slow exchange with the aqueous medium may occur at this site.

2. THEORY

The theory presented in this paper is based on Pink's model for lipid

hydrocarbon chain dynamics in bilayers [17]. This model has been used to

analyse several properties of tilayers, including the enhanced diffusion of Ha+

out of DPPC vesicles at Tc and the DMPC-dPPC phase diagram in the region of 300 K

[17]. The theory accounts for the temperature dependence of the Raman inten-

sities of the "1130 cm" " band of the hydrocarbon chain of pure bilayer

lipids [18]. The effects of intrinsic molecules such as cholesterol and

proteins have teen examined in detail [19]. In particular, the model leads to BILAYER MEMBRANES an understanding of the rigidifying effect of cholesterol in fluid bilayers and its fluidizing effect on bilayers in the gel state.

The model assumes that each site of a triangul^a: lattice is occupied by a single lipid chain as'shown in Fig. l(a). Each lipid chain can exist in a number of configurations, some of which are shown in Fig. l(b); an all °2 trans state with zero internal energy which projects an area of 20.4 A onto

a plane perpendicular to the long molecular axis, and a number of intermed-

iate degenerate states which can be excited below the main transition temper-

ature with kittle steric hindrance. AÍ1T other physically realistic^; config-

urations may be included in one excited or melted state of high internal

energy, Ey, high degeneracy, D , and "cross sectional" area of 34 A . The

internal energy., and degeneracy of each state are related to the formation

of 2-fold degenerate gauche states and depend on the number of CH' groups °2 per chain. Eight intermediate states each of area less than 26 A were used

by Pink et al. [18] to study the Raman data. The details of the excited state v .

are unknown and its parameters must be determined from experiment. For pure DPPC

bilayers E and D were found to be 2.74 x 10 ergs and 6x3 respectively

[17] . The hydrocarbon chains interact via a quadrupole-quadrupole interaction

J due to London dispersion forces as described by Wulf [20]. The interaction

between the polar head groups as modified by the surrounding water may be

represented either by an effective lateral pressure,H ,first introduced by

Marcelja [21] or by an effective Coulombic interaction,K . Positive values

of K correspond to attractive Coulombic interactions and negative values of

K^ correspond to repulsive Coulombic interactions. The Hamiltonian using

H is described in detail in several publications [17-19] and will not be

discussed here. 240 M. Zuckermann

Fig. 1 (a) A schematic representation of the 2-dimensional triangular

lattice. The dots (•) represent the sites of individual

lipid hydrocarbon chains or substitutional impurities and

the triangles (À) represent interstitial sites for anaesthetic

molecules. BILAYER MEMBRANES 241

Fig. 1 (t>) The all-trans state and some intermediate states for a saturated hydrocarbon chain with l6 carbon atoms. The internal energies 'E), areas (A) and degeneracies (D) are: (a) E = 0, A = 20.1* A2, D = 1; Ob) E = E , A = 23.5^ A2, D = 2;

(c) Kink. E = 2E , A = 21.86 A.2, D = k; (d) E = 3E , 6 6 A = 25.5 A2, D = 8; (e) E = 3E , A = 25.5 A2, D = h. Inter-

mediate states do not have gauche rotations around the two

bonds above the dashed line. E (=0.1(5 x 10 erg) is the

energy of a single gauche rotation (see [l?]). The next step is to model the site o£ interaction of the anaesthetic molecules in such a lattice. In this context it should again be emphasized

that cholesterol and anaesthetics have opposite effects on bilayers. For

example, anaesthetics do not effectively alter the heat of transition, AH,

at T whereas cholesterol alters AH drastically and destroys the phase tran-

sition at sufficiently high concentrations. Moreover, cholesterol rigidities

neighbouring lipids in the fluid state whereas most anaesthetics tend to flui-

dize them. Pink et al. [19] calculated such changes in AH and obtained a

theoretical phase diagram for DMPC-cholesterol mixtures assuming that a

cholesterol molecule can occupy a lattice site. Cholesterol is thus treated

as a "substitutional impurity" which interacts strongly via a quadruuole-quad-

rupole interaction with lipid chains in the all trans state and very weakly

with lipid chains in the excited state.

The insensitivity of the heat of transition to the presence of volatile

anaesthetics in the bilayer therefore indicates that such small anaesthetic

molecules cannot be considered as "substitutionai impurities" in the triangular

lattice model. Moreover, the results of Shieh et al. [11] indicate that the

effect of these anaestheticson bilayer dynamics is predominantly a surface

effect which weakens the electrostatic interactions between the polar head

groups of the phospholipid molecules. Hence,we place small anaesthetic mole-

cules at interstitial sites in the centre of each lattice triangle in our

model, allowing one anaesthetic site for every two lipid chains (vid. Fig.l(a)).

Any change in the Goulombic interaction due to the anaesthetics is modelled

by allowing the interaction, K^, between charges on nearest neighbour lattice

sites to be altered to KQ + K.^, if an anaesthetic site is occupied in a tri-

angle having both sites as vertices. Here K can be either positive or

negative. This allows either an increase or decrease in the interaction between BIIAYER MEMBRANES 243

neighbouring head groups due to the presencie of small interstitial impurities.

A complete description of this model requires the inclusion of a direct inter-

action, J,between neighbouring anaesthetic molecules. Positive J.. corresponds

to an attractive interaction. The interstitial model defined by the parameters

K , It, , J and JQ will be referred to below as Model I.

Certain local anaesthetics such as dibucaine and tetracaine have hydro-

phobic groups (e.g. aromatic rings) which will interact directly with the

hydrocarbon chains of the lipids via London dispersion forces if part or all

of the molecule penetrates into the interior of the biiayer. Since the experi-

ments of Papahadjopoulos et al. [5] indicate that the heat of transition is

independent of dibucaine concentration, we may assume that local anaesthetics

can also be considered to occupy interstitial sites in the triangular lattice

model at low concentrations. The hydrophobic interactions can then be modelled

by quadrupole—quadrupole interactions between the hydrophobic groups of the

anaesthetics and flexible hydrocarbon chains. These can also be written in

the form derived by Wulf [20]. Since the anaesthetics should interact

differently vith rigid and with fluid lipid chains, at least two interaction

parameters J._ and J^, must be defined. J.- describes the interaction between

a lipid chain in the ground state or in any of the eight intermediate states

with an anaesthetic molecule and J-j. describes the same interaction for lipids

in the excited or "melted" state. We can now construct two further models:

(a) Model II. In this case the interaction between the polar heads is modelled

by using Marcelja's lateral surface pressure He(= 30 dyne/cm ). In this model

electrostatic interactions in the surface of the bilayer are not taken into

account explicitly, i.e. only hydrophobic interactions between the anaesthetic

molecules and the lipid chains are considered. Model II then is an interstitial

model defined by IIe, JAG, J^, JM and JQ. In general Ite can be replaced by y.x

244 M. Zuckerr.ann

It + H'x, where x is the fraction of interstitial sites occupied by anaesthetic molecules and II' is the change in latent pressure per anaesthetic molecule. (b) Model III. When anaesthetic molecules are charged, e.g. dibucaine, both electrostatic and London dispersion forces need to be taken into account. j

Model III. is therefore an interstitial model which is described by the parameterr s Ko , K,1 , J._AG, J.,,AM , J.A.A and Jo . In order to calculate thermodynamic quantities and phase diagrams, the Hamiltonians describing models I, II and III were written in the molecular field (Bragg-Williams) approximation in terms of an order parameter, ?.„, for the hydrocarbon chains for all models and an order parameter, X , for the polar head region for models I and III. Next, an expression for the free energy»? , -'

is derived in terms of \,, X and x. F(X,T, X , x) is then minimised with "$ Ti' p V p' ~j' respect to the order parameters and self-consistent expressions for X and X jL. P w A£ ' <

are derived. The expressions for F, \ and Xw can now be used to calculate .'àíj the tb.ermodyn.amic quantities for a homogeneous mixture of lipids and anaes- j'l thetics. Let c be the anaesthetic concentration for the interstitial model described above. Then c and x are related by the equation: c = 2x/(l + 2x).

The calculation of phase diagrams for lipid-anaesthetic mixtures requires ;s a calculation of the free energy F'(c , c_) for two homogeneous mixtures each with different anaesthetic concentrations c and c_ and a fixed total concentration ,c ^uch that c.^ < c < c [17,19]. c^ and c^ are varied at a fixed temp-

1 erature ,T ,until a minimum in F (c , c ) is found. The values of c and c? at the minimum value of F' are limiting concentrations of a phase separation region at temperature T providing c. < c < Cp. Phase diagrams were obtained by numerical calculation for models I, II and III over a wide range of interaction parameters. The results of these and other calculations are discussed in the next section. The mathematical details will be published elsewhere [17]. BILAYER MEMBRANES 245

3. RESULTS AND DISCUSSION

The phase diagrams for lipid-anaesthetic mixtures found from numerical calculations for model I are shown in Figs. 2-5. The lipid was chosen to be

DPPC and in consequence the values for J" and K are those used to

fit Pink's model to experimental data: J"Q = 0.73 x 10 ergs and KQ = 0.6125

x 10 ergs [17]. Arbitrary values of the parameters K_ and J . were chosen

and varied systematically to cover a broad range of physical situations.

Since there is insufficient experimental data available, it is somewhat difficult

to match the theoretically obtained behaviour with physically realistic

situations. This point will be elaborated below.

In the first calculation, the direct interaction between anaesthetics

is taken to be attractive and weaker than the interaction J between the lipid

chains i.e., J = 0.3 x 10 ergs. The interaction IC was chosen to have a

negative sign and its magnitude was increased up to a value of -0.55 x 10

ergs. This corresponds to a weakening of the interaction K between polar

heads by the anaesthetic molecules. In this case a narrow gel-fluid phase

separation region was found in the vicinity of T for all values of ÍC, which

broadened as the magnitude of TL, increased (Fig. 2). The value of IC was then

fixed at -0.55 x 10~ ergs in order to maintain bilayer integrity and J was

increased in magnitude. Fig. 3 shows that the phase separated region retains

its shape, but broadens considerably for higher anaesthetic concentrations

(0.2 < c < 0.6) when J^ = 0.U5 x 10~ ergs. The nature of the phase diagram

changes completely when J.. is increased to 0.6 x 10~ ergs. This is shown

in Fig. h. The broadened region of Fig. 3 becomes a bell shaped curve with

three different phase separated regions: gel-gel, gel-fluid and fluid-fluid.

The phase diagram also exhibits a euteciic point and is similar to the phase

diagram found by Pink and Chapman [19] for small substitutional molecules

which interact strongly with rigid chains. The reason for the bell shaped 246 M. Zuckermann

ill 1 I _

310 - F UJ K G N - n: - \ UJ 300 \ a. u I- 290 t i i i i .V 11.0

9.0 I 0.0 0.1 0.3 0.5 0-7 CONCENTRATION

Fig. 2 (a) Phase diagram near the gel to liquid crystal transition in a

mixture of DPPC and anaesthetic molecules described

by Model I. The parameters used (in units of 10~ ergs) are

Jo =0.731*, KQ = 0.6125, 1^=2.78, JM= 0.3,

1^ = -0.55-

(b) The heat of transition,AH,as a function of anaesthetic concen-

tration,c,across the phase separation region. G and F stand

for the gel and liquid crystal phases respectively in all

phase diagrams. BILAYER MEMBRANES 247

11.0 r AHCkcal/mole)

9.0 0.0 O.I 0.3 0.5 CONCENTRATION

Fig. 3 (a) Phase diagram near the gel to liquid crystal transition in a

mixutre of DPPC and anaesthetic molecules describedtoy Mode l I.

The parameters used {in xmits of 10~ ergs) are 3 = °-13k' K = 0.6125, K = 2.78, J =0.1*5, = -0.55-

The heat of transition, AH,vs. concentration,c,across the

phase separation region. 248 M. Zuckermann

330 1 1 1 i i

F F 320 -- i ] /^

/ F"F 310 - V LU (T \ « G-F <300 \ Ul Ü_ L \ TC l - UJ G-G 290 1— —1 f- 1 \— 1' .AH (kcal/mole)

11.0

9.0 0.0 0.1 0.3 0.5 0.7 CONCENTRATION

Fig. h (a) Phase diagram near the gel to liquid crystal transition in a mixture of DPPC and anaesthetic molecules described hy Model I. 1 The parameters usea (in units of 10" ergs) are JQ = 0.73 *,

KQ = 0.6125, i^ = 2.78, JM= 0.6, = -0.55- T u and T refer to the upper and lower main phase transition

temperatures respectively.

Cb) The transition enthalpy,AH,vs. concentration,c,across the

phase separation region. BILAYER MEMBRANES 249 curve is that the interstitial impurities begin to form clusters at a given

anaesthetic concentration for high enough values of J^. This gives rise to

fluid-fluid, and gel-gel phase separation in the appropriate temperature range,

since the interstitial impurities are always surrounded by lipid chains. The

broad fluid-gel region beneath the bell-shaped curve occurs because the

interstitial anaesthetic molecules fluidize the neighbouring lipid chains in

the anaesthetic rich phase, which therefore has a lower main transition temper-

ature T . The phase with a lower anaesthetic concentration, however, has a

higher value T of T and therefore remains in the gel state to higher temper-

atures .VhenJJJ. is increased to 10~ ergs, the attractive interaction between

the anaesthetics dominates all other interactions. In this case phase

separation occurs for all but the lowest concentrations as shown in Fig. 5.

The calculated heat of transition only shows a small change as a function

of concentration across the phase separation region except in the bell shaped

portion (see Fig. 2 to 5). Fig. 6 shows that the entropy is a continuous

function of temperature which increases sharply across the phase separation

region when J^ = 0.6 x 10~ ergs, i^ = -0.55 x 10~13 ergs and c = 0.2.

This implies that the peak at T in the DSC measurements should broaden as a

function of anaesthetic concentration. This is indeed the case for DPPC-

volatile anaesthetic systems as described by Mountcastle et al.[U]. This does

not apply to the bell shaped phase separated region of Fig. U. (j = 0.6 x 10~

ergs), and to the phase separated regions of Fig. 5 (Jiyi = 1 x 10~ ergs).

The heat of transition is now distributed between a sharp lower phase transition

for the anaesthetic rich phase at T and a sharp upper phase transition for the

weakly concentrated phase at T^ (see Fig. T). Thus,there should be two separate

peaks in the DSC data. In this context it is interesting to note the shape of

Cater's heating curves obtained with DPMC and DPPC in the presence of the drugs

morphine MIY and amitriptyline [15]. The model here presented may provide the

explanation for the appearance of the two peaks in the DSC scans. 250 M. Zuckermann

320 r

290 -

O.O 0.1 0.3 O.5 0.7 CONCENTRATION

Fig. 5 The phase diagram near the gel to liquid crystal transition

in a mixture of DPPC and anaesthetic molecules described by

I. The parameters used (in units of 10~ ergs) are

J = 0.73U, K = 0.6125, o Q JAA = K_ = -0.55. ENTHALPY (Kcal/mole)

t-h Mi H to CD o H. 3" t o ^ X n> » • i-5 rr i-3 n C 3- 3- O ro re» w o o o 01 a i-n n> p. i 3 1 I 1 I 1 1 1 i ro rr o

(0 t— 1 3 -a ro ro *< t-h ••* - ro oo

c •c

13 3" W

1 O to Ki 3 in rt H- (t) ?• I o rb 3 * FREE ENERGY(10 ergs) ENTHALPY (kcal/mole) I ! I .b. .N -Ü O CD « en U) b CD o rT G- G ro 3* o in IT) ro W n w rr rt CD rt) - \ ro 3 I—» v; CL m n •i \ O a* m \ i ; /, \ "^ J> ÜJ s —\ O Oi m

II o / i 1 1 1 1 1 11 01

152 S3NVH9W3W H3AVHH 252 M. Zuckermann

In Model II, where the anaesthetic-lipid interactions are strictly hydrophobic, the phase diagrams are qualitatively the same as those obtained for Model I. Again, the nature of the phase diagram depends on the magnitude of J... Fig. 8 shows a typical "solid solution" phase separation

13 13 (JA(J = 0.0, Jm = 0.5 x 10~ ergs and JM = 0.1*5 x 10~ ergs (c.f. Fig. 2).

If J > J interstitial impurities prefer melted chains as neighbours resulting in a decrease in the value of T , corresponding to a negative K_ .

Similarly, the case J > J corresponds to positive ¥L , and T increases with increasing anaesthetic concentration. The phase diagram resulting from such an increase in T is shown in Fig. 9 for JA(} = 0.1 x 10~ ergs and

JAA = °"3 x 10~13 ergs#

Lastly Model III is of interest when the Coulombic interaction,!^,and the hydrophobic interaction between the anaesthetics and the lipid chains have

opposite effects onl , e.g. when K. > 0 and J < J.w. The phase separation

region nay then become much narrower because of the competition between the

two types of interactions. This is shown in Fig. 10a, for ÍL = 0.55 x 10~

ergs, J^ = 0.3 x 10~ ergs, JA = 0.0, and J = 0.3 x 10 ergs and in

Fig. 10b, for IC = 0.4 x 10~13 ergs, J = 0.1»5 x 10~13 ergs, J = 0.0 and

JAM = °*3x 10~ erSS- The heat of transition for Models II and III is

again weakly dependent on anaesthetic concentration, at lov values of J...

These results theoretically confirm the extreme sensitivity of the bilayer

to the presence of small molecules such as anaesthetics. Such molecules will

always cause lateral phase separations in the bilayer. When the anaesthetic

molecules are "interstitial", the phase separation has the "solid solution"

behaviour of Fig. 2 (i.e., a fluid-gel phase separation in the vicinity of T )

at low concentrations or when the direct intermolecular interactions J.. are AA veai. If J^ is attractive and increases in magnitude, the anaesthetic BILAYER MEMBRANES 253

.310 1 1 i i i 1

N F 305 LÜ

G \ G-F Ul 295 - V \

285 1 i i 1 1 0.0 0.1 0.3 0.5 0.7 CONCENTRATION

Fig. 8 Phase diagram near the gel to liquid crystal transition in a mixture of DPPC and anaesthetic molecules described by Model II. The parameters used (in units of 1-10 3 ergs) are

0.734, 2.78, 0.45, = 0.5, "AA JAM J = 0.0. The internal pressure II 30 dynes/cm. At» fi

i 254 M. Zuckerraanr.

I i I I > i AH (kcal/'mola) 9.0 -

8.0

6-0 1 1 i

310 0.0 0.1 0.3 0.5 O.7 CONCENTRATION

Fig. 9 (a) Phase diagram near the gel to liquid crystal transition in a

mixture of DPPC and anaesthetic molecules described

by Model II. The parameters used (in units of 10~ ergs) are

J* = Q.73^i ^ ~ 2.78 J = 0.3. 3 — 0.0. o AA AM (e) J.., = 0.1. The internal pressure II = 30 dynes/cm.

(b) The heat of transit ion, AH,as a function of anaesthetic concen-

tration ,c,across the phase separation region. BILAYER MEMBRANES 255

11.0

0.3 0-5 0.7 CONCENTRATION

Fig. 10 The phase diagram and the corresponding heat of transition ,AH ,

as a function of anaesthetic concentration ,c ,near the gel to

liquid crystal transition in a mixture of DPPC and

anaesthetic molecules described by Model III. The parameters

used (in units of 10~ ergs) are J = 0.T3U, E = 2.78,

JAG = °-°. J^ = 0.3. Fig. 10(i),(ii) corresponds to

values 1^ = O.U and JM = 0.1+5, and Fig. lC(iii) ,(iv> corresponds to

the values iC^ = 0.55 and J = 0.3 respectively. M. Zuckermann molecules tend to cluster. This results in broad fluid-fluid, fluid-gel and

gel-gel phase separations. The existence of a "solid solution" phase separa-

tion explains the temperature broadening of the main phase transition region

seen in the experiments listed in the introduction. Further, the insensitivity

of the heat of transition to low anaesthetic concentrations evidenced both

experimentally and in the calcualtions for Models I, II and III implies that it

is reasonable to treat the anaesthetic molecules as interstitial impurities.

It is now possible to correlate actual lipid-anaesthetic systems with

each of the 3 Models. Model I can be used to describe the behaviour of lipid-

volatile anaesthetic mixtures at low anaesthetic concentrations, vhere the

anaesthetic molecules only perturb the polar head group interactions. Model II

can be related to systems such as lipid-benzyl alcohol mixtures. In this case

the polar head region of the lipid bilayer should be much less perturbed than

the hydrocarbon chains provided that the benzene ring of the anaesthetic lies

in the bydrophobic region of the bilayer. The experimental results of Turner

and Oldfield [12] indicate that benzyl alcohol fluidizes the membrane vhich is

manifested by a decreasing order parameter at each carbon atom of the hydro-

carbon chains. This implies that J > J for this system. Calculations of

the order parameter as a function of position along the hydrocarbon chains

using Model II are in progress. Model III provides a description of lipid-

local anaesthetic systems such as lipid-dibucaine or lipid-tetracaine mixtures

where the anaesthetics are polarizable and the benzene rings lie in the hydro-

phobic regions of the bilayer. This situation corresponds to the second site

found by Boulanger et al.[l3] for tetracaine molecules in phospholipid bilayers,

where the anaesthetic molecules are oriented parallel to the lipid hydrocarbon

chains and the aromatic rings are located in the hydrocarbon interior of the

bilayer. BILAYER MEMBRANES 257

In general, the bilayer interfaces with an aqueous medium in the polar head region, resulting in the partition of the anaesthetic molecules between the bilayer and the surrounding medium. For single phase homogeneous bilayer- anaesthetic mixtures with a well-defined main phase transition temperature,

such partitioning implies that the anaesthetic concentration in the gel phase,

c_, is not equal to that in the fluid phase cp. Such a partition effect has

not teen included in the formalism leading to the results described above and

will now be theoretically analyzed for single phase homogeneous mixtures.

Previous authors [3,22,2-3] neglect lipid-anaesthetic and direct

anaesthetic-anaesthetic interactions and examine the change in TQ with

anaesthetic concentrations using the solution thermodynamics. This theory

gives the following well-known expression for the change,AT ,in T [3,22,23]

at low anaesthetic concentrations

where T is the transition temperature and AH is the heat of transition of co * o the pure "solvent" bilayer in the appropriate units. In expression (l) the

lipid-lipid interactions are the only interactions considered and are

implicit in the parameters T and AH . In order to investigate the general

case c # c^, the number of anaesthetic molecules in the bilayer was treated

as a variable and the anaesthetics were considered to be adsorbed molecules

on the bilayer surface. An extra term representing the free energy for a

single adsorbed anaesthetic molecule was therefore added to the free energy of

model I. The new free energy term F = F(X , X , NL, x) is now minimised with

respect to both the number of lipid chains, H , and the fraction of interstitial

sites occupied by anaesthetic molecules, x. The lipid components of F can now

be written as follows: F (c ,T) represents the free energy of the gel phase M. Zuckermann

lipid at a fixed anaesthetic concentration cQ and at temperature T and T? F:(c.-,,T) is the corresponding quantity for the fluid phase lipid. The

expression for the main transition temperature T is now given by the condition:

Equation (2) reduces to equation (l) if the anaesthetic-lipid and anaesthetic-

anaesthetic interactions are reglected, i.e. 1^ = J^ = 0.

Minimisation of the free energy F vith respect to x gives an expression

for the equilibrium value for x in terms of the partition function q,. for a

single adsorbed anaesthetic molecule and in terms of the parameters J^, IC^ and

X . When 1C = J = 0 this expression reduces to the Langmuir isotherm if q p X A A is identified as[C ]k,where[C ]is the concentration of anaesthetic molecules

in the aqueous medium in equilibrium with the bilayer surface and k is the

rate constant for the binding reaction.

Preliminary results of numerical calculations using the expressions for F^ and

F. with J. = 0 and K_ < 0 (i.e. decreasing polar head group interaction) are

shown in Table I. The parameters for DPPC were used and c^ was taken to be r

Q.I. Table I shows that (i) the maximum decrease in the main transition

temperature for c., = 0 occurs when KL = O and (ii) AT decreases in magnitude

with increasing values of IKJJ. Lee t23] points out that such a departure from

the result of equation (l) occurs for DPPC-n-octanol mixtures and ascribes this

departure to the existence of a phase separated region near T . However, the

results shown in Table I indicate that this behaviour can be explained by the

inclusion of interactions of the required sign between the lipid and the

anaesthetic molecules. These interactions will also cause a phase separated

region near T .

The phase diagram for any lipid-anaesthetic mixture must therefore be

adjusted so as to include the partition effect just described. Detailed BILAYER MEMBRANES 259

TABLE I Effect o£ Interactions and Anaesthetic Partition on the Main Transition Temperature

K (x 10~13ergs) 0,0 -0.2 -0.4 -0.55 -0.61

T,.*(K) 309.9 ' 311.8 313.6 315 315.6

) 3U-° 313.55 313.3 313 312.9

* Tc. is the main transition temperature for ananaesthetic-DPPC mixture with anaesthetic fraction x - 0.05, x*f = 0

**T£B is the corresponding temperature for x. = x| = 0.05. The parameters for 13 DPPC were used with KQ = 0.6125 x 10~ ergs and J, = 0.

calculations vill he presented in a future article. Preliminary calculations indicate that although the partition effect will cause quantitative changes in the phase boundaries, the qualitative behaviour of the phase diagrams described above should not change.

Wu and McConnell [2k] used spin labels to obtain phase diagrams for mixed phospholipid systems. In particular, they obtained a bell shaped curve with one eutectic point for dielaidoylphosphotydylcholine (DEPC) and dipalmytoyl phosphotydylethanolaaine (DPPE) mixtures, which has the same qualitative behaviour as the phase diagram of Fig. h. However,for DEPODPPC mixtures, only a broad solid-solution phase diagram was found. This implies that this bell-shaped phase diagram is due to the strong interactions between the polar heads of the DPPE molecules. This is precisely the reason for the bell shaped region of Fig.h , i.e., strong attractive anaesthetic-anaesthetic interactions.

The same formalism will therefore be used to study mixtures of charged and uncharged lipids with identical hydrocarbon chains, provided the impurities are now treated substitutionally. 260 M. Zuckermann

4. CONCLUSIONS

There are many possible extensions of the theory presented in this paper.

In particular, anaesthetic-lipid correlations are required for an analysis of the effects of short • range order in the bilayer. Calculations leading to the behaviour of the lateral compressibility are needed for an analysis of the

effect of anaesthetics on the passive ionic permeability of the bilayer near

T . Pink's model has already been extended to include the effect of proteins

on the physical properties of the bilayer (Pink, D.A. and Zuckermann, M.J.,

unpublished work; Pink, D.A. and Chapman, D., unpublished work). In this

context it is of interest to discuss again the near invariance of the heat of

transition with anaesthetic concentration.

All the models predict that the heat of transition should be nearly independent

of the anaesthetic concentration for simple mixtures of lipids and interstitial

small molecules - a prediction born out by Cater et al. experimentally [15]•

However, when proteins are included in the bilayer we must consider the

possibility of highly selective, stereospecific interactions between the

protein and. anaesthetic or drug molecules. It has been shown experimentally

by Hosein et al.[£5] that the heat of transition becomes a function of drug

concentration when treating brain lipid extracts containing the opiate

receptor with increasing amounts of morphine. This problem requires a totally

different theoretical approach in order to include receptor mediated phenomena.

Interactions in such systems are being modelled at present. From an experi-

mental point of view, more microscopic measurements designed to examine the

exact location of anaesthetics in model bilayers are needed and more detailed

investigation of the phase separations should also prove to be very useful. B1LAYER MEMBRANES 261

ACKNOWLEDGEMENTS

We wish to express our gratitude to Ante Padjen, Myer Bloom,

Jean Charvolin, Keith Miller, Don Patterson, Camille Sandorfy, Jim

Sheats and Ian Smith for interesting discussions and, in some cases,

availability of results prior to publication. One of us (MJZ) wishes

to thank Bernard Coqjblin and the Faculta des Sciences, Universitê

ds Paris-Sud for a visiting professorship in June 1979 and Anadi

Bhattarcharjee of the same institute for his suggestion of surface

adsorption. The research work reported in this paper was supported

in part by the National Science and Engineering Council of Canada and

the "Ministere de I1Education" of the Province of Quehec.

REFERENCES

1. Trudell, J.R. (1977) Anaesthesiology 1»6, 5-10.

2. Seeman, P. (1972) Pharmacological Reviews 2U, 583-655.

3. Miller, J.C. and Miller, K.W. (1975) in Physiological and Pharmacological

Biochemistry (Blashko, H.K.F., ed.), pp. 33-U6, Butterworths

and Co. Ltd. London, England.

h. Mountcastle, D.B., Biltoncn, R.L. and Halsey, M.J. (1978) Proc. Nat.

Acad. Sci. USA 75, U906-U910.

5. Papahadjopoulos, D., Jacobson, K., Poste, G. and Shepherd, G. (1975)

Biochim. Biophys. Acta 39U, 5Olt-519.

6. Veda, I., Tashiro, C. and Arakawa, K. (1977) Anaesthesiology 1»6,

327-332.

7. MacDonald, A.G. (1978) Biochim. Biophys. Acta 507, 26-37. 262 M. Zuckermann

8. HuVbell.W.L., McConnell, M.H. and Metcalfe, J.C. (1969) Brit. J.

Pharmacol. 35, 37>»-375.

9. Trudell, J.R. , Payan, D.G., Chin, J.H. and Cohen, E.H. (1975) Proc.

Nat. Acad. Sci. 72, 210-213.

10. Vanderkooi, J.M. , Landesberg, R. , Selick II, H.and MacDonald, G.G.

(1977) Biochim.Biophys. Acta U6U-1-16.

11. Shieh, D.D., Ueda, I., Lin, H-C, and Eyring, H. (1976) Proc. Hat.

Acad. Sci. 73, 3999-1*002.

12. Turner, G.L., and Oldfield, E. (1978) Nature 277, 669-670.

13. Boulanger, Y., Schreier, S. and Smith I.C.P. (1979) Biophysical Journal

25 (No. 2 Part 2) 12a, communication No. M-AM-B8.

lU. Pringle, M.J. and Miller, K.W. (1979) Biochemistry 18, 3314-3320.

15. Caber, B.R., Chapman, D., Haves, S.M. and Saville, J. (197U) Biochim.

Biophys. Acta 303, 5^-69.

16. Chapman, D. (1973) in Biological Membranes (Chapman, D. and Wallach,

D.F.H. , eds.). Vol. 2, pp. 91-11*1», Academic Press, New York.

17. Caille, A., Pink, D.A., de Verteuil, F. and Zuckermann, M.J. (1979)

Canadian Journal of Physics, submitted for publication.

18. Pink, D.A., Green, T.J. and Chapman, D. (1979) Biochemistry (in press).

19. Pink, D.A. and Chapman, D. (1979) Proc. Nat. Acad. Sci. 76, 15l*2-15lt6.

20. Wulf, A. (1977) J. Chem. Phys. 67, 225U-2266.

21. Marcelja, S. (1976) Biochim. Biophys. Acta UU5, 1-7.

22. Hill, M.W. (1978) N.Y. Acad. Sci. 1308, 101-110.

23. Lee, A.G. (1977) Biochim. Biophys. Acta U72, 285-3^.

2U. Wu, S.H-W. and McConnell, H.M. (1975) Biochemistry lU, 8U7-851*.

25. Hosein, E.A. , Lapãlme, M. andVadas, E.B. (1977) Biochim. Biophys.

Res. Comm. 78, 19U-201. 263

RECENT PROGRESS IN BIOPHYSICS *

G.Bemski Departamento de Física, Pontifícia Universidade Católica Cx. P. 38071, Rio de Janeiro, RJ, Brasil and Instituto Venezolano de Investigaciones Científicas Caracas, Venezuela

íiBSTRACT Recent Progress in Biophysics is reviewed, and three examples of the use of physical techniques and ideas in biological research are given. The first one deals with the oxygen transporting protein-hanoglobin, the second one with photosynthesis, and the third one with image formation, using nuclear magnetic resonance.,

1. HHRCDDCTICN I feel honored to speak at a Symposium on Theoretical Physics. I am sure that this is going to be the most experimental talk at this Symposium, and this for two reasons. In the first place Biophysics is still a strongly „' experimental discipline, and secondly my interest are in the experimental aspects. This is not the only bias - I will concentrate on Biophysics on molecular level which by no means implies a lack of progress on other "levels", but just a personal preference. Nothing similar to the DNA breakthrough of the 1950's has occurred in the last two decades in biophysical research. Spectacular achievements have given place to a gradual increase in the understanding of the biological processes. No new principles of physics had to be evoked to explain the workings of nature, thus the vitalist era seems to be behind us. Yet the sheer size of the biological systems, for instance the number of atoms in a typical protein - of the order of 15000 - allows for appearance of new phencmena absent in every day physics. One such important problem, is that of allostery - change in conformatic n of proteins produced by binding of small molecule to some well defined site in the protein. This is a good example of connection between structure and function, one of the main problems in today's i Biophysics. This fact raises important questions. Hew is the shape of a ' protein determined by its composition? Can we produce a protein with a given function by properly lining up (or having the DNA do it for us) the carponent *Vfork partially supported by FINEP and CNPq. 264 G- Bemski amtoo acids which constitute the building blocks of each protein?

There exists a tendency among the physicists to attack this and other biological problems from the atomic side. It is an outgrowth of success of solid state physics in the 50's and 60's. The progress in DNA research gave impetus to the molecular approach in Biophysics. Yet it is doubtful that this "small scale" approach will be sufficient to explain some "large" scale problems. There is certainly more to neurobiology than understanding of the working of a neuron. A difficulty in the "large scale" approach is the scarcity of models which correctly predict the mechanisms used by nature in solving a given problem. Nature surprises us, time and again, by inventing solutions to important problems which could hardly have been thought of by a theoretical physicist faced with them.It seems to me that experimentation remains the only avenue to the understanding of the fundamental processes. We will examine here three problems as examples of the use of physics in biologically important questions. I allowed myself not only to be biased in their selection, but also to leave out references of many who should be represented, keeping only a few directly connected to our discussion.

2. HEMOGLOBIN

The first problem deals with the oxygen transporting protein-hemoglobin. The understanding of how this transport is performed is of great interest in the general problem of allostery in proteins, mentioned in the Introduction. PROGRESS IN BIOPHYSICS 265

Hemoglobin is an almost spherical protein, of an o approximate diameter of 50 A, composed of about 10000 atoms which endow it with a molecular weight of about 65000. Among the component atoms, mostly H, C, N and P, there are also four iron atoms which even if numerically few, are fundamental in the oxygen transport, since O2 binds directly to them. The functioning of hemoglobin raises many questions. How is the shape of hemoglobin related to the characteristics of O2 transport? How does oxygen enter and leave the protein? Why are other similarly iron containing proteins (the so called heme proteins) performing functions different than hemoglobin? Myoglobin for instance stores oxygen in the muscles, while cytochrome c shuffles electrons back and forth in oxidation- reduction process. Studies of hemoglobin have attracted many physicists, in part because of the presence of iron and of the resulting magnetism. This protein has also a virtue of existing in large, easily available quantities for research purposes. Even isolated laboratory,is self sufficient in this row material. As a result the array of techniques employed in these studies is quite impressive, and becomes more and more sophisticated. Hemoglobin became for biology the equivalent of silicon in the studies of semiconductors in the 1950's. The three dimensional structure of hemoglobin has been established through a heroic effort of Perutz's by X-ray diffraction and become known in the 1960's [l]. This was the first determination of a three dimensional structure of any protein,and formed the basis for all the future studies. 266 G. Bemski

One of the exciting outcomes of the X-ray studies was the realization that the structure of a hemoglobin molecule while it is not transporting oxygen (deoxyhemoglobin) , is substantially different from the structure of hemoglobin saturated with

oxygen (oxyhemoglobin). The distances between various pairs of o iron ions vary as much as 6 A during this transition.

FIG. 1 Equilibrium curves showing the affinity for oxygen of hemoglobin (4 hemes) and myoglobin (1 heme). The presence of more than one heme in hemoglobin is responsible for the sigmoidal shape of the curve as opposed to the hyperbolic curve of myoglobin.

PARTIAL PRESSURE OF OXYGEN (MILLIMETERS OF MERCURY)

As this point one should have a look at the very important curve of oxygenation of hemoglobin. One plots in fig. 1, the percentage of the saturation of the four iron ions with oxygen, as a function of the partial pressure of oxygen. For comparison a curve for myoglobin, one iron containing heme protein, is also shown. The myoglobin curve is hyperbolic while the hemoglobin curve is sigmoidal. The former can be obtained on the basis of

a simple bimolecular reaction between iron and 02, the latter implies the notion of cooperativity. In hemoglobin the iron ions communicate, with the result that any iron "knows" whether

the other three have already captured their 02 molecules. The PROGRESS IN BIOPHYSICS 267 sigmoidal curve is obtained by assigning increasing values for affinity of oxygen to the yet unoccupied sites, as compared to the affinity of the first iron for oxygen. It can be seen in the figure that the sigmoidal shape of the curve is fundamental for the functioning of hemoglobin.

It allows for 0z to dissociate from iron and from the molecule at partial pressures existing outside the lungs, allowing for effective transport and utilization of oxygen by the body- something that myoglobin's hyperbolic curve does not permit.

Another experimental fact involves the magnetic properties of iron ions. As Pauling was first to show [2] they are paramagnetic with spin 2 in deoxy, and diamagnetic in oxyhemoglobin.

