A Study of Dual Models in the Theory of Strong Interactions

DOKTORSAVHANDLINGAR vid CHALMERS TEKNISKA HOGSKOLA Mr l>0

by LARS BRINK

G5TEBORG 1973

DOKTORSAVHANDLINGAR vid CHALMERS TEKNISKA HOGSKOLA

'•&

A Study of Dual Models in the Theory of Strong Interactions

by LARS BRINK

GOTEBORG 1973

The thesis consists or the following publications:

I A one-pion exchange model applied to the reaction pp-'pir (A )pir at 19 GeV/c Nucl. Phys. B26, 611 (1971) (together with S.O. Holmgren)

II Regge-pole fit to the Deck-peak in the reaction pp-*A pir at 19 GeV/c Physica Scripta in press (together with S.O. Holmgren)

III A comment on nucleon-antinucleon annihilation and the possibility to determine the spin for resonances in the direct channel of this process Physica Scripta £, 117 (1972)

IV Some consequences of resonance production according to the scaling hypothesis Phys. Letters 37B, 192 (1971) (together with W.N. Cottingham and S. Nussinov)

V The physical state projection operator in dual resonance models for the critical dimension of space-time Nucl. Phys. B56, 253 (1973) (together with D. Olive)

VI Recalculation of the unitary single planar dual loop in the critical dimension of space-time Nucl. Phys. B5j9, 237 (1973) (together with D. Olive)

VII Unitary single non-planar dual loops in 26 dimensions of space-time CERN TH-1773 (together with D. Olive)

VIII The gauge properties of the dual model pomeron-reggeon vertex: their derivation and their consequences Nucl. Phys. in press (together with D. Olive and J. Scherk)

IX The missing gauge conditions for the dual fermion emission vertex and their consequences Phys. Letters i+5B, 379 (1973) (together with D. Olive, C. Rebbi and J. Scherk

X A physical interpretation of the Jacobi imaginary transformation and the critical dimension in dual models Phys. Letters k3B, 319 (1973) (together with H.B. Nielsen)

XI A simple physical interpretation of the critical dimension of space-time in dual models Phys. Letters UjjB, 332 (1973) (together with H.B. Nielsen) XII Dual models with SL(2,c) symmetry Nucl. Phys. Bft6, 505 (1972) (together with A. Kihlberg)

XIII The nucleon form factors in a field theory on a homogeneous space of the Poincare group II Nuovo Cimento JOA, 533 (1972) (together with R. Marnelius)

XIV A study of the Hiltoert space properties of the Veneziano model operator formalism to be published in J. Math. Phys. (together with P.H. Frampton and H.B. Nielsen) Contents

1. Introduction page 1

2. Basic features of strong interaction 5 2.1 Nuclear reactions 5

2.2 Elementary particle reactions 6

3. Possible ways to construct a theory for the strong inter-

actions 10

3.1 Field theoretic approach 10

3.2 S-matrix approach 12

3.2.1 Postulates 13 h. Some simple phenomenological tests of the theory of strong

interaction 21

5. Dual Resonance Models 30

5.1 Four-particle Amplitudes 30

5.2 N-particle Amplitudes 36

5.3 The operator formalism of the Veneziano Model U0

5.^ The algebraic properties of the Veneziano model with

unit intercept k$

5-5 Unitarity corrections 51

5.6 The Neveu-Schwarz model 60

5.7 Fermions in dual models 65

5.8 The string interpretation of dual models 71

6. Attempts to construct new models 8k

7. Conclusions and speculations 92

Acknowledgement 95

-1-

1. Introduction

The human curiousity and interest in Nature led already the ancient Greek

philosophers to speculate about the existence of fundamental constituents of

matter. In modern philosophy and science this still remains one of the basic

questions; a question that hopefully can be answered in a near future. This

hope, however, did also permeate earlier generations of scientists. During

the 19th century it was believed that the atom played this role, as ths existing

chemistry could be understood fairly well on the basis of this concept. It was

not until 1911 that Rutherford did reveal that the atom in itself is built up

as a nucleus surrounded by a cloud of electrons. In the early 1930's thc> nuclei 2) had been understood to be built up by protons and neutrons . Beside these

e_ljem£nt_ary_pa,rt_i£l£_s one had found the electron and its antiparticle, the posi-

tron and some far-sighted scientists could also foreshadow the antiparticles

of the proton and the neutron even if the antiproton was not discovered until

some twenty years later . The goal then seemed to be reached. The electron,

proton and neutron together with the photon, the mediatior of the electromagnetic

interaction, were the natural candidates for the basic entities of matter. How-

ever, puzzling questions remained. The interaction between the proton and the

neutron was still to be understood and to offer an explanation of this, Yukawa''

postulated the existence of another elementary particle, the pion. Through ex-

tensive studies of cosmic rays the pion was eventually found , but, unexpected-

ly, other particles were also found. All these particles could be distinguish-

ed from the proton, neutron and electron in that they were unstable, decaying

with a life-time of about 10 seconds. With the advent of the high energy

accelerator physics some twenty years later new phenomena were discovered. A

seemingly endless number of very shortlived resonance states with a life-time -2-

-23 7)

of about 10 seconds occur in elementary particle processes . In many

cases they can be thought of as excited states of other more stable par-

ticles but still the fact remains that the study of the smallest distances

in Nature has revealed an enormous complexity. Even if one by now has found

many symmetries and regularities among all these particles, the concept of

elementary particles as the basic building block in Nature that seemed on

such a firm ground kO years ago is now an open question.

A natural way out would be to say that the particles that have been

found are not the fundamental ones, but are built up by another set of more

fundamental entities. In the late 195O's when the concept of internal symmetries grew out, one tried anew to find this set. The first attempt was made by Sakata , who tried the proton, neutron and theA-particle as fundamental entities, but he was not very successful. These attempts culmina- ted when Gell-Mann and Zweig found that all existing particles can be thought of as bound states of three new particles and their antiparticles.

These postulated particles were named quarks. This is a beautiful idea and an extensive search for quarks began immediately. However, so far one has not found any of these particles and if they exist as free particles their masses have to be quite large. The situation today is hence somewhat con- fusing. We do not know if Nature can be built up from a sma?l number of fundamental entities or if the set of possibly infinitely many particles that are seen in high-energy physics today constitute the fundamental set.

Closely related to the physical states that we see in Nature is the interaction between these states. As a matter of fact it is through their interactions that they reveal their often elusive existence. The gravitational and the electromagnetic interactions were found centuries ago and were thought to be the only ones in Nature, but with the increasing insight in the physics of nuclei it became clear that none of these interactions could bo responsible for the bindings in the nuclei. It was shown already in 1933 i\ by Wignor that the nuclear forces must have a very short range of action ] a 3 1i "- ] and that they necessarily are very strong within this range. These forces have

.1 hence no classical analogue as do the electromagnetic and gravitational forces.

1 Furthermore already in 1896 Becquerel had found a new phenomenon, radioactive

I decay of atoms and in particular thef&-decay, indicating another kind of inter- • 13) -< action. Not until some 35 years later Fermi tried to formulate a theory

li for this process. With the increasing knowledge of elementary particles it

;1 was found that this interaction is responsible for a huge number of processes. 5 J The difference between this interaction and the nuclear one is first of all I ;i that the nuclear forces are much stronger. Hence one obtained the division u /! into strong and weak interactions. Secondly and of tantamount importance was q 1 ^)

j the discovery of Lee and Yang 1956 that the weak interaction did not con-

>l serve parity, i.e. it distinguishes between left and right. It was soon found

: that this interaction violates most of the known conservation laws that the

•} strong interaction obeys. Since then one distinguishes clearly the four

types of interaction present in Nature, the strong, the electromagnetic, the

weak and the gravitational ones. It is to be remembered that while elemen-

, tary particle physicists today divide their field according to the four types

•>] of interaction only 20 years ago many physicists believed that the various

•.,'j d physical phenomena observed were only different manifestations of one general

J interaction. jI With the concept of elementary particles one has usually associated a I particle with no extension in space, however difficult this concept might be -;| to understand from a quantum mechanical point of view. If we believe that all

4 elementary particles are point-like then the ratio of the strengths between

the strong, the electromagnetic, the weak and the gravitational interactions

is approximately 1 : 10 : 10 : 10 . We are here comparing dimension-

less constants. As the nature of the weak interaction as well as the strong

interaction is not completely known, these numbers are somewhat model depen-

dent. Hence we can understand the clear distinction among the types of inter-

action. With the present experimental information one can, however, doubt if -k-

at least the strongly interacting particles, the hadrons, are point-like.

Data suggests that they have a distinct structure in space. Then the divi-

sion into different types of interactions might be at least partially trans-

muted into a division in different types of spatial extension for the ele-

mentary particles. Kinematics might play a major role in determining the

strength of a reaction. This is one of the deep unsolved problems in ele-

mentary particle physics and will be discussed later.

This thesis deals with some attempts to formulate a theory for the

strong interaction. It is certainly coloured by the author's personal

points of view and shall not be thought of as a review of all aspects of

strong interaction physics but rather as a logical conclusion of some work

in which the author has actively participated. The problem of understanding the strong interaction is one of the most complex ones that the human mind has tried to master. The amount of experimental information is by now enormous as is the amount of theoretical ideas proposed. The thesis will summarize the basic theoretical ideas that most physicists in the field agree on but will only cover those models with which the author has worked. This ;'> means that the thesis will give the reader a biased view, which, however, in 1 the author's belief is close to the right one. J

1 '•'•I 2. Basic features of strong interaction

While most of physics at the macroscopic level is governed by the electro-

magnetic and the gravitational forces, the strong interaction is of extreme

importance for the understanding of the microscopic properties of matter. In

nuclear physics the nuclei are kept together by the strong forces counteracting

the electromagnetic ones and in elementary particle physics most of the particle

reactions that one can study are due to the strong interaction. The two most

striking features of the strong interaction that clearly distinguishes it from

the electromagnetic and gravitational ones are its strength and its short range.