FIG. 2 Triggering mechanism for the transition between the 2 structures of hemoglobin is a movement of the heme iron into the plane of the prophyrin ring. This 0.6 angstrom movement produces a chain of structural changes in the entire molecule. There a small partion of hemoglobin is shown in the two structures (From: M. Perutz, Scientific Amer, Dec. (1978), p.77)

At the present time a certain amount of healthy controversy exists regarding the conformational transformation from deoxy to oxy form. Perutz has come to the conclusion, on the basis of X-ray data, that, starting with deoxyhemoglobin, the iron ion (Fe2+)is located about 0.7 A below the plane of 266 G. Bemski the heme, a plane formed by a square of four nitrogen atonswith iron at the center (or below,in this case). On oxygenation a series of events takes place: Fe2+ changes its spin state fran 2 to 0, decreases its ionic radius, moves into a new equilibrium position at the center of the heme and,as a consequence of this 0.7 A displacement,it pulls with it the attached residues (Fig-2) producing an overall change in the conformation of the entire molecule. The distant parts of the protein amplify this motion, with the result that an overall,large rearrangement takes place. As a result,the geometry around the positions of the other irons

changes in such a way that they become more acessible to O2, increasing the affinity of the irons for Oj molecules in agreement with observation. These changes can be fit by an allosteric theory of Changeux, Monod and Wyman £3] which is essentially based on the existence of two overall conformations of hemoglobin, with two different kinetic factors, k, characterizing them. This stereochemical approach has been challenged recently by Shulman et al [V] ,as a consequence of EXAFS experiments (Extended X-ray absorption fine structure). In these experiments the X-rays produced in high intensity by the linear accelerator, are monochromated by a crystal, and directed against the sample. Measurements of the absorption edge near iron K shell absorption, show a fine structure which reflects and its N. nitrogen neighbors which reflects the distribution of nitrogen atoms near iron, through a factor

proportional to t^.sin (2K Rpe + a) where R^ represents the distances between the iron atom and its N. nitrogen neighbors which modulate this absorption; k is the wave vector of the photoelectron, and a the phase (fig. 3). PROGRESS IN BIOPHYSICS 269

FIG. 3 X ray absorption spectrum of Fe in HbO2 measured using fluorescence EXAFS technique. Rapid rise near 7100 ev is the Fe, K-edge and the small modulation above the edge is the EXAFS (From: P. Eisenberger, R. Shulman, B. Kincaid, G. Brown, S. Ogawa, Nature 274, 30 (1978)). 7.000 7.200 7.400 7.600 7,800 8,000 Photon energy (eV)

The intense X-ray source allows to obtain full spectra in reasonable times compared to normal X-ray diffraction

Fe N spectrometers, and the determination of the "" n distances o has an accuracy of 0.01A, or 100 times better than in X-ray o diffraction experiments. The results show only a 0.07 A difference in these distances between oxy and deoxy forms of hemoglobin, which argues against a motion of iron in and out o of plane by 0.7 A. Thus the resulting rearrangements must be due to many small distortions in the neigborhood of the heme during the oxygenation process. A beautiful series of flash photolysis experiments performed by the Illinois [5] group dealt with a seemingly different aspect of hemoglobin behavior. Here a strong pulse of light has been used to dissociate CO from hemoglobin or myoglobin. CO, as well as O2, binds to iron, but its binding is much stronger-a fact responsible for a serious health hazard. After the light pulse, the rebinding of CO to the heme iron is studied. The kinetics of this rebinding indicate 270 G. Bemski that below 20 K CO hops back to its original position next to iron, by quantum mechanical tunneling (fig. 4). At higher temperatures a fraction of CO leaves hemoglobin and moves into the solvent, while the portion which remained within the molecule rebinds to Fe. Several barriers have to be overcome in this process and since it is no more a tunneling process the heights of these barriers are obtained (Arrhenius process). The innermost barrier has to do with heme itself, while the outermost, with hemoglobin-solvent interface. The former may be caused ny the accompanying motion of iron-the motions of Fe and CO being correlated. All the barriers reflect of course changes in the local geometry around the heme, hence these experiments provide microscopic details of the potential distribution in the region which is tied to the affinities of hemoglobin for ligand binding. CO has been chosen for purely experimental convenience.

Another aspect of these results throws some more light on the problem of hemoglobin's conformation. The barriers'

FIG. 4 a) Schematic showing positions of the histidine group of the fifth ligand, heme group, Fe and CO in states A and B; b) Potential energy surfaces of the system in B and A. The arrhenius and tunneling processes are indicated schematically in the figure (From: N. Alberding et al., Biochem. 17, 47 (1978)).

REACTION COORDINATE PROGRESS IN BIOPHYSICS 271

VoL 2SO 16 August 1979

FIG. 5 a) The possible conformations of the herae (conformational substates) correspond to the minima in the conformational potential Vc. Bottoms of the wells 1 are parametrized by power law Vc ax 2, and 3 cases are shown. b) Temperature dependence of confonnational z 2 ( c), vibrational ( v) and total mean square displacement as a function of temperature (From: H. Frauenfelder, G. Setsko, D. Tsernoglou, Nature 280, 561 (1979)).

UK) heights and widths had to be described by en enthalpy spectrum, indicating a distribution of conformational states (not all the heights and widths were equal). At low temperatures the relaxation time which describes the rate of change from one conformation to another is longer than the rebinding time, while at higher temperatures the converse is true. These results have interesting implications. They tell us that at room temperatures X-ray diffraction sees an average confoimation of the molecule which rapidly switches fran one configuration to another (fig. 5}, they all correspond of course to a group of states, Very close to that of minimum energy. This has actually been predicted theoretically by Karplus [6] and has now been also observed by X-ray diffraction experiments performed at different temperatures [7] (fig. 6).

These results concord with the intuitive feeling of surprise one has,when seeing that a protein,when disturbed. 272 G. Bemskl

FIG. 6 Backbone structure of myoglobin. The solid lines indicate the static structure, circles denote the Ca carbons. The shaded area gives the region reached by conformational substates with a 99% probability. Scale bar, 2R (From H. Frauenfelder, G. Petsko, D. Tsernoglou, Nature 280, 561 (1979)).

knows how to reconstitute its original 3 dimensional form. The protein simply goes back to one among the group of very similar conformations,characterized by the minimum Gibbs energy. Simultaneously with the experiments considerable amount of theoretical work goes into obtaining the stereochemistry of binding of ligands to Mb and Hb, calculating the conformations which reduce the hindrance due to Fe-ligand bonds, etc. [8] . Others [9i] calculate the electronic structure of Fe^porphyrin, Fe-ligand-porphyrin , to obtain the proper electronic ground state of these complexes, and to correlate the modifications in the protein with these states. I tried to show here that the very active field of heme protein research has not yet yielded the full description in physical or molecular terms of the detailed behavior of these molecules. Its dynamism is such that many new and fascinating results will be forthcoming. PROGRESS IN BIOPHYSICS 273

3. PHOTOSYNTHESIS

101"* kg. of organic matter is synthesized a year by the photosynthetic process in green plants and in certain bacteria.

This alone should make research in photosynthesis very active, as it actually has been, particularly since days of Calvin [JLO~]. In the last years, it has attracted physicists who study the first, fundamental step in the reaction between incoming photon and chlorophyll molecule. The overall,famous reaction is, in case of plants:

hv + nH2O + nCO •* (CH2O) + nO2

and a slightly different one for bacteria. While the outcome of the reaction, and the chain of events leading to formation of sugars has been known, the details of the first,photochanical step remained obscure. There exist several intriguing questions in this prodess from the physicists point of view. The photon is captured by anyone in a group of chlorophylls, but is transmitted to a privileged one, a so called reaction center, which is the only one to perform the photochemical reaction. What makes this center different from the remaining chlorophylls? Chemically no difference could be found.

Feher [ll] has studied the electron paramagnetic resonance (EPR) signal which appears during illumination. This signal is due to the following reaction:

DA + hv •+ D+A~ 274 C. Bemshi where D (donor) stands for chlorophyll and A is an acceptor molecule near by. Reaction centers in bacteria with which Feher has been performing his studies,can be separated from all other chlorophylls. It has been observed that the EPR signal from the extracted reaction centers has a signal /2 wider than signals from entire bacteria, where the reaction centers are in minority and are not seen (- 1%). Why is that so? The photo produced electron interacts with all the nuclei which possess a nuclear magnetic moment,and are encountered in its path (i.e. =re the electronic wave function is non-zero). This hyper interactions are responsible for the width of the Ei 1 i. . Chlorophyll is a conjugated molecule,and the unpaired electron is spread over nitrogen and proton nuclei. The width of the EPR line depends on the number of nuclei (N) and on the st igth of the interaction (A), in such a way that width A/N. : s compare two interactions, one in which the electron : acts with a monomer of N nuclei, and the other one where i ateracts with a dimer of 2N nuclei. The respective line widths,AW ,will be related:

AW A /N~ A /N~ -.m _ m / m _ m _ / "m AW d Aa ^ " 1/2 Affi

since N was doubled in the dimer, A was halved/because the electron spends only 1/2 of its previously spent time in interaction with each nucleus. This is exactly what Feher has observed, implying that the reaction centers are dimers of chlorophyll molecules. PROGRESS IN BIOPHYSICS 275

An even more persuasive experiment has been performed Plf] using the technique of electron nuclear double resonance (ENDOR). It consists of saturating the electronic transition,and applying RF frequencies corresponding to nuclear spin transitions at the frequencies which correspond to such transitions in the nuclei within the electron wave function. It can be shown that the ENDOR lines for a given nucleus are separated in frequency by the hyperfine constant A. The comparison of ENDOR lines in the bacteria and in an extract of reaction centers shows

A . . = 1/2 AD -, as required by the monomer vs dimer distinction (fig. 7).

-i 1 r—-l

BChl i»w.i 10'I

\ \ l\ '•— EIG. 7 Comparison of ENDOR spectra f\ r\^ c' c v ^ from bacteriochlorophyll (in vitro) K J (top) with chromatophores of ITI i i R. Spheroides R-26 (bottom) (From § D CHROHilOPHOHCS G. Feher, A. Hoff, R. Isaacson, T T L. Ackerson, Ann. N.Y. Acad. Sci. 244, A 244 (1975)). / \ » • »'

•^—^

« ( 1 W 12 14 II II 20 II I

Another interesting aspect of this work,particularly for physicists, has been that the length of time taken for the EPR signal to disappear, after photons stop arriving (light off condition) has been shown to be independent of temperature, at least between 4°K and 77°K. The only temperature independent process in quantum mechanical tunneling. This lead Feher to postulate the physical model shown in fig. 8 for the first step of the reaction of photosynthesis in bacteria: 276 G. Bemski

+ D - D - A D*- A D - A FIG. 8 Representation of the primary photochemical step. D - bacteriochlorophyll, A - an acceptor (From: G. Feher, "Electron Para- magnetic Resonance with applications to selected problems in Biology", p. 103, ed. Gordon and -t- Breach, 1970). (•) w

DA + hv + D+A~

It is interesting to note here that tunneling phenomenon is involved in several fundamental biological processes, such as first step in vision D-23 , photo association in heme proteins, previously discussed, etc. While this process occurs at temperatures well below physiological, where the Arrhenius, activated reactions take place (the particle moves over the potential hill, rather than through it) , it nevertheless may have more profound significance p.3] .

4. SPIN DENSITY IMAGING BY NUCLEAR MAGNETIC RESONANCE TECHNIQUE

In order to change somewhat emphasis, I would like to discuss now an experimental approach for formation of images by nuclear magnetic resonance-technique. This has been first suggested some 6 years ago by Lauterbur [14] and elaborated by Mansfied et al, in several papers £15] . The idea is relatively simple. Normal NMR is performed in a magnetic field which has to be extremely constant, both in space and in time, in order to obtain very narrow lines of PROGRESS IN BIOPHYSICS 277 true, undisturbed shape. If a magnetic field gradient is applied to a sample of water, the protons will be precessing at Larmor frequencies which will vary along the field gradient. Hence the total line width from protons in a water sample contained in a capillary will increase greatly. If we now have two capillaries somewhat separated in space and if the gradient is perpendicular to the capillaries, two resonance lines will be observed. If one now rotates the samples (or the direction of the gradient perpendicular to the samples), one can combine several projections and reconstruct the objects (fig. 9) in two dimensions. The size of the reconstructed objects is proportional to the number of spins of each, and depends on the relaxation times of the water protons. If the two capillaries contain water with two different relaxation times (i.e. by adding paramagnetic impurities to one of them) one notices •that applying high power levels one saturates the proton spins of the sample with longer relaxation time. This is due to the equalization of proton populations between the two Zeeman levels corresponding to the two projections of the spin with the result that the signal amplitude goes to zero. Hence, one can establish a situation where protons of only one capillary contribute to the signal (fig. 10).

After these first attempts a great deal of sophistication has been applied to this problem. The whole arsenal of NMR methods, accumulated over a period of 30 years has been let loose. The gradients can be consecutively applied in all three directions, pulsed methods can be used to selectively saturate slices of a large sample, etc. Biological materials contain regions of water protons and other, relatively dry. The distribution of hard 278 G. Bemski

FIG. 9 Relationship between a three dimensional object, its two dimensional projection along Y-axis and four one dimensional projections at 45° intervals in XZ plane. Arrows indicate gradient directions. (From: P. C. Lauterbur, Nature 242, 190 (1973)).

FIG. 10 Proton NMR zeugmatogram of two capillaries containing protons of different relaxation times: a) at low RF power, b) at high power. The high relaxation time capillary exhibits saturation behavior in b), hence there is no net resonance absorption (From: P. C. Lauterbur, Nature 242, 191 (1973)).

FIG. 11 Cross sectional view of mobile proton distribution in chicken wing bone. Light ring is the bone, dark region is the marrow (From: P. Mansfield, Contemp. Physics, Fig. 10, 17, 553 (1976)). PROGRESS IN BIOPHYSICS 279 matter and water is quite apparent in the chicken bone (fig.11). One can expect that with some technical progress pictures equivalent to X-rays could be obtained without radiation hazards involved. This will require low magnetic fields of very uniform gradients, and of large extension. It has been shown in the last few years that the nuclear relaxation times of protons and nuclei of phosphorous in malignant tumors are longer than in normal tissues p.6] . Hence there seems to exist a potentiality for detection of tumors by distinguishing them from the normal tissues. Other intriguing applications may involve the studies in vivo of water movement in plants. I have described this method, still in its infancy, as an indication of changing times. Physics, in great part through its techniques, has entered the field of biology. We have been talking in the above examples of quantum mechanical tunneling. EXAFS, pulsed NMR, ENDOR, etc. I believe that the progress in Biophysics will be due to the use of these and other sophisticated techniques taken over from the experimental physicists' arsenal. This is also true about the theoretical progress. Fifteen years ago the "Spin Hamiltonian biology" would raise many brows. Today,very serious attempts to describe electronic structures of at least some important regions of macromolecules meet with approval and hope. I am sure that physicists working both, on the experimental and theoretical aspects of biology,will see bright days ahead. 280 G» Bemski

REFERENCES

[I] M. Perutz, Ann. Rev. Biochem. 1£, 327 (1979). [2} L. Pauling and C Coyrell, Proc. Nat. Acad. Sci. U.S.A. 22, 210 (1936). [3] J. Monod, J. Wyman and P. Changeux, J. Mol. Biol. 12, 88 (1965). [4] P. Eisenberger, R. Shulman, B. Kincaid, G. Brown and S. Ogawa, Nature 27£, 30 (1978). [53 H. Frauenfelder, Conf. on Tunneling in Biological Systems. Philadelphia, 1977. [&] J. Mo Cammon, B. Gelin and M. Karplus, Nature 267, 585 (1977) . [7] H. Frauenfelder, G. A. Petsko and D. Tsernoglou, Nature 28£, 558 (1979).

[8] D. Case and M. Karplus, J. Mol. Biol. 1,2_3, 697 (1978). [9] B. Olafson and W. Goddard, Proc. Nat. Acad. Sci. U.S.A.

24, 1315 (1977) . [io] M. Calvin - "Light and Life". Baltimore, Johns Hopkins Press, 1961. p.317 - 355. [II] G. Feher, A. J. Hoff, R. Isaacson and C. Ackerson, Ann. N.Y. Acad. Sci. 244, 239 (1975).

£l2j K. Peters, M. Applebury and P. Rentzepis, Proc. Nat. Acad. Sci. U.S.A. 21* 3119 (1977). [l3] V. Goldanskii, Ann. Rev. Phys. Chem. 27, 85 (1976). [l4]. P. Lauterbur, Nature 242, 190 (1973). [l5] P. Mansfield, Contemp. Phys. r7, 553 (1976).

R. Damadian, K. Zaner, D. Hor and T. di Maio, Proc. Nat. Acad. Sci. U.S.A. 71, 1471 (1974). 281

ELECTRONIC ASPECTS OF ENZYMATIC CATALYSIS

Ricardo Ferreira (») and Marcdo A. F. Gomes (2) (') Departamento de Química, Universidade Federal de SSo Carlos, São Carlos, SSo Paulo, Brasil and (*) Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco, Brasil

Summary

Contrasting with inorganic homogeneous catalysis there is not an electronic theory of enzymatic catalysis. The reason for this is that enzymes are large aperiodic structures. This precludes both ab i.ni.ti.0 molecular orbital calculations and Bloch-type band theory approaches. However, because enzymes and substrates have evolved for two billion years, frontier orbital perturbation methods may be profitable used. It is proposed that the induced-fit conformational changes of enzymes, essential to their catalytic action, leads to a sharp increase in the electronic eigenvalue density at the active site, enhancing the catalytic orbital mixing. This effect is being tested computationally, and it is pointed out how it could be detected with pico-second sepec- troscopic techniques. As for óxidoreductases, the Mott-Anderson theory of amorphous semiconductors may explain important features of this class of enzyme systems.

1. Introduction

As we look at the current theories of chemical catalysis we are struck with a dichotomy with regard to the level of our understanding of this phenomenon. Inorganic catalysts, notably homogeneous ones but including heterogeneous systems as well, are described as providing suit able orbital pathways for electrons to flow between reactants, whenever direct flow is forbidden by symmetry restrictions and/or unfavourable eigenvalue differences (Johnson, 1977) . Enzymes, by contrast,are rationalized as binding the transition state tighther than the ground state of their substrates (Pauling, 1948, 1957; Lienhard, 1973). Details of the atomic arrangement of inorganic catalysts are, of course,important, and many processes are "structurally demanding" (Boudart, 1970). Once 282 R- Ferreira the geometry of the catalytic site is known (or surmized) the eigenfunctions of reactants and catalyst can be calculated, thereby provid ing a quantum-mechanical rationale for the catalytic activity (Johnscn , 1977; Abermark e.t at, 1977; Grabowski, Misono & Yoneda, 1977; Ugo, 1975; Somorjai, 1975). Enzymatic catalysis, on the other hand, is des_ cribed in terms of van der Waals, hydrogen bond, and electrostatic interactions between the enzyme active site and the transition state of the reaction, the two being complementary in Pauling's sense (Pauling, 1948, 1957; Jencks, 1966; Wolfenden, 1969, 1972).

This non-electronic approach to bio-molecular interactions has had a persistent critic in Szent-Gyflrgyi, whom, for almost forty years has stated his belief that electronic processes within protein molecules should be significant to the underlying sublety of biological phenomena (Szent-Gyôrgyi, 1941; Pethig and Szent-Gyôrgyi, 1977; Holden, 1979). The untractable character of enzymatic catalysis, from a quantum- -chemical view-point, stems from the fact that proteins are very large aperiodic structures. Schrfldinger (1944) firstly pointed-out that proteins are aperiodic solids, and even before that Jordan (1938) suggested that they are semi-conductors. The existence of valence and conduction bands in proteins is supported by experimental and theoretical evi- dences (Cardew & Eley, 1959; Eley & Spivey, 1960, 1962; Douzon & Thuillier, 1960; Vartanyan, 1962; Eley, 1962; Pullman, 1965). However, most enzymes work at about the same concentrations as their substrates in cells (Eigen & Hammes, 1963) and solid-state concepts should be applied with caution. At best enzymes are like amorphous te.e.,aperiodic) solids, and the absence os sharp Bragg reflections in amorphous systems makes it difficult to understand their semiconductivity. It was only in the late 1960's that Mott succeeded, on the basis of Anderson's early work (Anderson, 1958), to describe the behaviour of these materials (Mott, 1967, 1968, 1978).

It is convenient to our discussion to distinguish enzymes that catalyse electron-transfer reactions from those catalysing other processes. Some years ago Lõwdin (1964) suggested that the semiconductor properties of proteins may be significant to the former group, and we have recently put forward a model based on Mott's theory of non-crystalline

t Very recently Bolis, dementi, Ragazzi, Salvador! & Ferro (1978) tried a quantum-mechanical calculation of papain. CATALYSIS 283 semiconductors (Gomes & Ferreira, 1980) . As for the latter class of enzymes we develop here an orbital pertubation theory proposed a few years ago (Ferreira, 1973; Gomes, da Gama & Ferreira, 1978 a, 1978 b). Before proceding, however, we will briefly analyse the present state of enzymatic catalysis theory.

II. Enzymatic Catalysis and Transition - State Theory

For clarity we will consider the case of one-substrate reactions, which is the simplest case and as relevant to our discussion as the more general ones. For one-substrate reactions the transition-state theory predicts that the enzymatic catalytic factor, given by the ratio is of the ijrzymatic and the non-snzymatic rate constants, kg/kN, equal to the ratio KT/KS, where K^ and Kg are the binding constants for the processes E + S* -:—> ES*, and E + S -—> ES, respectively; S and S* represent the substrate molecules in the ground-state and in the transition-state (Lienhard, 1973) . Typical K^AN values fall in the range 10" to 10l"(Koshland Jr., 1956; Jencks, 1969; Wolfenden, 1972). Since Kg is in the range 103 to 10s, the expected values of K_, are from 10ll to 10ls. The theory receives its strongest confirmation from studies with transition-state analogs. The best transition state analogs divised 5 l so far have KT values of 10 to 10 ° (Lienhard, 1972). Se are still far from the observed kg/k^ rations, but the discrepancies are largely attributed to the "imperfect" structures of the analogs. It is signify cative that the analogs do not induce in the enzymes the same conformational change as the substrates themselves, which constitutes a strong indication that the induced-fit changes are essential for the catalytic action (Anderson, Zucker & Steitz, 1979).

The transition-state theory is undoubtly correct, but it is phenomenological in that the van der Waals and other such enzyme-substrate interactions reflect underlying electronic processes. Our proposition is that the electronic changes due to the induced-fit modifications (electron-phonon interactions, in solid-state parlance)ensure activation and contftoZ of enzymatic catalysis. This paper provides theoretical support for these effects and suggests experimental tests for them.

III. The Orbital Perturbation Theory of CataTysis

A predictive theory of catalysis will certainly come from ab 284 R. Ferreira ínítío calculations such as those of Abermark e.t a.1. (1977) on the reactivity of nickel-ethy-lene complexes. It is also clear that many catalytic processes cannot profitably be described by perturbation tech nigues. Thus, Armstrong, Novaro, Ruiz-Vyscaya & Linarte (1977) have made a M.O. study of the hydrogenation of styrene,with tetrachloropalladate ion as a catalyst. The H-H,Pd-Cl, H-Cl and Pd-H bond orders were calculated along the reaction coordinate and it was shown that in the initial step one hydrogen atom substitutes one chlorine atom of the tetrachloropalladate ion. Clearly, the involvement of the species PdCli,2 in the process is too severe for an orbital perturbation description to be useful. In other cases, however, the catalyst-substrate interaction is weaker,and perturbation theory can be used (e..g. Imamura & Hirano, 1975) . It is our contention that enzymes, being the (almost) finished products of 2 x 109 years of evolutionary history, belong to the latter class.

Our approach is based on the Frontier Orbital ory of chemical reactivity (Fukui, Yonezawa & Shingu, 1952, 1954;Fukui, 1971; Dewar, 1952; Ferreira, 1976; Gomes, da Gama & Ferreira, 1978 a, 1978 b) . Other interpretations of reaction pathways in terms of orbital symmetry considerations have been proposed (Bader, 1962; Woodward & Hoffmann, 1965; Longuet-Higgins & Abrahamson, 19.65; Klopman, 1968; Pearson, 1969, 1970, 1972). In particular Mango & Schachschneider(1967, 1969; Mango 1969, 1971), Caldow & McGregor (1971), and Klopman (1974), have considered catalysis as providing a perturbation potential acting on the frontier orbitais of the reacting species. Actual calculations have been made for some reactions, among others by Houk & Strozier (1973), Fujimoto & Hoffmann (1974) , Imamura & Hirano (1975), and Fukui & Inagaki (1975). It should be pointed-out that the Frontier Orbital description of chemical reactions is as sound as the molecular orbital method itself. For example, the interaction between two helium atoms can be described by a whole gamut of techniques; an acceptable one starts with the formation of a species with configuration (lag)2(lau)2 and can be refined to give a very detailed picture of the interaction (Margenau & Kestner, 1971).

In these approaches the key step in a chemical reaction is the flow of electrons from the HOMO of a reactant a to the LUMO of another reactant 0 (or to the LUMO of a itself in unimolecular processes). If the HOMO is a bonding orbital and the LUMO an antibonding one,the flow of electrons leads to a reactive collision. The symmetries of these highest occupied and lowest unoccupied orbitais, k° and Zi , should a p refer to the nuclear framework of the rising part of the U (Q) curve 6 CATALYSIS 285 leading to the transition-state. Because the interaction Hamiltonian is totally symmetric, once a process starts on a given reaction-path it must stay within the same point group until it reaches the transition state. Q transforms like the scalar representation of the given point group,

It is sometimes stated that this approach is limited to con- cetLttd Aea.ci-c.on4. By concerted it is meant one-step processes; hence the approach can describe the rate-determining step of any reaction. For example, the Ni(II)-catalysed cycloaddition of ethylene is though to occurr by the following mechanism (Traumüller, Polansky, Heimbach & Wilke, 1969):

Step 2 is a concerted process in which electrons from the HOMO of one ethylene molecule (of Ai symmetry) are allowed to flow into the catalyst-perturbed LUMO of the other ethylene molecule. The unperturbed

LDMO was of B2 symmetry but the Ni(II) catalyst mixed it with some

higherlying at orbital(s).

For brevity we will limit our discussion to bimolecular reactions. The transition moment is , where ft is the interaction a p Hamiltonian. ft transform like the identity representation of the transition-state point group; hence the reaction is symmetry allowed if the direct product r. o x V^ contains the identity representation. a B Even if this condition obtains, will be small if a p {e»o(Q) - e.o(Q)} is large. Since the rate constant is proportional to ||2, the reaction will be slow. 286 R- Ferreira A catalyst introduces a potential that perturbs k£ and/or l'. Supposing it is the LOMO of B which is perturbed we will have.

I = ll + Z c, m° (1)

where:

<£! H1 m°> £m B mB " £3

ft1 is the catalyst perturbation Hamiltonian. The transition moment for the perturbed system is:

U = (3)

•J ! The transition moment will not vanish if c» f 0 and simultaneously, there is at least one m° which has the same symmetry of k° .This means 1 ; that H must contain the direct product r.o x r.o (Ferreira, 1976). Ka ^B

Although first-order perturbation can break the symmetry restrictions to a reactive collision, it is easily shown that it cannot

change the difference {E£0(Q) - e^oiQ)}. Thus, the first-order perturbation energy is e'i)o= <£g H1 £g>r and this vanishes because H1 does not transform like the totally symmetric representation. The second-order energy term, however, is:

A = E ~ B (4)

2 Clearly e^0 ^ 0; because (e,o ~ e o)* 0, z po< 0, which means that •Cp í.g mg is E. = e»o + e'ffO is more negative than e^o- Since for most systems the H 4B ^B ^B (LUMO)„ is above the (HOMO) , the difference {e 0(Q) - eko(Q)1 is reduced by the perturbation of í'. B a

A detailed «..i.xysis of the interaction of an electron in £l CATALYSIS 287 with another electron in the catalyst j° orbital (Gomes, da Gama & Ferreira, 1978b) shows that the potential that mixes Zi with higher orbitais of £5 that transforms like k° is due to the tAa.nAi£Lon de.nt>i.ty, P-;o = fi-oD° (Longuet-Higgins, 1956; Bader, 1962). The significant term j Jo p C in H' is:

o(r") H'(r') = e dr1 (5, z

where p(r'') = e (see Figure 1) .

Figure 1 - The coordinate system of orbitais C° and jJ is the p C position vector of p.. relatively to the origin of SL; jx, t

It is easily shown that H{ has the symmetry of the direct product

r.o x r.o . The transition density, which corresponds to the integrand *e :c of the density matrix of Landau and von Neumann, does not represent any amount of charge, but gives instead a three dementional representation of the diAplacement of charge density.

Convergence of the terms in e

|<2.g R' m°>|<<|{ej,o - e 0}|. But" this means that the second-order energy correction will be large only if the summation in (4) contains many terms. Recalling that ft1 is the perturbation Hamiltonian of just one catalyst orbital, j°, it is seen that an efficient catalyst should have many closed-spaced orbitais of appropriate symmetry. 288 R- Ferreira IV. Enzymatic Catalysis

Enzymes very much conform with these requirements. As typical globular proteins enzymes have an almost band-like density of states (Eley, 1962; Pullman, 1965), and we are justified in substituting a Hamiltonian manyfold for H1. That is, the transition density is now given by:

p(r") =el' (r") f j° (r) S(e. ) de. (6) p J Jc Jc where S(e. ) is the distribution function of the highest occupied enzyme 3c orbitais. Under this condition, and considering further that the transition-state for an enzymatic reaction belongs to a pure rotational point group, with very few irreducible representations, one expects large negative values for eo0 , which will lower the activation energy 6 of the uncatalysed process, ie.o(Q) - e-o(Q)}- ^B a The induced-fit theory of Koshland (1959) is now firmly established and there is a growing body of evidence from various techniques for large substrate-induced distortions of enzymes in solution (Buttlaire & Cohn, 1974; Peters & Neet, 1978; McDonald, Steitz & Engeliran, 1979). These conformational changes seem to be essential to the catalytic action, since transition-state analogs do not produce the same changes as the substrates themselves (Anderson, Zucker & Steitz,1979) . These displacements of the nuclear framework (up to ± 15° in bond angles) affect the electronic levels. This effect corresponds to the electron-phonon interactions in solids, but is more convenient to describe it in terms known to the molecular spectrocopists. As shown by Walsh (1953) the electronic eigenvalues of molecules are altered in differing degree by a change of molecular geometry. In small molecules critical effects in the electronic properties caused by slight bending of the nuclear framework are known, such as the appearance of optical rotatory power in excited states of aliene (Rank, Drake & Mason, 1979) . The critical dependence of enzymatic catalysis on the induced-fit changes leads us to propose that the density of electronic states at the active site is drastically increased during a reactive collision (~10~ sec). The total change must, of course, be compatible with the induced-fit changes 0.1 e V). Of 9. is a conformational parameter, we can write:

|E I (^|i) AG.,1 < 0.1 e V (7) j dej J CATALYSIS 289 Provided there are both positive and negative terms in (7) the induced- -fit changes can produce a sharp increase in the density of states, enhancing the catalytic action.

Experimental evidence for this effect can eventually cane from pico-second techniques applied to the UV absorption spectra of enzymes in the presence of substrate molecules*. In the meantine the effect can be tested computationally. Mrs. F.G. Stamato at our laboratory is performing molecular orbital calculations of the active sites of a few enzymes, isolated for this purpose from the macromolecular framework . Calculations are done first for the geometries revealed by crystallogra_ phic data of the pure enzymes (which OAe not the active conformations) , and repeated for slightly different conformers.

It has always been a puzzle that the allosteric effects of small molecules on the catalytic action of enzymes are severe, comparable even with reversible denaturation effects, and yet only slight quatern&ry structural changes have been detected. This behaviour is,on the other hand, what one may expect if the allosteric control is due to small but critical changes in the density of electronic states caused by the binding, away from the active site, of effectors.

This is not to say that all control of biomolecular processes is electronic in character. For example, Perutz (1970) has shown that a substantial conformational change oE the polypeptide chains of haemo globin is responsible for the cooperative effect in the oxygenation of this important emphore (Pardee, 1968) . The same is probably true £or the binding of isozyme units. We feel, however, that one cannot dismiss the possibility of electronic contro] of the active forms (in Koshland's sense) of enzymes.

V. Oxidoreductases and Semiconductors

Most enzyme-catalysed reactions involve closed-shell molecules, but oxidoreductase systems are an exception in that they catalyse electron-transfer processes between open-shell species. Except for the Wigner-Witmer rules,• symmetry restrictions play no significant role in the kinetics of such reactions, which depends on the relative depth of

* Broad bands, originating frcm transitions from the highest occupied levels to the lowest unoccupied ones, would, upon concentration of both bonding and lerels, became narrower, without any shift in the band centers. 290 R. Ferreira the potential wells of electron donnors and acceptors, and the distance from one another (Hbby, 1963). Any theory of the electron-transfer chain must explain two features of the metalloproteins involved: the o transition metal ion (t.m.i.) is a few Angstroms away from the side of substrate attachment, and, secondly, the ratio of t.m.i. to aminoacid residues is small (10~l to 10~3). Built in such a theory must be the facts that globular proteins are aperiodic structures, and that electrons in organic semiconductors are localized (Boguslavskii & Vannikov,1970).

It is known (Pomonis & Vickerman, 1978) that the catalytic efficiency of t.m.i.-doped diamagnetic oxides such as A^(2-x)Crx^3 ' M 9(l_x)Ni Of etc., has the same dependence on the degree of doping as the decrease in the valence to conduction-band gap. Polaron hopping (Mott, Davis & Street, 1975) is particularly likely in proteins with their easily deformable structures. We think that an electron-hopping mechanism explains the characteristic features of oxidoreductases.

The fact that the t.m.i. is some distance away from the enzyme active site does not rule-out its participation in the pertinent redox processes, since the electron transport can occur through "variable range hopping" (Mott, 1968). In this mechanism, a phonon momentarily equalizes the energies of a pair of donnor and acceptor sites, and the electron can tunnel through the barrier separating the sites. In enzymes the induced-fit conformational changes may substitute for the phonons. The small (t.m.i.)/(a.a.r.) ratio (10~ to 10~ ) confers kinetic advantages to oxidoreductases if the desorption of the reduced species is a rate limiting step. Suppose desorption requires that the nega- tively charged species S~ looses its charge. Electron hopping between S and M (the oxidized form of the t.m.i.) competes favourably w'LL hooping from bulk M n ions as long as concentration of t.m.i. in the host macromolecule is low. If the concentration is 10-1 or higher a localized-non-localized transition occurs and S~ to M hcpping becomes very unfavourable. Details of such mechanisms will beoome clear as we known more about optical and transport properties of metalloproteins .

Acknowledgements

We thank Mr. A.A. da Gama for many helpful discussions. This work was supported in part by the CNPq and FINEP. CATALYSIS 291

References

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GEORGES BEMSKI Departamento de Física Pontifícia Universidade Católica 22451 - Rio de Janeiro - RJ, Brasil CARLOS G. BOLLINI Centro Brasileiro de Pesquisas Físicas Av. Venceslau Bras, 71 22290 - Rio de Janeiro - RJ, Brasil RAJAT CHANDA Instituto de Física Universidade Federal do Rio de Janeiro 22910 - Rio de Janeiro - RJ, Brasil SIDNEY COLEMAN* Harvard University and Stanford Linear Accelerator Center P.O. Box 4349 Stanford, California 94305, USA CARLOS 0. ESCOBAR Instituto de Física Teórica Rua Pamplona, 145 01405 - SSo Paulo - SP, Brasil RICARDO FERREIRA Departamento de Física Universidade Federal de Pernambuco Recife, Pernambuco, Brasil J.J. GIAMBIABI Centro Brasileiro de Pesquisas Físicas Av. Venceslau Bras, 71 22290 - Rio de Janeiro - RJ, Brasil BERTRAND GIRAUD Centre d'Etudes Nucleaires de Saclay Service de Physique Theorique Boite Postale n9 2 91190 - Gif-sur-Yvette, France HAIM HARARI* Department of Nuclear Physics Weizmann Institute of Science Rehovot, Israel JORGE S. HELMAN Centro de Investigacion y de Estúdios Avanzados Instituto Politécnico Nacional Apartado Postal 14740, México 14, D.F., México OSCAR HIPOUTO* Departamento de Física Universidade Federal de São Carlos 13560 - São Carlos - SP, Brasil SAMUEL W. MACDOWELL Department of Physics, Yale University New Haven - Connecticut 06520, USA EMERSON J. V. PASSOS Instituto de Física Universidade de São Paulo 05508 - São Paulo - SP, Brasil PAULO H. SAKANAKA Instituto de Física Universidade Estadual de Campinas 13100 - Campinas - SP, Brasil ABDUS SALAM International Center for Theoretical Physics P.O. Box 586 34100-Trieste, Italy BERT SCHROER Instituto de Física e Química de São Carlos Universidade de São Paulo 13560 - São Carlos - SP, Brasil H. EUGENE STANLEY Department of Physics, Boston University Boston, Massachussetts 02215, USA JAYME TIOMNO* Departamento de Física Pontifícia Universidade Católica 22451 - Rio de Janeiro - RJ, Brasil GERARD TOULOUSE Groupe de Physique des Solides de 1'Ecole Normale Supérieure 24 rue Lhomond, 75231 Paris Cedex 05, France CONSTANTINO TSALLIS Centro Brasileiro de Pesquisas Físicas Av. Venceslau Bras, 71 22290 - Rio de Janeiro - RJ, Brasil JOSÉ URBANO* Centro de Física Teórica Faculdade de Ciências e Tecnologia Universidade de Coimbra Coimbra, Portugal RUI VILELA MENDES* Centro de Física da Matéria Condensada Av. Gama Pinto, 2 1699 Lisboa Cedex, Portugal CHING S. WU Institute for Physical Science and Technology University of Maryland College Park, Maryland 20742, USA MARTIN J. ZUCKERMANN Department of Physics, McGill University Montreal, P.Q. H3A 2T8, Quebec, Canada

Not included in this publication. í •••

ÍP

Arte Serviço de Editoração do CNPq I mpressão Gráfica do Senado VI BRAZILIAN SYMPOSIUM ON THEORETICAL PHYSICS Erasmo Ferreira, Belita Koiller, Editors

Elementary Particle Physics and Relativity CG. Bollini, J.J Giambiagi, A.A. Salam, S.W. Mac Dowell, CO. Escobar, R. Chanda, B. Schroer, Contributors VOLUME III

CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTIFICO E TECNOLÓGICO u

&, J" li ¥?.

I 0 Ki ^.. Li

Erasmo Ferreira, Belita Koiller, Editors

Elementary Particle Physics and Relativity CG. Bollini, J.J. Giambiagi, A.A. Saiam, S.W. Mac DoweSI. CO. Escobar, R. Chanda, B. Schroer, Contributors VOLUME III

CONSELHO NACIONAL DE DESENVOLVIMENTO OBITtFICOE TECNOLÓGICO Coordenação Editorial Brasília 1981 VI BRAZILIAN SYMPOSIUM ON THEORETICAL PHYSICS

VOLUME III CNPq Comitê Editorial Lynaldo Cavalcanti de Albuquerque José Duarte de Araújo Itiro lida Simon Schwartzman Crodowaldo Pávan Cícero Gontijo Mário Guimarães Ferri Francisco Afonso Biato

Brazilian Symposium on Theoretical Physics, 6., Rio de Janeiro 1980. VI Brazilian Symposium on Theoretical Physics. Bra- sília, CNPq, 1981 3v. Bibliografia Conteúdo — v. 1. Nuclear physics, plasma physics and statistical mechanics. — v. 2. Solid state physics, biophysics and chemical physics. — v. 3. Relativity and elementary particle physics. 1. Física. 2. Simpósio. I. CNPq. II. Título. CDU 53:061.3(81) FOREWORD

The Sixth Brazilian Symposium on Theoretical Physics was held in Rio de Janeiro from 7 to 18 January 1980, consisting in lectures and seminars on topics of current interest to Theoretical Physics. The program covered several branches of modern research, as can be seen in this publication, which presents most of the material discussed in the two weeks of the meeting. The Symposium was held thanks to the generous sponsorship of Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq (Brazil), Academia Brasileira de Ciências, Fundação do Amparo a Pesquisas do Estado de . São Paulo (Brazil), Organization of American States, Sociedade Cultural Brasil- Israel (Brazil), Comissão Nacional de Energia Nuclear (Brazil), and the governments of France and Portugal. The meeting was attended by a large fraction of the theoretical physicists working in Brasil, and by about 30 scientists who come from other countries. The list of participants — about 250 — is given at the end of this book. / The Pontifical Catholic University of Rio de Janeiro was, for the sixth consecutive time, the host institution, providing excellent facilities and most of " the structural support for the efficient running of the activities. The community of Brazilian Theoretical Physicists is grateful to the sponsors, the host institution, the lecturers and the administrative and technical staff for their contributions to the cooperative effort which made the VI I Brazilian Symposium on Theoretical Physics a successful and pleasant event.

Organizing Comittee Guido Beck (CBPF) Carlos Bollini (CBPF) Hélio Coelho (UFPe) Darcy Dillenburg (UFRGS) Erasmo Ferreira (PUC/RJ) Henrique Fleming (USP) Í. Belita Koiller (PUC/RJ) I Roberto Lobo (CBPF) I Moysés Nussenzveig (USP) I Antônio F. Toledo Piza (USP) Ronald Shellard (IFT) Jorge A. Swieca (U.F. São Carlos e PUC/RJ) Zieli D. Thomé (UFRJ) VII

WELCOME ADDRESS

Ladies and Gentlemen: It is a great pleasure and honor for me to welcome to our University the participants of the Sixth Brazilian Symposium on Theoretical Physics. It is on such occasions that the true spirit of the university is manifested in the search for new knowledge and in the communication of new discoveries. Renowned physicists in several areas of Physics, and from leading institutions in Brazil and abroad, gather here to share their latest results of research, new hypotheses and unanswered questions. Through the exchange of new insights, suggestions and critical reactions, the academic community emerges, united in its commitment to truth and in its effort to contribute to the understanding of the world in which man lives. The understanding of Nature — which is the original meaning of Physics in Greek — is indeed the goal of this Symposium on Theoretical Physics. Events of recent years have taught us to be cautious in our dealing with Nature, since we do not understand all of its interrelations. The precipitate use of nature's forces, made possible by scientific progress, though intended to make the world more livable, threatens to destroy mankind itself. It is not possible to reduce nature to a single model and it is not possible to reduce history to a human project. The full understanding of nature demands humility and respect in dealing with it. Only under such conditions will the knowledge and use of nature be of serve to man. One of the basic goals of this Symposium is to show the fundamental unity of physical theory. The rediscovery of the deep unity of all the phenomena of nature is undoubtly a step forward in the understanding of nature, because understanding requires unification of many phenomena in one law or theory. But a unified theory, in order to be faithful to its object, must recognize the variety of levels through its use of analogy. It must respect the limits of its understanding of nature through the continual perfection and superseding of its own models. It must admit the existence of other dimensions and other approaches to reality, through openess to interdisciplinary dialogue. It is through the pluralism of viewpoints, consciously accepted as relative and interchangeable, that the global vision of reality progressively emerges, and converges to its final goal, which is man in all his complexity and his self-transcendence. May this attitude - so proper to the spirit of the university - be present in all the work of this Symposium. VIII

Rev. João A. Mac Dowell, S.J. Rector, Pontifícia Universidade Católica do Rio de Janeiro IX

INDEX

Volume I NUCLEAR PHYSICS, PLASMA PHYSICS AND STATISTICAL MECHANICS 1. The Generator Coordinate Method in Nuclear Physics- B Giraud- 2. Collective Motion and the Generator Coordinate Method - E. J.V. Passos 3. Some New Radiation Processes in Plasmas - C. S. Wu 4. Magnetohydrodynamics and Stability Theory - P. H. Sakanaka 5. Free Energy Probability Distribution in the SK Spin Class Model - Gerard Toulouse and Bernard Derrida 6. Connectivity and Theoretical Physics! Some Applications to Chemistry -HE. Stanley, A. Coniglio, W. Klein and J. Ferreira 7. Random Magnetism - C. Tsallis

Volume II SOLID STATE PHYSICS, BIOPHYSICS AND CHEMICAL-PHYSICS 1. Scattering of Polarized Low-Energy Electrons by Ferromagnetic Metals -J. S. Helman 2. Phase Separations in Impure Lipid Bilayer Membranes - F. de Verteuil, D. A. Pink, E. B. Vadas and M. J. Zuckermann 3. Recent Progress in Biphysics - G. Bemski 4. Electronic Aspects of Enzymatic Catalysis - R. Ferreira and Marcelo A. F. Gomes

Volume III RELATIVITY AND ELEMENTARY PARTICLE PHYSICS 1. Gauge Fields and Unifying Ideas - C. G. Bollini 2. On the Personality of Prof. Abdus Salam - J. J. Giambiagi 3. Gauge Unification of Fundamental Forces - A. Salam 4. Geometric Formulation of Supergravity - S. W. MacDowell 5. Weak Interactions - R. Chanda 6. Applications of Quantum Chromodynamics to Hard Scattering Processes - C. O. Escobar 7. Topological Aspects in Field Theory and Confinement - B. Schroer

AUTHOR INDEX - AT THE END OF THE VOLUME XI

CONTENTS

Carlos G. Bollini GAUGE FIELDS AND UNIFYING IDEAS 295

Juan J. Giambiagi I ON THE PERSONALITY OF PROF. ABDUS SALAM 305

Abdus Salam GAUGE UNIFICATION OF FUNDAMENTAL FORCES 1. Fundamental Particles, Fundamental Forces and Gauge Unification 312 2. The Emergency of Spontaneously Broken SU (2) x U (1) Gauge Theory 313 3. The Present and Its Problems 321 4. Direct Extrapolation from the Electroweak to the Electronuclear 324 j 5. Elementarity: Unification with Gravity and Nature of Charge 328 Appendix 1 — Examples of Grand Unifying Groups 335 ,i Appendix 2 336 I References 337

Samuel W. MacDowell GEOMETRIC FORMULATION OF SUPERGRAVITY I 1. Introduction 345 ..•'•:•'- 2. Unified Gauge Theories in Special Relativity 349 3. Unified Theories in General Relativity 354 4. The Formalism of Fiber Bundles 359 5. Fiber Bundle Formulation of Einstein's Theory of Gravitation 3 70 6. Fiber Bundles Over Graded Manifolds 374 7. Construction of Simple Supergravity 380

8. Construction of Extended SO2 and SO3 Supergravity 383 9. Critical Evaluation of Supergravity 393 ii: 10. Appendix A —Glossary of Notations and Definitions 394 1 11. Appendix B — Gauge Transformation of Vector Potentials 398 ; 12. References 400

RajatChanda WEAK INTERACTIONS .••_- 1. Introduction 403 \ . 3. Gauge Theory: Unification of E—M and Weak Interactions 404 4. Gauge Invariant Introduction of Masses for W ±, Z . Higgs Mechanism 405 XII

4. Effective Lagrangian 406 5, Introduction of Hadrons: Quark Mixing 407 6. Generalization of GIM Mechanism to n-doubiets: Flavor Conservation in Neutral Currents 407 7. Motivation for Including the Sixth Quark t 408

8. Limits on the Mixing Angles 62 and 03 409 9. Generalized Rules for Weak Decay of Heavy Particles 411 10. Charged Currents Neutrino Reactions 412 11. Violation of Bjorken Scaling: a Test of QCD 413 12. Neutral Currents: i>N, QV and eN Reactions 414 13. Important Unsolved Problem 416 14. Outlook 417 References 417

Carlos 0. Escobar APPLICATIONS OF QUANTUM CHROMODYNAMICS TO HARD SCATTERING PROCESSES 1. Introduction 419 2. The Parton Model 420 3. Quantum Chromodynamics 422 4. QCD Corrections to the Simple Parton Model 425 5. Drell-Yan Phenomenology 435 6. Beyond the Leading Order 438 7. Future Prospects 445 8. Conclusions 450 References 451

Bert Schroer TOPOLOGICAL ASPECTS IN FIELD THEORY AND CONFINEMENT 1. Introductory Remarks 453 2. Mathematical Methods 455 3. Models of Gauge Theory. Confinement and U (1) Problem 464 References and Footnotes 467

LIST OF PARTICIPANTS 469

SYMPOSIUM LECTURERS 295

GAUGE FIELDS AND UNIFYING IDEAS

C. G. Bollini Centro brasileiro de Pesquisas Físicas, Rio de Janeiro

First of all I would like to define the level of this talk. Some excellent lectures have already being given about unifying theories. And I think they were sufficient to cater even for the needs of experts in fiber bundles.

My intention is now twofold: to give more emphasis to the physi. cal and geometrical meaning of gauge fields, and to address my talk to the non expert, that is to say using an intuitive language or reasoning.

We all know that physical interactions can be divided into four classes or categories: The (classical) long range interactions : Gravity and Electromagnetism. The short range Strong interaction. And the Weak interaction responsible for 3-decay.

In principle all these interaction can have their own laws and have been studied independently of each other. However there is a widespread belief that they are but different aspects of a grand unified interaction, from which all the others can be derived as special cases.

Let us first consider Gravitation and the important step towards unification given by Einstein's "General Relativity".

Special Relativity is born when we impose invariance of physi. cal laws under Lorentz transformations. Such a transformation is a linear relation, with constant coefficients, between two inertial frames. We shall call it-a "Global Lorentz Transformation" or simply G.L.T..

When we think about G.L.T. we soon arrive at some interesting questions.

The necessary global inertial frame: does it exist? Or can we only have infinitesimal versions of inertial frames? - much in the same way'as only infinitesimal patches of curved surfaces can be said to be "planes". Further, why must we accept a transformation which is the 296 C. Bollini same everywhere? Will it not be more logical to permit the transformation to vary from point to point? Will it not be necessary to consider local Lorentz transformations (L.L.T.) instead of global ones ? All these points lead us to the consideration of Lorentz transformations whose coefficients are arbitrary functions of the coordinates. As soon as this possibility is accepted we are faced with an interesting problem.

For a G.L.T., two vectors a , b , which are parallel in one frame

a,, = X b, (1)

remain parallel in the other

a{, = * K • (2)

But when we allow the transformation to vary from point to point, the equality only holds at one and the same point. At different points a and b transform differently and in general when eg. (1) is true, eg. (2) is false.

This means that parallelism at different points looses its meaning; it is frame dependent. For the very same reason, the usual derivative of a vector field v (x) ceases to have any physical significance. Thus, the situation becomes unacceptable. We feel that someone or something should be capable of saying which is the vector parallel to a given one. That someone who, so to say, keeps us informed, at every point on local parallelism is, precisely, the "gauge field".

Let us write down some relations to formalize that point.

Suppose va is a "constant" vector for a given ("barred") frame of reference (the components are independent of position). After a local a - coordinate dependent - Lorentz transformation with coefficients L a(x), P we will have a vector field:

va(x) = La.(x) v6 , vB = (LS r1 va(x) (3) p a In a neighboring point we have:

a a w a a a , v (xtdx) = v (x) + dx a v (x) = v (x) + dx^ 3 L (x) v GAUGE FIELDS 297 Now, shall we say that va(x+dx) is the vector at x+dx which is parallel to va(x) at x ? Hardly! To accept that is to accept that the frame in which va(x) H va (constant) is a privileged frame. In other words, if eg. (4) is chosen to be the definition of parallelism, then a the transformation L o(x) to the barred frame is essential, and with it p the barred frame itself becomes privileged. Different frames are no longer equivalent and invariance under L.L.T. is lost. The only vay out is to accept the existence of an independent entity, unattached to any ct B —1 particular frame, not necessarily having the form 3 L „ (x) . (L ) Let us call it a "gauge field", or a field of "parallel transport"

r" (x) (5) y»p

This field is "in charge" of local parallelism. The vector va' ', at x+dx, which is parallel to the vector va at x, is: (compare with (4)) •

a a u a B V 'I(x+dx) = v (x) + dx r B(x) v (x) (6)

The gauge field (5) contains all what is needed to identify neighboring parallel vectors. In particular the law of inertia, which states that the tangent vector Ê0— , for the movement xa(T) of a point ax particle, keeps always parallel to itself, means that the tangent vector at T + dt

(r+dx) - f£ (x, + ^ dx , (7) coincides with (cf.(6))

1 dx! ' = «^ + dxH r« dx£ (8) dx dx uB dx K '

Thus equating (7) to (8), and dividing by dx, we get:

2 a u B d x _ dx ro dx ,„. 7F= dT ^ — (9) Which is Einstein's "geodesic" version of the law of inertia. The left hand side of èq. (9) is the acceleration of the particle. The right hand side is the "force" (inertial or gravitacional) produced by the gauge field in orderto keep the particle on an inertial movement; always parallel to itself. The gauge field ra „ unifies inertial and gravitational WP force. The particle does not know whether a gravitational field exists or not, it just obeys the order of the gauge field to move always in 298 C. Bollini the direction of its velocity four vector.

The important poperty of the gauge field of giving sense to the motion of local parallelism (see eq. (6) ) allows one to reintroduce the concept of physical or geometrical derivative of a vector field. In fact, if v1" (x) is a vector field, what is the physical meaning of 9 va(x)? None! To calculate the change of a vector when we go to a neighboring point we must first know what is "no change" and this inplies the knowledge of a parallel vector in the vicinity of the chosen point. In other words, given a vector field v (x), to calculate the actual change at x+dx, we must compare the first line of (4), namely:

a a U a v (x+dx) = x (x) + ãx ay v (x) with (6) , the vector at (x+dx) which is parallel to the one at x. Taking the difference we find:

va(x+dx) - vf, "(x+dx) = dxu 3 va(x) - dxp ra . v(x) II M UP or, dropping some indices:

v(x+dx) - vi,(x+dx) = dxu(3 -r ) v(x) = dxMD v(x) (10) where

D = 3 - r (ID u V u is the actual or real derivative. It is the only derivative with phys^L cal or geometrical meaning. It measures the actual change of the vector field.

It is interesting that the law of parallel transport implied by the gauge field is not necessarily integrable. This means that when the vector, defined by eq. (6), is again transfered by parallelism to another point x+dx+âx, the result does not coincide in general with the vector one obtains by transporting it in the reversed order, that is to say first by 5x and then by dx. The difference is oorportional to

The commutador GAUGE FIELDS [DU ' Dv] =yV

(there are some hidden indices not shown in the formula) defines the "curvature" tensor. When a vector, parallel - transported by two differ ent routes does not coincide at the meeting point, it is because we have curvature in between. When the tensor (12) is zero everywhere, par allelism is path-independent and the space is said to be flat, r then contains purely inertial forces. In all other cases the gauge field contains gravitational forces.

So much for Gravitation. Let us now consider Electromagnetism.

In quantum mechanics a particle is represented by a complex wave function (j>(x). We know that a constant phase factor e (a=cte) is physically irrelevant. That means that the transformation í -»• eia $ leaves physics unchanged. This is a global phase transormation (gauge transformation of the first kind). Can we repeat here what we said before about G.L.T. ? Certainly! ("mutatis mutandis").

Why must we only accept phase transformation which are everywhere constant? Will it not be necessary to consider local phase transformation? After all, physical effects are propagated locally, from a point to a neighbouring one. Again, as soon as we consider this possibility we face a problem similar to that of local parallelism. If phases are changed arbitrarily at different points, then no comparison is possible between the values (the phases) of a wave function a two different points. Even the notion of derivative loses its physical meaning. For example, for a pure coordinate dependent phase we have

eia(x+dx) = eia(x)+ ^ i°t( _ icc(x) ia (X) ,-ia(x) ^ia (x) y < ue _ ia(x) ia(x) e + djc i 3 a (x) £ (13) U

(compare the middle line of (13) with eq. (4)).

So any wave function $(x), which is gauge transformed with the factor e1 , suffers an additional change (in going from x to x+dx) proportional to i 3 ot(x) . This vector can have any value we please at any given point. Who is going to tell us the particular value needed to have "phase parallelism"? The answer is: the gauge field! This is an objective physical entity which is the messenger for local phase comparison. It is a vector field A , not necessarily of the form 3 a , C. Bollini which gives a meaning to the idea of phase parallelism.

For a wave function (x) we define the "phase parallel" wave function at x+dx, to be (compare with eq. (6) and eq. (13))

|,(x+dx) dxy i (14)

The actual change of a wave function is obtained subtracting (14) from:

Thus obtaining:

u 4>(x+dx) - (x+dx) = dx (3jj- i Ay) <|>(x) (15)

Or, with

Dp S V

(x+dx) (17)

Eqs. (16), (17) define the actual derivative of a wave function. It is the only derivative with physical meaning.

If the wave function obeys some deferential equation, for example Dirac's one, then the operator (16) should be the derivative operator appearing in the equation:

(iy.3 - m)

(iy.3 - (19)

We then immediately recognize the gauge field A as the electro magnetic four potential. The electromagnetic vector potencial is the messenger for phase parallelism. Just as for gravitation this parallelism need not be integrable. When it is not, we say that space presents "phase curvature" and this curvature is measured by the tensor (compare with (12)) . GAUGE FIELDS 301 v = [v °J - \ \ - \ \ (2o) which is nothing but the tensor of electromagnetic field intensities. In this sense, the electric and magnetic fields are the physical manifestation of phase curvature.

Let us now consider the theory of strong interaction. Nowadays ;•• - we think that the basic particles are "quarks" which can be found in i three different, so called "color", states. However, there is complete ! symmetry between these states. In other words, there is no physical change when we transform the quark states by any (3x3) unitary unimodular matrix. Such matrices have eight independent elements and can be represented in the exponential form: exp (i a G. ) , where G. (k=l,2,...,8) are eight independent "generators" for the given representation,and the ak are eight parameters. Instead of having a single phase, we now have " eight phases. The symmetry (SO(3)) is global if these phases are con- } stants. But just as for the cases so far considered, we can argue about :, the possibility (or necessity) of taking the phases to be eight arbitrary functions of the coordinates. Comparison of quark fields at different *j points no longer has any physical meaning; unless an independent entity, the gauge field, tell us how to calculate actual local variations. Sup- '£ pose, for simplicity, that the quark field q is constant, and we subject ij it to a SU(3) local gauge transformation:

;. q(x) = U(x)q ; q = u"1 q

r f. Then:

V 1 q(x+dx)= q(x) + dx" 3p Ü q = q(x) + dx 3y U.lf q(x) (21)

Now, the actual change for the quark field (in going from x to x+dx) is not ãxdxvu 3 U Uif 1. ThThid s quantity depends on a particular gauge, a particular matrix (U) .

í • If we want local invariance, no particular matrix should be y previleged. An independent entity, B (x) must replace the quantity |. 3 U(x) U~1(x). This object is the SU(3) gauge field: the messanger

5ii: for "color parallelism"

fp q'I (x+dx) = q(x) + dxu B (x) q(x) (22)

[-?' (compare with (6) and (14)). 302 C. Bollini The gauge field B (x) is a matrix in the color indices of the \,r quark fields. It can be expressed as a combination of the eight gener- pf- ators G, :

B (x) = i B* (x) G, (23)

B (x) is then composed of eight vectors fields B (x). The actual change in the quark field is obtained by subtracting

r.) = q(x) + dx 8pq(x)

from (22)

W q(x~dx) - q,*, (x-dx) = dx Oy-B^) q(x)

Or:

q(x+dx) - qi,(x+dx) = dxU D q(x) (24)

with

D = 3 - B (25) V U U D is the actual physical derivative operator to act on the quark field. In the equation of motion for the quark field D is the derivative operator to be used.

The gauge field B coaches the wave function to move in such a way that color is "inertially" transported. As in previous cases, the parallel transport of color is not necessarily integrable. When it is, the space is "color flat". If not, we.have "color curvature" measured by the tensor

which is also a matrix in color indices.

Let us now tackle the last of the interactions mentioned at the begining: the Weak interaction.

For the sake of clarity we shall consider a so called "lepton generation". For example the electron e and the corresponding neutrino vi. GAUGE FIELDS 303 We shall assume these particles to be massless without worrying about the process by means of which the electron adquires mass.

The neutrino is known to be left-handed; i.e. it has negative

helicity. Electron states can be divided into positive helicity eR, and

negative helicity eL- It seems natural to group together the left handed

states eL with the neutrino to form a two component left handed object I = V* I . The other state of the electron remains a single (right handed)

object eR.

At this stage there exists complete symmetry between the two components of SU. This means that we can transform & by any two by two unitary unimodular matrix without any physical consequence. Let us call

SU(2)L the group of such transformations. Also, the phase of eR can be arbitrarily changed without physical effects. The corresponding group

is U(1)R. Both transformations acting simultaneously, are simbolized by SU(2) x U(l). When the lepton fields are changed by any transformation of SU(2) x U(l), physics remain invariant. t The matrices of SU(2) can be represented in the exponential V" form using the three Pauli matrices: exp (i a. x.). The three phases jí.. a. (i=l,2,3) and the phase fi of the right handed state, form a set of (', four phases whose values are physically irrelevant. If these phases are i-r constants we have a global SU(2) x U(l) transformation. Once again, if f- we accept phases which are arbitrary functions of the coordinates, we f' need a gauge field to "parallel transport" these "chiral" phases.

I We need not enter into details. We shall have a vector gauge \{\{\ field for each of the phases. Three of these four fields acquire mass ';-•( simultaneously with the lepton (electron) . They form the so called : \ ±0 IS_ intermediate vector bosons W and Z . They are responsible for the :•:£•! "weak" part of the interaction. The fourth one remains massless and can çVsj be identified with the electromagnetic field A -We then recognize that ••) Electromagnetism and Weak interactions can be considered to be included in the SU(2) x U(l) case just examined, receiving thus a unified treatment. Summarizing: All known interactions are such that there are some properties which, so to speak, are parallel transported along the path of the particles. Those "inertially carried" properties are associ ated with a symmetry under a local group. Unification is achieved when these groups are succesfully included into a bigger group of local 304 C. Bollini transformations. Automatically the new group is married (till death to they part) with the corresponding gauge fields. They give sense to local comparison of the bases for the group. They instruct the particles on how to move in such a way that the property in question is compared and preserved from point to point. 305

ON THE PERSONAUTY OF PROF. ABDUS SALAM

J.J.Giambiagi Centro Brasileiro de Pesquisas Físicas

The invitation of prof. Erasmo Ferreira to make some comments on the personality of prof. Abdus Salam came to me as a great honour and as a great pleasure as he is an old friend of mine.

We physicists of Latin America are paying this homage to him as he has been honoured with the Nobel Prize. In celebrating his triumph we are celebrating the triumph of one of us, as he is a representative of the physicists of the third world. This remark about prof. Salam being a representative of the third world marks an important characteristic of his personality. He has lived most of his life in the first world, but he has remained loyal to the third world devoting a good part of his time to the effort of developing science and promoting progress in it. As a matter of fact, he is not a new-comer to the arena field of the Nobel Prize. He has been "dribbling" in it for a long time. After making a brilliant entrance in the world of physics through a series of papers on renormalization in field theories he made a famous report on Dispersion Relations in the Rochester meeting at CERN in 1956. It was some time later that he produced a paper on Y5 invariance of weak interactions theories. This, he sent to Pauli asking for his comments. As an answer, he received a post- card saying: Why do you waste your time? Why don't you think on something better? So, he dropped the manuscript in the drawer of 306 J.J. Giambiagi

his desk. The content of this paper was essentially the same as that of Yang and Lee on violations of parity in weak interactions, which appeared in Physical Review a few months later and won the Nobel Prize for its authors. When Salam saw it he rescued his own paper and sent it for publication in Nuclear Physics. But it was already too late for him. This is probably one of the reasons why he always gives encouraging words to anybody - particularly young physicists - who come to him to discuss nevi ideas. He is always helpful to push you to carry them on.

Few years later, when Abdus Salam was chairman in the Imperial College, Yuval Ne-eman came to him as a graduate student, wanting to work for a PhD degree. Then Salam suggested him to look into the representations of SU(3) and its applications to particle-physics. After many discussions, finally Ne-eman appeared at Salam's office with a draft of a manuscript and with the name of Salam at the top of it. To which Salam said: erase my name, you did most of the work. This paper came to be one of the most important of the decade. Ne-eman was seriously considered to share the Nobel Prize together with Gell-Mann who also published a paper on a similar subject. One can possibly argue that the reason why Ne-eman was not nominated was essentially that it was his thesis work: just the beginning of his career. But this was not the case of Salam. So, for the second time Salam was very near around the Prize. This is why I said that he was not a foreigner to the arena field of the prize. Finally, his success came with the unified theory of weak and electromagnetic interactions, success which we celebrate today. Of course, in between he obtained the most distinguished prizes from U.S., Europe and the Soviet Union. But his scientific activity, ON THE PERSONALITY OF A. SALAM 307 although deep and intense only shows one of the aspects of his complex personality. It is far from exhausting it.

As I said before, his loyalty to the third world is a making characteristic of his personality. One of the main

driving forces behind him.

It was mainly due to his own effort - and that of prof. P. Budini that the Center in Trieste was created. Salam was appointed its Director, and it was under his leadership that the Institute has acquired its present spirit of international cooperation between ail nations, irrelevant of religions or political system and also of helping mostly the developing countries. The idea is that science is an essential ingredient of progress and that no technological "know how" will be developed unless in parallelism with basic research. Both must go together, parallel, in order to have stable scientific institutions. This is an important philosophical attitude towards science in the third world. He has also stressed the urgent need for the scientists tc do something of immediate use in promoting welfare for so many millions of people living in proverty, besides their own basic research.

I am sure that prof. Salam will use all the prestige (and in a sense, power) that the Nobel Prize brings with it to promote peaceful international cooperation and the progress of science in the developing countries. I am also sure he will do whatever he can to diminish hunger and misery in the third world. 309

GAUGE UNIFICATION OF FUNDAMENTAL FORCES

Abdus Saiam International Centre for Theoretical Physics Trieste, Italy

In June 1938, Sir George Thomson, then Professor of Physics at Imperial College, London, delivered his 1937 Nobel Lecture. Speaking of Alfredo Nobel, he said: "The idealism which permeated his character led him to... (being) as much concerned with helping science as a whole, as individual scientists. ... The Swedish people under the leadership of the Royal Family and through the medium of the Royal Academy of Sciences have made Nobel Prizes one of the chief causes of the growth of the prestige of science in the eyes of the world ... As a recipient of Nobel's generosity, I owe sincerest thanks to them as well as to him."

I am sure I am echoing my colleague's feelings as well as my own, in reinforcing what' Sir George Thomson said — in respect of Nobel's generosity and its influence on the growth of the prestige of science. Nowhere is this more true than in the developing world. And it is in this context that I have been encouraged by the Permanent Secre tary of the Academy — Professor Carl Gustaf Bernhard — to say a few words' before I turn to the scientific part of my lecture.

Scientific thought and its creation is the common and shared heritage of mankind. In this respect, the history of science, like the history of all civilization, has gone through cycles. Perhaps I can illustrate this with an actual example.

Seven hundred and sixty years ago, a young Scotsman left his native glens to travel south to Toledo in Spain. His hame was Michael, his goal to live and work at the Arab Universities of Toledo and Cordova, where the greatest of Jewish scholars, Moses bin Maimoun, had taught a generation before.

Michael reached Toledo in 1217 AD. Once in Toledo, Michael formed the ambitious project of introducing Aristotle to Latin Europe , translating not from the original Greek, which he knew not, but from the Arabic translation then taught in Spain. Prom Toledo, Michael travelled to Sicily, to the Court of Emperor Frederick II. 3\o Saiam Visiting the medical school at Salerno, chartered by Frederick in 1231, Michael met the Danish physician, Henrik Harpestraeng — later to become Court Physician of Eric IV Waldemarsson. Henrik had come to Salerno to compose his treatise on blood-letting and surgery. Henrik's sources ware the medical canons of the great clinicians of Islam,Al-Razi and Avicenna, which only Michael the Scot could translate for him.

Toledo's and.Salerno's schools, representing as they did the finest synthesis of Arabic, Greek, Latin and Hebrew scholarship, were some of the most memorable of international assays in scientific collaboration. To Toledo and Salerno came scholars not only from the rich countries of the East, like Syria, Egypt, Iran and Afghanistan, but also from developing lands of the West like Scotland and Scandinavia. Then, as now, tiiere were obstacles to this international scientific concourse, with an economic and intellectual disparity between different parts of the world. Men like Michael the Scot or Henrik Harpestraeng were singularities. They did not represent any flourishing schools of research in their own countries. With all the best will in the world their teachers at Toledo and Salerno doubted the wisdom and value of training them for advanced scientific research. At least one of his masters counselled young Michael the Scot to go back to clipping sheep and to the weaving of woollen cloths.

In respect of this cycle of scientific disparity, perhaps I can be more quantitative. George Sarton, in his monumental five - volume History of Science chose to divide his story of achievement in sciences into ages, each age lasting half a century. With each half century he associated one central figure. Thus 450 BC - 400 BC Sarton calls the Age of Plato; this is followed by half centuries of Aristotle,of Euclid, of Archimedes and so on. From 600 AD is the Chinese half century of Hsiiah Tsang, from 650 to 700 AD that of I-Ching, and then from 750 AD to 1100 AD — 350 years continuously — it is the unbroken succession of the Ages of Jabir, Khwarizmi, Razi, Masudi, Wafa, Biruni and Avicenna, and then Omar Khayam — Arabs, Turks, Afghans and Persians — men belonej ing to the culture of Islam. After 1100 appear the first Western names; Gerard of Cremona, Roger Bacon — but the honours are still shared with the names of Ibn-Rushd (Averroes), Moses Bin Maimoun, Tusi and Ibn-Nafis — the man who anticipated Harvey's theory of circulation of blood. No Sarton has yet chronicled the history of scientific creativity among the pre-Spanish Mayas and Aztecs, with• their invention of the zero, of the calendars of the moon and Venus and of their diverse pharmocological dis_ coveries, including quinine, but the outline of the story is the same — GAUGE UNIFICATION 311 one of undoubted superiority to the Western contemporary correlates.

After 1350, however, the developing world loses out except for the occasional flash of scientific work, like that of Ulugh Beg — the grandson of Timurlane, in Samarkand in 1400 AD; or of Maharaja Jai Singh of Faipur in 1720 — who corrected the serious errors of the then Western tables of eclipses of the sun and the moon by as much as six minutes of arc. As it was, Jai Singh's techniques were surpassed soon after with the development of the telescope in Europe. As a contemporary Indian chronicler wrote: "With him on the funeral pyre, expired also all science in the East." And this brings us to this century when the cycle begun by Michael the Scot turns full circle, and it is we in . the developing world who turn to the Westwards for science. As Al-Kindi wrote 1100 years ago: "It is fitting then for us not to be ashamed to acknowledge truth and to assimilate it from whatever source it comes to us. For him who scales the truth there is nothing of higher value than truth itself; it never cheapens nor abases him."

Ladies and Gentlemen,

It is in the spirit of Al-Kindi that I start my lecture with ',f

a sincere expression of gratitude to the modern equivalents of the Uni- |fFv, versities of Toledo and Cordova, '•which I have been privileged to be associated with - Cambridge, Imperial College and the Centre at Trieste. 312 A. Saiam I. FUNDAMENTAL PARTICLES, FUNDAMENTAL FORCES AND GAUGE UNIFICATION

The Nobel lectures this year are concerned with a set of ideas relevant to the gauge unification of the electromagnetic force with the weak nuclear force. These lectures coincide nearly with the 100 death-anniversary of Maxwell, with whom the first unification of forces (electric with the magnetic) matured and with whom gauge theories originated. They also nearly coincide with the 100th anniversary of the birth of Einstein — the man who gave us the vision of an ultimate unification of all forces. The ideas of today started more than twenty years ago, as gleams in several theoretical eyes. They were brought to predictive maturity over a decade back. And they started to receive experimental confirmation some six years ago. In some senses then, our story has a fairly long background in the past. In this lecture I wish to examine some of the theoretical gleams of today and ask the question if these may be the ideas to watch for maturity twenty years from now. From time immemorial, man has desired to comprehend the complexity of nature in terms of as few elementary concepts as possible Among his guests — in Feynman's words — has been the one for " wheels within wheels" — the task of natural philosophy being to discover the innermost wheels if any such exist. A second quest has concerned itself with the fundamental forces which make the wheels go round and enmesch with one another. The greatness of gauge ideas — of gauge field theories — is that they reduce these two quests to just one;elementary particles (described by relativistic quantum fields) are representations of certain charge operators, corresponding to gravitational mass, spin, flavour, colour, electric charge and the like, while the fundamental forces are the forces of attraction or repulsion between these same charges. A third quest seeks for a unification between the charges (and thus of the forces) by searching for a single entity, of which the various charges are components in the sense that they can be transformed one into the other. But are all fundamental forces gauge forces? Can they be understood as such, in terras of charges — and their corresponding currents — only? And if they are, how many charges? What unified entity are the charges components of? What is the nature of charge ? Just as Einstein comprehended the nature of gravitational charge in terms of space-time curvature, can we comprehend the nature of the other charges — the nature of the entire unified set, as a set, in terms of GAUGE UNIFICATION 313 something equally profound? This hriefly is the dream, much reinforced by the verification of gauge theory predictions. But before I examine the new theoretical ideas on offer for the future in this particular context, I would like your indulgence to range over a one-man, purely subjective, perspective in respect of the developments of the last twenty years themselves. The point I wish to emphasise during this part of my talk was well made by G. P. Thomson in his 1937 Nobel Lee - ture. G. P. said "... The goddess of learning is fabled to have sprung full grown from the brain of Zeus, but is is seldom that a scientific conception is born in its final form, or owns a single parent. More often it is the product of a series of minds, each in turn modifying the ideas of those that came before, and providing material for those that come after."

II. THE EMERGENCE OF SPONTANEOUSLY BROKEN SU(2] X UQ) r> GAUGE THEORY

I started physics research thirty years ago as an experimen- *__ tal physicist in the Cavendish, experimenting with tritium — deuterium íp scattering. Soon I knew the craft of experimental physics was beyond Hi me — it was the sublime quality of patience — patience in accumulating £• data, patience with recalcitrant equipment — which I sadly lacked. Re_ %~ luctantly I turned my papers in, and started instead on quantum field theory with Nicholas Kemmer in the exciting department of P.A.M. Dirac. The year 1949 was the culminating year of the Tomonaga- Schwinger-Feynman-Dyson reformulation of renormali zed Maxwell — Dirac gauge theory, and its triumphant experimental vindication. A field theory must be renormalizable and be capable of being made free of infi nities — first discussed by Waller — if perturbative calculations with it are to make any sense. More — a renormalizable theory, with no dimensional parameter in its interaction term, connotes somehow that the fields represent "structureless" elementary entities. With Paul Matthews, we started on an exploration of renormalizability of meson theories. Finding that renormalizability held only for spin-zero mesons and that these were the only mesons that empirically existed then , Cpseudoscalar pions, invented by Kemmer, following Yukawa) one felt thrillingly euphoric that with the triplet of pions (considered as the carriers of the strong nuclear force between the proton-neutron doublet) one might resolve the dilemma of the origin of this particular force which is responsible for fusion and fission. By the same token, the so-called weak nuclear force — the force responsible for B-radioactivi1y A. Saiam

(and described then by Fermi's non-renormalizable theory! had to be '0, mediated by some unknown spin-zero mesons if it was to be renornalizable. fÇ- If massive charged spin-one mesons were to mediate this interaction,the '"•- theory would be non-renormalizable, according to the ideas then. Now this agreeably renormalizable spin-zero theory for the pion was a field theory, but not a gauge field theory. There was no conserved charge which determined the pionic interaction. As is well known, shortly after the theory was elaborated, it was found wanting The (4 , -a) resonance A effectively killed it off as a fundamental theory; we were dealing with a complex dynamical system, not "structure less" in the field-theoretic sense. For me, personally, the trek to gauge theories as candidates for fundamental physical theories started in earnest in September 1956- the year I heard at the Seattle Conference Professor Yang expound- his \_, and Professor Lee's ideas on the possibility of the hitherto sacred principle of left-right symmetry, being violated in the realm of the weak nuclear force. Lee and Yang had been led to consider abandoning X' left-right symmetry for weak nuclear interactions as a possible resol - 7 ution of the (T,8) puzzle. I remember travelling back to London on an & American Air Force (MATS) transport flight. Although I had been granted, j^i for that night, the status of a Brigadier or a Field Marshal — I don't y quite remember which — the plane was very uncomfortable, full of r crying service-men's children — that is, the children were crying, not |v the servicemen. I could not sleep. I kept reflecting on why Nature - should violate left-right symmetry in weak interactions. Now the hall- mark of most weak interactions was the involvement in radioactivity *- phenomena of Pauli's neutrino. While crossing over the Atlantic, came back to me a deeply perceptive question about the neutrino which' Pro- fessor Rudolf Peierls had asked when he Was examining me for a Ph.D. a few years before. Peierls' question was: "The photon mass is zero because of Maxwell's principle of a gauge symmetry for electromagnetism; tell me, why is the neutrino mass zero?" I had then felt somewhat uncomfortable at Peierls, asking for a Ph.D. viva, a question of which he himself said he did not" know the answer. But during that comfort - less night the answer came. The analogue for the neutrino, of the gauge symmetry for the photon existed; it had to do with the massless -

ness of the neutrino, with symmetry under the Y5 transformation (later christened "chiral symmetry"). The existence of this /symmetry

for the massless neutrino must imply a combination (1 + y5l or Cl - Y5) for the neutrino interactions. Nature had the choice of an aesthetic - ally satisfying but a left-right symmetry violating theory, with a neutrino which travels exactly with the velocity of light; or alternatively GAUGE UNIFICATION 315 a theory where left-right symmetry is preserved, but the neutrino has a tiny mass — some ten thousand times smaller than the mass of the electron. It appeared at that time clear to me what choice Nature must have made. Surely, left-right symmetry must be sacrificed in all neu - trino interactions. I got'off the plane the next morning, naturally very elated. I rushed to the Cavendish, worked out the,Michel parame -

ter and a few other consequences of ys symmetry, rushed out again, got onto a train to Birmingham where Peierls lived. To Peierls I presented my idea; he had asked the original question; could he approve of the answer? Peierls1 reply was kind but firm. He said "I do not believe left-right symmetry is violated in weak nuclear forces at all. I would not touch such ideas with a pair of tongs." Thus rebuffed in Birmingham, like Zuleika Dobson, I wondered where I could go next and the obvious place was CERN in Geneva, with Pauli — the father of the neutrino — nearby in Zurich. At that time CERN lived in a wooden hut just outside Geneva airport. Besides my friends, Prentki and d'Espagnat, the hut contained a gas ring on which was cooked the staple diet of CERN — Entrecôte ã Ia creme. The hut also contained Professor Villars of MIT, who was visiting Pauli the same day in Zurich. I gave him my paper. He returned the next day with a message from the Oracle; "Give my regards to my friend Salam and tell him to think of something better". This was discouraging, but I was compensated by Pauii's excessive kindness a few months later, when Mrs. Wu's , Lederman's and Telegdi's experiments were announced showing that left-right symmetry was indeed violated and ideas similar to mine about chiral symmetry were expressed independently by Landau and Lee and Yang . I received Pauii's first somewhat apologetic letter on 24 January 1957. Thinking that Pauii's spirit should by now be suitably crushed, I sent him two short notes I had written in the meantime. These contained suggestions to extend chiral symmetry to electrons and muons, assuming that their masses were a consequence of what has come to be known as dynamically spontaneous symmetry breaking. With chiral symmetry for electrons, muons and neu - trinos, the only mesons that could mediate weak decays of the muons would have to carry spin one. Reviving thus the notion of charged in - termediate spin-one bosons, one could then postulate for these a type of gauge invariance which I called the "neutrino gauge". Pauii's re- '• action was swift and terrible. He wrote on 30 January 1957, then on j 18 February and later on 11, 12 and 13 March: "I am reading (along the shores of Lake Zurich) in bright sunshine quietly your paper..." "I am very much startled on the title of your paper 'Universal Fermi inter - r action'... For quite a while I have for myself the rule if a theoreti-

'•'i 316 A. Saiam cian says universal it just means pure nonsense. This holds particularly in connection with the Fermi interaction, but otherwise too, and now you too, Brutus, my son, come with this word..." Earlier, on 30 January, he had written "There is a similarity between this type of gauge invariance and that which was published by Yang and Mills... In the latter, of course, no ys was used in the exponent." And he gave me the full reference of Yang and Mills' paper; (Phys. Rev. 96_, 191 (1954)). I quote from his letter: "However, there are dark points in your paper regarding the vector field B . If the rest mass is infinite (or very large), how can this be compatible with the gauge' transformation

B •* B - 3UA?" and he concludes his letter with the remark: "Every reader will realize that you deliberately conceal here something and will ask you the same questions". Although he signed himself "With friendly regards", Pauli had forgotten his earlier penitence. He was clearly and rightly on the warpath. Now the fact that I was using gauge ideas similar to the Yang-Mills (non-Abelian SU(2) - invariant) gauge theory was no news to 9) me. This was because the Yang-Mills theory (which married gauge ideas of Maxwell with the internal symmetry SU(2) of which, the proton -neutron system constituted a doublet) had been independently invented by a Ph.D. pupil of mine, Ronald Shaw ' , at Cambridge at the same time as Yang and Mills had written. Shaw's work is relatively unknown; it remains buried in his Cambridge thesis. I must admit I was taken aback by Pauli's fierce prejudice against universalism — against what we would today call unification of basic forces — but I did not take this too seriously. I felt this was a legacy of the exasperation which Pauli had always felt at Einstein's somewhat formalistic attempts at unifying gravity with electromagnetism —forces which in Pauli's phrase "cannot be joined — for God hath rent them asunder". But Pauli was absoluted right in accusing me of darkness about the problem of the masses of the Yang-Mills fields; one could not obtain a mass without wantonly destroying the gauge symmetry one had started with. And this was particularly serious in this context, because Yang and Mills had conjectured the desirable renormalizability of their theory with a proof which relied heavily and exceptionally on the masslessness of their / spin-one intermediate mesons. The problem was to be solved only years later with the understanding of what is now known as the mechanism, but I will come back to this later. Be that as it may, the point I wish to make from this change with Pauli is that already in early 1957, just after the set of parity experiments, many ideas coming to fruition now, started to become clear. These are: i GAUGE UNIFICATION 317 1. First was the idea of chiral symmetry leading to a V-A theory. In those early days ray humble suggestion of this was limited to neutrinos, electrons and muons only, while shortly after, that year , Sudarshan and Marshak , Gell-Mann and Feynman , and Sakurai had the courage to postulate y5 symmetry for baryons as well as leptons, *) making this into a universal principle of physics. Concomitant with the (V-A) theory was the result that :Lf weak interactions are mediated by intermediate mesons, these mesons must carry spin one.