The comparison of the strengths among the different interactions has been given before. The range is of the order of 10 cm beyond which the strong interaction force seems to fall off exponentially. Unlike the case of the electromagnetic and gravitational force it is very difficult to find an analytic expression for the strong force. For one thing it is spin-dependent and only approximate expressions for the force have been found.

2.1 Nuclear reactions

Nuclear reactions have been extensively analysed. When investigating nuclear reactions of the type a+N..-» b+Np, where N. refer to the nuclei and a and b are incoming and outgoing particles, one finds great variations in the cross section in the low-energy range {**> 1 keV). This phenomenon can often be understood within the so-called compound nucleus model , which says that the process occurs in two steps. In the first step a+N- form an intermediate compound nucleus with well-defined quantum numbers that in the second step eventually decays into b+Np. The probability amplitude T(i-» f) is well described by the well-known Breit-Wigner formula

g g . iR fR7 jr (2.1) i and f refer to initial and final states and g._ and g,™ are the coupling in IK -6-

strengths of the compound, which is called a resonance, to the initial and

final states resp. E is the total energy and ER the energy where T has the

maximum while P_ measures the width of the peak in T. We note the factori-

zation of T into one part just depending on the initial state and the reso-

nance and one part depending on the resonance and the final state together

with an energy denominator.

The resonances which occur in nuclear reactions are very abundant.

For lower energies they are well separated with widths of a few eV and

average spacing of about 10 eV. For energies in the MeV region the widths are of the order of 0.1 MeV and the density of levels is roughly 10 /MeV leading to completely overlapping resonances and a smoother cross section. Still it is meaningful to talk about resonances and a good description of the data can be obtained by superimposing terms of the type indicated in (2.1).

In a nuclear reaction it is intuitively easy to understand what happens.

The incoming particle joins the other particles in the nucleus which form a new state that is unstable and eventually decays. This explanation is built on the idea that the nucleons keep their identity within the nucleus, which we cannot prove. This is, however, consistent with all our knowledge.

2.2 Elementary particle reactions

After some 15 years of high-energy accelerator physics the most striking phenomenon in strong interaction particle physics is the richness of 7) the spectrum of stable and unstable particles . This is very similar to the case in nuclear reactions. One difference is the scale in energy.

These experiments are performed in the GeV-region and the widths of the resonances are of the order of 10 to 150 MeV. With energies around a few

GeV some resonances can be seen directly in the cross sections, but above this region the cross sections smooth out. However, if one carries out a partial wave analysis on these amplitudes one sees distinct resonances. If one plots the spins of the observed resonances versus their masses squared, the resonances seem to follow straight trajectories. Spins up to 10 have

teen observed among baryonic resonances and the excitation energy for each

extra unit of angular momentum is about 1GeV.

With such an abundance of resonances, how is it possible to determine

which resonances belong to one and the same trajectory? This is possible due to the classification of the elementary particles according to their quantum numbers and the conservation of these quantum numbers in strong interaction processes. Gell-Mann and others 17) found that the pattern of elementary particles follows approximately an SU(3)-symmetry (SU(3) is the group of special unitary 3x3 matrices). In this way the hadrons were given quantum numbers, the baryon number, isospin and strangeness. The particles are also assigned quantum numbers due to the symmetry under parity and charge conjugation transformations. A trajectory is distinguished in that all particles on it have the same quantum numbers except for mass and spin which vary. The importance of the other quantum numbers is that they are believed to be conserved under strong interaction, while under weak interaction this need not be the case. It should also be said that the concept of linear trajectories is so far only a hypothesis with only scant experimental verification. The spin of very massive particles is very difficult to measure.

The similarity between nuclear and particle reactions would be easier to understand if the elementary particles themselves were built up by more fundamental constituents like the nuclei are built up by nucleons. This leads rather naturally to the idea of quarks . But so far no quarks have been found. A more remote but equally deep idea is the so-called "bootstrap" 18) mechanism particularly emphasized by Chew . It says that no particle is more fundamental than the others and that they build up each other. There is a complete democrasy among the particles. This is philosophically a new idea in physics. The difference between a scheme with only a few fundamental particles and the bootstrap model is similar to the difference between the ancient Greek philosophies, from which the modern western ones originate and some Eastern -8- •} •j

, i ones in which no elements in-the world are fundamental. Each little thing j in the world is influenced by the rest of the world. ;; S It is an interesting observation that the leptons, the particles that "•] do not interact strongly, do not show the same type of resonance spectrum J "j as do the hadrons as previously discussed. This could be an experimental .] accident. It could be that the spacing between say the electron and a re- r\

currence of it is so large that it is not possible to find this recurrence ;j with the presently available experimental facilities. However, experiments j

indicate that the leptons really behave as point-like particles, while the ,«' hadrons most certainly have a spatial extension. This is most probably j part of the reason why hadrons can form resonances and imitate nuclei, 1 while leptons cannot. 1 A The idea that the hadrons are particles with spatial extension not .-1 divisible into point-like constituents is somewhat alien to quantum theory. •"•iI It definitely introduces complications in the theoretical description, but \

as long as no quarks or other fundamental particles are found, this seems ] an unavoidable situation. We will see later in the thesis that this idea ;> • j may even be very attractive. 'j Another important difference between hadrons and leptons is that for >H

high-energies the total cross sections for hadronic processes tend to a -j constant or at most rise logarithmically indicative of a black disc of fi- | — 13 + — '• nite size (°-M0 cm), while lepton induced processes like e e -«• hadrons, j which due to the smallness of the coupling constant at least for energies 1 below, say,100 GeV is dominated by e e -» one virtual photon-» hadrons are f expected to go as s at high energies (s is the invariant mass of e e~) The last important feature of the strong interaction to be mentioned is the strong damping of high transverse momenta (i.e. momenta perpendi- cular to the incident projectile) in the reactions. This is consistent with hadrons having a spatial extension and that the strong force only has a range of the same size as this extension. If we denote the "radius" -9-

of the particle with R, only partial waves with angular momentum t. = k«b * k-R contribute (krJ V"s* is the c.m.s. incoming momentum and "b the impact parameter),

This is a very qualitative statement but shows the origin of the cut-off in transverse momenta. -10-

3. Possible wavs_ to construct a theory_for_the strong interactions •-] 3.1 Field theoretic approach A A The analytic form of the electromagnetic interaction has been known »i i for a long time from the principle of minimal coupling. To introduce the j

interaction in the theory of a free particle one makes the replacement ;;] p^ •* pu-— A^, where pw is the four-momentum of the particle, A (x) the -\ electromagneti. . c field, e the charge and c the velocity of light. This 1> substitution can be done already at the Lagrangian level and one obtains J, 19) i the field theory of quantum electrodynamics . This field theory has j been tested for various physical quantities and the agreement between theo- % ry and experiment is overwhelming. It is a matter of taste whether the }l problem of quantum electrodynamics is considered to be ultimately solved •j1 or not. All physical quantities are calculated perturbatively and to get \{ •j rid of divergent contributions in the perturbation series one has to re- -/j normalize, which, however, can be done consistently. Questions one might -\ still pose pertain to the reliability of the perturbation series even though .'! the expansion parameter is small, and the introduction of infinite renorma- , j lizations . These are very ambitious questions, and from a practical point f of view we may regard quantum electrodynamics as a complete theory. With this success in mind it was natural to try the same type of field theoretic scheme for the other types of interaction as well. For the purely leptonic processes due to weak interactions an "effective" Lagrangian has 20) been written down . Computed to first order in a perturbation expansion one obtains results which are very close to observations but higher order terms diverge badly. Only recently progress has been made to improve this situation within the so-called unified field theory models for electromag- 21) netic and weak interactions . One can write down a Lagrangian which covers both quantum electrodynamics and weak interaction and has a sufficiently big gauge symmetry (we will define and discuss gauge symmetries at length later on) that higher order terms in weak interaction turn out to be finite. -11-

There is unfortunately a great deal of freedom in constructing these models

and so far no decisive criterion exists how to choose between different alter-

natives as long as they are consistent with actual observation within the limited

energy range so far available for experimental investigations.

The electromagnetic and weak interactions are well suited for a pertur-

bative field theoretic approach since the coupling constants in these theories

are small. In quantum electrodynamics one expands in the fine structure con-

stant , which in natural units is 1/137* and in weak interaction the coupling

constant is still smaller. In strong interactions, however, the corresponding

coupling constant is even greater than 1 which makes a perturbative local

field theoretic approach highly unreliable. With the increasing knowledge

about the hadrons this approach seems still more unlikely to lead to success.

How shall one incorporate the seemingly endless number of hadrons? One way

could be to start with a basic set of fundamental particles, whose fields build up the Lagrangian and then generate the other particles through divergencies

in the perturbation expansions. This seems mathematically untenable and besides that, everything points towards a "democrasy" among the hadrons, which in this case would be violated. Another way out would be to have infinite component

fields, but the arbitrariness here is huge and the guide-lines are few. Further- more with experimental data indicating an intrinsic structure of the hadrons, how can we ever believe in a field theory which is strictly local? Non-local

22) field theories have been tried since long ago"', but one needs definitely a better physical understanding of the type of non-locality of the particles to be able to make progress along such a line. Another difficult problem with theao non-local field theories is to understand and formulate the causality properties.

(Causality implies that signals cannot travel faster than the speed of light.)