2. Second, was the idea of spontaneous breaking of chiral symmetry to generate electron and muon masses: though the price which those latter- day Shylocks, Nambu and Jona-Lasinio and Goldstone exacted for this (i.e. the appearance of massless scalars), was not yet appreciated.

3. And finally, though the use of a Yang-Mills-Shaw (non-Abelian)gauge theory for describing spin-one intermediate charged mesons was suggested already in 1957, the giving of masses to the intermediate bosons through spontaneous symmetry breaking, in a manner to preserve the renormaliz - ability of the theory, was to be accomplished only during a long period of theoretical development between 1963 and 1971.

Once the iang-Mills-Shaw ideas were accepted as relevant to the charged weak, currents — to which the charged intermediate mesons were coupled in this theory — during 1957 and 1958 was raised the question of what was the third component of the SO(2) triplet, of which the charged weak currents were the two members. There were the two alternatives: the electroweak unification suggestion, where the electro magnetic current was assumed to be this third component; and the rival suggestion that the third component was a neutral current unconnected with electroweak unification. With hindsight, I shall call these the Klein 16) (1938) and the Kemmer 17) (1937) alternatives. The Klein suggestion, made in the context of a Kaluza-Klein five-dimensional space -time, is a real tour-de-force; it combined two hypothetical spin - one charged mesons with the photon in one multiplet, deducing from the compactification of the fifth dimension, a theory which looks like Yang-Mills-Shaw's. Klein intended his charged mesons for strong interactions, but if we read charged weak mesons for Klein's strong ones,one

*) Today we believe protons and neutrons are composites of quarks, so that y5 sym - metry is now postulated for the elementary entities of today — the quarks. 318 A. Saiam obtains the theory independently suggested by Schwinger (1957) , though Schwinger, unlike Klein, did not build in any non-Abelian gauge aspects. With just these non-Abelian Yang-Mills gauge aspects very much to the fore, the idea of uniting weak interactions with electromag^ netisra was developed by Glashow and Ward and myself in late 1958. The rival Kemmer suggestion of a global SU(2) - invariant triplet of weak charged and neutral currents was independently suggested by 211 Bludman (1958) in a gauge context and this is how matters stood till 1960. To give you the flavour of, for example, the year 1960,there 22) is a paper written that year of Ward and myself with the statement "Our basic postulate is that it should be possible to generate strong , weak and electromagnetic interaction terms with all their correct sym - metry properties (as well as with clues regarding their relative strengths) by making local gauge transformations on the kinetic energy terms in the free Lagrangiar. for all particles. This is the statement of an ideal which, in this paper at least, is only very partially realized". I am not laying a claim that we were the only ones who were saying this, but I just wish to convey to you the temper of the physics of twenty years ago — qualitatively no different today from then. But what a quantitative difference the next twenty years made, first with new and far-reaching developments in theory — and then, thanks to CERN, Fermilab, Brookhaven, Argonne, Serpukhov and SLAC in testing itI So far a theory itself is concerned, it was the next seven years between 1961-67 which were the crucial years of quantitative comprehension of the phenomenon of spontaneous symmetry breaking and the emergence of the SO(2) x U(l) theory in a form capable of being tested. The story is well known and Steve Weinherg has already spoken about it. So I will give the barest outline. First there was the realization that the two alternatives mentioned above a pure electromag_ netic current versus a pure neutral current — Klein-Schwinger versus Kemmer-Bludruan — were not alternatives; they were complementary. As was noted by Glashow and independently later by Ward and myself , both types of currents and the corresponding gauge particles (W~,Z° and y) were- needed in order to build a theory that could simultaneously accomodate parity violation for weak and parity conservation for the electromagnetic phenomena. Second, there was the influential paper of Goldstone in 1961 which, utilizing a non-gauge self-interaction between scalar particles, showed that the price of spontaneous breaking of a continuous internal symmetry vias the appearance of zero mass scalars — a result foreshadowed earlier by Nambu. In giving a proof of this theorem with Goldstona I collaborated with Steve Weinberg , GAUGE UNIFICATION 319 who spent a year at Imperial College in London. ,^ I would like to pay here a most sincerely felt tribute to ?'{ him and to Sheldon Glashow for their warm and personal friendship. I shaw not dwell on the now well-known contributions of Anderson , Higgs , Brout & Henglert , Guralnik, Hagen and Kibble starting from 1963, which showed the way how spontaneous sym metry breaking using spin-zero fields could generate vector-meson masses, defeating Goldstone at the same time. This is the so-called Higgs mech anism. The final steps towards the electroweak theory were taken by Weinberg and independently myself (with Kibble at Imperial College tutoring me about the Higgs phenomena). We were able to com - plete the present formulation of the spontaneously broken SU(2) x U(l) theory so far as leptonic weak interactions were concerned — with one parameter sin2 9 describing all weak and electromagnetic phenomena and with one isodoublet Higgs multiplet. An account of this development was given during the contribution to the Nobel Symposium ( organized by Nils Svartholm and chaired by Lamek Hulthin held at Gothenburg after , some postponements, in early 1968) . As is well known, we did not then, J,^ and still do not, have a prediction for the scalar Higgs mass. [t '" Both Weinberg and I suspected that this theory was likely to -'•', be renormalizable. Regarding spontaneously broken Yang-Mills - Shaw : theories in general this had earlier been suggested by Englert, Brout \ 29) and Thiry . But this subject was not pursued seriously except at Utrecht, where the actual proof of renormalizability was given by •t Hooft 33) in 1971. This was elaborated further by that remarkable physicist the late Benjamin Lee 34) , working with Zinn Justin, and by 't Hooft and Veltman . This followed on the earlier basic advances in Yang-Mills calculational technology by Feynman , DeWitt , Faddeev and Popov , Mandelstam , Fradkin and Tyutin , Boulware 415 , Taylor 42> , Slavnov 43) , Strathdee 44' and Salam. In Coleman's eloquent phrase " 't Hooft's work turned the Weinberg-Sa'.am frog into ~~ an enchanted prince". Just before had come the GIM (Gla3how,Iliopoulos and Maiani) mechanism , emphasising that the existence of the fourth *) When I was discussing the final version of the SU(2) x U(l) theory and its possible renormalizability in Autumn 1967 during a post-doctoral course of lectures at Imperial College, Nino Zichichi from CEEN happened to be present. I was delighted because Zichichi had been badgering me since 1958 with persistent questioning of what theore - tical avail his precise measurements on (g—2) for the rauon as well as those of the muon lifetime were, when not only the magnitude of the electromagnetic corrections to weak decays was uncertain, but also conversely the effect of non-renormalizable weak interactions on "renormalized" electromagnetism was so unclear. 320 A- Saiam charmed quark (postulated earlier hy several authors) was essential to the natural resolution of the dilemma posed by the absence of strangeness-violating currents. This tied in naturally with the understanding of the steinberger-Schwinger-Rosenberg-Bell-Jackiw-Adler anomaly and its removal for SU(2) x U(l) by the parallelism of four quarks and four leptons, pointed out by Bouchiat, Iliopoulos and Meyer and inde - 47) pendently by Gross and Jackiw . , If one has kept a count, I have so far mentioned around fifty theoreticians. As a failed experimenter, I have always felt envious of the ambience of large experimental teams and it.gives me the greatest pleasure to acknowledge the direct or the indirect contributions of the "series of minds" to the spontaneously broken SU(2) x U(l) gauge theory. My profoundest personal appreciation goes to my collaborators at Imperial College, Cambridge and the Trieste Centre, Paul Matthews, John Ward, Jogesh Peti, John Strathdee, Tom Kibble and to Nicholas Kemmer. In retrospect, what strikes me most about the early part of this story is how uninformed all of us were, not only of each other's work, but also of work done earlier. For example, only in 1972 did I learn of Kemmer1s paper written at Imperial College in 1937. Kemmer's argument essentially was that Fermi's weak theory was not globally SU(2) invariant and should be made so — though not for its own sake but as a prototype for strong interactions. Than this year I learnt that earlier, in 1936, Kemmer's Ph.D. supervisor, Gregor Wentzel , had introduced (the yet undiscovered) analogues of lepto - quarks, whose mediation could give rise to neutral currents after a Fierz reshuffle. And only this Summer, Cecilia Jarlskog at Bergen rescued Oscar Klein's paper from the anonymity of the Proceedings of the International Institute of Intellectual Cooperation of Paris, and we learnt of his anticipation of a theory similar to Yang-Mills-Shaw's long before these authors. As I indicated before, the interesting point is that Klein was using his triplet, of two charged mesons plus the photon, not to describe weak interaction but for strong nuclear force unification with the electromagnetic — something our generation started on only in 1972 — and not yet experimentally verified. Even in this recitation I am sure I have inadvertantly left off some names of those who have in some way contributed to SU(2) x U(l). Perhaps the moral is that not unless there is the prospect of quantitative verification, does a qualitative idea make its impress in physics. And this brings me to experiment, and the year of the 49) Gargamelle . I still remember Paul Matthews and I getting off the train at Aix-en-Provence for the 1973 European Conference and foolishly deciding to walk with our rather heavy luggage to the student hotel GAUGE UNIFICATION 321 where we were billeted. A car drove from behind us, stopped, and the driver leaned out. This was Musset whom I did not know well personally then. He peered out of the window and said: "Are you Salara?" I said "Yes". He said: "Get into the car. I have news for you. We have found neutral currents." I will not say whether I was more relieved for being given a lift because of our heavy luggage or for the discovery of neu - tral currents. At the Aix-en-Provence meeting that great and modestman, Lagarrigue, was also present and the atmosphere was that of a carnival — at least this is how it appeared to me. Steve Weinberg gave the rapporteur's talk with T.D. Lee as the chairman. T.D. was kind enough to ask me to comment after Weinberg finished. That Summer Jogesh; Pati and I had predicted proton decay within the context of what is now called grand unification and in the flush of this excitement I am afraid I ignored weak neutral currents as a subject which had already come to a successful conclusion, and concentrated on speaking of the possible decays of the proton. I understand now that proton decay experiments are being planned in the United States by the Brookhaven, Irvine and Michigan and the Wisconsin-Harvard groups and also by a European collaboration to be mounted in the Mont Blanc Tunnel Garage n9 17. The later quantitative work on neutral currents at CERN, Fermilab. , Brookhaven, Argonne and Serpukhov is, of course, history, but a special tribute is warranted to the beautiful SLAC-Yale-CERN experiment of 1978 which exhibited the effective Z -photon interference in accordance with the predictions of the theory. This was foreshadowed by Barkov et al's experiments at Novosibirsk in the USSR in their exploration of parity-violation in the atomic potential for bismuth. There is the apocryphal story about Einstein, who was asked what he sould have thought if experiment had not confirmed the light deflection predicted by him. Einstein is supposed to have said, "Madam, I would have thought the Lord has missed a most marvellous opportunity." I believe, however, that the following quote from Einstein's Herbert Spencer lecture of 1933 expresses his, my colleagues and my own views more accurately. "Pure logical thinking cannot yield us any knowledge of the empirical world ; all knowledge of reality starts from experience and ends in it." This is exactly how I feel about the Gargamelle-SLAC experience.

III. THE PRESENT AND ITS PROBLEMS

Thus far we have reviewed the last twenty years and the emergence of SU(2] x U(l), with the twin developments of a gauge theory of basic interactions, linked with internal symmetries, and of the 322 A. Saiam spontaneous breaking of these symmetries. I shall first summarize the situation as we believe it to exist now and the immediate problems.Then we turn to the future.

1. To the level of energies explored, we believe that the followingsets of particles are "structureless" (in a field-theoretic sense) and, at least to the level of energies explored hitherto, constitute the ele - mentary entities of which all other objects are made.

SUC(3] triplets

IV V l Family I quarks leptons SU(2) doublets

3R' CY' ' Family II quarks leptons

3R, sv, s,B]

IV V Si Family III quarks leptons

|bR,

Together with their antiparticles each family consists of 15 or 16 two- component fermions (15 or 16 depending on whether the neutrino is massless or not). The third family is still conjectural, since the top

quark (t,,, tv, tn] has not yet been discovered. Does this family really follow the pattern of the other two? Are there more families? Does the fact that the families are replicas of each other imply that Nature has discovered a dynamical stability about a system of 15 (or 16) objects , and that by this token there is a more basic layer of structure under- 52) neath?

2. Note that quarks come in three colours; Red (R), Yellow (Y) and Blue (B) . Parallel with the electroweak SU(2) x U(l), a gauge field *)

theory (SUC(3)) of strong (quark) interactions (quantum chromodynamics,

*) "To my mind the most striking feature of theoretical physics in the last thirty-six years is the fact that not a single new theoretical idea of a fundamental natuse has been successful. The notions of relativistic quantum theory....of a great number of talented physicists. We live in a dilapidated house and we seem to be unable to move out. The difference between this house and a prison is hardly noticeable" - Res.Jost C1963) in Praise of Quantum Field Theory (Siena European Conference). GAUGE UNIFICATION 323

QCD) has emerged which gauges the three colours. The indirect discovery of the (eight) gauge bosons associated with QCD (gluons), has 54) already been surmised by the groups at DESY.

3. All known baryons and mesons are singlets of colour SU (3). This has led to a hypothesis that colour is always confined. One of the major unsolved problems of field theory is to determine if QCD - treated non- perturbatively — is capable of confining quarks and gluons.

4. In respect of the electroweak SU(2) x U(l), all known experimentson weak and electromagnetic phenomena below 100 GeV carried out to date agree with the theory which contains one theoretically undetermined parameter sin29 = 0.230 ± 0.009. The predicted values of the asso ciated gauge boson (W~ and Z°) masses are: IIL. = 77-84 GeV, m_ = 89-95 GeV for 0.25 i sin26 5 0.21.

5. Perhaps the most remarkable measurement in electroweak physics is 2 that of the parameter p = Í w KI I . Currently this has been detar- l 2 J mined from the ratio of neutral to charged current cross-sections. The predicted value p = 1 for weak iso-doublet Higgs is to be compared with the experimental p = 1.00 ± 0,02.

6. Why does Nature favour the simplest suggestion in SU(2) x U(l)theory **) of the Higgs scalars being iso-doublet? Is there just one physical Higgs? Of what mass? At present the Higgs interactions with leptons , quarks as well as their self-interactions are non-gauge interactions. For a three-family (6-quark) model, 21 out of the 26 parameters needed, are attributable to the Higgs interactions. Is there a basic principle,

*) The one-loop radiative corrections to p suggest that the maximum mass of leptons contributing to p is less than 100 GeV.

**) To reduce the arbitrariness of the Higgs couplings and to motivate their iso- doublet character, one suggestion is to use supersymmetry. Supersymmetry is a Fermi-

Bose symmetry, so that iso-doublet leptons like (ve>e) or (v ,v) in a sypersymmetric theory must be accompanied in the same multiplet by iso-doublet Higgs. Alternatively, one may identify the Higgs as composite fields associated with bound states of a yet new level of elementary particles and new forces (Dimopoulos & Susskind , Weinberg and 't Hooft) of which, at present low energy, we have no

cognisance and which may manifest themselves in the l-100NTeV range. Unfortunately , both these ideas at first sight appear to introduce complexities, though in the con - 324 A. Saiam as compelling and as economical as the gauge principle, which embraces the Higgs sector? Alternatively, could the Higgs phenomenon itself be a manifestation of a dynamical breakdown of the gauge symmetry.

7. Finally there is the problem of the families; is there a distinct SU(2) for the first, another for the second as well as a third SU(2) , with spontaneous symmetry breaking such that the SU(2) apprehended by present experiment is a diagonal sum of these "family" SU(2)'s? To state this in another way, how far in energy does the e-y universality (for example} extend? Are there more Z°'s than just one, effecti - vely differentially coupled to the e and the y systems? (If there are, this will constitute mini-modifications of the theory, but not a drastic revolution of its basic ideas.) In the next section I turn to a direct extrapolation of the ideas which went into the electroweak unification, so as to include strong interactions as well. Later I shall consider the more drastic alternatives which may be needed for the unification of all forces (including gravity) — ideas which have the promise of providing a deeper understanding of the charge concept. Regreetfully, by the same token, I must also become more technical and obscure for the non-specia. list. I apologize for this. The non-specialist may sample the flavour of the arguments, with the next section (Sec. IV) ignoring the Appendi- ces and then go on to Sec. V which is perhaps less technical.

IV. DIRECT EXTRAPOLATION FROM THE ELECTROWEAK TO THE ELECTRONUCLEAR

4.1 - The Three Ideas

The three ideas which have gone into the electronuclear — also called grand — unification of the electroweak with the strong nuclear force (ar * which date back to the period 1972-1974), are the following:

1. First: the psychological break (for us) of grouping quarks and leptons in the same multiplet of a unifying group G, suggested by 60) Pati and myself in 1972 . The group G must contain SU(2)xU(l)xSUc(3); must be non-Abelian, if all quantum numbers (flavour, colour, lepton ,

**) cont... text of a wider theory, which spans energy scales up to much higher masses, a satisfactory theory of the Higgs phenomena, incorporating these, may well emerge. GAUGE UNIFICATION 325 ,„. quark and family numbers) are to be automatically quantized and the re- tâ suiting gauge theory asymptotically free. ri'

2. <• Second: an extension, proposed by Georgi and Glashow (1974) which places not only (left-handed) quarks and leptons but also their antiparticles in the same multiplet of the unifying group. Appendix I displays some example of the unifying groups presently considered. Now a gauge theory based on a "simple" (or with discrete sym metries, a "semi-simple") group G contains one basic gauge constant This constant would manifest itself physically above the "grand unific- -*' ation mass" M, exceeding all particle masses in the theory — these themselves being generated (if possible) hierarchially through a suit - able spontaneous symmetry-breaking mechanism.

3. The third crucial development was by Georgi, Quinn and Weinberg (.1974) who showed how, using renorraalization group ideas , j/ ;; one could relate the observed low-energy couplings a(y) , as(jj) (u ~ lOOGeV) | to the magnitude of the grand unifying mass M and the observed value of 17,, sin29(p); (tan6 is the ratio of the U(l) to the SU(2) couplings). |.!

*) !' 4. If one extrapolates with Jowett , that nothing essentially : new can possibly be discovered — i.e. one assumes that there are no '<.. new features, no new forces, or no new "types" of particles to be dis - covered, till we go beyond the grand unifying energy M — then the Gaorgi, Quinn, Weinberg method leads to a startling result: this featureless "plateau" with no "new physics" heights to be scaled stretches to fant- astically high energies. More precisely, if sin26(y) is as large as 0.23, then the grand unifying mass M cannot be smaller than 1.3 x 1013GeV 63> . (Compare with Planck mass nip = 1.2 * 1019GeV related

*) The universal urge to extrapolate from what we know to-day and to believe that nothing new can possibly be discovered, is well expressed in the following:

"I come first, My name is Jowett I am the Master of this College, Everything that is, I know it If I dont't, it isn't knowledge" -

The Balliol Masque. 326 A- Saiam *) to Newton's constant where gravity must come in.) The result follows from the formula '

2 z lia ln M = sin 8(M) - sin 6(y) (I) 3lr v cos2e (M)

if it assumed that sin29(M) - the magnitude of sin28 for energies of the order of the unifying mass M - equals 3/8 (see Appendix II). This startling result will be examined more closely in Appendix II. I show there that it is very much a consequence of the assumption that the SU(2) x U(l) symmetry survives intact from the low regime energies y right upto the grand unifying mass M. I will also show that there already is some experimental indication that this assumption is too strong, and that there may be likely peaks of new physics at energies of 10 TeV upwards.

4.2 - Tests of electronuclear grand unification

The most characteristic prediction from the existence of the ELECTRONUCLEAR force is proton decay, first discussed in the context of grand unification at the Aix-en-Provence Conference (1973) . For "semi-simple" unifying groups with multiplets containing quarks and leptons only, (but no antiquarks nor antileptons) the lepto-quark com - posites have masses (determined by renormalization group arguments), of the order of = 105 - 10G GeV . For such theories the characteristic proton decays (proceeding through exchanges of three lepto-quarks) con 29 31 serve quark number + lepton number, i.e. P = qqq •+ III, Tp - IO - 10 * years. On the contrary, for the "simple" unifying family groups like SU(5) 61' or SO(10) 67^ (with multiplets containing antiquarks and anti^ leptons) proton decay proceeds through an exchange of one lepto-quark into an antilepton (plus pions etc.) (P •+ T). An intriguing possibility in this context is that investi - gated recently for the maximal unifying group SU(16) - the largest

*) On account of the relative proximity of M ~ 1013 GeV to m_ Cand the hope of eventual unification with gravity), Planck mass EL is now the accepted "natural" mass scale in Particle Physics. With this large mass as the input, the great unsolved problem ^of Grand Unification is the "natural" emergence of mass hierarchies (mp, amp, a2nip,...) or

nip expC-c^/a), where cn's are constants. GAUGE UNIFICATION 327

group to contain a 16-fold fermionic family (q» $» • Jl" + ir+ + ir+) and P + 3i (e.g. N + 3v + TT°, P -f 2v + e+ + ir°), the relative magnitudes of these alternative decays being mcdel-deDendent on how precisely SU(16) breaks down to SUC3) x SU(2) x UCD• Quite clearly, it is the central fact of the existence of the proton decay for which the present generation of experiments must be designed, rather than for any specific-type of decay modes. Finally, grand unifying theories predict mass relations ,.. 68) like:

m, m m. — - s - p - n a m m m e p T

for 6 (or at most 8} flavours below the unification mass. The Important remark for proton decay and for mass relations of .the above type as well as for an understanding of baryon excess in the Universe ' , Is that for the present these are essentially characteristic of the fact of grand unification - rather than of specific models. "Yet each man kills the thing he loves" sang Oscar Wilde an- guishedly in his famous Ballad of the Reading Goal. Line generations of physicists before us, some in our generation also (through a direct extrapolation of the electroweak gauge methodology to the electronuclear) - and with faith in the assumption of no "new physics", which lead to a grand unifying mass - 1013 GeV - are beginning to believe that the end of the problems of elementarity as well as of fundamental forces is nigh. They may be right, but before we are carried away by this pros - pect, it is perhaps worth stressing that even for the simplest grand unifying model (Georgi and Glashow's SU(5) with just two Higgs (a 5 and

*) the calculation of baryon excess in the Universe - arising from a combination of CP and baryon number violations - has recently been claimed to provide teleological arguments for grand unification. For example, Nanopoulos has suggested that the

"existence of human beings to measure the ratio n_/n, (where nfl is the numbers of baryons and n, the numbers of photons in the Universe) necessarily imposes severe bounds 2 on this quantity: i.e. 10 "~ (tne/m )^neutrinos, (3) from asymptotic freedom and (4) from numerous (one-loop) radiative calculations. It is clear that lack of acceler^ ators as we move up in energy scale will force particle physics to reliance on teleo- logy and cosmology (which in Landau's famous phrase is "often wrong, but never in doubt?). 328 A- Saiam 24]), the number of presently ad hoc parameters needed by the model is still unwholesomely large -— 22, to compare with. 26 of the six-quark mo

del based on the humble SUC2) x u(l) x SUC(3). We cannot feel proud.

V. ELEMENTARITY.:. UNIFICATION WITH GRAVITY AND NATURE OF CHARGE

In some of the remaining parts of this lecture I shall be questioning two of the notions which have gone into the direct extrapolation of Sec. IV — first, do quarks and leptons represent the correct elementary ' fields, which should appear in the matter Lagran- gian, and which are structureless for renormalizaibility; second, could some of the presently considered gauge fields themselves be composite ? This part of the lecture relies heavily on an address I was privileged to give at the European Physical Society meeting in Geneva in July this year 64).

; 5.1 - The Quest for Elementary, Prequarks (Preons and Pre-Preons)

While the rather large number (15) of elementary fields for the family group SU(5) already makes one feel somewhat uneasy, the number 561, for example, proposed in the context of the three-family tri - bal group SU(ll) or 2048 for SOÍ22) (see Appx. I Cof which presumably 3 x 15 = 45 objects are of low and the rest of Planckian mass) is posi- .;: tively baroque. Is there any basic reason for one's instinctive revulsion when faced with these vast numbers of elementary fields. ;' The numbers by themselves would perhaps not matter so much . .;' After all, Einstein in his description of gravity , chose to work with 10 fields (g (x)) rather than with just one (scalar field) as 72) ** Nordstrom had done before him. Einstein was not perturbed by the multiplicity he chose to introduce, since he relied on the sheet-anchor of a fundamental principle — (the equivalence principle) — which per mitted him to relate the 10 fields for gravity g with the 10 compo - nents of the physically relevant quantity, the tensor T of energy

*) I would like to quote Feynman in a recent interview to the "Omni" magazine: "As long as it looks like the vay things are built with wheels within wheels, then you are looking for the innermost wheel - but it might not be that way, in which case you are looking for whatever the hell it is you find I" In the same interview he remarks "a few years ago I was very sceptical about the gauge theories... I was expecting mist, and now it looks like ridges arid valleys after all." GAUGE UNIFICATION 329 and momentum. Einstein knew that nature was not economical of structures: only of principles of fundamental applicability. The question we must ask ourselves is this: Have we yet discovered such prin ciples in our question for elementary, to justify having fields with such large numbers of components as elementary. Recall that quarks carry at least three charges ( colour , flavour and a family number). Should one not, by now, entertain the notions of quarks (and possibly of leptons) as being composites of some more basic entities *' (PRE-QUARKS or PREONS), which each carry but one basic charge . These ideas have been expressed before but .they have become more compulsive now, with the growing multiplicity of quarks and leptons. Recall that it was similar ideas which led from the eight-fold of baryons to a triplet of (Sakatons and) quarks in the first place. The preon notion is not new. In 1975, among others, Pati , Salam and Strathdee ' introduced 4 chromons (the fourth colour corresponding to the lepton number) and 4 flavours, the basic group being SU(8)

— of which the family group SUp(4) x SUC(4) was but a subgroup. As an extension of these ideas, we now believe these preons carry magnetic charges and are bound together by very strong short-range forces, with quarks and leptons as their magnetically neutral composites . The im portant remark in this context is that in a theory containing both elec- trie and magnetic generalized charges, the analogues of the well-known Dirac quantization condition gives relations like, f^ = § for ttle strength of the two types of charges. Clearly, magnetic monopoles v; (g = 4^"4 - , fr- = -jisi ) of opposite polarity, are likely to bind much more £ tightly than electric charges, yielding composites whose non-elementary % nature will reveal itself only for very high energies. This appears to .;!•£ be the situation at least for leptons if they are composites. ;f-v; In another form the preon idea has been revived this year by :-Á Curtright and Freund 52', who motivated by ideas of extended supergravity ": I (to be discussed in the next subsection), reintroduce an SU(8) of 3 chro- ;• mons (R, Y, B), 2 flavons and 3 familons (horrible names). The family i;| group SU(5) could be a subgroup of this SU(8). In the Curtright-Freund

Cf(| scheme, the 3 * 15 = 45 fermions of SU(5) can be found among the

*1 One muse emphasise however that zero mass neutrinos are the hardest objects to conceive of as composites.

**) According to 't Hooft's theorem, a monopole corresponding to the SUT(2) gauge symmetry is expected to possess a mass with the lower limit — . Even if such mo t-, * nopoles are confined, their indirect effects must manifest themselves, if they exist . ""if (Note that — is very much a lower limit for a grand unified theory like SU(5) for which the monopole mass is a a times the heavy lepto-quarks mass.) 330 A. Saiam 8 + 28 + 56 of SU(8) (or alternatively the 3 x 16 = 48 of SO(1Q) among the vectorial 56 fermions of SU(.8)I. (The next succession after the preon level may be the pre-preon level. It was suggested at the Geneva Conference that with certain developments in field theory of comp - osite fields it could be that just two pre-preons may suffice. But at this stage this is pure speculation). Before I conclude this section, I would like to make a prediction regarding the course of physics in the next decaUe, extrapolating from our past experience of the decades gone by:

DECADE 1950-1960 1960-1970 1970-1980 1980

Discovery'in The strange The 8-fold Confirmation w, z, early part of p.articles way, fi of neutral Proton decay the decade currents

Expectation for SU(3) Grand Unification, the rest of the resonances Tribal Groups decade

Actual Hit the next May hit the preon discovery level of level, and composite elementarity structure of quarks with quarks

5.2 - Post-Planck Physics, Supergravity and Einstein's dreams

I now turn to the problem of a deeper comprehension of the charge concept (the basis of gauging) — which, in my humble view, is the real quest of particle physics. Einstein, in the last thirty-five years of his life lived with two dreams: one was to unite gravity with matter (the photon) — he wished to see the "base wood" (as he put it ) which makes up the stress tensor T on the right-hand side of his equation R - -^g R = -T transmuted through this union, into the "marble" of gravity on the left-hand side. The second (and the complementary) dream was to use this unification to comprehend.the nature of electric charge in terms of space-time geometry in the same manner as he had successfully comprehended the nature of gravitational charge in terms of spacetime curvature. GAUGE UNIFICATION 331 In case some one imagines that such, deeper comprehension is irrelevant to quantitative physics, let me adduce the tests of Einstein's theory versus the proposed modifications to it (Brans-Dicke ' for example). Recently C1976] , the strong equivalence principle (i.e. the proposition that gravitational forces contribute equally to the inertial and the gravitational masses) was tested to one part in 1012 (i.e. to the same accuracy as achieved in particle physics for 781 (g-2)e) through lunar-laser ranging measurements . These measure - merits determined departures from Kepler equilibrium distances, of the moon, the earth and the sun to better than ± 30cms. and triumphantly vindicated Einstein. There have been four major developments in realizing Einstein's dreams: 1) The Kaluza-Klein 791 miracle: An Einstein Lagrangian (scalar curvature) in five-dimensional space-time (where the fifth dimension is compactified in the sense of all fields being explicitly independent of the fifth co-ordinate) precisely reproduces the Einstein- Maxwell theory in four dimensions, the g ^Lv = 0,1,2,3) components of the metric in five dimensions being identified with the Maxwell field . A . From this point of view, Maxwell's field is associated with the í extra components of curvature implied by the (conceptual) existence of

,; the fifth dimension.

\\ 2) The second developement is the recent realization by Cremraer, Scherk, Englert, Brout, Minkowski and others that the compactification of the extra dimensions — (their curling up to sizes perhaps smaller than Planck length < 10~ cms. and the very high curvature

; I associated with them) — might arise through a spontaneous symmetry •' —43 breaking (in the first 10 seconds) which reduced the higher diraen - sional space-time effectively to the four-dimensional that we apprehend directly.

*) The following quotation from Einstein is relevant here. "We new realize, with, special clarity, how much in error are those theorists «ho believe theory comes indue tively from experience. Even the great Newton could not free himself from this error (Hypotheses non fingo)." This quote is complementary to the quotation from Einstein at the end of Sec. II.

**) The weak equivalence principle Cthe proposition that all but the gravitational force contribute equally to the inertial and the gravitational masses) was verified by Eotvos to l:108 and by Dicfce and Braginsky and Fanov to l:10E . 332 A. Saiam ,

3) So far we have considered Einstein's second dream, i.e. the unification of electromagnetism (and presumably of other gauge forces) with gravity, giving a space-time significance to gauge charges as corresponding to extended curvature in extra bosonic dimensions. A full realization of the first dream (unification of spinor matter with gravity and with other qauge fields) had to await the development of supergravity ' 82' — and an extension to extra fermionic dimensions of superspace 83^ (with extended torsion being brought into play in addition to curvature). I discuss this development later.

4) And finally there was the alternative suggestion by Wheeler andd Schemberg that electric charge- may be associated with space-time topology — with worm holes, with space-time Gruyère-cheesiGri - *) ness. This idelea 1has recently been developed by Hawking and his 85) collaborators

5.3 - Extended Supergravity, SU(8) preons and Composite Gauge Fields ;

f Thus far I have reviewed the developments in respect of FC Einstein's dreams as reported at the Stockholm Conference held in 1978 ':•' in this hall and organized by the Swedish Academy of Sciences. A remarkable new development was reported during 1979 by Julia and Cremmer which started with an attempt to use the ideas of Kaluza and Klein to formulate extended supergravity theory in a higher (compactified) space-time — more precisely in eleven dimensions. This development links up, as we shall see, with preons and composite Fermi fields — and even more important — possibly with the notion of composite gauge fields. Recall that simple supergravity is the gauge theory of 88) supersymmetry the gauge particles being the (helicity ±2) gravitons

*) The Einstein Lagrangian allows large fluctuations of metric and topology on Planck-length scale. Hawking has surmised that the dominant contributions to the path integral of quantum gravity come from metrics which carry one unit of topology OfL\ per Planck volume. On account of the intimate connection (de Rham, Atiyah-Singer) of curvature with the measures of space-time topology (Euler number, Pontryagin num - ber) the extended Kaluza-Klein and Wheeler-Hawking points of view may find consonance after all. \ GAUGE UNIFICATION 333

and (helicity + J-) gravitinos ' . Extended supergravity gauges supersymmetry combined with SOCN} internal symmetry. For N = 8,the (tribal) : supergravity multiplet consists of the following SO(8) families: '

Helicity ±2 1 ±§ S ±1 28 I- * Í I' ''•J. 0 70

As is well known, S0(8J is too small to contain SU(2) x u(l) x SUC(3) . Thus this tribe has no place for WT (.though Z° and y are contained) and no places for p or T or the t quark. This was the situation last year. This year, Cremmer and Julia 87} attempted to write down the N = 8 supergravity Lagrangian ex- plicity, using an extension of the Kaluza-Klein ansatz which states that extended supergravity (with SO(8) internal symmetry) has the same Lagrangian in four space-time dimensions as simple supergravity in (compactified) eleven dimensions. This formal — and rather formidable I ansatz — when carried through yielded a most agreeable bonus. The jl supergravity Lagrangian possesses an unsuspected SO(8) "local" internal J~' symmetry although one started with an internal SO(8) only. Í The tantalizing questions which now arise are the following.

I. 1) Could this internal SU(8) be the symmetry group of the i?. 8 preons (3 chromons, 2 flavons, 3 familons) introduced earlier? •? " 2) When SU(8) is gauged, there should be 63 spin-one fields. " > The supergravity tribe contains only 28 spin-one fundamental objects which are not minimally coupled. Are the 63 fields of SU(8) to be 1 identified with composite gauge fields made up of the 70 spin-zero ob- . '•.-. jects of the form V~ 3 V; Do these composites propagate, in analogy ' with the well-known recent result in CPn~ theories , where a com - ^ posite gauge field of this form propagates as a consequence of quantum •; '.! effects (quantum completion) ?

••' Í ip- The entire development I have described — the unsuspected '"• *} Supersyrometry algebra extends Poincare group algebra by adjoining to it super- = T ie •Vj symmetric charges Qg which transform bosons to fermions. {Q , Qfi} ûûâB' * \; '•] currents which correspond to these charges CQ and P ) are J and T - these are '•';-! essentially the currents which in gauged supersymmetry Ci.e. supergravity) couple to -'; the gravitino and the graviton, respectively. 334 A. Saiam extension of SO (.81 to SUC81 when extra compactified space-time diigensions are used — and the possible existence and quantum propagation of composite gauge fields — is of such crucial importance for the future prospects of gauge theories that one begins to wonder how much of the SU 3 extrapolation which took SU(2) x U(l) * OC ) into the electronuclear grand unified theories is likely to remain unaffected by these new ideas now unfolding. But where in all this is the possibility to appeal directly to experiment? For grand unified theories, it was the proton decay What is the analogue for supergravity? Perhaps the spin •=• massive gravitino, picking its mass from a super-Higgs effect 90) provides the answer. Fayet91 1 has shown that for a spontaneously broken globally super symmetric weak theory the introduction of a local gravitational inter - action leads to a super-Higgs effect. Assuming that supersymmetry breakdown is at mass scale nv., the gravitino acquires a mass and an effective interaction, but of conventional weak rather than of the gravitational strength — an enhancement by a factor of 103" . One may thus search for the gravitino among the neutral decay modes of J/i|i -the predicted rate being 10 - 10~ times smaller than the observed rate for J/IJJ •* e e . This will surely tax all the ingenuity of the great 92) men (and women) at SLAC and DESY. Another effect suggested by Scherk is antigravity — a cancellation of the attractive gravitational force with the force produced by spin-one gravi-photons which exist in all extended supergravity theories, Scherk shows that the Compton wave length of the gravi-photon is either smaller than 5 cms. or comprised between 10 and 850 metres in order that no conflict wii-h what is pre - sently known about the strength of the gravitational force. Let me summarize: it is conceivable of course, that there is indeed a grand Plateau — extending even to Planck energies. If so,the only eventual laboratory for particle physics will be the Early Univer- se, where we shall have to seek for the answers to the questions on the nature of charge. There may, however, be indications of a next level of structure around 10 TeV; there are also beautiful ideas (like, for example, of electric and magnetic monopole duality) which may manifest at energies of the order of a~ IIL. (= 10 TeV) . Whether even this level of structure will give us the final clues to the nature of charge, one cannot predict. All I can say is that I am for ever and continually being amazed at the depth revealed at each successive level we explore. I would like to conclude, as I did at the 1978 Stockholm Conference with a prediction which J.R. Oppenheimer made more than twenty-five years ago and which has been fulfilled to-day in a manner he did not live to see. More than anything else, it expresses the faith for the GAUGE UNIFICATION 335 future with which, this greatest of decades in.particle physics ends "Physics will change even more ... If it is radical and unfamiliar ... we think7"that' the future" will be only more radical and^not'less, only more strange and not more familiar, and that it will have its own new insights for the inquiring human spirit.". J.R. Oppenheimer, Reith Lectures BBC 1953).

APPENDIX I

EXAMPLES OF GRAND UNIFYING GROUPS

Semi-simple groups Multiplet Jxotic gauge particles Proton decay (with left-right GR l Lepto-quarks -*• (qS.) jepto-quarks •* W symmetry) MV [k +(Higgs) or Jnifying mass = Example [SU(6)F x G G- x G_ = 106 MeV Proton=qqq * ^Í

Simple groups Diquarks * (qq; qq + qi i.e. q Dileptons •* (IV) Examples l Lepto-quarks (q£) , (qJ.)Proton P=qqq * £ Family groups*ÍSU(5) or (S0(10) G * Also possible, q Unifying mass Tribal groups*|SU(10 or (SO(22) = 1013- 1015 GeVP •+ t,P * 31, I P •* 3?. L

*) Grouping quarks (q) and leptons (Í.) together, implies treating lepton number ar 93) the fourth colour, i.e. SU (3) extends to SUc(4) (Pati and Salam) A Tribal group, by definition, contains all known families in its basic representation. Favoured representations of Tribal SU(ll) (Georgi) and Tribal S0(22) ( Gell - 9 5} Mann etal.) contain 561 and 2048 fermions! 336 A. Saiam APPENDIX II

The following assumptions went into the derivation of the formula CD in the text. a) SUT(2) x OT -Cl) survives intact as the electroweak symmetry group from energies = p right upto M. This intact survival implies that one eschews, for example all suggestions that i) low-energy SUL(2) may be the diagonal sum of SO? (2), SU*X(2}, SV*11(2) , where I, II, III Jj Li J_I refer to the (three?l known families; ii) or that the UL R(l) is a sum of pieces, where UR(1) may have differentially descended from a (V + A) -symmetric SU_(2) contained in G, or iii) that U(l) contains a piece from a four-colour symmetry SUC(4) (with lepton number as the fourth colour) and with SUC(4) breaking at an intermediate mass scale to

SUQC3) x uc(l). b) The second assumption which goes into the derivation of the formula above is that there are no unexpected heavy fundamental fermions, 2 which might make sin S(M) differ from -5 — its value for the low mass *) fermions presently known to exist. c) If these assumptions are relaxed, for example, for the three 2 family group G = [SUF(6) x SUGC6QL + R, where sin e(M) = ^ , we find the grand unifying mass M tumbles down to 106GeV. d) The introduction of intermediate mass scales Cfor example , those connoting the breakdown of family universality, or of left-right symmetry, or of a breakdown of 4-colour SUC(4) down to SUC(3) x UC(D)

*) If one does not know G, one way to infer the parameter sin28(M) is from the formula:

2 ZT f 9 N + 3 Nn sinz6(M) ZQ2 [ 20 N + 12 N

Here N and No are the numbers of the fundamental quark and lepton SU(2) doublets q X. (assuming these are the only multiplets that exist). If we make the further assumption that N = N. (from the requirement of anomaly cancellation between quarks and leptons) we obtain sinz9(M) = -^ . This assumption however is not compulsive; for QQ\ example anomalies cancel also if (heavy) mirror fermions exist . This is the case

for [|U(6)14 for which sin29CM) = — . 28 GAUGE UNIFICATION 337 will as a rule push the magnitude of the grand unifying jnass M upwards. 96) • In order to secure a proton decay life, consonant with present em 97) pirical lower limits (-1030 years} this is desirable anyway. (T . for M - 1013GeV is unacceptably low - 6 x 1023 years unless there are 15 Higgs.) There is from this point of view, an indication of there being in Particle Physics one or several intermediate mass scales which can be shown to start from around lO^GeV upwards. This is the end result which I wished this Appendix to lead upto.

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89) A. D'Adda, M. Liischer and P. Di Vecchia, Nucl. Phys. B146,63 (1978). 90) E. Cremmer et al., Nucl. Phys. B147, 105 (1979); See also S. Ferrara, in Proceedings of the Second Marcel Grossmann Meeting, Trieste, July 1979 (in preparation), and references therein. 91) P. Fayet, Phys. Letters 7QB, 461 (1977) ; ibid. 84B, 421 (1979) . 92) J. Scherk, Ecole Normale Supérieure preprint, LPTENS 79/17, Sep - tember 1979. 93) J.C. Pati and Abdus Salam, Phys. Rev. D10_, 275 (1974). 94) H. Georgi, Harvard University Report no. HUTP-29/AQ13 (1979). 95) M. Gell-Mann (unpublished). 96) See Ref. 64 above and also Q. Shafi and C. Wetterich, Phys. Letters jS5B, 52 (1979) . 97) J. Learned, F. Reines and A. Soni, Phys. Letters 4_3, 907 (1979) . 98) J.C. Pati, Abdus Salam and J. Strathdee, Nuovo Cimento 26A, 72 (1975) ; J.C. Pati, Abdus Salam, Phys. Rev. Dll, 1137, 1149 (1975); J.C. Pati, invited talk. Proceedings Second Orbis Scientiae ,Coral Gables, Florida, 1975, Eds. A. Parlmutter and S. Widmayer,p.253.

í \j 345

GEOMETRIC FORMULATION OF SUPERGRAVITY *

Samuel W.MacDowell Yale University, New Haven, Ct. 06520

I. INTRODUCTION

A basic problem in constructing a field theory of elementary particles is to find out what are the underlying symmetries of the interactions of the fundamental fields.

A symmetry is an invariance of a physical law, under a group of transformations of the fields. This group is called the symmetry group.

A classical theory of fields, in general, is a theory in which a group of symmetries is realized on a set of functions, called fields, defined on a four dimensional space-time manifold. The time evolution of these fields is governed by differential equations, invariant under transformations of the symmetry group. These equations are derived from an action principle, where the action is a functional of the fields, defined in terms of a local Lagrangian density which is invariant under the group of transformations.

These symmetry transformations are of two kinds: i) space-time symmetries in which the transformations correspond to a change of the space-time coordinates,

ii) internal symmetries involving degrees of freedom other than the space-time coordinates.

If the parameters of the transformations of a internal symmetry group are space-time dependent, the theory is said to have a local gauge invariance and the symmetry group is called a gauge group.

•Research supported in part by the U.S. Department of Energy under Contract No. EY-76-C-02-3075. 346 S. MacDowell

The lnvariance of Maxwell's theory of Eiectronar.netism under Lorentz transformations led Einstein to formulate the theory of Special Relativity. Electrodynanic3 is also invariant under a local Ui group of gauge transformations.

A gauge invariant theory with a non-abelian SU2 gauge group was first formulated by C. N. Yang and R. L. Mills .

The principle of Equivalence of local frames of reference was the basis for the development of General Relativity in which gravitation is described by a symmetric tensor g , the metric tensor; the field equations are invariant under general

coordinate transformations. The group of space-time symmetries

is the Einstein group of general coordinate transformations with

non-vanishing Jacobian. A formulation of General Relativity as

a gauge theory with the Lorentz group as gauge group was first

given by Utiyama .

In Special Relativity the group of space-time symmetries

is the Poincaré group, which is the semi-direct product of the

Lorentz group S£(2C) and a four dimensional Abelian group of

translations. It can be obtained from the de Sitter groups

30(4,1) and SO(3,2) - Sp(4) by a Wigner-Inonii contraction3.

The total symmetry group is the direct product of the

Poincaré" group and of the internal symmetry group. The fields

are then classified according to irreducible representations

of this group. Fields that belong to a tensor representation of

Sl{2C) (that Is,a representation of S0(3,l)) are called bosonic;

those belonging to a spinor representation are called fermionlc.

An irreducible representation of the Poincaré group contains

either boson or fermlon fields.

Supersymmetry is a generalization of the concept of symmetry

which allows for an invariance under transformations in which SUPERGRAVITY 347 I) boson and fermion fields transform into each other. The generators of infinitesimal supersyir-ietry transfcrmaticns and the corresponding infinitesimal parameters belong to a splnc-r representation of the Lorentz group. The algebra of the generators is a

L with the superalgebra, a Z2-graded alcebra L = i-zvift « odd' following Lie products:

The fermionic generators of supersynmetry transformations r belong to L ,,. In the supersyrametric model of Wess and Zunlno the Lie product of two' supersymmetry generators is a translation. Therefore, in this model the Poincare* group is embedded in a supergroup. If a theory is invariant under local supersymmetry transformations, and the algebra of generators is some generalization of the Wess-Zunino algebra, then it is also invariant under local infiniteSinal translations, that is, under general coordinate transformations. Because such a theory would incorporate general relativity, it is called Supergravity. If, in addition to the local supersymmetry, it has local f.auce invariance under an internal symmetry group, i.t is called Extended Supergravlty. A Quantum Field Theory is obtained from the classical theory by r.eans of canonical quantization or by Peynman's path integral nethod. In the canonical quantization, the classical fields become operator-valued distributions and Poisson brackets are replaced by commutators for two-boson or one-boson one-fermion fields, and anti-commutators for two fermions. Equivalently, in the path Integral method 348 s- MacDowell boson fields are treated as comnutinc functions whereas

fernion fields are treated as anti-commutinc ones. The commuting or anticommutinp; property of the fields determines the statistics, namely, Bose-Einstein for bosons, Permi-Uirac

for fernions. Physical states are interpreted as rays in a complex

KUbert space and scalar products as probability anplitudes.

The symmetries of the classical theory are carried over to the

quantum theory, but they may be spontaneously broken if the

state of lowest energy, the vacuum, is not invariant under

transformations of the symmetry croup. Spontaneous symmetry

breaking; is expected to occur as an essential feature of

unified gauge theories.

In order to preserve the correct statistics for forrcions

and bosons, the fermionic parameters of a supersyrr.r.etry

transformation must be anti-comr.utin£ numbers (Grassmann numbers)

or antl-comnutin£ functions.

The probabilistic interpretation of scalar products in the

quantum theory requires that the fields transform as a module

for a unitary representation of the internal sy rase try p;roup.

Hence, if the number of fields is finite, the internal symmetry

croup is compact, since only compact groups have finite dimension

unitary representations.

These lecture notes are organised as follows:

Section II is a survey of the main steps in the development

of a field theory of elementary particles in which the internal

symmetries are incorporated as local cauge invariances.

In Section III we discuss some of the attempts towards a

unification of the elementary interactions within the framework

of General Relativity, and give a brief review of Supercravity

and Extended Supergravity. SDPERGRAVITY 349

The main scope of these lectures is to present a formulation of supergravity and extended supergravity in which both the pauge invariance and the invariance under General coordinate transformations are Riven a unified geometrical interpretation.

The appropriate mathematical formalism for this purpose is the theory of Fibre Bundles which is described in Section IV and applied to Einstein's theory of gravitation in Section V.

In Section VI the geometry of fibre bundles is generalised in order to accommodate the new ir.variance under supersymmetry transformations. This is done by considering fibre bundles over graded manifolds.

In Section VII we show how one can actually construct simple supergravity as a realisation of the symmetries of a bundle of frames, on the space of functions consisting; of the vector

components of the connection and "vielbein".

In Section VIII this method is applied to the construction of extended supergravity, taking as its starting point a hypothesis of minimal coupling. Complete results are given for models with SO2 and SO3 internal symmetry.

Finally , in the last section, we make some remarks on the prospects for a realistic description of the known interactions of the elementary particles within the context of supergravity.

II. UNIFIED GAUGE THEORIES IM SPECIAL RELATIVITY

A difficulty that arises in the quantization of a classical

field theory is the appearance of divergent integrals giving

rise to infinities. In order for the theory to make sense,

these infinities must be removed in a way that preserves the

symmetry. The process by which the infinities are removed, by 350 S. MacDowell r.eans of a redefinition of physical parameters, is called

Renormalization. A theory for which this process can be carried

through is called renormalizable. Renormalizability has been

a powerful guiding principle in the selection of models for the

description of elementary interactions.

The foremost example of a renormalizable theory is Quantum

Electrodynamics. Here, renormalizability depends crucially on

the local gauge invariance, together with the hypothesis of

minimal coupling.

On the other hand, in quantizing general relativity, one

finds that the power counting criterion for the convergence of

integrals in the perturbation expansion of the S-matrix is not

satisfied. Therefore, the renormalizability of this theory

depends on the cancellation of infinities in every order of the

perturbation expansion. It has been shown that, for the self-

interaction of the gravitational field alone, such cancellation

occurs in terms corresponding to Feynman diagrams with one

closed loop. For two or more loop diagrams, one has not been

able tr> show that all divergences cancel.

In view of this obstacle with the quantization of general

relativity, the field theory of Elementary Particles has developed

within the framework Of Special Relativity. This is justified

since, at the level of processes involving elementary particles

at energies as produced in the laboratory, the gravitational

interaction does not play any significant role and can be

completely ignored.

The lntractions known to date can be classified into four

categories: 1. Electromagnetlsm 2. V/eak interactions SUPERGRAVITY 351

3, Strong interactions

4. Gravitation

Of these, the Electromagnetic and Gravitational interactions are the best understood.

They are lone-range interactions mediated by massless spin-1 and spin-2 fields respectively.

The source of the Electromagnetic field is a conserved vector current, that of the gravitational field is a conserved energy momentum tensor density. In contrast, the Weak and Strong interactions are of very short range and the fields that mediate these interactions are not well known.

The experimental evidence on the weak interactions shows

that they give rise to current - current couplings of vector and axial-vector currents. In the first model of the weak interactions proposed by 6

Fermi , the interaction Lagrangian was taken as a sum of products

of vector and axial-vector currents. This model was later extended

to a universal theory of the weak interactions when it was realized

that the strongly interacting part of the strangeness preserving

vector current coincides with the isotopic spin current, which is 7 conserved by the strong interactions. As subsequently modified o by Cabibbo , this theory accounts for many of the experimental facts about the weak interactions.

However, a serious problem with this theory is that it is

not renormalizable; thus, it should be considered as a phenomeno-

logical model. The local gauge invariance of Electrodynamics,

together with the minimal coupling hypothesis, are essential

features for its renormallzability. Since in the weak inter-

actions we also have couplings of vector (and axial-vector),

currents, a natural solution to the problem of renonnalizability

would be to assume, in analogy with Electrodynamics, that 352 S. MacDowell the weak interactions are also gauge invariant and mediated by vector bosons. However, since this interaction has very short range, the intermediate vector bosons have to be very massive.

As already mentioned, one can have a gauge theory with a non-abelian compact symmetry group and a set of spin-one gauge fields belonging to the adjoint representation of the symmetry group. If the symmetry were unbroken, as in Electromagnetism, all of these gauge fields would have to be massless. If, on the other hand, the symmetry were spontaneously broken, then according 9 to a theorem due to Goldstone .there should exist massless particles called the Goldstone bosons. The absence of such

Goldstone bosons was a serious obstacle in interpreting the weak interactions as a non-abelian gauge theory with spontaneous

symmetry breaking. This difficulty was resolved by Higgs who

showed that Goldstone bosons do not emerge in a gauge theory if

the symmetry is spontaneously broken because, in the vacuum state,

certain scalar fields have non-zero expectation value. This

process of symmetry breaking in gauge theories is known as the

Kiggs mechanism.

The discovery of the Higgs mechanism paved the way for a

unified formulation of the Electromagnetic and Weak interactions

as described by a non-abelian gauge theory. This was done by 11 12 S. Weinberg and Abdus Salam who proposed such a theory with

a gauge group SU2 8 Ui. A large array of experimental tests lends support to this theory. The success of this unification

has encouraged physicists to extend the concept of gauge invariance

to the strong interactions. It should be pointed out that if

the SU2 8 U, theory is restricted to the interactions of leptons, 13 one finds that it has non-vanishing Adler-Jackiw-Bell anomalies, connected with the coupling of the gauge bosons to axial currents, SÜPERGRAVITY 353

Which preclude renormalization. It is only when the Quarks (which are thought to be the building blocks of the strongly interacting particles) are incorporated into the theory that these anomalies cancel and the theory becomes renornalizable. >:

In the quark theory of strong interactions one postulates ? an exact SU3-symmetry (called SUa of color). If again we follow r the analogy and assume that this SU3-group is a group of local ;- gauge invariances, then we have the corresponding gauge fields is, which are called Gluons, as the vector mesons that mediate the ' , strong Interactions. This theory of strong interactions is $r called Quantum-Chromodynamics (Q.C.D.). It has been rather '* ". successful in the understanding of the physics of the newly *-• discovered particles. Moreover, an important feature of a |..- non-abellan gauge theory of the strong interactions is asymptotic m. freedom.which gives a natural explanation of the scaling laws % of form factors at large momentum transfers (Bjorken scaling). -if

Therefore, present experimental evidence favors a description f( of the Electromagnetic, Weak and Strong interactions in terms ^~ of a gauge theory with gauge group SU3 S SU2 8 Ui,with SUj • Uj j'- spontaneously broken down to the electromagnetic U, gauge symmetry. Since this group ls not simple, such a theory has (at least) three independent parameters, the strong coupling constant, the electric charge and the weak interaction angle (the Weinberg angle). A unification of all the interactions would require a sinple group as a gauge group. The group SUS is the smallest sir.ple group which has SU3 8 SU2 8 Uj as a subgroup. A grand unified theory with SUS gauge group has been proposed by Glashow and Georgi. The known leptons and the quarks required to account for the symmetries of the strong interactions fit, so far, within three copies of two irreducible representations of SUS 354 s- HacDowell of dimensions 5 and 10 resnectively. If one embeds the group

SUS into an S0,0 gauge group, these representations (together with a singlet) belong to an irreducible spinor representation

of SOl0 of dimension 16.

Other unification schemes have also been proposed; in 16 particular, a model suggested by F. Gürsey et ai. uses the

exceptional group E6 as gauge group. These unified theories are supposed to account for all the interactions of elementary

particles with the exception of gravitation. In the next section

we review unified theories which build upon General Relativity.

They are based on Einstein's idea of extending the symmetry

group so as to obtain all the interactions of the physical space

as manifestations of its symmetries.

III. UNIFIED THEORIES IN GENERAL RELATIVITY

After the discovery of General Relativity and the first

few experimental astronomical observations confirming its

predictions (bending of light rays, advance of perihilion of

Mercury, red shift), Einstein and nany co-workers in the field

devoted much of their scientific research effort to extending

the geometrical self-coupling theory so as to Incorporate other

interactions. Particularly sought after was a unified description

of the electromagnetic and gravitational fields in geometrical

terms. 17 In the conformally invariant .theory proposed by Weyl ,

an invariance under local dilations was assumed, which was

appropriately called gauge invariance. The connection associated

with this one dimensional group of transformations was identified

with the electromagnetic gauge potential. As it turns out, the SÜPERGRAVITY 355 quantum theory of Electrodynamics is invariant under a phase transformation or a local Ui group of transformations, not under the non-compact group of dilatations. However, in spite of its inadequacy, the name of gauge invariance continued to be used to describe the invariance under a compact group of local transformations and the symmetry group has since been called a Gauge group. 18 Another interesting proposal was made by Kaluza .

He considered a five-dimensional Riemannian space with the

de Sitter group SO(i|,l) as the group of local invariances.

He postulates that the metric n:ijc(i,h=l.. .5) has cylindrical symmetry, that is, there exists a coordinate system on which

the metric is independent of the fifth cOUl'Uitjate. Kaluza also

ir.poses the constraint ri =1. These conditions are preserved by

a transformation

with g = n - n t-nV(- left invariant. Also one finds that

det|nlk| = det|g I is invariant under this transformation. The metric g is identified with Einstein's metric. The

transformation (1), which has the form of a gauge transformation,

suggests the interpretation of n ^ as a gauge potential. Further

more, calculating the scalar curvature corresponding to the

metric ilk, one finds that it coincides with the Lagrangian density of the Einstein-Maxwell generally covariant theory.

Kaluza's theory, albeit ingenious, does not, as pointed 19 out by Pauli , solve the problem of unification. First one can criticize the ad hoc assumption of cylindrical symmetry.

Secondly, it does not address itself to the more general problem

of unification in the sense of incorporating the sources of 356 S. MacDowell the electromagnetic and gravitational fields.

The fact that matter is basically described by fermion

fields raises some general questions concerning unification.

First, in order to incorporate these fields into a theory of

general relativity, it is essential to use the "vierbein"

formalism. This implies that the metric tensor is no longer a

fundamental field. Instead we have g = h h^,, with the

1 "vierbein" h as the fundamental field, and g.1 the Minkowski metric.

Furthermore, in order to preserve the local Lorentz

invariance of the theory, the condition of zero torsion has

to be abandoned: the fermions appear as a source of torsion.

Thus one is led to consider non-Riemannian geometries. Although

it might be possible to include in the theory other bosons

beside the spin-2 field g , simply by enlarging the group

of symmetries, a true unification with the fermions cannot be

achieved in this way.

The new concept, that at least in principle allows for

the inclusion of fermions in a unified manner, is the concept

of supersymmetry, in particular local supersymmetry

A theory with local supersymmetry was first formulated 20 by D. Freedman, P. van Nieuwenhuizen and S. Ferrara . It is a locally supersymmetric theory of a spin-2 field, the

"vierbein" h and a Majorana spin-â field ha. A more concise

formulation using the first order formalism was given by Deser

and Zumino . The Lagrangian is simply the sum of the Einstein

Lagrangian and the generally invariant form of the Rarita-

Schwinger Lagrangian for a Majorana spin-|- field with the

ordinary derivative 3 replaced by the covariant derivative

+ o w erc s 3U h [iJ]* ^ u * the Lorentz connection and SUPERGRAVITY 357 °rin tne Dlrao~a matrix in the Najorana representation. A first attempt at a geometric interpretation was given 22 by P. Mansouri and myself using a four-dimensional space- tine manifold and the supergroup OSp(l|4) as gauge group.

The action integral is of the form: S where: k 1J] a b Cij:i -EU] _ Ml + h AhV + h *h f R " R Einstein + n h 1W + h " ^6

Hence, when h = 0, S reduces to the Einstein action with a cosnological term plus the Eula?-Poincaré invariant of general relativity.

The sane approach was used by P. van Nieuwenhuizen and 23 P. Townsend to construct SO2 supergravity. The action turns out to have the same form as before, plus a term corresponding to the generally covariant Maxwell action. Extended supergravity oil PS has been constructed with internal symmetry groups S02 , SO, , SO»26 and SO,27. 28 It has been shown that a linear representation of a supersynmetry algebra with SO., internal symmetry, which does not contain representations of the Lorentz group with spin higher than two, exists only for Ii £ 8. Therefore, in this

sense, the SO8 theory is the most general supergravity theory that can be constructed having an orthogonal group as the internal

synmetry group. This would not be large enough to construct a realistic grand unified theory of all interactions. One would need at least an SOl0 internal symmetry group. Remarkably, 27 the SOg-theory constructed by Julia and Crenmer has an additional local SU8-symmetry; however, the gauge potentials 358 S. MacDowell associated with this symnetry are not elementary but composite _-

; fields made out of the spin-0 fields in the model. The SUa " group is large enough to contain the known symmetries of the strong, weak and electromagnetic interactions, but In this model the gluons and intermediate vector bosons would not be elementary vector fields.

In the latter part of these lectures I shall discuss the. geometric structure and give an alcebraic.construction of these theories, based on what is generally referred to as the superspaee 29 formalism. This formalism has been used by several authors with a variety of objectives and techniques.

Although I shall be concerned only with the construction of the classical theory, I would like to mention that an early and major motivation for developing supergravity was the expectation that a consistent quantum theory could also be formulated.

In general, the existence of a symmetry is responsible for elimination of divergences. Thus, electrodynamics and the Yang-

Mills theories are renormalizable because of gauge invariance.

Also it has been noticed that certain cancellations of divergences do occur'in supersymmetric theories.

In quantizing supergravity one finds the same kind of difficulties already encountered in the quantization of general relativity. However, it has now been shown that, in simple and 30 • j. SOí supergravity , one and two loop diagrams are finite.

In order to investigate the convergence of diagrams with more loops, new and more powerful techniques using superfields are being developed. SUPERGRAVITY 359 31 IV. THE FORMALISM OF FIBRE BUNDLES A common mathematical framework for a possible grand unification, which would encompass gravitation, is the theory of fibre bundles . Within this framework one can give a geonetrical formulation of gauge theories and, also, one can cast general relativity into the form of a gauge theory, with the Lorentz -group as gauge group. A Principal Fibre Bundle is a set (P,M,G) with the following structure: 1. P is a topological apace called the bundle space. 2. M is a manifold called the base. 3. G is a Lie group called the structure group or gauge group, which acts (freely and differentlably) to the left on P: GxP+P. 4. M is the quotient space of P by equivalence under G, and the projection II: P+M is c". 5. P is locally trivial, that is, for meH there is a neighborhood 17^ of m and a diffeomorphism l H~ (U1)*U1fiG given by p+(n(p),*1(p)), for

pen" (U^), where ^ is a C™ map: H~ (U1)+G such that 4*(SP) • g^Cp). for any gcG.

6. Let us denote by $., m the map $, at 11** (m) where

If meU,nu, the map , o4$ ~ :G*G is an element g(m) 1 J J J™ I,m of GjC* in U.OU.. For meM, n" (m) is called the fibre over m. Let F be a manifold ana let there be a right action of F: fS~leF for all geG and feF. An associated bundle to the principal bundle (P,M,G) with 360 S. MacDowell standard fibre P Is the quotient space of P8F by equivalence under G. Let {U.} be a complete set of coordinate neighborhoods in M. Por each U. one defines a coordinate basis as a set of vector fields {3 } satisfying the conditions:

C3y,av] «o (2) and such that they form a basis for the tangent space M of M at every The dual of the basis {3 } is the set of one-forms {dxw} such that by contraction dxu(3 ) * 6**. Let us now consider the tangent space at a point p in the bundle. Displacements along the fibre passing through p are generated by tangent vectors called vertical. Their projections onto the base manifold vanish. One can choose a complete set of vertical vectors {P. }, such that, under confutation, they form an algebra lsomorphic to the Lie algebra of G:

where {f. ') is a set of structure constants for the Lie algebra of G. The set {V. } is also a module for the adjoint representation of G. Now in order to obtain a complete basis for the tangent space at p, vie have to specify a complete set of horizontal vectors which generate displacements in the tangent space orthogonal to the fibre. The horizontal lift of a vector 3 in a coordinate basis is a vector SOPERGRAVITY 361 whose projection onto the base manifold is 3 , and which is "equivariant", that is, it commutes with the vertical vectors

This Implies that the functions h '(p) transform under G as the components of a vector in the adjoint representation of the Lie algebra of G. A. A, C. °B h,, (P} " fB C \

<6> ao W o « ^ The set of vectors V form a basis for the horizontal tangent space at p. The functions h (p) are called connection coefficients. The set of one-forms which is dual to the basis {p. ,V. } A A is {« °,dxw} where w a is a vertical one-form called the one-form w of the connection.. Iff {iJ{UA °,dx°,dxu} iss the dual of the basis '. ,3 } then, from the condition: A, U

it follows that:

A, A ,+ n A rp A, A, u • (1 u dx = B + h (8)

The curvature R is a horizontal two-form taking values in Aa the algebra {X. } of G, that is, R-R X. where: n* " 0

A « o J Ao l Bo COj, A, X «dill - r 01 AID fn . (9)

Then, since u is vertical, it follows that:

Therefore, since [P ,f ] is vertical, we have: 362 5. MacDçrçell

Hence,

The bundle of bases B(M) of a manifold M of dimension d, with structure group G£(d,R),is a principal bundle defined by

specifying a point pelT'CU) as p = {ra^.. .ed},where meU and

{eA } is a basis for Mm. The projection is n(p) • m, and the 1 B B action of the group is given by gp = {m,g JeR . ..g^e,, }, where fleG-£(d,R) is represented by the d-dimenslonal non-singular ~ matrix (g *). Ai Let {e,,. } be a d-dimensional set of independent vector I- i I * fields in U. Then we have e. = g.1 el (m) where (g.1) is a %• non-singular matrix and therefore represents an element.of J Bi ;' Gi(d.R). The map * can be defined by

(e' (m).e' Cm)) - gA - (13) *i °i AiBi where g. n is a constant non-singular matrix. Aiai A subspace of B(M), which is obtained when the matrices Bi ga are restricted to belong to a subgroup H of &£(d,R), is _ "i also a principal bundle with structure group H. In particular, if H is a subgroup of G£(d,R) which preserves the metric

g. B , that is: AiBi

for all gell, then the bundle B(H,H) is called a bundle of frames. The relevance of bundle of frames for physics lies in the SUPERGRAVITY 363 fact that in a bundle of frames the parallel transport of vectors along a path in the base manifold preserves the scalar products. Let Y(S) be a continuous, piecewise smooth curve in M,

parametrized by s and passing through a point mQ=Y(0). The

horizontal lift 7(s) of Y(S) through some point p0 in the fibre over m,, is a curve such that n(7(s)) = y(s) and a vector tangent to 7(s) at arW °F its points is horizontal. Now, in a bundle of bases, a point in ^(s) is given by p = ^(s),e,(s)...e (s)}.

l If t(0) = ta e. (0) is a tangent vector in M,, the parallel 1 A transport of t(0) along Y(S) 1S the vector t(s) = t e. (s).

The scalar product of two vectors tt and t2 parallel transported along -y(s) is given by:

l l (t1(s),t2(s)) = t^ t^ (eAi(s),eBi(s))

A, Bl. C1, D1, t, t2 gA (s)gB (s)gc D (15)

Thus, on account of (I1*), it follows that in a bundle of frames the scalar product is preserved. Aq Ai Let (Í2 ,h ) be the set of one-forms dual to the basis f« »ea )• The horizontal one-form h = {h '}, called the solder form, is completely specified by the choice of vector fields

Indeed, let e^ be given by:

Í i A i where (h Mm)) considered as a matrix is non-singular. Then 364 s• MacOowe11

The functions h l(p) are called "vielbeins". Since every group element g.1 has an inverse and (h (m)) is non-singular, then h l(p) has an inverse which vie denote by hJJ (p).

The metric tensor in M is defined by:

Al Bl g_= gA B h (m)-h (m) (18)

or, in a coordinate basis:

u U The scalar product of two vectors ti = t?3 » t2 = t2d

is then given by:

<*i.t«> " 8^*2 Í20)

We notice that in a bundle of frames, because of (1M),

we also have

A B £ - EAiBih i(p)-h i(p) , (n(P) = m) (21)

Let {X. } be a basis for the matrix representation of the

Lie algebra of H, acting as a group of transformations on the

vector basis {e. }. We introduce the connection one-form A * u=u °X in B(M,H) and define the covariant basis [V. ,f>. } Ro Ao Ai in the tangent space to the bundle B as the dual of the set of A. A. one-forms tw ,h }. Then one can easily verify that the

horizontal lift V of the vector 3 of a coordinate basis is given by:

so that SUPERGRAVITY 365 J J J (23)

The torsion T Is defined as the set of horizontal two forms ÍT *} given by:

Al A Bl dh - «o °Ah fA £\ (24) Q 1 where f. £l - (X. )°1.. A,B, A, B, -V. Since u B is vertical, the components T * * T ^n'^v^ of Ai T are readily obtained:

- »„*;• -

We shall consider, in particular, bundles of frames

honeomorphic to a group space G, with structure group a sub- 32,33 group H of G and base manifold homeomorphic to G/H . Let ÍX.} be the set of generators of the Lie algebra L A t of G in the adjoint representation. We have:

[xA,xB] = -fAg xc (26)

where f^ = UA)£. Also ÍXA> = {XA } 9 ÍXA }, where {XA }

Is a set of generators for the Lie algebra Lo of H and ÍX,.}

Is a basis for the vector space L, = L/L9. The Killing metric of L Is defined by:

f f "AB " - A? BS (27)

^We follow here the notation of reference 33 with some changes in convention which are listed at the end of Appendix A. 366 S. MacDowe11 with the following property: r

rA£nDC - fCAnBD " ° (28)

We Introduce a one-form $, taking values in t, defined by:

4 - u + h (29) where u • u °X. Is the connection one-form and h » h *X. *0 "I Is the solder form. The set of one-forms ( } is the dual of the covariant basis (P.). A 3«l

It is convenient to introduce the two-form . :

A F - F XA - d* (30) taking values In I, whose components in the covariant basis are given by: v

c • c where PAB = F (1>A,V^). Thus:

F P f P R F A,B, 'A.B.^A.BL " BtAt A,Bl« AlBl AlBl» AlB1

The Bianchi identities, which result from the associativity of the differential operators V., are equivalent to :

dF - 0 (32) which, in component form in the covariant basis, gives:

D (ABC) - I [OAPBD • P^g] - C (33) (ABC) where the sum is over cyclic permutations of the indices. In the standard formulation of gauge theories, the physical SUPERGRAVITY 367 fields are defined on a cross section of the bundle. A cross section of a coordinate neighborhood U of M is a c", one-to-one map x* U • P, such that n(x(m)) • m for all meU.

A cross section through a point p(eP with It(p() * n>,eU, f: can be defined by the set of first order differential equations: js

p 1 ft •• flp *(x)dx (3 !) • r r where the functions n (x) are C in U (and satisfy the required t:v integrabllity conditions), and the Initial data is x(m9) " pt. |, Then,, on a cross section, we have: If

p w u td ' - Q ' +.hj*dx - (tlj° + hp°)dx - Vu°dx (35) y. A. which defines the vector potential V (x) on the cross section x of the bundle. Notice in particular that, in the trivial cross section x(m) • m, V ° and h ° coincide. A gauge transformation corresponds to a change of cross fr section. In an infinitesimal gauge transformation we have: ':

B0C,

The derivation of this result is given in Appendix B. We shall here follow a somewhat different approach. We shall consider a class of equivalent bundles of frames 8 • {B(M,H)} over a common base manifold M (covered by the same set of coordinate neighborhoods) and with the same structure group H. Two bundles B and D1 in this class have connections 368 s« MacDowell and "yielbelns" related by:

fA A h (p)U'A(p) = h (p)UA(p) (37) where

BB CC J(p) - exp{XX (p)l>(p)l>(B(P)} l?l?(pA(p) expi-AexpiA ((p)Pc(p)} (38) with XB(p) satisfying the condition:

P. XB(p) - f. I XC(p) (39)

These relations may be considered as transformations of the connections and the "vlelbelns", with the coordinate basis Í3 } and the vertical basis ÍI>A } left unchanged. ^ In an Infinitesimal transformation with paramaters e j. one obtains:

A B «hj - 3ue + hJe FB; (40) and

These transformations close under commutation, [6i,623 'i» with the composition law :

Transformations with X '(p) " 0 are equivalent to gauge transformations. Given a cross, section x of the bundle B(M,H)e8, there exists a bundle B'(M,H) such that h|A»(m)

The composition law (42) has been incorrectly given in references 33 and 34 where the first two terms were omitted. SOTljRGRAVITY 369 and h'Al(m) coincide respectively with ^'(xtm)) and hAl(x(m)). Moreover, h1 (p) and h (p) are related by a transformation (37-39) F! with XBl(p) - 0. L On the other hand, transformations with parameters

XA(p) - Xu(m)hA(p) , m - lt(p) are equivalent to coordinate transformations In a single bundle. In a general coordinate transformation {{xw} + lx'v • fM(x)} '*•'- we have h|A(x«) - hA(x), P'A(x') - PA(x), or:

dxuhj(x) (43a) L and

F^(x') - P*c(x) (43b)

For an infinitesimal transformation x|U • xv + cu(x) we have:

A v b» .(x»> - hj(x) - -OuE (x))h*(x) (44) Then: A v v A h'J(x) - h (x) - -[(3ue (x).hJ(x) + c (x)3vh (x)] (45)

Similarly:

PBC(x) -

Now an infinitesimal transformation in the set of equivalent A V bundles (B(M,H)} with parameters e (p) - -E (m)hv(p), m - n(p), gives: 370 S. Macoowell and i\. >h F (18) fiFBC E X BÍ; " "e 3vFBC i

Therefore: 6hJ- h'J(x) - hJU)

which establishes the correspondence between infinitesimal general coordinate transformations in a single bundle and infinitesimal transformations within the set 8.

A choice of gauge in our formulation corresponds to the' selection of a subclass of 8 whose elements differ by transformations (37-39) with parameters XA(p) - Xv(m)h*(p).

V. FIBRE BUNDLE FORMULATION OF EINSTEIN'S THEORY OF GRAVITATION

General Relativity can be formulated within the formalism developed in the preceding section. For pure gravity one should look for a ten parameter group G, which contains the six paramete subgroup H « S£(2C) to be taken as the gauge group, and a four dimensional coset space G/Sl(2C) for the base manifold. Among the simple groups only two satisfy these conditions; namely, the covering groups of the deSitter groups S0(4,l) and SO(3,2). The covering group of SO(3,2) is the symplectic group Sp(4). These two de Sitter spaces differ in their topology: the coset space G/S£(2C) has three compact and one non-compact dimensions for the former,, one compact and three non-compact dimensions for the latter.

A bundle of frames B(M,H) is defined by a set of vector SOPERGRAVITY. 371 fields {e°(m),l>1...4} which are local Lorentz frames. The metric gl1 defined by (13) Is the Mlnkowskl metric n^. The structure constants {fAg} are those of either de Sitter group with A, • [1J], a pair of Indices with i<3„ and Ai « i. The elementary fields are the connection coefficients hCiJ] for the eauge group Si(2C) and the "vierbeins" h* defined by (16-17). The metric tensor g is defined by:

(50)

The covariant derivative of a vector field t*t (x)3x alonp the direction of the vector 3 is defined by:

u Vutdx - t||(x+dx) - t(x)

where U(x+dx) is the parallel translation of t at x+dx to x. The components of V t are given by.