A step forward within this framework has recently been taken with the formulation

we mean by a local field theory, a field theory for objects with no spatial extension. The above mentioned field theories are of this type. -12-

23) . y of a quantum mechanical model of certain one-dimensional objects and it ?j seems possible now to formulate field theories for these objects with very |j interesting properties. This will be discussed in a later section. x| 2k) '1 There exist some model Lagrangians of the so-called Yang-Mills type p; that could be thought of as describing some mesons. One interesting feature -i| of these models is that if one builds in isospin mvariance the theories fc| become very restrictive. This symmetry (gauge symmetry) makes also the ,|j models rather well behaved; they are for example renormalizable. These mo- r| dels have probably little connection with the strong interaction. They || might be thought of as some limiting case of a real strong interaction :^

model, but so far their main use has been as a basis for constructing the 'J 21) i unified models of electromagnetic and weak interactions . a

3.2 S-inatrix approach The fields in a field theory are no physical observables but are convenient means to construct scattering amplitudes. Another approach to the strong interactions is to forget about the fields and consider directly the scattering matrix (S-matrix) and try to deduce as much as possible about it from general physical principles. Historically, this idea was proposed around 19^0 by Wheeler and Heisenberg 25). More recently it has been much explored by Chew , who also introduced the bootstrap idea mentioned before into this scheme. Hence in this approach one starts directly with the S-matrix elements,

which we write =Sfi, where Sfi is related to the probability for the reaction i -»• f. More precisely the probability for this process is |s_. | . One now imposes physical postulates on the S-matrix. Just as in a Lagrangi- an formulation one first introduces all kinematical constraints that the matrix should fulfil. Then in order to single out a physical model one has to make additional dynamical assumptions. In a field theory this is done already at the Lagrangian level, while in S-matrix theory this is a con- -13-

% strainton the S-matrix elements. We now list the main S-matrix postulates

;| from which a theory of the strong interactions is to be constructed. We

'•>}, tacitly assume postulates about quantum mechanics. Ii ii 3.2.± Postulates

I Poincare invariance

II Connectedness structure or cluster decomposition

III Unitarity

IV Causality or analyticity of the first kind i»| V Crossing

"'I VI Asymptotic behaviour or analyticity of the second kind y) VII T, C and P invariance

'•••i These postulates should be combined with the experimental knowledge about

,| the particles and their couplings. Whether this is enough to single out a

| unique physical model in agreement with experiments is difficult to say. We 1 1| will discuss the postulates and see that they impose at least very strong re- J strictions. In order to find the relevant postulates one has used field theory t as a guide and extracted the most general and nonperturbative features. We •-i consider the postulates each separately.

I It is a kinematical constraint that S_• has to conserve energy-momentum.

This means that one can write

We also demand that S». be independent of in which frame it is measured. A

convenient way to implement this is to write !Sf. in terms of Lorentz invariant

quantities. Let us in this context define the kinematical variables to be used

in the ensuing discussion. We consider a scattering process (P.T.O.) a, 8, Y,Kand n denote particles.

Fig. 3-1

We write all momenta p^ as incoming. If p° = E sin units when 1i=c=1,is nega- >| tive, it means that the corresponding line represents an outgoing antipartic- ;'| le K in the process with p_^ = - p V. A convenient set of Lorentz invariant '? K K 4 27) * quantities are the so-called Mandelstam variables which are defined as >| follows: Define a channel as a partition of the W particles with at least -|

2 particles in it. Add up all four-momenta of the particles in this channel !| and square this sum. This is a Mandelstam variable often denoted as -''%

si- k = (?£ + P- + ... + Pfc) (3.2) i To each partition of the N particles there is one Mandelstam variable J defined in this way. There are then obviously more Mandelstam variables for a certain reaction than there are degrees of freedom, which leads to certain relations between them. It is, however, convenient to use all Mandelstam variables and distinguish the various channels by their corresponding Mandel- stam variables.

We can hence assure that S». is Lorentz invariant by writing it as a function of the Mandelstam variables. Spinning particles introduce further complications but can be handled by writing Sf. as a sum of amplitudes with the correct spin factors multiplying scalar functions built up by Mandel- stam variables. j The next question is whether S_. is free of further 6-functions and this -:! 'j is answered by the next postulate.

•-i II As strong interaction has a very short range we can decompose the S-matrix •j elements into different categories according to if the particles miss each other 1 or if they interact. We write this dia,grammatically in the elastic scattering fjj of two particles as

-)«

Fig. 3.2

(In these diagrams we associate particles to the left of the blob as incoming, and those to the right as outgoing.)

The first diagram on the R.H.S. indicates the part of the amplitude which arises when the particles miss each other and the second one when the particles really interact. The second one is called the connected part. Both diagrams are assumed to conserve energy-momentum separately, so that the first one has

&li in out fil n Ut) while the second one has ha - Pa ) W " Pb° a b

out out \

One can now generalize this and postulate that

q _ T „ q i (3.3) Sfi " Z 7 Sc C 1 where the sum runs over all possible groupings of initial and final particles such that each group can provide a possible process allowed by conservation of four-momentum. The product is then over the disconnected pieces that each grouping consists of. This means that each S 1 carries a 6-function indicating conservation of energy-momentum of this connected part. Apart from this 6-function the S x is assumed to be a S-funetion free function of Mandelstam variables. -16-

One now usually defines the scattering amplitude A^f by the relation

if U U in OUt Sc - - i (2,) 6 (SP - Sp ) A.f

where the constant in front is chosen for future convenience.

III Conservation of probability forces S to be a unitary matrix

SSf = SfS = 1 (3.5)

Together with postulate II this implies severe restrictions on the matrix

elements. Consider Eq. (3.5) between two states |i> and |f> and insert a

complete set of intermediate states |n>. Then Eq. (3-5) becomes

f Z = <5fi (3.6) n

If we decompose and according to the cluster decomposition

postulate II and identify terms with the same connectedness structure, a comp-

licated set of equations results such as

+ f + = Z < n|S |i> (3-7) n

where in the R.H.S. the individual matrix elements need not be connected,

but the products must be. The matrix elements with S • will be discussed

in the next section.

IV A crucial assumption about the S-matrix is its analyticity properties

as a function of ingoing and outgoing particle momenta. These properties are usually assumed to follow from macrocausality, i.e. the principle of

special relativity stating that signals cannot travel with velocities greater than c. It is also a typical feature extracted out of field theory. -17-

The simplest singularities that an analytic function can have are poles. In a specific channel the amplitude is assumed to have poles in its Mandelstam variable, s . , related to all particles that can appear as intermediate states in en this channel. These poles appear at values of 5 equal to the mass squared ch

of these particles. A stable particle has a real mass while a reson.ance has a

small negative imaginary part in its mass the size of which is related to its

width. A very important feature of the residues of the poles is that they fac-

torize in one part depending on the initial particles that create the resonance

and the resonance and one part depending on the resonance and the outgoing

particles. These parts correspond to the couplings of the resonance to the

initial state and final state respectively. This is reminiscent of the con-

ventional Breit-Wigner description.

In addition to particle poles a second type of fixed singularities appear

in the Mandelstam variables. These are the branch points associated with

channel thresholds. The term threshold describes the fact that for real values

of se above the "threshold" s > the channel is physically open, so that the

process can proceed through this channel. Every channel has a branch point start-

ing at the threshold. A rough explanation why this is so, is that at threshold

something singular happens as the channel opens up. That this singularity

should be a finite branch point is dictated by the unitarity equations. An im- i I portant difference between these branch points and the poles is that the branch

points do not factorize.

By drawing the cuts from each normal threshold branch point in a Mandel-

stam variable along the positive real axis to +°° we define a sheet in the

complex plane which is designated as the physical sheet. As the branch points may not be encircled casually and because they divide the physical region into sectors, intervals of the real axis between two succesive thresholds, it is necessary to prescribe how two adjacent physi-

cal sectors in s are connected. The rule is that one shall always approach ch the real axis from above, i.e. one gives s . a small positive imaginary part. -18-

This rule is often called "the it prescription".

In the unitarity equations (3.7) we encountered the matrix elements of

S ^. We now assert that these matrix elements can be identified with the c

analytic continuation of the matrix element of S to a non-physical point reached by encircling in a counterclockwise sense, all threshold branch points lying on the physical sheet to the left of the physical sector in question.

The sign convention used is actually such that when means the value just above the cut, - is the corresponding value just below the cut. Inserting this into the unitarity equations (3-7) we see that this means that the discontinuity of a certain matrix element of S is related in a non-linear way to an infinity of other matrix elements of S. Hence it is seen that the unitarity equations constitute a system of equations of enormous complexity and imply severe restrictions.

As a by-product these equations show that there are further singularities in the S-matrix, for example, the so-called Landau singularities.

V A further restriction on the S-matrix is the demand of crossing symmetry. The scattering amplitude (we call it an N-point function, when K particles participate in a reaction) has to describe all possible scattering processes between these N particles. Hence one analytic function is used for all these processes and to go from one reaction to another, it should be possible to make an analytic continuation from the physical region of the first reaction to the physical region of the second one. This is a further very strong restriction, since it means that the scattering amplitude has to have all singularities in all channels demanded by unitarity.

VI The restrictions above together with the experimental knowledge of poles and branch points completely determine the singularity structure of the

S-matrix. To get complete knowledge of the S-matrix, one also needs the a- symptotic behaviour. Through the analysis of the analytic structure of scattering amplitudes projected onto angular momentum eigenfunctions and conti- -19-

nuing in angular momentum, one has concluded that the amplitude is governed

by so-called Regge-poles in the high-energy limit . This is most easily

described by the U-point function for scalar particles t PO *

Fig. 3.3

and = + For lar e s with Mandelstam variables s = (p1 +p^^ * (Po Po) • S

t fixed and negative the amplitude should behave like s ' t where OC(t) is a so-

called Regge trajectory, it is determined by the resonances in the t-channel, where OL measures the spin of the resonances. With the present experimental

information it is consistent to believe that oC is a linear trajectory with universal slope ck « 0.95 (GeV) . This postulate VI is clearly a dynamical assumption. It connects the high-energy behaviour in one channel to the singularities in another channel.