X y X X X v p :vwt) dx •= t (x+dx) - t (x) + r j]V(x)t (x)dx (51)

where r uv(x) is the familiar Christoffel symbol. Vie shall now express 1< F wv(x) in terms of hv'- '-' and h^. Accordinp: to the definition of parallel transport alonp: a curve y(s) we have:

v v 1 Al t (s)3v - t (s)h* eAi - t (O)eAi (52)

so that alonp Y(S) we have:

But, since Y(S) is horizontal, then: 372 s- MacDowell

On the other hand if t Is parallel translated alone Y(S) we nave VVfc fs~ * °an d (51)pive3 ! |f (55)

Taking (5»O and (55) into (53) one obtains:

U p A p A AB P A

where specifically A, • [1J] and A, - 1. From (56) and (25) one immediately obtains

relating the torsion to the antisymmetric part of ru, . If T • 0, then h^ ^ can be determined in terms of h^ and the space is a Riemannian symmetric space. Then one can show that the only second order invariant equation for h that can be derived from an action principle is:

which is equivalent to Einstein's equation in the absence of external sources,with an arbitrary cosmological constant \. A solution with constant Riemann tensor RjÇ11^ • *fijn^ corresponds to a homogenous de Sitter space. Performing a Wigner-In6nu contraction -of the de Sitter, group down to the Poincarl group, the cosmologlcal tern is eliminated. Equation (58) together with the torsion constraint

constitute the unique complete set of 16 + 2k Invariant

\ i equations for tit» 16 components of the "vierbein" and 21 y SUPERGRAVITY 373 conponents of the connection, which are linear in the curvature and torsion. So far we have only described general relativity, within this framework, for the gravitational field alone. One possibility of incorporating other fields using this formalism is to consider an associated bundle whose fibre is a carrier space for a representation X' of the Lorentz group. The connection in the principal bundle induces a connection In the associated bundle with the covariant derivative defined by:

where the matrices {x! } are the generators of the Lie algebra of H in the representation X'. However, a truly unified theory, incorporating fermions, can only be achieved by embedding the Lorentz group in a supergroup containing in addition the subgroup of internal symmetries. In order to accomplish this goal within a geometrical framework, it will be necessary to consider more general spaces with the structure of supergroups, and to develop a formalism for fibre bundles defined over graded manifolds. This will be done in the next section, in which we use the concept of 35 graded manifold as developed by Kostant . We would like to point out that this concept is not sufficiently broad to allow for the notion of Superfield as Introduced by Salam and Strathdee However, It seens to be adequate for the construction of any classical supergravlty theory. 374 S. MacDowell VI. FIBRE BUNDLES OVER GRADED MANIFOLDS

We shall give here an overall description of fibre bundles on graded manifolds, without being concerned with mathematical rigor or completeness. We start with the notion of graded 35 manifold according to Kostant Consider a manifold M of dimension d and a vector space V of dimension n, over the reals, and define a Grassmann algebra G on V. Given a complete set of neighborhoods {U,} in M, for each IL one introduces a "sheaf" in {U^G}. as an algebra A which is the direct product of G and the space of C - functions on U.. An element acA is then a sum of elements of G whose coefficients are C°* - functions in U,. A graded manifold is a set (M,A) in which a coordinate system has been Introduced in the following way: For every neighborhood U* of M take a set {uv} of d elements of A, called the even coordinates, which are even functions of the generators of G, and a set {ua} of n elements of A, called the odd coordinates, which are odd functions of the generators of G, with the following properties: i) The Jacobian of the coefficients of the Identity of the even coordinates íuu} is non-vanishing throughout U.. il) The product of the n odd coordinates {ua} is non- vanishing throughout UJ. We remark that the even coordinates are commutative, the odd coordinates anti-commutative and the product of more than n factors of odd coordinates always vanishes. Hence any function of the coordinates can be expanded as a polynomial in the odd coordinates with coefficients which are functions of the even coordinates. SÜPERGRAVITY 375 The concept of differentiation is introduced in the following ,£, in way: " i) For even coordinates let xv be the coefficient of ,

the identity of A in U^ and let fu(x) - 3yf(x). Then the partial derivative of a function f(u) with respect p f to the even coordinate u is defined by 3.,f(u} = u(u). ii) The partial derivative 3 with respect to an odd coordinate ua is defined in the following way: The derivative of a sum of terms is the sura of the derivatives of each term. The derivative of a term which does not contain u° as a factor is zero. The left (right) derivative of a term that has u° as a factor is obtained by displacing u>a to the left (right) of the product of , { odd coordinates and removing it from the term. f b- These derivatives have the following properties: f:(

(61)

We say that the set (3u»3a) form a coordinate basis for the tangent space to the graded manifold. a 1# Vie attribute to the indices fti,a)a signature a = °* a ~

A vector e. » (h¥ 31( + h? 3 ) has a definite signature a. • 0(1) according to whether the functions hV are even (odd) and h? are odd (even) functions of the odd coordinates ua. Ai Vie shall only consider bases {e. } for the tangent space to the graded manifold in which there are d elements of even signature and n elements of odd signature in the set {e. }. Let us next define differential forms in graded manifolds. Let (t,...t ) be a p-fold set of tangents t-j^M- of given

signature a*. Denote by vP a set whose elements are (ti...t ). 376 S. MacDowell A p-form Is a p-linear map w: vP •*• A such that:

1) w(t1...t1...t1+j...tp)

(62)

11) if t± = hJ?K then: ... w - H 1 °J

(63)

A contraction of a p-form w with a vector t, Is a (p-1)- form w«. such that: •V «(t,tt...tp) - wt (ta...tp) (61)

The wedge product of a p-form w and q-form w' Is a (p + q)- form w" = WAW1 defined by:

v"(tl...t p+._q ) = E(semr)w(t AA ...tA. )w'(tAft ...tA , ) TI i p p+l p+q (65)

where IT is a permutation (A,...A A +,...A + ) of (l...p+q)

such that (A,

(66) r Mai nn +1 j .i (sgnir)

where n. and nu are respectively the" number of indices SUPERGRAVITY 377

>A A.(iíp) with odd and even signatures and such that Ai p+j' The exterior derivative of a p-form w is a (p+D-form dw defined by:

1 1 i 3 + I (-1) w([t1,tJ},t1...ti_1,t1+1...tJ_1,tj+1...tp+1)

(67) where n* is the number of elements of odd signature with index less than 1.

The definition of fibre bundle may be taken- over with

appropriate changes. In general one can have fibre bundles

where the base is a graded manifold and the structure group 1 is a super-group. Here, we restrict our analysis to the k cases of bundles over graded manifolds with a Lie group

as structure group.

If Ü = (uX,ua) is a coordinate system in (M.A), the

set (3i»3 ) is a coordinate basis for the tangent space to

the base manifold.

We consider a bundle of frames G(M,H) where G Is a

supergroup, H a Lie subgroup of G and M a graded manifold

homeomorphic to G/H. The Lie algebra L of G is a graded

algebra L = L ® ^add with generators {X,,}, whose Lie products are given by:

AB [XA,XBJ = -(-l) [XB,XA} = -fA°xc (68)

where ifAg) is a set of structure constants for the superalgebra

L and oA is the signature of the generator XA:aA = 0,1 according

£l L We to whether ^pL -evint 0(i(i' remark that the signature of a 378 S. MacDowell

generator XA of the Lie algebra £.„ of K Is o^ =0 because H is

a Lie group; but the signature of X, belonging to L, = L/l0

depends on whether Xft eLevcn or Xft EL^.

A carrier space for a linear representation X1 of G is

also a direct sum V = Ve ® Vo of subspaces of even and odd signature satisfying the conditions:

Let {U«|} be a basis for V of vectors of definite signature

o.,, andd (XI)., be the matrix representinrepresent g the generator XA of L. Then, for a simple supergroup, we have: {• °D» = (-1) - X&J&\ = -f(X')nAB (69) . I)

where n.B is the metric of i (Independent of the representation) and f(X') is the index of the representation. For simple Lie

groups one can always set f(X.1) = 1 for the adjoint representation c c ^XA^B = fÀB» 3O that nnji coincides with the Cartan-KIUIng metric. However, for simple supergroups, in some special instances the

index of the adjoint representation vanishes. In general, one

can use the fundamental, representation to define the metric.

The metric n»r, satisfies the following important Identity:

fAB"DC- fCAnBD=Q

Since the functions defined on graded manifolds depend on

antlcommutlng variables, some care must be taken in preserving

the order of factors. The connection coefficients arid "vielbeins" SUPERGRAVITY 379

h? ({A} = {X,ct}) are assured to te even or odd functions of the

odd coordinates ua according to whether (o^ + o ) = 0 or 1.

Likewise, the parameters XB in the transformations (37-39) or

the infinitesimal Daraneters eB in (40-41) are also even or odd

functions of ua according to whether Og = 0 or 1.

The symmetry of the metric (13) is Riven by

_ i 1N°A1°B1 e . (71)

The algebra of the vectors {V.} in the covariant basis for the tangent space to G(K,H) is as in (3D, with the commutator replaced by:

where F ^ = fC(.V.,V~).

The Bianchi identities (33) are replaced by:

" (Jc) ^^ CVBS + FABFE^ = °

The relation between the components of F in a coordinate

basis and in the covariant basis is piven by:

a +a B c A °l(E A A B n ) i i FEA = (-D i H\

where P£J = Av^VJ. In general, the concepts introduced and results obtained

in the ordinary theory of fibre bundles can be carried over to

bundles on p-.raded manifolds. One noticeable exception is that

the concept of Integration of forms, so useful in the geometry 380 S. MacDowell of manifolds, cannot be generalized to graded manifolds.

Althouph it is possible to define an invariant integral in a graded manifold, the invariant measure is not a form.

In the next section we' use the formalism developed here to construct a Supergravity model which is a locally super-

symmetric generalization of Einstein's theory.

VII. CONSTRUCTION OF SIMPLE SÜPERGRAVITY

In analogy with the formulation of general relativity as

the geometry of a bundle of frames, in a simple supergravity

model without internal symmetry one should look for a simple

supergroup which contains either SO(l!,l) or SO(3,2) as its

r.aximal subgroup. The only such supergroup is the ortho-

symplectic group OSpCljA) with 14 generators.

Vie shall then consider a bundle of frames with structure

group II - S£(2C) and base supermanifold M homeomorphlc to

OSp(l|M)/S£(2C). This is an eight-dimensional graded manifold

with four even coordinates {xv} and four odd coordinates {ea}

The bundle of frames G(M,H) is defined in terms of a set of

vector fields {e°(m),e (m)} which are, respectively, local

vector and spinor Lorentz frames.

The metric (13) is (j^, ,Cafa) where n.j* is the Minkowski

metric and Ca{, (the charge conjugation matrix) and its ab inverse -C are real and antisymmetric, with the property:

£b Ca6C « «a* (75)

The matrices Ca^ and C are used to lower and raise spinor indices. SOPERGFAVITY 381

The Lorentz connections h*- •'•' and the "vlelbeins" (h ,ha) have, in a coordinate basis, two sets of components: i) Vector components: [jp

II) Spinor components: ha i0^»*1*-

Since the physical space is really four dimensional, the question now arises of interpreting the new fermion-like coordinates 8°.

We remark first that one wants to formulate a theory, based on an action defined as an Integral over the space-tine manifold, invariant under general coordinate transformations.

A necessary and sufficient condition to enforce the assumption *i that (h, ) is a non-singular matrix is that the matrices (h t ) A • u a and (h ) be themselves non-singular. Therefore, one can use det(h )d x as the invariant measure.

The Lagrangian may depend on the vector components of h and their space-time derivatives. But since the infinitesimal transformations of the spinor eomnonents contain 3 E and the a odd variables are not Integrated over, it follows that the only dependence of the Lagrangian on h must be through the covarlant C C A components F. n of F . Also, the only dependence on h in the 11 A C "

transformation laws of h Is through P. B • Since derivatives of a field with respect to 6n have to be considered as new

independent fields, it follows that all the components F. H Aitíi must be so constrained that they can be expressed by means of

(7*0, in terms of F^* ^yv'^uv^ and a subset of covarlant

components to be considered as independent fields.

We shall thus obtain a realization of the superalgebra

(72) In the space of functions h plus some additional fields ÍF} taken from {F „ }. The remaining components of {F V, } BjCn , ° BjCD , have to be expressed in term3 of F and the new fields F. 382 S. MacDowell An inspection of (7*0 with AE • pv shows that, in order to carry out this program, one has to impose a certain set of constraints which relate the components F,.»,c, of Fc to the independent AaO| fields {F} and, linearly, to the components F.^. Assuming no new fields F, these constraints take the form given in Table 1, which has been reproduced from ref. 33 for the general case of extended supergravity to be discussed in the next section.

TABLE 1. Covariant constraints for simple (and extended) supergravity. The t's are covariant parameters constructed out of the structure constants of.G> The X's are constants restricted by: ^ii^ii*^*^ = aiifi2jfiik5»4 for alx Permutations

(ijk ) of (1 2'3 *») and XM = X2l » X,, = \+% = 1. Note the slight change in notation (a,b,c,...-»•(*,B,Y) , (a,b,c,...*a,b,c...).

Win »"/,<„ •,

which reproduce those obtained in ref. (40), and are equivalent to the equations of motion in ref3. (22,23). Prom the equations of motion one can reconstruct the Lagrangian. It should be mentioned that only the equations of motion are invariant under the supersymmetry transformations (40-41). The Lagrangian is invariant under a set of transformations which coincide with (40-41) when the equations of motion are taken Into account. However, this new set does not close under commutation. In order to obtain a Lagrangian invariant under the transformations (40-41) one has to include some new independent fields P as auxiliary fields. These fields vanish as a result of the equations of motion. Although the problem of auxiliary fields is essential for the quantization of this theory, we shall not go further into a discussion of this subject here.

VIII. CONSTRUCTION OF EXTENDED S02 AND SO, SUPERGRAVITY

The extension of the construction of the previous section to a model with internal symmetry is conceptually straightforward. One has to consider an orthosymnlectic supergroup OSp(Il|4) whose Lie subgroup is Sp(4) 6 SO(H), where SO(N) will be the internal symmetry group of local invariances. The bundle of frames has structure pr°up H » S£(2C) 8 EO(N) 384 S. McDowell and base supermanifold M homeomorphic to OSn(N| 1J)/(S£(2C) « SO(N)), which has four even coordinates íxV) and *»! odd coordinates {6°}. It is defined in terns of local frames {e!(ra),eò (m)}.

The metric (13) has symmetric components r\i, and anti-

Tne w symetric components C .n b. * ° sets of components of the connection have, in addition to the Lorentz components h ^ and possibly new sets of fields F belonging to Irreducible repre- * sentations of Pn _ with respect to the gauge group H. Following the procedure described in the previous section, one finds that the constraints in Table 1 for a meael without additional fields F are consistent with the Bianchl identities If the internal symmetry group Is S0(2) but not consistent, for symmetry groups SO(N) with N>2.^ For the S0(2) r.odel one obtains the sane set of equations of motion (76) as for simple supergravity plus the equation for the S0(2)-gauge field:-

=0 , ([ab] = [12]) (76a)

For SOdO-supereravity with N>0 one does not expect the constraints on Table 1 to *? consistent with the Bianchi identities. In fact, the linear representations of the (Poincaré contraction of the) superalgebra OSp(H|4) in a Rilbert space with a state of highest helicity s, singlet with respect to SO(II), contains a set of state vectors with helicity (s-n/2) in the completely antisymmetric tensor representation of SO(N) of rank n, for all values SUPERGRAVITY 385 of nSH . Thus, for s-2 and N-3 one expects a set of spln-if fields belonging to the representation of SO(N) antisymmetric in 3 indices. One can show that this field comes from the torsion component T 5^. Therefore, the constraints in Table 1 should be modified to include the dependence on the new fields. In analogy with the models of simple and S0(2) supergravity 31» we shall set

1 • 1 m J aa'bb ab ab * aai

as a starting point for the construction of extended supergravlty with N>2. The physical meaning of these constraints is that, if the equations of motion are derived fron an action principle in space-time, the Lagranglan 'contains the Parita-Schwinger term with a covariant derivative of h*a, and no other term depending on h*J6 which involves derivatives of fields belonging to nontrivial representations of the gauge group S£(2C). The constraints (77) are then an assumption of "Minimal Coupling". Next in the construction of SO(N)-models, we take the constraints (77) into the Bianchi identities and determine the components of the curvature and torsion that have a lower spinor index,in terms of the subset R.,« ,T..* and a set of independent and auxiliary fields. The Bianchi identities should also give the super- syrxietry transformations of the new fields, that is, their

first order covariant derivatives Pfta. When the auxiliary fields are expressed in terms of the independent fields, the Bianchi identities also give equations of motion for the latter, as well as equations of motion for the gauge fields. Most of this program involves decomposing each Bianchi identity (A,B,C) - 0 in terms of irreducible representations of S*(2C) 9 SO(N). The only technical difficulty is in finding 386 S. MacDowell the relation between auxiliary and independent fields for j. N>3 supergravity models. ' We outline here the results obtained from each Bianchi < identity1"1"1":

1. (aa,bb,cc) It determines the torsion T £? to be of the form:

The spin-is field ^^r^cji which belongs to the irreducible representation of SO» anti-symmetric in three indices, is an T ls a independent multiplet In the extended theories; aa(be) spin-% auxiliary multiplet, and one must impose the constraint:

T«Cbc] " ° (79)

wt/'ch forms a set of covarlant equations for the spinor components hj; c-' of the internal symmetry connection.

2. (aa,bb,cc)dd

i) It gives the supersymmetry transformations of the irreducible components of.T £jj. For N>3 one finds that T transformation of aa/bc).contains terms quadratic in This analysis follows closely ref. 34; however, there are several misprints and some mistakes in that reference which have been corrected here. SUPERGRAVITY 387 Therefore, It becomes a non-vanishinp auxiliary field. As for

N • 1,2 one has to impose a trace condition on Ttta/b )'•

CT + T + ab aa(bc) b(ac) xTc(ab))n - 0 (80)

where x Is a parameter. The simplest choice is x = which, for N - 1, yields the conventional model26.

T and T contaln II) The transformations of jr-hon nfbc) new multlplets of scalar and pseudo-scalar fields, which are irreducible components of R '•?. ^ frc'd'lc 6lven by

R°»ss - i I p + R \ K[abcd°» ] " F tabc ) flRaa,fabCcd] + Rac,6d[ab] (81)

K[abc]d " I (abc)v aa,bb[cd] " Rac,bd[ab]nc )Ys > (82)

The first is a set of Independent fields, the latter a set of auxiliary fields.

For N=8, Rfahcdi ls not lrreduclble» In order to obtain an irreducible realization of the supersymmetry algebra one has to impose the self-duality conditions:

REabc«i] " <* Í)EabcdalbI°tdI REá'b'c.d'] (83) ut! which turn out to be consistent with the Bianchi identities. 1 \ ill) The remaining Irreducible components of Rfla I; are determined in terms of Taa £jj and T^ 388 S. MacDowell

3. Ua,()b,i)J~

1) It determines R a ^ ^ in terms of and ii) It restricts to be of the form:

(84)

+ T'(ac)

where

i(ab)f[pq]J ab and

(86)

It determines the curvature °f ., and the torsion T?..

c ln erms of I) It defines the curvature •• frbtci]b t torsion components and the supersymmetry transformation

Ti[bcy

II) It gives the supersymmetry transformations of the bosonlc auxiliary fields ,T«ab) .TJab) One has to impose further constraints on the auxiliary fields,

compatible with their transformation laws, in order to determine

them in terms of the independent fields. Once this is done, the

following become equations of motion:

ill) for the spin-| field SUPERGRAVITY 389

" Ti[ce] +

3" (afie) a Cbc] Y*tTa(bd)T [cé] " Ta(cd)T [be])n ír[kí]

(87)

lv) for the spln-| field h*a:

^'i Jo) T.tVaCbe) - VTl(bo) + ? Ti(bc)> + TiaTd(bc)

(I Tj fi • i T^f^fjn^ - 0 (88)

dd 6. («a.if

which, combined with. (85) and (86) can be solved for T^. This enables one to determine the Lorentz connection hj" ^.

ii) It gives the equation of motion for the spin-2 field h :

(90) 390 S. MacDowell

R Cd i 111) It relates the Yann-Mllls curvature R±J a = 1j "' '[cd]a to T.p . -j and the auxiliary fields:

(OiTk[ab]

T fV

= 0 (91)

This equation determines the auxiliary field in terms of the Yann-Hills field strength Rlk and the independent fleld Ti[ab]'

iv) It gives the equation of motion for the spin-1 gauge field hjab]:

0 fTCrs] m,Crs].f ,1 „, I frnCrs] m,Crs],_ J VT[ab] " T[ab] )f[rs]k + TJk (T[ab] " T[ab] )f[rs]£

+ TJk fJ T[abc] " TTjk f

c Tke " Tkc

(92)

The constrairts (77,79) together with the Bianchi identities A A (1-6) determine all the components of R ° and T ' with at least one A A lower spinor index in terms of (R..°, T^.1) and the following sets:

independent fields: Td[abc]f R'[aJcd], T1[ab].

Auxiliary fields: Ttta(bc), H'^ T^], SÜPERGRAVITY 391

rp!; in' ips r" 5. mti

For N>4 (with the exception of N=8), new independent fields

will emerge as covariant derivatives V of ~ "

For N=2,3 one can set

T«a(be) " 0 (93>

and the remaining auxiliary fields become determined in terms of

the Independent fields by:

Rtabc]d °

[ab] f [poji ^Jk - I Rika "be + l(1?iTkCab] " Vl[ab3>

+ è(Ti[ac]Tk[bd] " Tk[ac]Tl[bd])l)C

nab tor zero; , T(ab) = u |

m5 - T'* . 1 ~ m „ -p mCode] mil Ti(ab) ~ TKab)- 12 ^ab A[cde]Y5flT • T[ab]

where TÍ bx = n . or zero corresponds to the values for de Sitter

or Poincaré supergravity. In order to obtain an irreducible realization of the symmetry

one has to specify the independent field T.r .-,. Again we invoke

the hypothesis of "minimal coupling" to establish the constraint

that determines this field, which is

Then for N=2,3, the equations (87), (88), (90) and (92) reduce to: 392 S. MacDowell

(92')

(89')

- [rs] _ J . 1 _, f0 m[abc] ., ij fCrs]k + 7 T[abc] fkpiT + Xt n

These are the eauations of motion for SO(2) and SO(3) super- ;| gravity. Prom them one can derive the space-time action, by integration of the functional equation: • í

\\ h3 5S r,l p [rs] _ J to - [rs] ,, J, . 19 h n = {( R f n n R f } + 12 n Jk r £j [rs]n ik " ij [rs]k ik

5ija ÈWb^ Vt(hJ) (96)

where the covariant components of the curvature and torsion have to

be expressed in terms of Trabe-| and the vector components h of h by means of the relation (7*0, with A£ •= pv. For N>3 we have not yet obtained a complete determination of the auxiliary fields. Once this is done one expects that a space- tine action may also be derived from a functional equation analogous to (96). f- SÜPERGRAVITY 393 IX. CRITICAL EVALUATION OP SÜPERGRAVITY

The most attractive feature of supergravity and extended supergravity lies in its potential for achieving a complete geometric unification of all the Interactions, Including gravitation, with all the fermionic and bosonic fields appearing as members of one supersymmetric raultiplet. The SOe-model is singled out, not only because it is the most general model in which all the elementary fields have spin<2 (and the gravitational field is the only one with spin-2), but also because all the fields with spin>|- are gauge fields. This last condition is important for the removal of states of zero or negative norm.

However, the phenomenology of the elementary particles

gives evidence for an underlying SU, 8 SU2 8 U, gauge group,

which is not a subgroup of S0e. Thus SO, will not be a

sufficiently large gauge proup unless the SUa symmetry of the weak and electromagnetic interactions is a dynamical symmetry. Moreover, since all the fields in this theory are real, charge conjugation is trivial. Thus It appears that supergravity models with internal symmetry based on the orthogonal group do not qualify well as realistic theories. An alternative is to consider models based on the supergroup A(M|N). For I-M, in one of its forms A(1||N) has SU(2,2) 9 U(N) as maximal Lie subgroup; SU(2,2) is the conformai group which contains Sl(2C) as a subgroup. These theories have been investigated 36 by P. van Nieuwenhuizen et al . They correspond to generalizations of Weyl's theory and have an intrinsic difficulty with causality. Hence, although supergravlty contains the necessary features of a truly unified theory, some elements are still missing for the construction of a realistic theory. 394 S. MacDowell

APPENDIX A - GLOSSARY OF NOTATIONS AND DEFINITIONS

P - principal bundle; p - an element of P: (peP)

H - base manifold or graded manifold; m - an element of 11: (meM)

G - Lie group or supergroup; 3 - an element of G: (geG)

H - Lie subgroup of G

V - Vector space

B(H) - Bundle of bases over M

B(K,H) - Bundle of frames with structure group H

Ad.G- Adjoint representation of a group

I - Lie algebra of G; Lo - Lie algebra of H; Lt - Vector space L/Lt A,B...E - group indices for the generators of L.

AB,BB...EQ - group indices for the generators of LQ

AliB1...E1 - group indices for the generators of I, {f.g} - set of structure constants for L

{f, R'} - set of structure constants for Le f ' C c {XA> - set of generators of 1 in AdX3: (Xft)B = fAB" {X. } - set of generators of L. in Ad.G: (X. )?." = f. ?." Ao A» Bo AoBo

1 {XA } - set of generators of L, in Ad.G: (XA )B» = fft g

{Pft } - basis for vertical tangent space in a fibre bundle which

under commutation form an algebra isomorphic to Lo IV.) = {0. ,0. } - Covariant basis-in a bundle of bases

A Afl A, D • u °Xft - one form of the connection (vertical) A, h • h X. - solder form (horizontal) A • - in + h = XA A A A {$ } «= i(o ,h } - basis of one-forms dual to {V.} r,A,u,v,p,o - indices of even coordinates in a supermanifold M

iCfi - indices' of odd coordinates in a supermanifold M

- coordinate basis in tangent space M SUPERGRAVITY 395

3 3 {V ,V } - horizontal lifts of Í u» a^í basis for horizontal tangent space P to the bundle P at p. t - vector in a tangent space,

{e,} - set of vector fields forming a basis for the vector

representation V-i,h) of S£(2C)

{e } - set of vector fields forming a basis for the spinor

representation [(2s,0) + (0,^)] of S£(2C)

i,j...r,s - vector indices for a representation of S£(2C):(i = I...1*)

a,b...e,d - spinor indices for a representation of S£(2C).(a = 1...4)

a,b...e,d - vector indices for a representation of the group

S0(N)(a = 1...N)

[ab] - pair of indices with a

index for the adjoint representation of SOfi, or vector indices a,b = 1...N, when it osnotes components of a

tensor antisymmetric in a,b

(ab) - vector indices denoting components of a tensor symmetric

in a,b

h - connection coefficients for ,'3£(2C)

" connection coefficients for S<">(N) h •* - "vierbeins"

h* - Rarita-Schwinger field (Majorana representation)

^Ci]^ ~ 3tructure constants for S£(2C)

" structure constants for SO(N)

^fCiJ]Ck£]»f[iJ]k>fki i ~ structure constants for

The structure constants for OS (M|il) -may.be written in terms of

the structure constants for 0SD(l|i)) and S0(N + 1): 396 S. MacDowell

[pq] r I f [ij] - b _ b f [1J] f i, \íkf]' í±3lktlkl »1[ij]a»Iia'Ittb »xabJ constants for OS,

structure constants for 80(11+1)

Cpq3 fa b bb bfib C ]]'r[ij]kflk »r[ab][«wi] •1CiJ]aa

. dã _ . d.d [lj] [ij] f i _ f i f [cd] _ fCcd] f r s r 1 n I xf [ab]cc ~ [ab]c £' aa16b b " ab ab» aa,bb aabb

structure constants for OS (ill1!)

C , = -C, ,(abC - -CCfaafaa:CbcbcC = Sb) = Charge conjugation matrix; C , a a. CLO a a CLC 0. antisymmetric metric used to lower and raise spinor Indices.

Definition of ys'.~

v 6 e Ysa f[iJ]b

.a.b be 1 sac

The netric of OSp(N|il) may be written in terms of the metric of OSp(l|t) and of S0(>5 + 1). Definine the metric through the fundamental representation with index f = *1, we have:

l3'cab) = Metrlc of OS

(nCab][cd3'nab) = Metrle of

Metrlc of e [c'd'3 f[cd]aneb " fab n[c'd'][cd]

Notice the following differences in notation in this paper and in reference 33:

The Lie algebra of a group G is denoted by L, instead of g. The elements of the group G are denoted by 9, instead of {?. tit The elements of H in the adjoint representation are denoted by g, . «0 ft The elements of H in the co3et representation G/H are denoted by g. » Mi Dirac spinor indices are denoted by a,b,c..., instead of a,B,y.... The structure constants fBA and functions Ffij? have been replaced

*>y -fj^g and -FAB which is equivalent to multiplication by the o.on signature factor (-1) * .

The generators Xft of G have been replaced by -XA. 398 S. MacDowell APPENDIX B - GAUGE TRANSFORMATION OP VECTOR POTENTIALS

Let us consider a Lie group G, whose elements p(x) are

obtainened by eexponentiatio: n of the elements X XA of the Lie algebra of G: AA g(X) - expXX XA (B.I)

Let us denote the composition law for the parameters X, corresponding to the operation of group multiplication:

by:

X* - **(*,,X) (B.2)

The generators t>A of the Lie algebra of G, acting as differential operators on G considered as a topological space, are given by :

- ujcxj -Is (B.3) x -o ií x

The basis of forms {0 } dual to the set of basis vectors

{PA> is then:

tlA - dXB U~Q

Therefora, on a cross section xi G*Rn we have:

8*,^^-;'.^* (B.5)

In an infinitesimal change of cross-section, corresponding to a local gauge transformation with infinitesimal parameters V. SUPERGRAVITY 399

£ (x), we have;

.A . B \A •• AXA (B.6) X + C e-0 so that &XA - cBUg.

Therefore:

3U,B A nCE,,D (B.7)

But from (3) and (B.3) one gets:

3U B 3U L r ,-iA _ - A (B.B) 3X

Therefore:

(B.9) 400 S. MacDowell

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WEAK INTERACTIONS

RajatChandow Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil

1. Introduction

The traditional charged-current weak interaction Lagrangian which accounts for low-energy weak processes, especially non-charmed par ticie. decays is written as

where the weak current J, has a leptonic part I, and a hadronic piece h,. A " " i. can be explicitly written down in terms of leptonic fields

Z, = I YiCl-Ys^ + V Y}Cl-Y5)v,,- (2)

The hadronic currents are known to have left-handed V-A structure and possess., strangeness-conserving and strangeness violating pieces according to the Cabibbo parametrization

=0 As =1 hx = hf cosec + 4 l sinec (3)

The Cabibbo angle 8 = 139. Only 'first class1 AS=0 currents with the following G-parity transform - ation properties are known to occur in nature G Vf=0 G-1 = Vf=0 (4) G Af=° G"1 = -Af=° .

Hadronic currents satisfy conservation laws called conserved vector current (CVC) and Partially Conserved Axial-Vector Current (PCAC) hypothesis. They also satisfy Equal-Time Commutation (ETC) rules pro - posed by Gell-Mann . Consequences of these general rules of conservation laws coupled with the algebra of currents were worked out about fifteen years ago. 404 R. Chanda Although the Lagrangian Cll leads to many successful predictions at low energies, we can not accept it to form a basis for Lagrangian Quantum Field Theory for Weak Interactions. To see the problems associated with it, let us consider the purely leptonic process vT +e~-*v+u" at high energies. Since the currents have dimension 3 in mass units, the coupling constant Gp has dimension -2 (Gp = 1.05xl0~ /mM. Therefore the total cross-section

- G/S (5) Jtot where S is the usual Mandelstam variable. On the other hand unitarity sets and upper limit on it:

'tot « I (6)

Hence above a certain energy given by &\ = —\- or E^ = 300 GeV. , the linear rise given by (5) can not continue and one meets the 'unitarity crisis'. Also, the renormalizability criterion requires the coupling

constant to have positive dimension in mass units. Since GF has negative dimension, the Lagrangian (1) is non-renormalizable and one can not canpute

higher order weak processes such as KL - Kg mass difference.

To dampen the growth of a^ tiintroduce massive boson exchange W~. However, the longitudinal modes raay grow with energy cancelling out the damping coming from the propagator of r

2. Gauge Theory; Unification of E-M and Weak Interaction

Local gauge invariance requires massless gauge fields. Renorm- alizable Yang-Hills theory handles internal degrees of freedom (4)

Choice of Gauge symmetry; The currents in the electron sector are given below. Inclusion of other leptons is trivial.

The chiral projections are

+ j (Leptonic) WEAK INTERACTIONS 405 Eq. (7) can be rewritten as

M J^ = L Vu T3 L + |(L Vp L + R Yu R)

(13-current) (Y - current)

where t's are the usual isospin matrices. These current structures show that the minimal gauge group containing weak and electromagnetic inter - actions is SUW 8 U*. Vector gauge fields associated with them are res - pectively W* and B (i1 = 1, 2, 3) :

^ * P y (8)

Note that W^'' can not be the photon. The photon field A is a suitable linear combination of w ' and B ]i V

1 A u will have pure vector coupling to e~ with strength "e , the electronic charge, and decouple from v if

;f e = " . ; sin 6 r~T~k '9 = ~±— {10) 12 5 w 12 ; (g^g ) (g^g )^ sin8w

3 The orthogonal combination of w t and B is

Zw = oosew Mi!31 + sin8w By-

This couples to both e~ and v . Electroweak model requires neutral weak currents coupled to Z°.

The parameter &w is to be determined experimentally.

3. Gauge Invariant Introduction of Masses for W+, Z°. Higgs Mechanism

i9(3Í Introduce a charged scalar field 4>(x) = *, (x)+i*2(x)=p(x)e in presence of gauge fields:

L = - 7 Fyv + \ 'V'2+ po21*'' % 406 R- Chanda /'< where F is the gauge invariant gauge-field strength and D is the co- ~, variant derivative. If y* < 0, V(i))I = U2|| + M "f1 nas 3 minimum at i£? | (ji | = p = /-u§" /x = v ft Q. The ground state is SU2 doublet

pi =

then mf = tíos8w

A more complicated Higgs structure would change this relation.

4. Effective Lagrangian j-;

In the region of momentum transfer q2 << mA which corresponds fi' to present experimental situation we have an effective current - current '•', interaction with

= — (14) /2 (14) , (13) and (10) tells us

m» = I "/?e2 I 2 = 79 GeV (15)

and m- = 90 GeV. We have used the experimental value sin28 =.23. The neutral coupling strength of any Fermion f to Z° is

Note that the couplings depend on l3 assignment. WEAK INTERACTIONS 407 5. Introduction of Hadrons ; Quark Mixing.

AS = 0 neutral currents were found experimentally in v-induced reactions but JAS..J = 1 neutral currents are highly supressed as seen from the absence of processes like K° •*• ]iVi K •* •n e e etc. The Cabibbo currents can not account for these suppressions. Glashow et al (GIM) provided an explanation introducing the charmed quark C(Q=2/3=OP . In GIH scheme the charged current couplings among quarks are

u d1"

1 1 where d d cose + s sin6c d eose (17) s1 =-d sine + s cose or s1 -sin6

In this scheme, the neutral currents do not have the |AS| = 1 terms: ds" + 3s pieces cancel out. Experimentally the Cabibbo angle 8c = 13?. Discovery of the charmonium, charmed mesons D, F, charmed baryon í and the decay properties of D obeying the predicted selection rule AC S has led to the acceptance of this picture (7) .Experimentally r a\ • we have an upper limit on AC j4 0 neutral current

•+ v C <.O2. (18)

6. Generalization of GIM Mechanism to n-doublets: Flavor Conservation in Neutral Currents

Consider n-doublets of quarks (i = l...n)

di where d! =

+ A A (19)

The n 'up1 quarks have electric charge Q = 2/3, the down quarks have 2 Qd = -1/3. The matrix of unitary transformation 'A' needs n parameters for its characterization. Among the 2n quark fields there are 2n - 1 unobservable relative phases. Hence the number of observable 408 R. Chanda parameters are n2 - (2n-l) = Cn-11 For n=2 (GIM case) we have just one parameter 9 , the rotation angle in d-s plane. For n=3 case (Kobayashi-Maskawa) we have 4 observable parameters 3 of which may be chosen to be the Euler angles Q^i = 1/ 2, 3) in the d-s-b space; we are left with an observable phase :6. Thus, in K-M six quark model one can naturally incorporate the CP violation. In the general n-doublet case (n > 3) the number of phases 6^ which can not be removed by redefinition of fermion field phases is fn n2 n(n-l) _ Cn-1) (n-2) (n-l) - 2 jr •

7. Motivation for Including the Sixth Quark t.

The reasons, for expecting an yet undiscovered sixth flavor of quark called 't' are as follows:

(i) The spin \ T-lepton (1782 MeV) and the associated v,p are

established. Hence there are six leptons. mv < 100 MeV. The limit on

my will improve soon via the ir end-point energy in x •+ it v transition. The Michel parameter p from the decay electron momentum spectrum in x •* e v gives e expt = "72 ± "15

in good agreement with V-A coupling for T-\> i.e. p ~A = .75. The T-decay branching ratios also check with V-A coupling. Renormalizability of Salam-Weinberg model demanding cancell - ation of Adler-Bell-Jackiw anomalies require a quark doublet (t,b) with V-A coupling. The fifth quark flavor b is already well established .

(ii) The natural incorporation of CP violation is an attractive feature of 6-quark models. Fermion contents of the 6-quark 6-lepton model are the follow - ing left-handed SU^ doublets

c (20) s1

The quark mixings are givsn by WEAK INTERACTIONS 409

1 d S1C3 S!S3

s' 8 c C C C +S S e10 C C C S S eÍ6 (21) - l i 1 2 3 2 3 1 2 3- 2 3

b" -S,Sa cisac«-C2SselS ClS2C3+C2C3e10

where c. = cos6. and s. = sin9.. 6j = 9 = 139. Note that for 82 = 6. =6 = 0 the mixing matrix reducex to the GIM case:

c, 8, 0

0 0 1

All right handed Fermions are Suw singlets.

8. Limits On the Mixing Angles 92 and 93.

Sirlin has calculated radiative correction to the allowed vector decay of O1"1 (u -»• d) . Together with the experimental results on K. decay (u + s) , the experimental check on Cabibbo universality for u-d and u-s couplings

z 2 li-s- v =^u g- leads to

c2 + s2c2 = 1.004 ± .003 (22) with s-j = .05, we have

(s3) = .28 ± .21 (23) which determines the mixing between u and b quarks.

Bound on 92 is obtained indirectly by considering t-quark contribution to Kj^ -*• yjl or to KL-Kg mass difference:

The mass difference is a complicated function of slf s2, m and

f s2, mc, 410 R. Chanda U,C

u,c,t

f 9) Limits on |s2| depends on m^ ':

0 < |sj < .25C.19) (24)

for mt = 15(30) GeV.