VII It is experimentally known that the strong interactions are invariant under the time reversal, the charge conjugation and the parity transformations within experimental accuracy. No violation of these symmetries has been seen.

The feasibility of implementing all these restrictions together with the experimental information about poles and branch points to find the full

S-matrix is definitely very small. Even in quantum electrodynamics one cannot write down a function describing a certain reaction in closed form but one must be content with a perturbation expansion. Similarly the approach most likely to lead towards a full theory of strong interactions ought to be based on some kind of perturbation expansion. This means that one should try to find a rather simple and unique first-order S-matrix to which one could find perturbative cal- culable corrections. We do not know so far whether there exists such a scheme or not. The Nature might be so complicated that it can never be comprehended.

We shall, however, propose a way to find such a scheme in a later section and i I describe some models which are quite close to reality, the hope being that some -20-

changes in these models could lead us onto the right track. -21-

! k Some simple phenomenological tests of the theory of strong interactions j The simplest approximation to the analytical structure of a scattering

amplitude is to assume it to have only one pole. This is clearly a great ' simplification of the problem, since we believe there are whole trajectories i i with possible infinitely many resonances that can occur as poles. However., if a process can proceed via the exchange in one channel of a pion , the corresponding pole is outside the physical region, but very close to it since the pion mass is very small. We recall that for a physical scattering process the momentum transfer is always spacelike while poles corresponding to i 1 exchanged particles appear for timelike values of the momentum transfer. For j processes where the pion pole contributes it may in fact dominate the scatter- i ,;: ing amplitude at least at small momentum transfers. This idea of single-pion I exchange was introduced in the late 1950's ,by several authors, e.g. by Chew i 29) \ and Low and Goebel . The predictive power of the model is clearly small, •i ' but for some processes one can achieve reasonable agreement with experiments. j One such process was examined in paper I, namely pp -*• p" i*~p where we restric- i ted the investigation to incident protons with a momentum of 19 GeV/c in

I order to compare with data from the experiment performed by the Scandinavian Colla- .jj boration. We also restricted the energy of pir+ to the A++(i236) region. We I picture the process as in fig. k.^

Fig, lt.1

We define the Mandelstam variables s. • = (p.-+Pi) and t = (p -p ) •'•o * J iCl X. 1 As the pole factorizes we can write the scattering amplitude A as -22-

where A v± v± denotes the scattering amplitude for the appropriate prr

scattering and one of the pions is off the mass shell. One can use the

experimental scattering amplitudes for pir ->• pir and pn •+ pir respectively,

if one corrects for the exchanged pion being off the mass shell. We assume

that this can be done by simply introducing form factors. The amplitude

so obtained is at best a reasonable approximation to the real amplitude if

t,, is small. To implement this latter condition, we consider only the o part of the phase space t^ > -0.2 (GeV) and further we specialize to o Sj_ ^2.56 (GeV) in order to simplify one of the form factors.

In this restricted domain of phase space the amplitude (U.1) gives a

rather good description of the process. The distributions in Mandelstam

variables related to one of the vertices, s, = S^13?' SU5' *13 and *2"5 are

well reproduced while those related to both vertices like s.^ cannot be

reproduced satisfactorily. This is related to the approximation of having

only one pole between the vertices. If one measures the distribution of

the so-called Treiman-Yang angle

COS |P5 x P2| |P3 x p1

where is measured in the rest system of (U5), this angle measures the an-

gular correlations between the two vertices. A spinless particle propagat-

ing between the two vertices cannot mediate any information about angular

distributions, so in a one-pion exchange model the Treiman-Yang angle

distribution is flat. Experimentally this distribution is never flat even

if it gets flatter the more one restricts the t.--region around the threshold.

This analysis shows that the one-pion exchange approximation is nowhere in phase space able to reproduce the full amplitude completely,but in a restricted domain of phase space it provides a decent description of the reaction. !.| -23- ,jj

| We regard this work more as an evidence that there is a pion pole in the •I| scattering amplitude that dominates the other poles for small momentum transfers | between the vertices, rather than as an attempt to find the complete functional I form of the scattering amplitude. I The distribution in s_. which could not be well reproduced by the one- * pion exchange model above shows a big enhancement above threshold. There is .1 probably a resonance, N(iU7O), contributing part of this peak. It was, However, found by Deck that even models without this resonance show a kinematical enhancement just above threshold, just as we observed in our analysis above. I (Deck actually investigated ir p -»• n pir IT , but the same reasoning goes through I for pp •+ pir pjr .) The true origin of the peak has been a subject of debate •t I for a long time, but still this problem is not fully resolved. It is probably is a mixture of a kinematical and a dynamical effect, but it is difficult to prove the presence of a resonance in the peak since one cannot perform a phase shift analysis for these processes. These peaks appear in almost all three particle channels and it is sometimes difficult to associate resonances to them 32). In order to investigate this distribution further, one might analyse the

process in terms of a Reggeized model, i.e. one assumes an exchange of a Regge

pole instead of the simple pion pole. This was done in paper II. This means

that one uses the high-energy behaviour of the amplitude and extrapolates that to the energies under consideration. The so-called finite energy sum rules 33),

which partly led to the dual models to be described in the next section, imply that this extrapolated amplitude represents the true amplitude averaged over the low energy region. If the true amplitude contains a resonance pole for the Pir it -system, the effect of this will be averaged out, hence such an analysis cannot answer the question whether the Deck-peak is of kinematical or dynamical origin. Still it is of interest to see how well it can reproduce the s,^- distribution. 3U) This type of analysis has been carried out before, for example by Berger The difference between previous work and our analysis is that we do not use a -2k-

Monte Carlo method to simulate the process tout calculate the physical quan-

tities directly by numerical methods. This leads to more accurate results but has the disadvantage that each physical quantity has to be calculated

separately. As we are mainly interested in the effect of Reggeization on the s .-distribution, we found this approach more appropriate.

In this analysis we find that full agreement between the theoretical and the experimental distribution cannot be achieved though the agreement is much improved as compared to the calculation with one-pion exchange.

The calculation has some inherent approximations but still it seems legitimate to conclude that the predicted peak is necessarily too broad. This is an indication that there can be a resonance in the amplitude, since an average over a distribution containing a resonance peak tends to broaden this peak. However, the origin of the Deck-peak still awaits its final clarification.

We have above investigated the pole structure in crossed channels.

Next we consider the pole structure in the direct channel. This is an easy way to find resonances. One simply looks at the mass distribution of the final state to distinguish peaks in it. At these peaks one can investigate if the final state particles originate from an intermediate resonance or not. The most efficient way, however, to find resonances is to perform a partial wave analysis of the amplitude. In this latter way one has found 7) many of the established resonances . This is a much more accurate method than to look for peaks in the mass distribution. Often the effect of a resonance is washed out in a mass distribution, but it will easily be detected by a partial wave analysis. The latter method has, however, clear limitations. It is difficult to perform the analysis for other amplitudes than l+-particle amplitudes and it is also difficult to go high up in energy, since one then has to take very many partial waves into account and the numerical calculations become very lengthy. It is for this reason particularly difficult to establish resonances with high spin and high mass with -25-

certainty. A possible way to see such resonances was suggested by Rubinstein

who considered the process pn -»• IT'S. He pointed out that if this amplitude

proceeds through the 7r-trajectory in the direct channel as in fig. k.2

•n Fig. k.2

one ought to be able to see the intermediate resonances X in the mass distribution of the IT'S. One should look in the isospin -1 channel with an odd number of outgoing pions, (an even number of pions cannot couple to the ^-trajectory by conservation of G-parity) and vary the mass of the pn-system around the possible poles which should occur for

a = a1 (s-m 2) » 0.9s - 0.02 = n (integer) 7T IT

One nice feature of such an experiment is that the initial energy is so high due to the large mass of the nucleons that the first possible resonance already has spin k. It was also believed that the contamination would be rather low so that the resonance peaks should be possible to see.

In paper III we have examined the possiblity of measuring high spins. 36) For low spins there are many sophisticated methods ,but it seems much more difficult to measure high spins. We find that one way is to measure the distributions as functions of the polar angles of the outgoing particles. These are shown to be sums of terms of the type cos n8, where n ^ J, the spin of the resonance, and 8 is the polar angle. Experimentally one has to plot do/d cos9af(®) which we now know may be' written as

2 2J f(9) = a + cos 9 + ... + Q cos 6 -26-

However, it is better to expand f(e) in a complete set of functions on

the unit circle and use its Fourier properties to investigate the highest

power in f(e). We can then use the system {cos n6} and write

f(6) = b + b_ cos 26+ ... +b2J cos 2J6 (U.5)

bOT can be obtained from the data as

C0S 2J9

As there will be background in the experiment, this will contribute to

even outside the resonance, so the idea is then to plot b as a function 2J OT of energy. Hopefully one will get a peak in the resonance region, while bOTiO will not give a corresponding peak. The great difficulty with this

approach is, of course, that very good statistics is needed. One can, however, improve on the situation, if one adds all the independent angular correlations in all channels with odd number of pions into one distribution and investigate this according to the method outlined above. Clearly a method like this will not provide a conclusive determination of the spin, but it can give some understanding of the process.

The above mentioned experiment has so far not been performed since there is one obvious snag with it. We assumed that the process proceeds through the leading ir-trajectory in the direct channel. This trajectory, we assumed, is a linear relation between the spin and the mass squared. Do we have enough angular momentum to create the resonances on the leading

*• -trajectory? If we think of the strong interaction to have a certain radius R, the maximal angular momentum would be k*R ** V*s"« This heuristic argument shows that it is not at all clear that the process provides enough angular momentum for the resonances on the leading 7r-trajectory to be created. Not until we understand the basic theory for the reaction better, -27"

will the experiment become meaningful. We remark that the models to be

described in the next section, the dual models, have linear trajectories in all

I possible channels. For high energies the effect of the resonances on the leading

f trajectory is exponentially damped out.

: | In an experiment one usually measures processes with two incoming particles

I producing a certain number of final particles for which all kinematical vari-

;„*! ables are determined. These types of reactions are called exclusive reactions.

, I An alternative way is to only measure the variables of a certain number of

I outgoing particles. This is called an inclusive reaction. The complete knowledge

of all exclusive reactions is the same as the complete knowledge of all inclu-

sive reactions. The simplest experiment to do is the inclusive reaction in

which one only measures one outgoing particle. Correspondingly it gives the

least information, buc since it is easy to perform it is a very appropriate

experiment at new accelerators and was among the first experiments to be per-

formed at the new intersecting storage rings (ISR) at CERN, when these were

put into operation 1971. It was an experiment very much looked forward to. 37) Feynman in 1969 had proposed that this inclusive distribution should scale

at high energies, i.e. in the distribution

at high enough energies the function F is a function only of x=-Ek and p S J. and p_ denote the transverse and longitudinal part of the momentum of the

outgoing particle, E is the energy, o the cross section and x is the Feynman

scaling variable (0 < x < 1). The formula (U.7) implies that at high energies

the distribution only depends on p. and s through the ratio x. The meaning of

the term high energies (or asymptotic energies) is very vague and Feynman og\ could not specify it. The experiment at ISR showed that one can indeed use o the same function F(x, p_) (h.j) for reactions pp -> TT + anything for protons -28- \

••4

;| of incident momentum from about 20 GeV/c up to 1500 GeV/c. It looks there- |

fore as if asymptotia starts already at ordinary accelerator energies. This |

result arose much attention and the field of inclusive reactions became a j very vivid and popular one. Since then no real spectacular results have "1

emerged, but the study of inclusive distributions is a good complement to ,:|

the ordinary study of exclusive ones. ^

In this context we studied in paper IV the scaling behaviour if the -4

..'if detected particle is a decay product of a resonance. This is another test | A of the pole structure of the amplitudes. We investigated the specific ex- '.-\ ample of a resonance decaying into two pions. It was found that the corre- '•-,* 4 sponding distribution eventually scales like in Eq. (U.7)» if the distribu- v| i tion of the resonance scales. The scale is essentially determined by the .\A mass of the resonance. This shows that different pieces of an amplitude 1 might scale for different energies, which can lead to substructure of a | scaling distribution. |

We also found that if one restricts to pions with small transverse mo- | mentum, the distribution from the resonances will be the sum of two functions with quite different scales. For example for pi = 0 one gets from the P's with no transverse momentum where F (x) is the scaling distribution for the p's. This effect remains, but is not so sharp when we allow the p's to have nonvanishing transverse momenta, (it is recalled that most outgoing particles have transverse momenta less than 0.3 GeV/c.) By varying the cut-off in pZ one changes the scale in the term F(29x) which means that if one has abundance of p's, one should be able to see a fall-off at x=1/29 or the corresponding value.

To see this effect requires good statistics, but will probably be possible at ISR in the near future. So far there seems to be some signs of fall- -29-

offs but these are still within the experimental errors39) -30-

5 Dual Resonance Models -A A natural way to seek a first-order approximation to the S-matrix is j

to retain as many of the postulates of section 3 as possible, but to viola- I te some of them. There is a great deal of arbitrariness in making such a \ choice, but the important thing is to choose the Born-term (first-order j approximation) in such a way that it is possible to handle it and that it '\ 1 agrees fairly well with experiments. There have been many attempts to

5.1 Four-particle Amplitudes Consider first the U-point function for scalar particles with mass m. t

Fig. 5.1 -31-

.; The Mandelstam variables are defined as

! s • (p,+po) • ( *

2 2 t = (P2+P3) = (P^P^) (5.1)

2 u • (P1+P3) • ^2+Vkf

These variables are linearly dependent since s+t+u=Um . The physical region for the scattering of particles 1 and 2 into particles 3 and k is s ? km , t, u ^ 0. Consider the scattering amplitude, A(s,t), in this region and expand it in a partial wave expansion

A(s,t) = I (2A+1) A (s) P£(COS6) (5.2)

I Every partial wave I can be thought of as being built up by resonances with i, spin I. If we continue analytically to the region t £ km and s,u S 0, the \ | physical region for scattering of particles 2 and 3 into particles 1 and U, the is | i term gives a behaviour for t -*• », A (s) t , which violates a rigorously de- % rived result of unitarity, namely the Froissart bound . The way by which a I disaster may be avoided is if the series diverges for t > km . Then one can- "| not calculate the asymptotic behaviour term by term. Hence the conclusion is that resonance dominance together with crossing can be reconciled if and only if the amplitudes have an infinite number of resonances. To construct the it-point function for scalars we start by seeking a function A(s,t) with poles in the s- and t-channels such that

A(s,t) = A(t,s) (5.3)

To implement the full crossing symmetry we define the complete amplitude to be

F(s,t) H A(s,t) + A(s,u) + A(t,u) (5.M -32-

This choice of splitting the amplitude into three parts is very natural.

A good way of describing the lov-lying mesons is to say that they have the

quantum numbers of a quark-ant iquark pair". This means that only certain

quantum numbers are possible. Hence some amplitudes can only have reso-

nances in certain channels. Consider for example TI it •* it it . If we call

the s-channel, the channel through which the process proceeds, the amplitu-

de can only have poles in the t- and u-channels. Thus an amplitude which

lacks singularities in one channel is needed. In field theory this is a

property of planar diagrams, so dual models with this property are called

planar. This property will be naturally extended to any N-point function.

To find an expression for A(s,t) we write it as a fixed -t dispersion

relation

(5.5)

We choose o(s) to be a linear trajectory

ot(s) • aQ + o's (5.6)

One solution of the Eqs. (5.3) and (5.5) is given by

ex (5.7)

This is an Euler B-function ' and it represents the U-point function in the

Veneziano Model or the Conventional Dual Resonance Model .

Let us explore some of its properties. By expanding (i-x)""^'"1 in powers of x in (5.7) the amplitude can be written as

A(8,t) = f n=o - ->-*) n« -33-

where a » wi s (a+n-1) ... (a+i)a is called a Pochhammer polynomial

For <*(•) x n the process pioceeds through resonances of spin from 0 up to

n. The trajectory ct(s) is called the parent trajectory and the trajectories

I a(s)-k, where k is a positive integer, daughter trajectories. It was for a

fj long time debated if daughter trajectories are present in Nature. Now the I 7) i experimental data clearly indicate that they are . | Using Sterling's formula it can be seen that

A(s,t) * r(-o(t)) (-a(s))a(t) (5.9)

t fixed

which shows the Regge behaviour. In taking the asymptotic limit one must keep

a reasonable distance from the s-channel poles in order that Sterling's formula

be valid. One anticipates that when a perturbation series is summed to give a

unitary amplitude, the poles will move onto unphysical sheets and the asympto-

tic analysis will then be valid throughout the physical sheet, including the

physical region itself.

The same limit applied to F(s,t) gives

F(s,t}-> r(-a(t)) (i+e""iTO(t)) (a(s))a(t) 3+CO t fixed

which displays a so-called even signature Regge pole.

Another physically reasonable feature of the amplitude is the behaviour

at a fixed angle. For example, at 90° in the center of mass frame and for

large s one has

at 32.1 *- p- s(21n2) dt e »*«"*' (5.11) Jon90°

This is in qualitative agreement with data at moderate energies. At higher

energies the experimental cross sections appear to fall less rapidly, although

.•;;] -3k

1+3) this is not yet completely clear

The scattering process involving only scalar mesons is not of great ex-

perimental interest because there are no prominent low mass scalar mesons

in Hature. Still it is an interesting process as a theoretical laboratory to investigate the general features of these types of models. When we discuss N-point functions» it turns out to be most easy to generalize the scalar it-point function and this generalization will be discussed «-.„ ^.ength.

The hope is, of course, that the features that are missing, such as certain quantum numbers, can be incorporated eventually into the model, without destroying the general properties of the model.

There is a way to introduce isospin for isospin-1 particles, in such a way that the amplitude is isospin invariant. That is to use Pauli matrices T to represent the charge states and multiply the amplitude by the hk) trace of these t-matnces taken in order and then sum over all permutations

This approach can be generalized to include an SU(3)-symmetry, if one instead multiplies by the corresponding SU(3) matrices. This would, however, lead to a complete conservation of SU(3) and it is most probably a too simple- minded an attempt in view of the fact that experimentally the SU(3)-symmetry is slightly violated. 1*2)

The original amplitude written down by Veneziano dealt with the process inr ->• TTWJ in which he could take care of the quantum numbers. A corresponding formula for TT TI -»• IT IT (where a,b,c and d denotes the charges) was soon after the work of Veneziano proposed by Lovelace and ShapiroUs ). It is based on a function C(s,t), which is not quite identical with a B- function

t p rd-o U)) rd-an(t)) c(s t) = g ' 2 r(A» (s)-a (t)) ^.12) p p -35-

f where now the coupling constant S is explicitly introduced. Here a (s) is the P o-trajeetory which experimentally is well described by

9 +a's

4 We note that the first poles occur for <* = 1, i.e. for the P-meson itself.

I With the isospin convention from above the amplitude can be written as

= \ Tr( V feT cx d) c(B.t) + \ Tr(

C(s,u) + — Tr(xa TC T^ xd) C(t,u)

(5.1U)

For the process TT IT -> TT IT this means that

(s,t,u) = 2 C(t,u) (5.15)

which is easily seen to have the correct pole structure. This amplitude can be

shown to provide a very good description of low-energy 7rir-scattering. It is for

example compatible with the requirement of PCAC (partially conserved axial cur-

rent) and current algebra. For a review of these concepts we refer the reader k6) to the book by Adler and Dashen . One can show that when the four-momentum

of one of the pions approaches 0, the amplitude vanishes. This is a require-

ment of PCAC. One can further calculate the scattering lengths to find them in

good agreement with the predictions from current algebra. Hence it seems that

the amplitude (5«1'+) provides a good candidate for a first-order approximation

to the process.