From the analysis of CP violation in the kaon system

+ M(KT -*• TT IT~) n . = +~Z 0 1 M(KL + IT IT TT J

- 2xlO-13 The six-quark model yields I (25) In. | = s2s3 sinfi. ,-29 Also, the electric dipole moment of neutron - 10 e.cm. is well below the experimental limit. Therefore, for kaons the results are similar to the superweak theory of CP violation ^ '. However, in this model CP- violation should be much larger in B° - B° system. This prediction may be tested at Cornell and may throw new light on the CP-puzzle. It must be remembered that there is a problem of how to eliminate or control the QCD induced strong CP-violation via

(26) *QCD

where Fyv is the gluon field strength. The arbitrary angular parameter 9 < 10~9 experimentally; otherwise it leads to too large electric dipole moment for neutron. T2ie axion or m,, = 0 option for eliminating this does fill not work . An ad hoc proposal to controll 8 exists: WEAK INTERACTIONS 411

6 eff = 6 + 8Q = 0 where 9Q = arg det M

(10) MQ being the quark mass matrix

9. Generalised Rules for Weak Decay of Heavy Particles

(27) 192TT3 where 9 is the x-y weaeak mixing angle factor. The kinematic factor F fe]= 1 m < 1 for 0 < 1. m

Special cases of weak decays of hadrons containing c, b and t quarks can be worked out. m__ Example: Decays of b-flavored particles involve working out 8, , ebcandFn^j- The result is:

r(b + ux) = sz s2 x .8 x 1015 sec"1. 1 3 (28) ex) =(s2 + sz + 2s s cos6) x .25xlO15 sec"1. Z 3 2 3

Using the limits on the angles 6. and 5

T(b) > 10" sec"1 or r(b) é-IO"11 sec.

Note r (b -v CX] » r (b + ux) . (29) 412 R- Chanda

Also, if 62 = 63 = 0, x(b) -+• "• or b becomes stable! Determination of b life-time (at Cornell) would provide new constraints on the mixing angles. XM is to be noted that in the 6-quark model b-decay and CP-violation become related. jl< One can calculate the decay rates of t-hadrons in a similar way. Dominant decay chains are:

t -* b •+ c •+ s (30) u -*• d

Mixing may also occur in the lepton sector but present inform-

ation on leptonn flavorr conservationn (pp 7•/* ey, v -V+T ,T —A ey.) and T life-time does not require such mixing.

10. Charged Current Neutrino Reactions

Consider v N -»• y~X and UN * p+x, where X is any hadronic sys- tern consistent with conservation laws. Let E be the incident energy , M and p be the mass "and 4-momentum of the target and q the momentum 2 2 transfered to the hadron. Defining.-q = Q i 0 and v•= Ei2, we intro duce the scale variables

2 x = -^jO and y = M-;0íxíl,0íyíl.

Differential cross-section can then be written down in terms of the nu - cleon structure functions:

d2gv'v 2 dx dy ± xF3) + |(P2 ? XF3) (1-y) ] (31) 3

All the dynamics of the process is contained in the structure functions í\^(x, Q2) which can be extracted from the experiments as follows:

(32) 2 v z v VP n d g d a xt3 a dx dy dx dy

In the quark-parton model Bjorken Scaling in the deep inelastic collision becomes, natural: F^s are expressed in terms of quark-parton distributions where x becomes the longitudinal momentum fraction carried by the WEAK INTERACTIONS 413 struck quark; for example x F3(x) = x(q(x) - q(*i) • v — Experimental results show that the y-distribution of -—— does not require right-handed couplings of u and d to heavy quarks and consequently charge-symmetry is obeyed. Also, the linear rise of the total cross-section t^ot with incident neutrino lab. energy Ey — (s = 2ME,, -) up to E, - ~ 200 GeV. tells us that in + > 30 GeV.

11. Violation of Bjorken Scaling: A Test of QCD.

2 2 Experimentally F2(x, Q ) and xF3(x, Q ) show, a small but significant Q2 dependence even after removing target mass and threshold effects. QCD predicts scale breaking because of the dynamics of strong interactions.

Consider moments of the structure-function xF3 :

í1 7 t1 n 1 M (Q2) = dx xn~z xF.Cx, O2) = dx x (q(x,Q2)-q(x,Q2)) n JO >Q ,1 , NS = dx x q (x, Q2) Jn Within the framework of QCD one can compute the Q2 dependence of the non singlet quark distribution function qNS(x,Q2). The result is(13)

2 2 dn Mn(Q ) = Cn(lnQ )~ ; dn > 0 for n 5 2 (34) or In M (Q2) = -d lnQ2 + c (341) n n n 1 Comparing logs of a pair of moments (n, n ) we find dn/dn" experimentally to be compared with the QCD result

1 +, 1 22 y 1 d 2 nCn+U I j Vdn' = 3zL_ (35) n • z 2 n'{n'+3J j£2 j

for various values of n and n1 one obtains remarkable agreement of (35) with BEBC measurement of M (Q2I. This constitutes the best available quantitative test of QCD with spin 1 g.luons . 414 R. Chanda 12. Neutral Currents: VN, ev and eN Sectors

Hi (i) \)-N Reactions; Space-time and isospin structure of hadronic neutral currents have been investigated using large amount of experimental data on

(a) Inclusive: v N+VUX, v N + \7 X (quark distributions needed)

(bl Elastic: v N -*%N, v H •+• v N (Elastic form factors needed)

(c) Pion production: vuN + vp IT X ; v^N •*• vy ft N V H + V T X i V N + \J TT N (Adler model dependent ) .

Analysing these data one can determine the couplings of the neutral currents to u and d quarks (main constituents of the nucleon):

(Hu) C 35 07) Neutral G | L " ' * - CÜu)Rx(-.19 ±.06)1 ^06) (dd] x (-.40± .07) (dd) x(0 ± .1) I TL Rn '

This is to be compared with the Salam-Weinberg predictions for the 2 couplings gT D « (I3L n - CQ)T „ sin 6 with standard I3 values and 2 sin ew = .23

(uu)LM.35) (uu)Rx (-.15)

(37)

(3d)T x (-.43) (Hd)x (.07 ) li K

The new reactor experiment at Irvine ' on v d + v np determines the

relative signs of the couplings uL and uR and confirms the opposite signs found above.

Comments: (a) Space-time structure is a mixture of V-A and V+A. fl4) (b) Isospin structure v of the neutral current is determined from v p •*• ~v pir" and %f n ->• \7 pit processes at CERN. The A (1232) peak in pir° and pir mass distribution shows that the neutral current has a 1=1 part to transform I = 3j nucleon to I = 3/2 A.

(ii) v - e Scattering: Pure leptonic processes are free from strong interaction effects and provides direct confrontation with theory. Expe- WEAK INTERACTIONS 415 rimental difficulty arises because of extremely low cross sections. New Fermilab experiments (-16) on v e~ + v e~ C46 events) yield sinz9 = ,25 + .05, The small error is quite an achievement,

>J e~ •+ v e~ and v e" + v e" CReactor) are also consistent with the standard model within experimental errors.

(iii) e~ - N Neutral Current Reactions: Parity violation in el d •+ e~X comes from the interference of neutral weak and eletromag - • (17) netic interactions. The asymmetry parameter A -,(x,y) is defined by

d^ da I dlT5 y + alTdyJ A j(x,y) f 0 when there is parity violation. The target deuteron being isoscalar, the x-dependence in A is cancelled out. An order of magnitude estimate for A can be carried out in the following way

0L-°R~ 0(GFe2)

2 aT + dp - 0(e") + 0(G e )

-10"4

In general we can write

A _ _ , _ 1 J-- \-J--Jf / [ ^Qj

For isoscalar targets al and a2 are constants related to the neutral current couplings of the electron and the light quarks u and d. Recently y dependence of A/Q2 has beccrae available in the range

0-1 S y S °-4 , enabling one to separately determine at and a2. Results agree well with S-W prediction with

sin2e = .224 ± .012 ± .008 w stat syst.

and rules out more complicated models proposed earlier. Parity violation experiments in atomic transitions are still not in agreement with each other. Note, the heavy atom parity violation experiments determine a different combination of e-q neutral couplings

than that appearing in eT _d experiments. 4ie R. Chanda (iv) - e+ - e~ neutral current effects should show up in y-Z" interference in Bhabha scattering. N-N and e~ - e~ neutral current effects are more difficult to see.

13. Important Unsolved Problem

(i) The age-old AI = \ rule in non-leptonic weak interactions is still with us. L,N L has AI = \ and 3/2 pieces with equal weight. W flRI It has been suggested that the dynamical enhancement of Al = \ piece is due to QCD correction to the weak amplitude. V w-

Effective |As| =1 quark scattering operator

Perturbative estimate in the lowest order in a gives an enhancement fac tor of 4-5 for the AI = Jj piece. Experimental enhancement factor is about 20. It has been pointed out that a new set of pure M = \ terms are introduced by QCD, generically called penguine diagram:

gluoni

Still the enhancement is not enough.. WEAK INTERACTIONS ' 417 (li) Attempts have heen made to calculate the weak, mixing angles 6^ in tetras of masses inspired by the. empirical relation

tane c =

However, the mass spectrum is not understood. It may be remembered that electron mass and charge are not calculated in QED.

(iii) Number of generations of families of leptons and quarks remain unknown.

Civ) Higgs structure of the theory and Higgs meson masses are not known.

14. Outlook

(a) CESR machine at Cornell will explore in detail the.b-particle fa mi lies. (b) Search for t quark will continue at PETRA, PEP and LEP. (c) PP Colliding beam at CERN may find W* or even Z° via Drell - Yan mechanism qq +r , Z°. (d) At LEP, CERN,2° resonances should appear at E^ - 90 GeV. (Z°- factory, - 1986). (e) Evidence of proton' decay may set the scale for the 'Grand Unific ation'. (£} Cosmic ray studies will play important role in elucidating particle physics at high energies.

References 1. S. Weinberg, Proc. 19tíl Int. Conf. High Energy Physics Tokyo 1978. 2. S.L. Adler and R. Dashen, Current Algebras and Applications to Part- icle Physics, W.A. Benjamin (1968). 3. R. Marshak, Riazuddin and C. Ryan, Theory of Weak Interactions.Wiley_ -Interscience, 1969. 4. C.N. Yang and R. Mills, Phys. Rev. 9£ (.1954) 631. 5. Por details see S. Coleman, These Proceedings. 418 R. Chanda

6. S. Glashow, J. Illiopoliis and L. Maiani, Phys. Rev. D2 (1970), 1285. 7. K. Berkelman, Proc. 19th Int. Conf. High Energy Phys., Tokyo, 1978. 8. K. Winter, Proc. Int. Symp. on Lepton-Photon Interactions, Fermilab, 1979. 9. R. Shrock et.al., Phys. Rev. Letters 42_ (1979), 1589. 10. R.N. Mohapatra, Proc, 19 Int. Conf. High Energy Physics, Tokyo , 1978. 11. R. Pecci, ibid. 12. K. Tittel, Proc. 19th Int. Conf. High Energy Phys., Tokyo, 1978. 13. See, For example, J. Ellis in Proc. SLAC Summer Inst., 1978. 14. C. Ealtay, Proc. 19th Int. Conf. High Energy Phys., Tokyo, 1978. 15. P. Hung and J. Sakurai Phys. Letters 88 B, 91 (1979). 16. K. Winter, Proc. Int. Symp. on Lepton-Photon Interactions, Fermilab, (1979). 17. C. Prescot et.al., Phys. Letters 77 B (1978) 347; ibid 84 B (1979), 524. 18. M. Gaillard, Proc. Int. Symp. on Lepton-Photon Interaction, Fermilab (1979).

I 41S

APPLICATIONS OF QUANTUM CHROMODYNAMICS TO HARD SCATTERING PROCESSES

C. O. Escobar Instituto de Física Teórica, São Paulo, Brasil

I. Introduction

It is our aim in these lectures to review the applications of Quantum Chromodynamics (QCD) to hard scattering processes .Throughout the lectures we shall keep in mind the basic results and general features of the simple parton model , so that we can examine which of these jfea- tures survive after taking into account the interactions of QCD.

We start the lectures by reviewing the philosophy and basic formula of the parton model, applied to processes like lepton-nucleon deep inelastic scattering (DIS), e e~ annihilation into hadrons, production of massive lepton pairs in hadron-hadron collisions (Drell-Yan process), etc. We then present a very brief summary of QCD, the main purpose of which is to set notation and to suggest QCD as the leading candidate for a theory of the strong interactions

The main topic of these lectures are the modifications introduced by QCD to the parton model. We begin by examining the leading order scaling violations in DIS, using the approach of Altarelli and Parisi ' . The next step is to consider processes which are not light-cone dominated (e.g., the Drell-Yan process). It is for these processes that great theoretical advances have been made during the last couple of years , regarding the applicability of perturbative Quantum Chromodynamics. A very simple recipe for parton model calculations in the framework of QCD emerges as a result of these theoretical advances.

As a first test-ground for perturbative QCD off the light-cone, we consider the phenomenology of the Drell-Yan process*9* . But it has to be remarked that there are also important corrections implied by QCD, beyond the leading order in the colour coupling constant ' . This is our next subject and we illustrate the importance of higher-order corrections by considering DIS and the Drell-Yan mechanism again. Finally we devote a section to further applications of perturbative QCD and consider some recent developments which have to do with the resummation of double logarithimic terms in the perturbative serie, namely the small-p dis- 420 c- Escobar tribution of lepton pairs (7) and the prospect of measuring the triple gluon vertex by looking for a marked rise with energy, in the multiplici- + — ( 8) ty of heavy flavours produced in e e annihilation . II. The Partorv Model

In the parton model picture of a hard collision (i.e. cne involving large energy and momentum transfers), we view a rapidly moving hadron as an assembly of collinearly moving partons (point-like constituents) one of which undergoes the hard scattering with the probe, may it be an electromagnetic or weak current or even another energetic parton belonging to some other hadron (as is the case in large-p_ processes) . For the validity of this impulse picture, it is crucial that we can talk about two time scales, a short time scale determined by the large momentum transfer in the collision (T ~ 5) and a long time scale set by the soft forces responsible for the binding of the constituents into hadrons (T - hadroni^mass scale) • Ifc is the clean separation of time scales that enables us to write the cross sections for hard processes in a convolution form, involving cross sections for collisions between partons and structure functions which have to do with the binding and are therefore -') not accessible to perturbation theory. A typical parton model expression for an inclusive cross-section of a hadron with momentum P is of the form, • i parton I-- da(P) ="Í t ÍI dB f,(B)do. (BP) (2.1) The functions f^B) give the distribution of partons of type i inside the hadron, carrying a fraction B of its momentum (this is governed by the long time scale) and dc^arton is the hard scattering cross section involving partons.

In the following we will apply the general form (2.1) for parton model cross sections to several hard processes so as to abstract their common features for later comparison with the QCD modified parton model.

1. Deep Inelastic Electron-Nucleon Scattering (Fig.l)

Dropping obvious kinematical factors, we have,

f r _ aH(x) = I eflx qj(x) + x qj(x) (2.2)

n where x = 3— , e, is the electric charge and(qJH are the distri- 2P.q x - x bution functions of (anti-) quarks of flavour i inside the hadron H (f HARD SCATTERING PROCESSES 421 is total number of flavours). So in DIS we directly measure the momentum distribution of quarks.

2. Semi-inclusive e+e~ •+ H+x (Fig.2)

Here we have (again dropping kinematical factors) ,

dg (z) = I e£ zfõg U) + z D3 (Z)1 (2.3)

2PH H where z ^ —^- and D_ (z) gives the probability of a quark i fragmenting into a hadron H, carrying a fraction z of its momentum.

3. Massive lepton-pairv ' (Fig.3)

where, T = M2 /s and

What are the common features in (2.2-2.4)? We immediately notice the following characteristics. ti a) They involve quark distributions xq.(x) or/and quark fragmentation H functions zD (z) which exhibit Bjorken scale, i.e. they depend solely i o of x and/or z but not Q .

b) Factorization: In x, and x2 in the case of (2.4) , in x and z for eH. •*• e + Hj + x, semi-inclusive electroproduction (Fig.4) ,

F H H IS 2 a° (X,z) = I e? xq^ix) z Da (z) + q H1'H2 l 1L qi J c) The distribution and fragmentation functions are process independent i.e. universal, we may use a quark distribution extracted from DIS in the calculation of the Drell-Yan cross section.

d) No mention is made in the above formula to partons other than quarks. This cannot be the whole story since there must be other quanta in order to bind quarks into hadrons. How can their presence be felt? 422 C. Escobar

Fig. 1 Parton model diagram Fig. 2 Parton model diagram for DIS. for semi-inclusive e+e~ -> H + x.

Fig. 3 Parton model diagram Fig. 4 Parton model diagram for for massive lepton semi-inclusive lepto- pair production. production.

e) The expressions above have a very simple probabilistic interpretation but in quantum mechanics we talk about amplitudes, so where are the interference terms?

After this brief presentation of the parton model we now turn to a quick review of QCD.

III. Quantum Chromodynamics

We will not review here the reasons leading tc the introduction of a new degree of freedom for the quarks, called colour, such that quarks come in three colours*3' (solving the spin-statistics problem in the quark HARD SCATTERING PROCESSES 423 model for baryons, improving the ratio of TT°->-2Y by a factor 3, improving the ratio R = a (e+e~ -»- hadrons/a(e e~ -> p+u~) by a factor 3 again) . Once the evidence for colour is accepted it is natural to construct a field theory in which this new quantum number plays a dynamical role and therefore to introduce colour dependent forces. The simplest possibility for achieving this is to make colour an exact local gauge symmetry with a gauge group SU(3) , and introduce eight coloured massless vector mesons as the carriers of colour forces (the so called gluons). As the gluons themselves are coloured we expect self-coupling among them and thus the theory must be qualitatively different from QED, as actual computations have indeed shown it to be.

yí Quantum Chromodynamics has the following Lagrangian density, |J<

where f labels the quark flavour (u,d,s,c,b, ?) , a = 1,...8 labels S the colour of the gluons and a,g = 1,2,3 label the quark's colours. jfeL

Ga the gluon field tensor is given by, j|f

Ga =3 Aa - 3 Aa + gf3^ Ab AC (3.2)

and D11., the gauge covariant derivative, acting on the quark colour components, is given by

Da8 " *a»\ ~ ig Ta6A P (3"3)

T . are the eight hermitean traceless matrices which generate SU(3) , , abc colour with f the SU(3) structure constants, we have b T ] - TC (3.4)

The Feynman rules from this Lagrangian are indicated below (in the r Feynman gauge) gluon propagator k2 C. Escobar 424

ghost propagator k2 + ie

i.—^ + m 6 quark propagator k2-mz+ie af5

Vertices

-ig (Ta) quark-gluon vertex 6

-g fabC[g)JV(k-q)X + gvX (q-r) V

+ gXv(rr-k)- v three gluon vertex

g fabc gluon-ghost vertex

fcdg(gvXgvp-gupgvX)

facg

four gluon vertex

The new features in this theory, at the level of Feynman rules is the appearence of self-couplings between the gluons and of colour factors in the vertices (we shall never worry about the ghosts, a proper choice of gauge will eliminate then) .

It is quite simple to deal with the colour factors, since they factorize outside the loop integrals and are expressed in terms of the eigenvalues of the quadratic Casimir operator C_ of the fundamental (quarks) re- HARD SCATTERING PROCESSES . 425 presentation and CA of the adjoint (gluons) representation. Next we give examples of such colour factors,

fiaBCF= 2 —- 6 o= 7 s o

aCd bcd 6a b. CA = f f = N fiab .

C. = 3 (N=3 colours) . A

There are also factors proportional to the number f of quark flavours, which are independent of Cp and Cft,

f TaB

Sab For the purposes of these lectures, the most important property of QCD is its asymptotic freedom , i.e., the vanishing of the effective coupling constant as we go to higher and higher momentum. This property immediately raises the hope of understanding why the parton model is such a good description of hard processes. It is, however, not a trivial task to obtain the parton model from an interacting field theory like QCD, even allowing for the fact that it is an asymptotically free field theory (we could think that at really large momentum transfers the interactions between quarks and gluons could be switched off). It is our main objective in the following to show why this is not trivial and how this goal can be achieved.

IV. QCD Corrections to the Simple Parton Model

i. Scaling Violations in DIS

Instead of applying the formalism of the Operator Product Expansion (OPE) plus Renormalization Group Equations (RGE) in the study of scaling violations in DIS, we shall follow a more intuitive treatment, 426 C. Escobar closer to the parton model language, due to Altarelli and Paris! (AP) , ;; which is however equivalent in leading order to the formal approach. ;T>

Remember that in the parton model Bjorken scaling was obtained thanks ,, to the fact that we neglected the interactions between partons and assumed a clean separation between short and large time scales. So for DIS (Fig.l), it is clear that the cross-section will be proportional to a ~ 6((ZP4q) ) - 6(x-z), thus leading to Bjorken scaling. However, in an interacting field theory we have to consider the radiative corrections to the Born cross section (fig.5). 2 Naively we would think that these corrections are down by order a (Q ) '., and since QCD is asymptotically free, t»g(Q ) << 1 for high enough Q and thus the radiative corrections would be negligible. Unfortunately this is not true because when calculating the cross section we get a factor log ^ (p is the momentum the incoming quark or gluon, p^ ~ 0 (300 MeV )) from the integral over the loop momentum and so the true ex- 2 9 O^ ' pans ion parameter is not a (Q ) but is instead a (Q^) log — which is •: s s p2 _"< 0(1). P ,' We will now show how the log Q factor comes about. When calculating cr- the cross section we square the amplitude shown in Fig.5, so for ; instance the term 5a when squared gives (Fig.6) a loop integral of the form, ç 4 2 2 2 d k fi(k )6((p+q-k) - m ) f(k) •í Rp-k)2 - m2l

4 2

but d k = dk0 k dk d(cos )d$, where 8 is the angle between k and p.

The dkQ and dk integrals are easily done using the two 6 functions and using, 2 2 (p-k) = m - 2EÜS + 2 |p| u cos 8 where a = k = |k| , the e integration is

1 d Í

But Ew ~ Q and 2Eu> - 2|p|iii - p , so we obtain a term proportional to log Q /p . We see that the logarithm comes from the angular region where 8=0, that is, from the region where the gluon is almost paralell to the quark. This is a well known example of mass singularity (emis- HARD SCATTERING PROCESSES «27

Fig. 5 Sane radiative corrections of order g , to the Born cross-section for DIS.

l-z

Fig. 6 A diagram which gives rise Fig. 7 The vertex q-Kj+g, responsible 02 to a log ^=2- term. for the scaling violations in leading order for non-singlet I distributions (see ec. 4.4).

Fig. 8 Radiative corrections which only contribute at z = l. 428 c- Escobar r.

sion of hard collinear massless quanta) dealt with by Kinoshita, Lee \(:_ and Nauenberg^ ' . fit.

We come to the conclusion that the quark distribution must depend on 9 2 Q and therefore it does change when we change Q , the resolving power of the probe, Q2 2 q(x,Q2) = 0(

p z Following Altarelli and Parisi, we introduce a function a+_ ( ) , which will govern the evolution in Q of the quark distribution and which has a simple interpretation as the probability of a quark emitting another quark with longitudinal momentum fraction- z (Fig.7).

From now on we consider flavour non-singlet distributions such as F^" eP eN " and F2 - Fj so that we do not have to worry about the gluon contribution to the evolution of the quark distributions (Fig.5c) . For this non-singlet distribution we can write with A.P , ' j Q2X,qN-S-(x,Q2) = "f

It is, at this stage, convenient to introduce moments of the distri- Í butions, defined as follows,

1 ( dx x11"1 qNS(x,Q2) (4.5) 0 Since the right hand side of (4.4) is a convolution of qNS and P taking the moment of the left hand side, we obtain,

" =- = — MK,(Q2) * K, (4.6) d log Q2 2TT ^ « where N 1 AJJ = I dz z ~ P_MI(x) (4.7) 0

2 I But ag(Q ) = — j to leading order i'" (11 - I f) log S_ HARD SCATTERING PROCESSES 429 therefore,

2 ^N = 2_ d log Q

2 2 Integrating this equation from Q to Q , we obtain,

(4.8)

where

(4.9)

So, we arrived at the same expression for the moments as obtained via the more formal approach of the OPE plus RGE . Here the d^, are the anomalous dimensions and once we know P-^-tz) they can be calculated q-»q using (4.7). So our next task is to find the function P (z). n (12) In QED, the Wei zs acker-Williams ' equivalent photon approximation, permits the calculation of the photon density inside an electron,

(4.10)

where

(4.11)

for z^O. The only difference in QCD, is a colour factor «- = E Ta T D pa so that we obtain (4.12)

and by z + 1-z, Wz) • I (4.13)

How do we deal with the singularity at z=l . It is reminiscent of infra-red singularities due to soft gluon emission and as usual they should be regularized by radiative corrections (Fig.8) which do only contribute at z=l. With these corrections we replace 1/1-z by a distribution denoted by — defined in the following way. c 430 - Escobar

dz dz (4.14) for any f(z) regular at z=l.

Now we examine the constraints which must be satisfied by the function P (z). Since quark number is conserved in the bremsstrahlung of gluons, 111*1 P (z) has to obey the following constraint, q->q

dz (4.15) but using the regularized form of P .(z) ((4.3) with y^ replaced by q-Ka+Cj ) we see that (4.15) is not satisfied and therefore we must include a term fi(l-z) in P _(z). So

WZ) = 3 the coefficient a is easily found by imposing (4.15) and we find a = 3/2, so finally we obtain 4 ll+zj (z) «L. + 1 6(1-2)1 (4.17) L-z)+ * J

With this we can easily compute the anomalous dimensions for the non- -singlet case according to (4.7), finding, N "2 I X j=2

To summarize, leading order QCD predicts for the non-singlet moments,

-id. N (4.19) log 4 where fl = - -— and A., as given by (4.18) .

The above form suggests two tests of scaling violations. Since 2 2 log ^(Q ) - - djj log Q + const, we can compare the logarithms of two moments Mj, and My and should find, if QCD is correct, a straight line with slope given by HARD SCATTERING PROCESSES 431

N 1 j=2 5" (4.20) [~ 2+ M(M+1) " 2 ^2 3" J

We show in Fig.9 the data from the BEBC*13' and CDHS(ll>) collaborations which are in good agreement with the QCD prediction. This test is sensitive to the spin of the gluons, a scalar gluon would give results which are in disagreement with the data, as indicated in Fig. 9.

A second possible test of QCD, based on (4.19), is to compare logarithmic versus power lav scaling violation. Consider the quantity [Mj^0- )~}~ N' leading order QCD predicts that it should vary linearly with log Q2. The intercept of this straight line gives the fundamental parameter of QCD A2 to leading order (see later for higher order corrections). Fig. 10 shows the data^13'11** which is again consistent with the QCD prediction and not consistent with a power law violation of scaling.

We have only considered the non-singlet structure functions, there are also possible and useful tests of QCD involving the singlet structure functions 16, but unfortunately we do not have time to go into these. We have also skipped several technical problems like the question of higher twist effects, Nachtmann versus Cornwall-Norton moments etc. We refer the reader to the literature, for detailed analysis of these problems(17).

ii. QCD Off the Light-Cone: A Recipe for Parton Model Calculations

We have just seen how to describe the scaling violations in DIS using a simple parton model language (the Altarelli-Parisi approach). However, for DIS this is not new since by using the OPE and RGE formalism we already knew how to sum the leading logarithms. The question is: can we do the same for processes which do not have an OPE, in particular processes with two initial hadrons, like PP •+ v~V" + x, PP •+ ir+ x at large - P_? The answer is yes 5 and the essence of the proof is to 2 2 show that the mass singularities (log Q /p ) factorize and can be absorbed into the definition of scaling violating distribution and fragmentation functions of incoming and outgoing hadrons, respectively. These singularities are universal in the sense that they are associated to the incoming (or outgoing fragmenting) quarks and gluons and thus do not depend on the particular hard process under consideration. O.I 250 1 ' 1 11 / 1

/ 200- / / z 0.01 / / / UJ / / / 5 150- / \/ N =4 i is o 0.01 h dN=0.775 A 4/100- r , 2 X/ _ Scolõr A _V. Gluons 1 50 - Si 0.001 i i i i I .. i' i i i I 0.01 0.1 0.1 - N =3 1.0 dN=0.6l7 LOG OF MOMENT $)> T , 1 i l 10 100 2 (u l) 2 2 Pig. 9 Logarithmic plots of moments of F3 (x,Q ) '" l Q (GeV ) The solid lines are the QCE predictions {eq. 4.20), the dashed lines are the results from a scalar gluon Fig. 10 Plots of Clog theory. Data fran BEBC (Ref. 13). Dashed lines are power-law fits. o

W to O 8- tu HARD SCATTERING PROCESSES ' 433 Most of the calculations ' are done in a special gauge, the so called transverse gauge (n-A = 0, n2 < 0) in which the sum over the polarxza- tions of the gluons is given by

k.Ti (n.k)!

We see that this gauge is a physical gauge, since it does only involve , the two transverse polarizations, i.e. for k2 = 0, -g Pvv = 2 and | k Pvv =0. In this gauge the diagrams giving leading mass singularities ; are of the ladder type (Fig.11) where the dots indicate vertex and r self-energy insertions. These diagrams are amenable to a parton-like description, while the diagrams involving the crossed ladders (Fig.12), which would invalidate the parton model interpretation, do not contri- : 'v. bute to leading order.

It is not difficult to see why this happens, if we remember that in a physical gauge, the vertex for emission of a gluon by a massless quark •.'- vanishes as B , the angle between the quark and the gluon, goes to zero. The propagator l/(p-k)2 , in Fig.13 behaves as 1/82 and the phase space ,| is proportional to 6 de. Thus in a transverse gauge, a diagram like ' Fig. 13 behaves as

(r f e ae log^i (4.22) •:J • J. r. P2 e2 e2 p2

- _ while a diagram like Fig.14 behaves as il • f H yi' 6 de — = finite (4.23) ]•: J e2 This simple argument has been generalized to all leading logarithmic orders '.

Another important remark is that the effect of vertex and self-energy ; insertions is to enable us to put at each vertex the effective coupling '•• •• 2 2 constant g(k^) , where k^ is the largest four momentum squared at the t vertex, [it should be stressed that the dominant region of integration .-V ' for the ladder diagram comes from the ordering of four momentum squared, ) ' p2 « k2 « k2 ... << k2 << Q2 ] .

', l" ; ; The up-shot of these studies is the following recipe for calculating --:% hard scattering cross sections for which there is one large invariant 434 C. Escobar

Fig. 12 A crossed-ladder diagram which does not contribute to the leading logarithm approximation. Fig. 11 The dominant diagram, whithin the leading logarithm approximation, for a hard process (represented by the cross) involving a momentum transfer Q . The dots indicate self- energy and vertex insertions.

Fig. 13 A diagram contributing to the Fig. 14 A diagram, «iiich in a leading logarithm approximation. transverse gauge, does not contribute to the leading logarithm approximation. HARD SCATTERING PROCESSES 435 Q2 » p2. A2. ). a. Use the parton model formula R*i b. Replace the scaling distribution and fragmentation functions by non- -scaling ones (at mass scale Q ) c. Convolute these with the lowest order cross-section for the parton

subprocess, calculated using the effective coupling constant as(Q ) .

For example, the Drell-Yan formula is modified as follows, da 4ira2 „ I ãQ' 9Q4 i (4.24)

Notice that to leading order in QCD, a simple feature of the parton model is still valid, namely, factorization in x^ and x2 (the factorization in x and z for semiinclusive electroproduction. Fig.4; is also valid after allowing for leading order QCD corrections).

In the next section we will study the phenomenology of the Drell-Yan process.

V. Drell-Yan Phenomenology

Looking at (4.24) which is a very simple expression, we can imediately devise some tests of the Drell-Yan model.

The first test is the approximate scaling of the Drell-Yan cross-section, that is Q4 ^ = F(t) (5.1) dCT where we have neglected small scaling violations in the quark distribution. We show in Fig.15, data from the CFS collaboration at FNAL , por Pj^ = 200, 300 and 400 GeV/c. The data supports the approximate scaling behaviour shown in (5.1). The best test for scaling would undoubtedly be to compare the FNAL data at /s = 20-27 GeV with the ISR data at /s = 60 GeV, the trouble being that there is a small overlap in T between FNAL and ISR.

A second test has to do with the beam-type dependence of (4.24). For beam particles containing antiquarks as valence constituents (IT'S, k's, M/Vs 1 1 02 0.3 0.4 05 0,6 o a 200 Ge V/c I 1 1— ü? 1 u - A 300 » . .1 * GeV/c - 3 400 GeV/c C> 11o- ' • 11 < > io2r O g LJJ J CM

E 10-32 _ TI ! 1 -o 43 1 -

- - D < 33 % IO" W- O -

CM •o 1 1 I • it 9 L4 0 1 b

à 1 11 ** 8 * CM t "O 0.6 1 1 1 . 1 > 1 .1 I 0.5 • 0.2 0.3 0.4 6 8.10 12 2 Fig. 15 Approximate scaling of the Drell-Yan cross- M (GeV/c ) section (eq. 5.1). Data from Ref. 19 at three n energies, lhe figure at the botton displays the ratio Fig. 16 The ratio of it induced to proton induced y y. pair of the cross-sections. cross-sections, as a function of the mass of the pair. Data fran Ref. 19. o & sv H

• -r- HARD SCATTERING PROCESSES 437 p's), the cross-sections for lepton pair production should be much larger than those involving sea-valence annihilation (nucleon-nucleon collisions). This intensity rule should be even more marked at large values of T , where we probe the quark distributions at large x. We show in Fig. 16 the ratio ir~P -»-p+i/" + x/PP ->-y%~ + x , which exhibits the qualitative behaviour expected in the Drell-Yan model. We should mention that the study of massive lepton-pair production enables us to determine the (20 21} quark distribution in hadrons other than the nucleons ' A third test of the Drell-Yan picture is to check the spin -1/2 nature of the quarks, by looking at the angular distribution of the leptons in a suitable frame(22) .

In the centre of mass of the lepton pair, assuming that they are produced via one-photon exchange, the angular distribution is given by,

2 2 — = WT(Q) (l+cos e) + WL(Q) sen e (5.2) -j dQ d cose

where Wm and WT are transverse and longitudinal structure functions, .1 T Jj | respectively. ' In order' to define the angle e, we must choose an axis in the lepton ']• pair center of mass frame. One familiar definition is called the Gottfried-Jacks on (G-J) fratie , the z axis being chosen along the ': beam direction. This frame would be a very convenient one, had the quarks no transverse momentum in which case the beam direction would -:.{ coincide with the quark direction of flight. We know however that ;•"• quarks do have a certain transverse momentum (more on this later) so another frame was introduced by Collins and Soper {-O-S) , in order to minimize the effect of the quark's transverse momentum. For our purposes there is little difference between the G-J and the C-S frames. If quarks have spin- 1/2, the longitudinal structure function W_ is zero, so we are left with A- O 45.3)

•'; The data is usually fitted with the form 1 + X cos 8 and for ir-nucleon

j ; collisions at Plab - 200 GeV/c and 3.5 GeV < Q < 9 GeV, the result of ';• the CIP collaboration(20) is X = 1.30 ± 0.23, using the C-S frame, while ' the NA3 collaboration*21) at the SPS finds X =0.80 ± 0.16 using the G-J frame and X = 0.85 ± 0.17 using the C-S frame. We see that these results support spin-1/2 quarks.

\l 1 438 C. Escobar The deviations from X=l, as well as the dependence of X on the rapidity of the lepton pair are important, since they could signal the presence of higher order effects, which we will discuss later on.

We conclude this very brief discussion of the Drell-Yan phenomenology, to which we shall return when discussing higher-order corrections in QCD, by saying that the data supports the naive parton model picture. The main task is to look for the QCD corrections in the experimental data.

VI. Beyond the Leading Order

i. Deep Inelastic Scattering

We have seen that in lowest order QCD predicts for the non-singlet moments, dxxN 1 dN " J " (6.1)

with -±_-2 I i N(N+1) j=2 3

But B and A^ are just the first terms in the expansion of the B function and of the anomalous dimension in powers of g,

(6.2) 2 3 5 •B(g ) -Bog - ,g +

Going to the higher orders in QCD imply calculating the next terms in (6.1). A typical diagram contributing to B, is shown in Fig. 17. The calculation of the anomalous dimensions to order g also involve two-loop diagrams (2"),

Caswell and Jones.(25K25>) first calculated ^ and found, Fig. 17 Typical two-loop diagram contributing to the. Bx = 102. - — f . B-function to order g5 (eq. 6.2). Due to the lack of space, we will illustrate the importance of higher order corrections by means of one simple example, the ambiguity in ;the HARD SCATTERING PROCESSES • 439 determination of A in lowest order QCD. (26)

When higher orders are considered, (6.1) gets modified.

(6.3)

where the C.T involve A»T and BT and were calculated by Flor atos et ai. and Bardeen et ai. A leading order calculation consists in neglecting the O(—=—j- ) term relative to the 0(logQ ) terra. But this is dangerous log ÇT , since the definition of A depends on the neglected terms 0( j). This is most easily seen by rescaling A, Og

= a A (6.4)

Then, using the leading order expression (6.1)

2 2 A"

(1 + .. ) (6.5)

But since we are neglecting the terms of order Q2//,2 we find that MJJ(Q ) = Mi(Q ) ! So we come to the conclusion that to leading order, different A's lead to the same Q dependence for the moments, so it is impossible to determine A unambigously to leading order.

To eliminate this ambiguity, we have to include higher order terms. r - ~\ Following Bardeen et al , we define a leading order parameter AT _,(N) , where -C» AL.0.(N) " A (6.6)

with this definitions the moments ') in (6.5) can be written as,

o2 "dN C.. N (1 + + .... ) - dog a.) b

(log (1 + 0(- ) L.O Q2 -dv = (log (6.7) L.O. 440 C. Escobar which is the same as the lowest order expression but now AL Q depends on N in a calculable way (6.6). This conclusion is very important for the phenomenology of QCD, since fits using the same A for different N are in principle wrong. The correct procedure is to either consider the higher-order corrections directly in the expression of Mg(Q ) (6.3),

or introduce these corrections into the definition of AL Q (N).

Fig. 18 shows the dependence of A on N, as given by equation (6.6), compared with an analysis of combined muon and electron deep inelastic data (see reference (27) for more details). Strictly speaking only

ratios A. n (N)/AT n (M) are significant, due to the fact that the coeffi cients Cj, depend on the subtraction scheme employed when regularizing

the theory. CN is of the form,

where y is an arbitrary number, independent of N but dependent on the (ZB) * * subtraction scheme . Another correction which is of some interest is introduced in the /29\ log MJJ vs log M^ plots. When higher-order terms are considered we obtain

2 2 dN F as 1 log MjjiQ ) = const + log W^(Q ) * •£• 1 + C^ •£ + (6.8)

where

and dM N are two-loop anomalous dimensions. The following table shows values of C^ for low values of M and N

H N CMN 2 4 0.42 4 6 0.21 6 8 0.15

These numbers show that we do indeed have a good perturbative expansion for the quantity log K^ vs log M^, with the corrections to the slope being of the order of 10% at moderate values of Q2.