The four-point functions so far written down are not unique. One can add

to them terms of similar type with almost the same pole structure and Regge

behaviour, so-called sattelite terms. These can be obtained by changing the ar-

guments in the r-functions. The reason why we do not discuss such terms here

is that when considering factorizable N-point functions those are found to corres-

pond to the it-point functions mentioned above. -36-

5.2 N-particle amplitudes

In case the H-point functions mentioned above really represent first- order approximations of an S-matrix there have to be N-point functions with essentially the same properties. These have been found in various ways and we shall start by considering the amplitude in the formulation of Koba and

Nielsen , which is an amplitude for scalar mesons. Let us fxrst generalize the concept of planar duality. Diagrammatically we can write the four-point function A(s,t) as

Fig. 5.2a Fig. 5.2b with the internal lines indicating the channels in which resonances occur.

When we later introduce vertices and propagators, fig. 5.2b is said to be obtained by a duality transformation of fig. 5»2a. The equality in fig. 5-2 shows that the basic term is invariant under a cyclic or anticyclic permutation of all external momenta. This can be generalized to an N-point function. We write diagrammatically an N-point amplitude as N-1

Pig. 5-3

For all four lines that constitute a Ij-point amplitude (we also count internal lines) we demand that the equality in fig. 5.2 be valid. This shows the pole structure of the N-point amplitude. For example, by a duality transformation, Fig, 5.3 can be written as -37-

N-1

Fig. 5.U

An N-point amplitude with this type of pole structure is said to fulfil planar duality. To get the full amplitude one has to add all permutations of the external legs. If these external legs carry isospin the amplitudes should be multiplied by the trace of the product of the corresponding -r-matrices.

In the Koba-Nielsen form of the N-particle amplitude, to each external particle i is associated a complex variable z. that falls on the unit circle

(see fig. 5.5)

Fig. 5.5

Let us define the anharmonic ratio of four complex numbers as

/ -u = \ a-c b-d (a, b, c, d) = — • -- (5.16) and let us define as o. .the trajectory associated with the channel

... +p.) , that is to

(5.17) -38-

and a channel variable u.. called the Chan variable for that channel as

uii = ^zi' zi-1» zi'

Then the N-point function is H 7T dZ. W _, • —1

~dv i=1 1+ X G,«. J where the product IT extends to all possible planar channels. The amplitude can have only (W-3) simultaneous poles. To obtain this one can choose

dz dz, dz dv = , a w C T? r (5.20)

It can be checked that one can choose any z , z, and z . The amplitude is ct u C independent of the specific choice. The integration in (5-19) is performed in such a way that the order of the z.'s on the unit circle is maintained. An important property of the amplitude is evident in this form. It is well-known that anharmonic ratios are invariant under protective trans- - '' formations

One can easily convince oneself that also the extra factors in the integrand are invariant under these transformations, which shows that the amplitude has a built-in symmetry. This is not so strange since the amplitude is constructed on the premises to have the dual symmetry. It is instructive to see how the poles of the N-point function can arise in the Koba-Kielsen formulation. As in the case of the U-point function, poles in the invariant mass s.. located at a-•=!! (n=0,1,2...) will occur when u^. ->• 0, which is achieved when z. -> z., that is to say, when all ths KoTaa-Hielsen variables associated with the external particles of f * -39-

the channel under consideration overlap. This shows that the amplitude

satisfies planar duality.

It is possible to pass to another form of the Veneziano formula, the

Bardakci-Ruegg representation , by choosing a specific projective frame,

i.e. by making a particular choice of the three fixed Koba-Nielsen variables.

Introduce first a projective transformation that maps the unit circle onto

the real axis and fix %. = 0, zw-1 = 1 and a = °°. Then .Introduce the variables

(i=2, ..., N-2) (5-22)

which will have integration regions

0 * x• $ 1 (i=2, ..., N-2) (5.23)

3 With this choice the Bardak

A= /|n dx. 1J ^ I i=2 x

(5.2U)

"|1 where x. . is the product

(5-25)

|fl From now on we write the Regge trajectories as

o(s) = aQ + - s (5-26)

The beauty of formulae (5-19) and (5.2U) is obvious, while the complexi-

ty is somewhat hidden. The exact calculation of differential cross sections

turns out to be an unsurmountable task if the number of external particles

exceeds 5. In the 6-point function one has to perform a 3-dimensional integral

for all points in the phase space, while the phase space has 8 degrees of freedom. 5-point functions have, however, been calculated and used phenomeno- k9) logically with considerable success . In this context we have to stress

that there is another type of singularity in Nature, the Pomeronchuk singu-

larity or the Pomeron, which is associated with diffraction dissociation.

It is not fully clsar what type of singularity it is, but it is usually

assumed to be a pole with intercept unity and slope about half of the

slope of the ordinary trajectories and with the quantum numbers of the

vacuum. The high intercept leads to a dominance of the Pomeron over the or-

dinary trajectories at high energies. The Pomeron is assumed to be dual

to background and not to other resonances, so the trajectories in (5.19)

and (5.2*0 cannot be associated with Pomerons. Hence the A^'s of (5.19)

and (5.2^) can only be used for processes that lack channels with the quantum numbers of the vacuum.

If the different N-point functions correspond to matrix elements of the same S-matrix, they should have the same particle poles as intermediate

states. To check if this is the case one has to factorize the amplitudes, and investigate the structure of the poles. This can be done already for

(5.19) and (5.2*0 but is complicated. Instead we will write the amplitude in an operator form in which factorization is evident.

5.3 The operator formalism of the Veneziano Model

In the Koba-Nielsen form of the H-point function (5.19) it is seen that the factors in the integrand mix the z.'s in such a way that the integrand cannot directly be written as a product of terms, each term depending on only one external particle. There is, however, a chance to accomplish factorization by introducing operators, whose commutators are c-numbers, because of the relation (Baker-Hausdorff)

A B B A fA.B] . e e = e e el ' •» (5.27) m i I which holds for arbitrary operators A and B provided their commutator

[A,B] is a c-number. In the L.H.S. the expression is factorized, while in

;. • th& e R.H.S. the corresponding expression is not. It has, indeed, been found that by introducing an infinite set of

i.-f d-dimensional annihilation and creation operators, where d is the dimension

of space-time (we keep d arbitrary for future purposes), with the commutation

m relations

]s, v = 0,1,2,...d- -g"V 6 (5.28) nm m, n = 1,2,3,

that a factorization is possible.

Since (£* - diag (1, -1, -1, •.., -1) the space components a • are normal

creation operators for a (d-1)-dimensional harmonic oscillator whose level

spacing we take to be n. We assume that the time components act similarly,

though we must recognize that since la °, a o+ I = -1 the state a o+| o»

has negative norm. These negative norm states, or "ghost" states, create diffi-

culties for us. In order to have a correct first-order S-matrix we demand that

only states with positive norm couple in the theory. The ghost states are in-

troduced in order to have a consistent covariant factorization and if they r '-3: contribute at the poles, the model is n?t acceptable.

The fundamental constructs that we introduce are the Fubini-Veneziano 1 50) fields

«V - £nz - i ! / ft (5.29) n=1

n . u -n BP(Z) = iz z +an z n (5-30) n=1 -k2-

Besides the creation and annihilation operators we have introduced "zero

mode" operators p11 and q^ with the commutator

(5-31)

We may now write the Koba-Nielsen formula in the following way

xOlO-1

, z2) (5.32) where V(k,z) - : (5.33) and : : implies that the operator is normal ordered.

To further simplify the structure and for future purposes we introduce the Virasoro generators L defined by

(5.3U) where the contour encircles the origin. These Ln satisfy the algebra

L L L + (n3 n) 6 [ n> m] " <»-«> n+m fe " n+m,o (5.35)

The commutators of LQ with V(k,z) can be calculated to be

n, V(k,8)] f^-n^) V(k,z) (5.36)

The commutator with L leads to o

L L V(k,z) = z ° " ° (5.37) -1*3-

An explicit calculation of L further gives

na + a (5-38) Lo = f2 - E n s n H

K i where H is the total mode number operator. By considering a protective frame z. = 0, z™ .. = 1 and = °° the ampli-

tude can be written as

ikik 11 ik q =

(5.39) Ki where V(k) s V(k,0 (5.1*0)

and ?'-.3i a 1 D = fl dx (1-X) °-

1^' The factorization properties are now obvious. It is easy to see that the

poles occur for a(s) = 0,1,2, ... By writing the propagator as

D = E ( lo ) (5.U2)

the amplitude can be written as

(5.U3) «,o=o

if we define the "physical ket" |p> by

and similarly for the "physical bra" vector , where '•'?1 -hk-

(only a finite number of *• 's are different from zero!) These states satisfy

{1} (5.U6)

H | {A} A = Z nx, n=1 n

Since (L +£ ~a_) is diagonal in the occupation number space can be written as

(5.W)

From this form the residues at each pole can be found. However, it is not convenient to check that all residues are positive from this expression and thereby establish that no ghost states couple. In fact one will find some ghost states by direct calculation of the residues at lower levels when

The degeneracy D(j) of the level characterized by a(s) = J is given by the number of ways one can partition J in the form J = I + Znfc with m o n integers &Q and %^. This is not fully correct since this counts all the states in the Fock space a,nd some states may be decoupled in the scattering amplitudes but it gives a rough idea of the number of resonances that is p needed asymptotically in order to have duality. For J and m -*• « one gets

m/mo D( J) ** m""2~ e -U5-

with m = •

The quantity m may be interpreted as the maximum possible temperature

of hadronic matter. If one tries to heat hadrons to higher temperatures, 52) particles (mostly pions) are "boiled" off. Hagedorn was led to postulate

a density of states formula of this type on the basis of "statistical boot-

strap" considerations, well before the advent of dual models and decided on

empirical grounds that m should be about 160 MeV. This would require d

be 5 or 6. We shall not take this estimate too seriously and it is further

to remember that it is rather the total number of degrees of freedom for hadronic matter that counts and not the dimension of space-time. In any case it is interesting that dual resonance models have the same sort of degeneracy as

statistical bootstrap models. If such a degeneracy is correct, it could have played an important role in the development of the very early universe when the temperature was close to m "'. The statistical bootstrap models also appear to favour m in front of the exponential indicating d=5 consistent with the estimate from the exponential in (5.U9).