The conclusion on higher-order corrections in DIS is that they are small, generally within the errors of present data (for more details see a HARD SCATTERING PROCESSES 441

p-n AN • F

1.0

0.5

Fig. 18 Experimental results for A.(N: )' compared.with the definition (6.6). From an analysis of Duke and Roberts (Ref. 27). 442 c- Escobar recent paper by Para and Sachrajda '). As we will see next, this conclusion is, unfortunately, not true for other processes, like massive lepton pair production. ii. Higher Order Corrections to the Drell-Yan Process.

In Section V we presented a brief review of the phenomenology of the Drell-Yan process, but neglected the' QCD corrections. In this section we consider these corrections and compare them with the available data.

The first obvious QCD effect is a small scaling violation to the simple 2 2—2 scaling behaviour (5.1), through the Q dependence ofqix^Q )q(x2rQ ). The available data is not precise enough to see this small violation. A second manifestation of QCD effects, is through the transverse momentum distribution of the lepton pair . In the simple parton model the leptons have a small and limited transverse momentum, = 300 MeV/c, due to the Fermi motion of the quarks inside the hadrons (this is sometimes called the intrinsic transverse momentum of the quarks). But in QCD there is another source of large transverse momentum, coming from the diagrams of Fig.19, for the processes, q'+g •+ Y*+ 5 and _ 2 q+q -»• g+Y*, which are down by O(CL (Q ) relative to the qq annihilation. They should however be important when we look at lepton pairs with large 2 2 transverse momentum, P* ~ O(Q ) • QCD makes an unambigous prediction for the behaviour of the average p+y- transverse momentum of the pair. At fixed T = M + , , < P,

a) b)

Fig. 19 Diagrams giving rise to large-PT muons pairs. a) Compton diagram: g+q •+• y*+q b) Annihilation diagram: qq -+ Y*g. HARD SCATTERING PROCESSES 443 increases with the energy, being given by the following expres- (32) 1 sion,v '

v 2 2 = as(Q ) /I" f (T,as(Q )) + non-perturbative term (6.10)

It is clean from phase space considerations, that at T-0 and 1, f(t) must vanish." In order to compare with the data, we should keep x fixed and vary the energy, unfortunately as we commented before, there is a very small overlap in T when going from FNAL (/i~ = 19-27 GeV) to ISR (/s~ = 60 GeV) energies, so we make use of the flatness of with T, which is confirmed experimentally and average the data over x, obtaining the plot of Fig. 20 showing versus /s~ . It is difficult to make a quantitative comparison with the QCD predictions for the slope due,

r CFS CHFMNP 200I 30I°t 400

=0.6 + 0.022 VS GeV/c 0 I 10 20 30 40 50 60 Vi (GeV) Fig. 20 Average transverse momentum versus /s" .

among other things, to the unknown gluon distribution. Also there is 2 2

an ambiguity in the value of a (Q ), in particular, do we use as(Q )

or ag(s)? The last question can only be answered when higher order corrections are included (see section VI i) .- Anyway, there is defi- < > nitely an. increase of PT with energy which is evidence for QCD effects in the Drell^Yan process. We will leave the discussion of the

small-PT region for the next section. Of course the perturbative cal-•

culation based on Fig; 19, cannot be extended to Pm=0 since the cross 2 section diverges there as 1/P_. We now consider a third and more disturbing effect of QCD in the Drell- -Yan process. The result given in (4.24) was obtained in the leading 444 c- Escobar log approximation. When the first order corrections to the leading loq calculation are considered, it is found that they are very large (3 3) and modify it as follows' ,

~ , Air dx,dx_ P - H. , H9 - 1 2 2 2 2 d£_= 4*«i _±_l j e2 q (x,,Q )qi (x2,Q ) ± dQ2 9s J xL x2 [U

2 + l+-*2 6(l-z) + as(Q )8(l-z)fq(z)J [ H H "I 1 2 2 2 j^ F2 (x1,Q )G (X2,Q ) + 1 ++ 2 *

2 x as(Q )e(l-z)fG(z)> (6.11)

where

— , F*(x,Q2) = x L e2[q?(x,Q2) + 2 x,xXl*2n i L and

4 S /i . 4TT2» . /i • 2 osfq(«) - % £ Id + ^)«(l-«) + 2(l+z )

+ —5 6 - 4z j (6.12) (1-zK

2 2 2 a f (z) = I Ü1 (z + (l-z) )log(l-z) + - z - 5z + 2. | (6.13) sG 2 2, 2 2

We show these corrections in Fig.21. The first term in f (z), 4 a 4ir2 " j2* (1+ -j—)6(l-z) comes from the vertex and soft gluon corrections shown in Fig.20.a. ' The other terms in f-iz) are due to hard brentnstrahlung

of gluons (Fig.21.b), while the term fG(z) corresponds to Fig.21.c. The gluori corrections are not very large, however the quark corrections f (z) are really large, the most important term being that proportional to 5(l-z) which renormalizes the usual Drell-Yan cross section (the term proportional to u2 comes from the continuation from space-like to HARD SCATTERING PROCESSES 445

* «1

Fig. 21 Corrections of order ag to the Drell-Yan process: a) and b): virtual and soft gluon corrections c) hard gluon bremsstrahlung d) gluon contribution.

time-like Q of (log S_) terms, a continuation that is necessary when p2 comparing DIS and the Drell-Yan process) . We show in Fig.22 the effect of f_(z) on the Drell-Yan cross-section. As we see, their effect, in the present range of x is to simulate a large change in the normalization of the "zero order" cross section. It is possible that this effect is being seen in experiments with pion beams at FNAL(20) and at the CERN-SPS(2l). It is however premature to reach any firm conclusions on this. If the first order corrections are already so large, can we believe that the next corrections are indeed negligible? Of course more study has to be devoted to this subject and a proper phenomenology of higher-order corrections has to be developed .

VII. Future Prospects

i. Summing Double Logs.

a. Small-Pt distribution of lepton- pairs

Up to now we have considered the production of massive lepton pairs with either P2 * Q2 (section VI) or P2 - 0(p2) << Q2 (section V). In both cases there is only one large invariant and therefore dangerous 446 C. Escobar

0.01 0.05

Fig. 22 Effect of O(a ) corrections to the ir~N cross section. From Altarelli et al (Ref. 33).

2 2 logarithms of the form log PÍ/Q could be neglected in the first case 2 o2

(PT - Q ), while in the second case the log *£_. terms were absorbed into the quark distributions giving rise to the QCD modification of the simple Drell-Yan expression (sec.eq. (4.24)). In this section we consider a new development in the perturbative approach to QCD, which consists in examining situations where there are more than one large invariant'. Consider, for instance, the production 2 2 of lepton pairs with transverse momentum P* , much smaller than Q but larger than P (or A2),that is, HARD SCATTERING PROCESSES 447 Q2 » P2 » P2 (7.1)

In this case we have to keep track, to all orders, of terms like log Pr/Q , with the additional difficulty that now it is not obvious which mass scale will govern the violation of scaling of the quark 2 2 2 2 distribution, do they come from log Q /P or from log PT/P ? This problem was investigated by Dokshitser, D'Yakonov and Troyan (DDT) , who arrived at the following expression for the cross section for pro- 2 2 ducing a massive lepton pair of mass Q , with transverse momentum PT at rapidity y, such that Q2 >> P2 >> P2 ~ A2,

2 H H O2 do _ 4™ a (_2 l( 2. - 2, 2. Q - y *• e. q.= \x. ,FTJ q. \x~,f-) 2 2 2 dQ dP dy 9sP 3 log PT i L + 1 «-+ 2| t T2(Ê- ) (7.2) a2

where,

2 T(X) = f dzz expp - -\ 3I-- asalog X (7.3)

We notice the following novel features in (7.2). First of all, it is

not Q but PT which sets the mass scale for the scaling violations in the q, q distributions, this results from summing the terms , eT ,n ( o nlog -j- ) s Pz Secondly, there is a Sudakov form factor T2 (Q2 /P_) 2 , which measures the probability of producing a virtual photon of mass Q2 , without emission of gluons having transverse momentum greater than P . The Sudakov form factor comes from the exponentiation of double logarithms of the form

Parisi and Petronzio ' have made a careful study of the DDT calculation

and concluded that as Q increases the PT distribution of the lepton pair is less and less sensitive to the intrinsic transverse momentum of the quarks. They claim it is possible to calculate the P distri- 44g C. Escobar bution.perturbatively, down to PT=0 • The physical reason behind this is shown in Fig.23, As Q increases it also increases the probability of multiple bremmstrahlung, which washs out any memory of the initial transverse momentum quarks may have. Then, asking for a pair with 2 1 PT >> A , is something that can be calculated in perturbation theory .

We show in Fig. 24, the result of a calculation by Parisi and Petrcnzio17 , 2 of the Pm distribution down to P™=0 (the parameter M is chosen such o p 2 as to avoid the singularity in s( T)) •

( 8) ii. Heavy flavour multiplicities at very high energies

The authors of Ref.8 have shown that in an transverse gauge (n.A=0), the main contribution to the heavy flavour multiplicity in e e~ annihilation comes from the ladder diagrams shown in Fig.23, where the final 2 2 gluon is off shell by QQ > 4m_. The summation of double logs log2 Q2)n yields for the multiplicity, 2 /2~ 2 1/4 Nn(Q ) - (log Ry)~ exp /const log %• (7.4) Q Q2 / Q2 which rises faster vJian any power of log Q but slower than any power 2 of Q . The above mentioned authors stressed that this result is very sensitive to the three gluon vertex and in Fig.25, taken from their paper, it is shown the behaviour of the multiplicity N (Q ) (eq.7.4) compared with the exact lowest order calculation, which does not involve the three gluon vertex. So, it is possible that at LEP energies this marked rise in the heavy flavour multiplicity becomes apparent, thus signaling the presence of the three gluon vertex, an unmistakable characteristic of ''

Among the more recent developments of perturbative QCD, we should also mention the interesting attempts that have beem made to extend it to exclusive processes, which for long have been thought to be outside the region of applicability of perturbative methods. We do not have the time to go into this important new branch and instead refer the reader to the literature(3S'. HARD SCATTERING PROCESSES 449

Big. 23 Multiple gluon bremsstrahlung in Drell-Yan process.

Fig. 24 Transverse manentum distribution of the lepton pair, theoretical predictions fran Parisi and Petronzio (Ref. 7) cairoared with the data of Yoh et al. (Ref. 18). (S = 750 Gev\ Q2 = 56 GeV2).

Fig. 25 The Qz dependence of the charm multiplicity in e e~ annihilation (from Ref. .8). The dashed line is the exact lowest order calculation. 450 C. Escobar

VIII. Conclusions

In closing these lectures we would like to call your attention to several points awaiting further clarification and for which we can expect that a great deal of work will be done in the coining years.

1. It would be nice to perform (or improve) higher order calculations for several processes (large-P_, Drell-Yan, etc) and see if they are indeed so large (see Section VI). Also a more careful phenomenology of these higher-order corrections has to be developed.

2. Can we finally forget about the intrinsic transverse momentum of partons, which has unfortunately vitiated much of the phenomenology

of QCD (in Drell-Yan, high-PT etc)? See Section VII i.

3. Is it possible to isolate the three gluon vertex, maybe at future machines (LEP, hadron colliders-multiplicity of heavy jclusters?) ? See Section VII ii.

4. We have not touched several other topics of great interest like Y-Y collisions 36 (a very clean place where to study QCD), three jet production in e e~ and lepton scattering 37 , etc.

Acknowledgements

I would like to thank Ronald Shellard for his careful reading of the manuscript. This work was supported by PINEP, Rio de Janeiro, under contract 522/CT. HARD SCATTERING PROCESSES 451 References 1. In preparing these lectures I have made extensive use of the following excellent lectures and reviews: C.T.Sachrajda: Parton Model Ideas and QCD, lecture given at the XIII Rencontre de Moriond (1978). Applications of QCD to Hard-Scattering Processes, lectures given at the CERN ACADEMIC TRAINING COURSE (1979). A.J.Buras: Asymptotic Freedom in DIS in th« Leading Order and Beyond, Fermilab preprint PUB-79/17-THY (1979), to appear in Rev.Mod.Phys. J.Ellis: Status of Perturbative QCD, invited talk at the 1979. Inter- national Symposium on Lepton and Photon Interactions, FNAL, Batavia, Illinois (1979) - CERN-TH-2744. Yu.L.Dokshitser, D.I. D'yakonov and S.I.Troyan: Inelastic Processes in QCD, SLAC Translation 183 (1978) . 2. R.P.Feynman, Photon-hadron interactions (Benjamin, New York, 1972). P.V.Landshoff and J.C.Polkinghorne, Phys. Rep. 5c, 1(1972). ' 3. For reviews, see, H.D.Politzer, Phys. Rep. 14c, 129(1974). W.Marciano and H.Pagels, PKys.Rep. 36c, 137(1978). 4. G.Altarelli and G.Parisi, Nucl.Phys. B126, 298(1977) . b. D.Araati, R.Petronzio and G.Veneziano, Nucl.Phys. B140, 54(1978) and B146, 29(1978). R.K.Ellis, H.Georgi, M.Machacek, H.D.Politzer and G.C.Ross, Nucl. Phys. B152, 285(1979). . A Yu.L.Dokshitzer, D.I.D'Yakonov and S.I.Troyan, Proc.of the 13 / Winter School of the Leningrad Institute of Nuclear Physics (available as SLAC translation N9 183 (1978)). : C.H.Llwelyn Smith, Acta Phys. Austriaca Suppl. XIX, 331 (1978) . H- 6. For a review of QCD beyond the leading order, see A.J.Buras (Ref.1). : 7. Yu.L.Dokshitzer et al (Ref.5) and Phys.Lett B79, 269 (1978) G.Parisi and R.Petronzio, Nucl.Phys. B154, 4T7~(1979) . 8. W.Furmanski, R.Petronzio and S.Pokorski, Nucl.Phys. B155, 253 (1979). 9. S.D.Drell and T.M,Yan, Phys.Rev.Lett. 25, 316(1970). •• t 10. For recent reviews of the OPE plus RGE approach to DIS, see A.Peterman, Phys. Rep. 53C, 157(1979). A.J.Buras (Ref.l). 11. T.Kinoshita, J.Math.Phys. 3, 650(1962). T.D.Lee and M.Nauenberg, PKys.Rev. 133, 1549(1964). 12. C.F.von Weizâcker, Z.Phys. 88, 612(1934). E.J.Williams, Phys. Rev. 45_~729 (1934) . 13. P.C.Bosetti et al., Nucl.Phys. B142, 1(1978). 14. J.G.H. de Groot et al., Z.Phys. Cl, 143(1979); Phys .Lett 82B, 292, 456 (1979). — 15. J.Ellis, Current trends in the theory of fields (eds.J.E.Lannatti and | P.K.Williams) (A.I.P., N.Y., 1978), p.81. I 16. E.Reya, Phys. Rev. Lett., £3, 8(1979). .;] 17. See, for instance, the following review I J.Ellis, Status of perturbative QCD, invited talk at the 1979 Interna- " tional Symposium on Lepton and Photon Interactions, FNAL, Batavia, Illinois (1979) - CERN- TH 2744. 452 Escobar IS. D.C.Hom et al.,Phys.Rev.Lett. 37, 1374(1976); J.K.Yoh et al., Phys. Rev.LettTTTT 684(1978) and 4J7 1083(1978) . 19. K.J.Anderson et al., Phys.Rev.Lett. ^2, 944(1979). 20. CIP Collaboration, Ref.19. 21. NA3 Collaboration, J.Badier et al., CERN preprint, EP 79-67,EP.79-68. 22. For a review see R.Stroynowski, SLAC-PUB-2423 (1979). 23. J.C.Collins and ü.E.Soper, Phys.Rev. D16, 2219 (1977). 24. E.G.Floratos, D.A.Ross and C.T.Sachrajda, Nucl.Phys. B129, 66(1977), 139, 545 (1978) and B152, 493(1979). W.A.Bardeen, A.J.Buras, D.W.Duke and T.Muta, Phys.Rev.D18, 3998 (1978). For reviews see Ref.10. 25. W.E.Caswell, Phys .Rev .Lett. 33_, 244(1974). D.R.T. Jones, Nucl.Phys. B75, 531(1974). : 26. M.Bacê, Phys.Lett. 78B, 132(1978). 27. W.A.Bardeen et al., (Ref.24). D.W.Duke and R.G.Roberts, Phys.Lett. 85B, 289(1979). 28. See A.J.Buras, ref.1 for more details on this point. 29. M.R.Pennington and G.G.Ross, Oxford Univ,preprint 23/79 (1979). '-• 30. A.Para and C.T.Sachrajda, preprint TH.2702-CERN (1979) . 31. G.Altarelli, G.Paris! and R.Petronzio} Phys.Lett. 76B, 351 and 356 (1978). si K.Kajantie and R.Raitio, Nucl.Phys. B139, 72 (1978). i F.Halzen and D.Scott, Phys .Rev.Lett.~W7 1117(1978). H.Fritzsch and P.Minkowski, Phys.Lett." 7_3B, 80(1978). (~ 32. See G.Altarelli, G.Parisi and R.Petronzio (Ref.31). I 33. J.Kubar-André and F.E.Paige, Phys.Rev. D19, 221(1978). G.Altarelli, R.K. Ellis and G.MartinellTT~Nucl.Phys. B143, 521(1978). Erratum B146, 544(1978) and Nucl.Phys. B157, 46K1979T7" J.Abad and B.Humbert, Phys.Lett. 78B, 627(1978); 80B, 433(1979). K.Harada, T.Kaneko and S.Sakai, NücT.Phys. B155, T5¥(1979) and ; Erratum to be published. B.Humpert and W.L.van Neerven, Phys.Lett. 84B, 327(1979) and 85B, 293U979); CERN preprint TH 2738(1979) and"1rratum. 34. For other proposals for measuring the three gluon vertex see: E.Reya, Phys.Rev.Lett. 43, 8(1979). I-. De Rüjula, B.Lautrup and R.Petronzio, Nucl.Phys. B146, 50(1979). 35. S.j.Brodsky and G.P.Lepage, SLAC preprint SLAC-PUB-2294 (1979); Phys.Lett. 87B, 359(1979); Phys.Rev.Lett. 43, 545(1979). G.Farrar an

TOPOLOCICAL ASPECTS W FIELD THEORY AND CONFINEMENT

Bert Sduoct Instituto de Física e Químki de Slo Carios, Unnenidade de Sio Paulo

1. Introductory Remarks

The hypothesis that the model of QCD is the correct description for strong interaction has been steadily gaining strength ever since nonabelian gauge theories became theoretically respectable as viable renormalizable field theories. The popularity which QCD enjoys is primarily related to its conceptual clarity, its simplicity in terms of a Lagrangian description and the conspicuous absence of alternative models which qualitatively explain as many experimental facts. By some divine justice the Lagrangian of QCD is easily written down, but the exploration of its physical consequence turned out to be one of the most difficult problems in theoretical physics up to now. The main reason is the complicated relation between particles and fields. The fact that this relation is more involved than in the "old" QFT is of course related to the confinement problem.

Let us remind ourselves briefly of the historical development of thoughts on the particle-field relation. In "ancient" days particles used to be identified with field quanta, i.e. Fourier-components of fields at a particular time. This led to confusion in the computation of radiative correction to lowest order perturbative processes. Clarity was restored by incorporating the S-matrix concept of Wheeler and Heisenberg into QFT, an achievement which goes under the name of LSZ (Lehmann Symanzik Zlmmermann) theory. The essential idea was that the remormalized Lagrangiean field describing dressed field quanta have a simple, asymptotic behaviour for large times.

LSZ A(x) • A±n (x) , ,i out

:(. The assumptions under which this asymptotic property was derived , were: 454 B. Schroer (a) spec V E forward light cone, with a finite gap between vacuum and one particle states as well as a gap to the continuum

<0| A(x) )p> * 0

(b) locality, i.e. spacelike (anti)commutation relation for the basic field.

Formally also bound states |b,q> with say <0| A(x) |b,q>=0 but <0|polyn.(A) |b,q> + 0 were incorporated into the LSZ frame, although no systematic approximation method for processes involving bound states had been proposed. It was believed that such a framework leads to complete description of particles (asymptotic completeness) and serves to describe strong interaction in which no physical zero mass particles play a role.

Bya slight modification one could also incorporate particles with permanent infrared-clouds so called infraparticles, i.e. the electron of QED. However, according to our best knowledge the LSZ framework in the above form is not valid for QCD. The reason is two-fold:

(a) there may exist particles (or infraparticles) whose interpolating fields are not constructable in the polynomial algebra of Lagrangian fields. (b) the quark resp. gluon fields may not have any accompanying particles: quarks may be confined and color may be screened.

The first phenomenon is not typical for gauges theories.lt has been first observed in the Sine-Gordon model. There are particles, called topological solitlons and kinks whose interpolating fields (though mostly local with respect to themselves) are non local with respect to the Lagrangian fields. The usually carry a (hidden) symmetry group of topological origin which may be continuous (U(l) in the case of the topo- 11 4 logical Sine-Gordon soliton ) or discrete Z- for the A_ kinds or the 2) t& Ising,-kinks ). Historically these particles have been treated first quasiclassically, but nowadays it is an accepted fact that the duality3) of statistical mechanics (Kramer-Wannier duality) is. the most adequate framework for soli tons and kinks. According to Mandelstam 4) and 't Hoof*5t ) there are dual fields (order-disorder fields) in QCD. It is not known whether they lead to new particles, i.e. particles which cannot be in- terpolated by Lagrangian fields and their gauge invariant composites. TOPOLOGY AND CONFINEMENT 455 The problems of confinement and screening has presently not ,/. been understood in the QCD4 model. Attempts have been made to obtain p insight into static quark confinement of corresponding lattice gauge theories . The use of the topologically significant order-disorder op- "' erators (similar to the lattice Ising model discussion) and topological boundary conditions ' was important for this recent progress. Another line of research is concerned with the physical properties of string models 9)'. Here differential geometric methods have been quite powerful tools . In those investigations the relation to the QCD. Lagrangian remains somewhat unclear. The most explicit picture has come from the n study of two-dimensional models (QED2, QCD2 and the CP models). In two dimensions the color neutrality of physical states is (kinematically) guaranteed. One can use such models for the study of the "quark screen ing versus the quark confinement problem". Furthermore the chiral U(l) symmetry breaking which is topological in origin and leads to the massive n - plasmon has been completely understood in such models•

In a condensed form the present frontiers of and problems of '• /, theoretical (non-lattice) QCD research may be described in the following ] way: r 1) Attempts to understand functional integrals. What is the (topological) structure of the field manifold over which the functional integral ; is extended? How does one obtain an insight into the differential geometry including the integration theory on such manifolds? 2) Systematic expansion methods: 1/N expansion. 3) Dual structures: functional integrals for order-disorder fields. 4) Study of particles and fields in two dimensional "solvable" models as a source of dynamical intuition.

In most of these problems topological ideas and methods play an important role. Here we will only review attempts concerning the ~~ understanding of continuous QCD or related continuous field theory models.

2. Mathematical Methods

In a situation in which the basic Lagrangian is simple and physical information is difficult to extract, the progress depends very much on the improvement of the mathematical machinery. The starting point of most mathematical investigations is the functional integral 456 B- Schroer representation of Euclidean correlation function, i.e. in QCD

e"SX

X = product of ijifij) and A 's

= 1 F2 + i F D 4 J pv J YVYV. UU* V V V

4 For perturbation theory one has to follow Faddejev and Popov's '{•• procedure of adding in addition to a gauge breaking term a Faddejev-Popov ghost contribution: S+S + S-,-, + S_ _ =S__ , u.r. r.Jr. r.Jr.

From S_ p one may obtain Feynman rules to which the process of renormalization may be applied: a redefinition of the coupling constant can be compensated by changes in the wave-function renormalization Z•:• . (and possibly the gauge fixing parameters) . The corresponding parametric differential equations contain a function 3(g) " 14) whose negative sign leads to asymptotic freedom : perturbative theory .:. is converted into an asymptotic expansion for small distances.The tacit v assumption here is that 6(g) in a small region around g = 0 really has •, a global (non-parturbative) meaning. However, the coupling constant g j- plays no role in the physical parametrization in terms of say the lowest ( . partial mass, all physical quantities in particular the partial spectrum do not depend on g. In the so called mass transmutated physical theory

:;. the meaning of B(g) is less clear. Nevertheless, asymptotic freedom f has had great importance for the structural understanding as well as |. for phenomenological applications. All attempts to obtain a hint about : the structure of the mass transmutated correlation functions from the perturbative infrared aspects have failed up to now. We will in the following briefly discuss several non-perturbative attempts to understand functional integrals. (1) Structural investigation of the field manifold

. QCO4 is as any gauge theory a geometric theory. There are ;• other models as for example the class of two-dimensional Sigma-models | which share with gauge theories a rich topological structure of their I ; field manifold. Consider for example the CPn model. Classically it •1 15) -) can be described in terms of a complex n-component vector field .

(x) = j • j with [ifi]2 =i()+ $ = 1 TOPOLOGY AND CONFINEMENT 457

The field manifold consists of a equivalence classes Ú ] : J."

.(.'(x) = eiA(x)

The classical CPn~ Lagrangian

*- D

depends only.on the equivalence classes M . It is renormalizable and conformaily invariant. The R can therefore be one-point compactified 2 2 and we may think of R as a S (use stereographic projection) .The field c manifold «-• , Lll n_x 5 : Sz CPnX

is an infinite dimensional manifold (a natural topology in terms of Sobolov norms may be introduced), which decomposes because of

iM .o-*, - m.

into disjoint submanifolds.

r- p rk

The existence of this decomposition, as it is well know, leads :; to an ambiguitam y in the quantization which can be parametrized by a phase angle 6 . In each sector 5^ there exist local minima: the- instantons

which form a finite dimensional manifold fk •

According to a theorem of G.B. Segal these instanton-mani-

folds for large k asymptotically approach í» k in the sense of homotopy theory

On the other hand the right hand side is

2 1 TTq (S + CP"'?") = irg+2 (CP"" ) q > 0 458 B« Schroer independent of k.

Via a standard fibre bundle argument applied to the fibring: S211"1 + CP11"1 we obtain Va (cpn"1] " V2(s2n "1)

We therefore have

IT (£) = (0 , q+2 2n-l q K \ < , q+2 = 2n-l [unknown but in general nontrivial for q+2 > 2n-l

From these mathematical remarks we collect the important facts

(a) The field manifold has a complicated homotopy (and homology) structure, it is far removed from the structure of a linear, i.e. free field manifold which is homotopically flat.

(b) The instanton manifolds f^ homotopically approximate -Fk for large k. However (at it becomes clear from a more detailed examination): Morse theory of cristical points is not applicable; one meeds a generalization including "quasi-critical points".

(c) For n + M we have

i.e. 3r becomes asymptotically flat.

This is the topological prerequisite for finding in the leading order 1/N a free field theory. There have been similar investigations of the QCD4 manifold space for the QCD4 manifold for n •* °° .This limiting manifold is known, but complicated.

The simplest model of topologically rich field manifold ' is the rigid body. The action to be used in a Feynman path integral description of the rigid body is an action on a path space in SU(2). In this case (and its SU(N) generalization) the Morse theory of extrema give a complete description of the homology groups and (via "spectral sequences") on the homotopy structure of the field manifold. However, the Morse theory does much more: It gives the exact value ( via a quasiclassical approach including all saddle points) of the functional

integral even though the action is not gaussian. One can use the SU(N) TOPOLOGY AND CONFINEMENT 459 rigid body, i-iorse theory to understand the Bott-Atiyah "theorem conjecture": the Yang Mills action on two dimensional surfaces is a perfect Morse function. Using formal intuitive arguments of Migdal one may expect that two-dimensional (renormalizable) sigma models bear a similar relation to nonabeli^n 4-dimensional gauge theories as the SU(N) rigid body to the two dimensional (superrenormalizabie) SU(N) gauge theory. At this exciting speculative remark one should, however, not surpress a sobering note: a "good" Morse theorv does not yet exist for 19) renormalizable two dimensional field theories as sigma models

We now comment on the 1/N expansion technique.

(2) The 1/N technique

As a typical example we take again the CPn model

Here the only change to the classical Lagrangian is the normal ization of (j); | |2 =~ ; one has to allow for wave function renornal - ization. Now _ — r _ -A [d$] [d<(>] [dx]e = [d$] [d"J [dx]

with <£A = £+(j3-iA)2

A z = iDy

D A =3 -i A V V V

This is the trick of introducing a "dummy" field in the functional integral, a standard method to convert a quadrilinear interaction (between the "s) into trilinear Yukawa form. It is now easy to see that

-N Seff

Seff =to ln 460 B. Schroer The minimum of S .. is easily determined: it occurs at ±L C 2 2 N . Aj = 0 , X = m = A e~ * A = cutoff. In lowest order we see the famous mass transmutation happening in this model. We now write

X = XC + X

and obtain Feynman rules in terms of < A A > and propagators. To obtain the parameter in the correlation functions one has to compute with

8 Q e 3 A S = Seff -i 98 , ^ | uv v u

The A-propagator is in lowest order:

. C

f This is the so-called "secret long range force" related to the confinement of the 's. This method works for all models in which the interaction can be cast (by the trick of dummy functional variables) into the form

3 í or 3

(D = (pseudo) scalar, 3 = (pseudo) vector)

where the 's are isoscalar.

The simplest model with pion isoscalar interaction is

I i where $ J are fields transforming, i.e. under the adjoint representation |. ] of U(N) . In that case, by integrating over the and using Fierz i-'oj identities the interaction takes the form20'

i'-'l

7 , S/.y TOPOLOGY AND CONFINEMENT 461 g2 [[ 2 _ 2 - - "I lnf 4 JL ç y u J g* ff _ — 2 2

It consists of a Gross-Neveu part and a nonlocal part.Now both parts are isoscalar, though the second part is nonlocal. One plays the trick with the dummy fields. In addition to the local dummy fields it, a and A for the Gross Neveu and Thirring coupling one has to introduce a nonlocal fields V (x,y) . Again the Fermion integration brings into the form V Sinf = N Sinf V J

but now the variational equation for the minimum has in addition to the constant value for a a nontrivial value for

V (q) = q V(q)

This is given bj a nonlinear integral equation which precisely corresponds to the summation of all planar two point function graphs for . Lack of time prevents me form saying more about the results obtained by this 1/N method for Yukawa type nonisoscalar interactions.

The method cannot be applied directly to QCD2 because of the occurance of trilinear and quadrilinear selfcouplings of the gluon field. It can formally be applied to the Euclidean functional integral in the Z = x-iy = 0 gauge (the Euclidean "light cone" gauge) however a gauge transformation into such a gauge is a GL(N,C) father than a SU(N) transformation. This creates some complication21' .

(3) The dual variable method

We first illustrate this method in a simple example (Kadanoff- 221 -Ceval) ': the Ising model A = - J Z a. a. a^ {±1}

•a(xn)>=f ií, "(«,)••.• 462 B. Schroer

The dual "disorder-variable" y is introduced with the help of an antiferromagnetic coupling on strings connecting the u's

A 1 u(y2)> -i t e" r

y. = dual lattice points,

p = A/p : i.e. sum over those links which are crossed by r.

Despite the apparent r dependence in their definition they are actually r-independent. The y's are very util in the understanding of the phase transition. They are of crucial importance in the construction of the scaling limit of the theory. The results of the scaling limit , i.e. the limit T •+• T , a •* o, such that correlation length Ç = const. 2) c are: (1) The product of order x disorder (at adjacent points) are described by the components of a massive free Majorana field ty = I|JC .. The dynamical variables a and y are given in terms of kink operators: u (x) My) = eiir9(x-Y) My) o(x)

V (x) = short distance limit U(x) i|i(y) y •+ x

With,, a minimaluty assumption on short distance behavior U and v are unique. U is of the form 0 = exp. bilinear (a, a ) and they are both local Bose fields of scale dimension 1/8.

(2) The order (disorder) variables a and y are given by:

T " Tc "* °+ u(x) = U(x) a(x) = V(x)

T - Tc -+• 0_ y(x) = i(c+c ) V(x) a(x) = (c+c+) U(x) where {c, c } = 1 , {c* , i|j*} = 0 , c = c* or c+

c is a (x independent) spurion operator which describes the two fold vacuum degeneracy of the 0_phase. The mixed correlation functions are double valued in the Euclidean region: one needs to introduce a ramified covering in order to talk about the Euclidean field theory of dual fields, TOPOLOGY AND CONFINEMENT 463 i 24) i.e. one is outside the Wightroan framework

It has up to now not been possible to take that scaling limit directly in the functional integrals: this is a common difficulty of all Zy theories for which the field values of the lattice model are discrete. However, in the case of U(l) duality the functional integrals of dual field theory can be directly discussed in the continuum limit Followinswing Marino and SwiecSwie a let us briefly consider such an example: free massless field

-ib f EWV 3 * (z)dz a(x) = ei%altl , u(x) = e * v u

They lead (application of canonical commutation relations) to | the dual commutation relations Iy y(x) a(y) =a(y) u(x) e2iTi 9(v"x) jf' s= fr |

The correlation function of a and p are represented by A Euclidean functional integrals with a bilinear action. Such integrals Eft are saturated by the classical confirgurations. An interesting corre- K lation function is the mixed correlation: m !r + + 2 = j I [d<(i] exp-J [ 3*<|> + (a+B) if»] d x r

a(Z) = -ia [ç(Z-x,)- ç(Z-x2)] y2 < P(Z) = -b j ey'a^ ç(Z-Z") dZ"

yir This is a system of (imaginary) charges and monopoles at the -*_ ends of I. As long as the o's do not cross F ,the functions are univalued; under a crossing of the string the interaction energy of the system 1 changes by 2ir is • If the number of a and a+ as well \i and y+ are not the same the correlation functions vanish because of boundary contribution lin e~c nR . The string is the position of the Euclidean branch cut. This string position is fictitious as long as a charge does not cross. Apart from the above mentioned crossing rules it always remains | fictitious. How define the dyon-operators 464 B. Schroer

ípj = lim CJ(X) y(y) y+x

i|>2 = lim a (x) p(y) y+x

with a suitable convention for the limit.

If we take the limit of the electric charge approaching the monopole in opposite angles,

then there remains only the ambiguity depending on whether the limit is taken on the same sheet or whether a a(relatively to the other a) has crossed the line of ramification. The two point function ijJ> has as many possible values as there are sheets, i.e. finitely many (if S is rational) or •» otherwise. The order of i() and Tp has e priori no meaning !• in the Euclidean regime. However for S = 1/2 one may link the order fi-, uniquely with the two classes of path-limits and on obtains the relation ;i

r = - -!

Hence two-dimensional fermi statistics has a topological origin. This U(l) duality structure exists in all two-dimensional models involving complex fermions. Therefore the bosonization of ferminons and the Coleman equivalence between the Sine Gordon model and the massive Thirring find their ultimate understanding in the framework of Euclidean duality: they are just special cases of the grand duality scheme.

The ZN duality is more diffici-lt to handle than the U(l) duality: one has not yet understood to write down continuous functional integrals although in special cases, as the Ising model, there are other methods to construct the scaling limit. In D = 4 dimensions there are formal difficulties (renormalization problem) in the U(l) case.

r ! 3. Models of Gauge Theory, Confinement and U(l) Problem

Interesting regorous results have been obtained by studying TOPOLOGY AND CONFINEMENT 465

two dimensional gauge theories. There already abelian gauge theories r-x ave 8-vacua and lead to confinement. It has been known for some time that massless QED, screenes and massive QED_ confines quarks. The confinement versus screening properties are most conveniently exposed by 26) introducing flavour into the model

I|J * \|). , i corresponds to the up(N) flavour group

screening; colorless states with fundamental flavour appear in i ("bleached" quarks, if they are one particle state)

confinement: there are no states with fundamental flavour in Tit . .

A solvable model with very realistic confinement properties is the SU (NJJJ- - "torns" model: it is obtained from QCD_ by only integrating over diagonal configuration, i.e. A .. ,. = 0 • for __ y #OII—oxag. mq = 0 3fpnys . contains only trivial representation of the discrete Z (N) subgroup (apart from certain 6 values for which confinement changes into screening). Most of the results can be obtained on the basis of the functional integration over fermions. In the presence of 9-vacua one has:

Fermion integration: -/-+• Mathews-Salam rules —*• modification due to Atiyah-Singer zero modes '

This method does not work if a fractional winding number : -i— I F ?£ integer becomes important. In that case uses the method of duality (dyonisation of f ), This method also _gives a nicer physical insight: the magnetic monopoles through the * A. become coupled to a background of magnetic sources, the fictitious Dirac string is converted via condensation of monopoles into a physical string. The effective potential computed with Wilson's formula:

•:. = c e - ^ 28) -; behaves different in the screening and confining case 1 466 B. Schroer However it does not reveal the tramatica difference between the two situations in the particle structure of ^ * If

Tn two dimensional models the fermion mass term is essential for confinement. However, even with this mass term, there exists a discrete set of 6 angels at which confinement goes over into screening .

It is currently believed that in 4 - dimensional teories the Wilson area-law expressing quark confinement will insure the (light) quark confinement of the full QCD theory.

There is a curious aspect of screening versus confinement if one considers quark propagation in relativistic gauges:

n2 í.n(x-y) 2 u x - y •+• °° e q

i.e. has the secret long range from behavior only in a gauge where - g to all orders in part one loons this pathological behavior.

If in QCD. by some non-perturbation method one can establish

then (by using sceleton expansions or Dyson equations) it seems very plausible that quark propagation increases exponentially for large distances. Again the logical connection of this observation with the structure of

In QCD^ there have been many controversies concerning the so called U(l)-problem: why and how the anomalous U(l) . . , current in the limit n = 0 leads to an y-plasmon (instead of a Goldstorian bosan)

A very neat discussion has been given recently by Roth and Swiça . They comple the U(l) chiral SU(N) Gross - Neven model

! l L -g |OFÍ) - (TTr*) l inf e TOPOLOGY AND CONFINEMENT 467 with QED,. Their findings are: (1) The infrared cloud of the chiral G.N. Model (in D = 2: Goldston bosons infrared cloud) is eaten up by massless gluous via the Schwinger-Higgs mechanism. The resulting massive cloud destroys chiral selection rule, the corresponding massive particle is the y (which appears already in lowest order jj) .

(2) The relevant gauge field configurations which lead to ^ 0 have fractional winding number = - . So the controvery represented 25) 26) in its extreme version by the names of Crewther and Witten is superficial at least in this model. Both of them are partially right: Witten on point (1) and Crewther on point (2).

I thank the organizers of the 6 Brazilian Symposium in Theoretical Physics, in particular Prof. Erasmo Ferreira for the invi, tation and the hospitaly. This manuscript was written during a breef stay at DESY. I thank the DESY theory group in particular Prof. H. Joos for inviting me.

References and footnotes

Here we give only those references wuich were directly used in preparing this talk. Via these references the reader may back to the often more important earlier papers.