5.k The algebraic properties of the Venegiano model with unit intercept

As mentioned before one can find ghost states if o • 1. In the future study we will hence restrict ourselves to the case of unit intercept, even though it is unphysical. As the model describes scattering among scalar states, p it means that these states have m s -2 and are so-called tachyons. The existence of this state in the theory violates our postulates, since a tachyon always travels faster than light. The algebraic beauty of this case is, however, so great that it is worthwhile and hopefully instructive to investigate the model carefully even under these unphysical premises. The hope is that in a future better model, an unphysical state like this will be decoupled.

We note that for this particular case the propagator (5.^1) simplifies greatly and takes the form -U6-

1 D=/ dx Lo-1= _1_ (5.50) a Jo x x T V-"1

With the help of (5.35) and (5.36) we can now prove that

(L-L+1) xL°~1 V(k) = xLo+n"1 V(k) (L -L +1) - n o n o dx

L 1 °~ V(k)] (5.51)

which leads to

L 1 (VLo+l)/o1 ^ x o" V(k) KLo+n-1 v(k) (Ln-Lo+1)

(5-52) n The operator (L -L +1) essentially commutes with the product of a propagator

and a vertex. There are infinitely many of these operators, so we can con-

clude that there is a huge symmetry inherent in the theory. In analogy with

field theory a relation like (5-52) or (5.51) is called a Ward identity.

As a consequence of this symmetry one easily proves that

(L -L 1) DV(k ) DV(k ) V(k _ ) n o+ p p+1 N 1 |0>=o

(5.53)

V(k ) DV(k ) ... 2 3 -n o

We want to investigate the properties of the states that couple in the residues of the poles. The Eq:s (5.53) and (5.5U) show that an arbitrary state |(f>> occuring as a pole in an N-point function necessarily satisfies the conditions

n = 1,2,3, ... (5.55) -1*7-

= 0 (5.56)

Eq. (5.56) is a wave-function, analogous to the Klein-Gordon equation giving

the condition for the particle to "be on its mass-shell. The L subsidiary

conditions are further restrictions determining which states are actually

physical, in the sense of non-zero couplings to the states in the N-point

functions. States satisfying the wave equation hut not the subsidiary conditions are called "spurious". A typical spurious state is<<)>| L_ which is

seen to decouple in the residues. Hence we see that the large £auge_ jsymmetry that the model possesses is responsible for the decoupling of an infinity of states. This provides for having only positive-norm states couple in the residues. It is a fundamental question to understand if the gauge symmetry de- couples all negative-norm states introduced "by the factorization procedure.

As a preliminary we first look for an expression for the states that couple in the residues.

Consider again Eq. (5.32), now with « = 1, in the protective frame z1 = 0> = 1 and z.. =

N-2 o>

(5.57)

1 0 1 We choose •% (k..+k9) + 1 = n and insert a complete set of states |n>

dz. n> n i=2

N-t H (5.58) i=3

The pole occurs for z2=0 and the residue can he obtained by Cauchy's theorem -1*8-

Res Aj. = I - n> 2 n (k1+k2) =2(n-D

(5.59) i=i33 zi

It is now easy to extract the physical state

(5.60) with 2 2 k^ = k "2 (5.61) • 2(n-D

Note that the requirements (5.61) are necessary to make the integral (5.60) well-defined.

In this way the physical states that couple to two ground state particles can be constructed. It is now possible to iterate the procedure to get the physical states coupled to m ground state particles 1idz

dz r (5.62) with the conditions

k.2 '- -2 for 1 < j < m J (5-63)

... + kj for 1 < j < m-1 -1*9-

The physical states (5.62) can be seen to satisfy

n J L »0 n > 0 m-11 -n

so these states are indued physical. One can in principle construct all states

that couple in the residues in this way, but it is difficult to say anything

about the ghost problem because the states (5.62) are not orthogonal.

In order to find an orthogonal set of states we note that with the technique outlined above it is possible to construct vertices for emission of any physical state. This is actually a trace of the democrasy among the particles, since knowledge of the vertex of emission of one state is enough to determine the vertices of emission of all the other states in the model.

The vertex we will use is the vertex for emission of a spin-1 particle.

We will redo the calculations that led to (5.62) with this vertex. The way to get an orthogonal set of states is to restrict all momenta in the vertices to be collinear. This leads to the physical state operators first found by

Del Giudice, Di Vecchia and Fubini'* '

(5.6k) -n where

(5.65) k«p = 1 p is the momentum of the state that A acts on. The operators A1 satisfy

n,m - 1,2, (5.66) which shows that these operators create a set of orthogonal physical states.

These states can, however, not be all physical states since they only have transverse degrees of freedom. This is a reflection of the limitation to colinear -50-

momenta. Note that the definition of A is frame-dependent.

With the knowledge of these states and the gauge conditions, Goddard

and Thorn could prove that for d-26 these transverse states together with

a set of zero-norm states are the only states that couple. Hence there are

no ghosts for d=26 and consequently not for d < 26. 57) Brower proved this no-ghost theorem independently by constructing

operators that also create longitudinal physical states. For d < 26

these operators create positive-norm states and for d = 26 zero-norm states,

showing that the Hilbert space of physical states has positive metric if

d < 26• For d > 26 these operators can create negative-norm states, so

d=26 is the maximal number of space-time dimensions to obtain a model free

of ghost states. Therefore, d=26 is called the critical dimension.

With the knowledge of the transverse physical states one can construct

a projection operator from the full Fook space onto the space of transverse

states. This was done in paper V. Consider the following operator

where the integration contour encircles the origin and further

,£0= ? I A 1+(k) A\k) (5.68) n=1 1=1 n n and H was defined in (5.38). H sums the mode numbers of all oscillators

in the Fock space while JC sums the mode numbers of the A fs. Hence it is o n seen that £ -H is a Hermitian operator such that

(i) its eigenfunctions span the full Fock space

(ii) its eigenvalues are integers <0

(iii) its eigenvalues are zero only on the transverse subspace.

This shows that 7" (k) is a projection operator onto the subspace of transverse states. -51-

The form (5.6?) is not a very useful one, but from the definition of

A 1(k) (5.6*0, X, -H can by a lengthy calculation be seen to be identical with n o

1 where

i B = t ~- zn \ . (5.70) y ;-if n 2taz k'P(z) wi^y • 11 I As usual the integration contour encircles the origin. •i

I It is seen that «t -H is much simplified for d=26, and we restrict our fur- -I ther investigations to this case. It will be more evident later in the text why I we make this highly unphysical choice. For d=26 we write -?!

E E Xo-H = (DQ-1) (Lo-1) + I (D.nLn+L_nDn) (5-71) n=1

E is essentially built up by gauge operators. Hence it is an easy task to see

that the effect of 3"*(k) in a residue in an W-point function is unity. This

is the third type of no-ghost theorem.

5.5 Unitarity corrections

We now have a full handle on the dual W-point functions with the knowledge

of the spectrum and the couplings. It is of immediate interest to see if these functions can be interpreted as first-order approximations to the full S-

matrix elements. The N-point functions have the structure of free graphs and

to implement unitarity one has to construct also loop-graphs. The simplest

attempt to construct such graphs is to consider a tree graph and connect two

external states with a propagator' . This is called to "sew" together these

states. With the single loop we want to construct the right two-particle -52-

discontinuities. To obtain this it is essential to have only physical states coupling at the poles. The naive sewing procedure described above leads to ghost states coupling at the poles and hence wrong analyticity properties. The reason for this is that the propagator also propagates negative-norm states. These are decoupled in a tree graph by the gauge conditions and the fact that all external particles are on the mass shell. In a loop graph these ghost states can propagate around the loop without ending up at a state on the mass shell. This problem is similar to what

happens in field theories with gauge symmetries. For these field theories

the problem was solved by Feynman 59), who was able to construct single loops. Later the procedure was refined by Fade'ev and Popov who gave the complete Feynman rules. For dual models the construction of single loops were solved in papers VI and VII. ••£

A difference between the conventional field theories and dual models is the lack of Lagrangians for dual models (so far). This means that higher-order terms have to be constructed from the first-order terms that

we master. This is similar to Feynman's approach to gauge field theories. 59) Let us sketch the idea of Feyxunan's tree theorenr7 , which gives single loops in field theories in terms of tree graphs. We refer the reader to paper VI for a complete account of the theorem.

Consider a x

(Ap(x)) and A+ Green's function by

with the contours C^ in the k°-plane orientated with respect to the poles of the integrand as shown in Fig. 5.6 (P.T.O.) -53-

A C+ * ©> + • & •

Fig. 5.6a Fig. 5.6b

These functions are related by

AR(x) - Ap(x) +A+(x) (5-73) or pictorially

Fig. 5.7

We remember that in a perturbation expansion the Feynman propagators should be used. Inserting (5-73) into the integral

k fd x AR(x) leads to the diagrammatic equation

Fig. 5.8

If we forget about problems of ultraviolet divergence (which disappear when we consider single loops with three or more internal lines), (5-71*) can be shown to vanish by the retardedness property, since one or other of the two factors in the integrand is always zero. A line with a A -propagator is always on the mass shell so the diagrams with such an internal line correspond to tree diagrams. In this way we have related a single loop to sums of tree graphs. This is Feynman's tree theorem.

The real importance of the tree theorem is revealed when we consider

gauge-field theories or dual models in which the tree graph expressions

are known for external states on the mass shell and physical and in which

the physical states satisfy a bootstrap property: the totality of physical

states completely factorizes the residues of the poles in the tree graph amplitudes. As we said above, the propagators in such theories propagate a larger spectrum than the physical spectrum, but the unphysical part is eventually annihilated when it hits an external state. This is the case in a tree graph and the idea underlying Feynman's tree theorem is hence rather natural. Feynman's proposal is now to use the loop expression from the tree theorem for x -theory to define the single loops for gauge-field theories, since it will satisfy the desired analyticity properties, provided when one joins legs of trees, one sums only over physical i states, i.e. uses a physical state projection operator.

The same idea was followed in paper VI to construct a single planar loop. In dual theory we have to distinguish between planar and non-planar graphs. The reason is that the dual vertex has only cyclic symmetry which means for example that diagrammatically 2 3k 2 k 2 k 5

Fig. 5-9

The order 1, 2, J, kt 5 can be restored by introducing a twisting operator

(denoted diagrammatically by a x on the propagator), which reverses the order of the particles (P.T.O.) -55-

3 U

Fig. 5-10

Hence the diagram in Fig. 5.9 can "be written as 2 %k 2 ,3 J _ A

Fig. 5.11

For tree graphs the twisting operator is superfluous, since it only means

a permutation among the external particles, but when going to loop graphs it is

needed in order to write all topologically different loops. We call a loop with p, twists a non-planar loop. There exist at least four different operators in the literature that

accomplish the operation in Fig. 5'10' The reason why different operators can

do the same job is due to the gauge symmetry. For tree graphs the difference

.4 between the operators can be gauge transformed away. By sewing together twisted •i tree graphs, however, one will get different results depending on which twisting

operator is used. This is another indication that the tree theorem is natural

to use for the construction of loops since the effect of all four twisting ope-

rators are the same in tree graphs. We only have to decide which is the most

suitable twisting operator to use. To get calculations which are as similar as

possible for the construction of planar and non-planar loops it is found in

paper VII that the best twisted propagator to use is

,1 tox10"1 (-1)H (5.75) o x

This twisted propagator is originally due to Susskind

With this choice one has, however, to sum all diagrams that can be reached

by dualizations of the twisted lines, which diagrammatically means, that one has ;<:| -56-

to count diagrams

k Fig. 5.12 separately. The tree graph is defined to have the right poles in all planar channels. With this twisting operator this is only accomplished after the above mentioned summation. This leads to more complicated Ward identities. We remember that Eq.. (5-52) is valid for untwisted propagators and vertices and leads to Ward identities for each graph separately, which is convenient when the tree theorem is used. With the twisted propagator

(5.75) the situation is more complicated since the corresponding Ward identities are only valid when the sum over all dualizations of twisted lines m is performed. This is similar to the case of a Yang-Mills field theory in which the Ward identities are valid only order by order in a perturbation expansion and not graph by graph.

The expressions for the single loops can now be evaluated. The calculations are typical for many calculations in dual models. They are very involved but by the great underlying symmetry the final results turn out to be relatively simple due to many cancellations in the final steps. The final results differ from the results from the naive sewing procedure by two inverse powers of a certain partition function and for some twisted loops also by a change in the integration region. The planar single loop is strictly speaking infinite, so the reliability of the calculations 'i J might be questioned, but it is hoped that it will be possible to remove this divergent contribution.

There is one twisted loop of particular importance. That is the loop

s •*•

Fig. 5-13 -57-

The s-channel can be seen to have quantum numbers of the vacuum and carries also a new singularity. The properties of this singularity nafce it very tempting to identify it with the Pomeron singularity. For arbitrary dimensions of space-time this singularity seems to correspond to a unitarity- violating cut, but for d=26, the procedure above instead gives a set of faotorizable poles corresponding to a trajectory

= 2 + | (5.76)

62) This result was actually conjectured some time ago by Lovelace . We remark that the slope of this trajectory is half the slope of the ordinary Regge trajectories in rough accord with the experimental information that one has. The ratio of the intercepts of this Pomeron trajectory and the ordinary Regge trajectory is two which also seem to be correct. However, toth these intercepts are too high.

The fact that this singularity seems to violate unitarity unless d«26 is one of the main reasons why we only consider d=26.

I The diagram in Fig. 5.13 can now be dualized into

Fig. 5. 1 \ It was found,63 )' that this diagram can be factorized as

Fig. 5.15

where the wiggly line corresponds to a Pomeron propagator coupled to the external legs via reggeons. This Pomeron propagator is very similar to the propagft- »

tor of the ^irasprp^ShaEirp^ model6h) , which is a dual model very similar to

the Veneziano model, but which does not have planar duality. It has poles

in all channels, so the model could only be used to describe scattering of

particles without quantum numbers such as the particles on the Pomeron tra-

jectory. By using the projection operator for the Virasoro-Shapiro model, 55) Olive and Scherk found that the Pomeron sector was completely factorized i,

by the transverse states of the Virasoro-Shapiro model. Hence the residues i '

at the Pomeron poles are built by positive-norm states and these poles are [i'.

fully compatible with unitarity. >:

When constructing the loop graphs, it was only possible to ensure that [: the right states couple at the poles of the propagators that can be seen I in the graph. This means that we know little about the duality properties [;- of these loops. We have seen that the dualized form Fig. 5.15 has the correct Pomeron poles, but it must also be checked that the Reggeon poles factorize correctly. This was done in paper VIII. The idea how to do this is simple. We insert a physical state projection operator in a residue and check that it gives unity. It is, however, technically very difficult.

The reason is that the gauge conditions for physical states involving Pome- ron propagators are much more complicated than those appearing in normal tree graphs. These gauge conditions were found in paper VIII. While an ordinary tree state on the mass shell satisfies

llll ; = ° (5.77) n the state with a Pomeron propagator satisfies

(L + ! c L ) >s n , n -ffl I «•—»•• MM ° (5.78) m=1 where the coefficients cm are quite complicated numbers. It is somewhat surprising that there are different gauge conditions for different physical states. This is a new feature that does not occur in gauge field theories. -59-

'' With (5.78) it is, however, possible to prove that the projection operator in ; :| the residues of the reggeon propagators acts as the identity operator so that I-'. 1 the pole structure of the dualized diagram Fig. 5»15 is compatible with uni- ' :4 tarity. This is a check that duality and unitarity are compatible with each

[/-f other. r4 There are still some difficulties in the case where a Reggeon mass shell !-:-ij coincides with a Pomeron mass shell. To get a rigorous proof of no ghosts in

[:*f this case one has to find a regularization procedure, which has not been found j;.| so far. We feel, however, optimistic and believe that this can be done. T','3 f I These studies of loop-graphs have shown that there exists most certainly I a perturbation expansion for dual models and that duality and unitarity seem 1 . ... to be compatible with each other to each order in the perturbation expansion. There are still much more one could check. We have not yet examined if all channels in the loops have the correct analyticity properties but everything we have checked so far indicates this be the case. It would also be interesting to construct multi-loops with the correct analyticity properties, but this has not either been done yet. Although technically very complicated we feel that all this is doable. We have to remember, however, that the Veneziano model is not the ultimate model for strong interaction. We believe it is close to reality and hope that the general features will survive in a future better model. Hence the motivation for the work we have done is more to acquire confidence in the approach rather than to investigate all details of the model. There exists yet another good dual model . The interesting thing now is to investigate if this other model has the same general properties as the Veneziano model. If this is the case it will strengthen our belief in the approach. In the next section we shall, therefore, analyse this other model more in detail. -6o-

5.6 The Neveu-Sehwarz model

It took almost three years of intense studies of the Veneziano model

until a new dual resonance model with planar duality was found. We have

seen that the states in the Veneziano model are constructed with a set of

operators with commutation relations. It was then natural to ask if one

could enrich the spectrum and improve the model by introducing operators

with anticommutation relations. Neveu and Schwarz found that this

was possible and the model they obtained is surprisingly close to what

might be a correct model for strong interaction.

The model has a Fock space generated by the operators (5.28) and in

addition a set of operators that commute with the a's and satisfy

V : - - «T V "•V = °-1 «-' (5.T9) n,m = 0,1,2,

The model is constructed entirely in terms of these operators, i.e. one constructs a vertex and a propagator with the a- and b-operators in such a way that these operators have the same symmetries as in the Venezi- ano model. The scattering amplitudes constructed in this way can then be shown to satisfy the general requirements we imposed on a dual amplitude.

Besides the Pubini-Veneziano fields Q (x) and ^(x) one introduces a new field.

• b" z-Q-1/2] (5.80)

Furthermore, in order to construct Virasoro operators L we split these operators into two parts

Ln " Ln& + Lnb (5-81) d "61"

j i The operator L a was defined in (5-3*0 and L is found to be ! n n

We also introduce another set of operators

' G = #1^ zr P(z) • H(z) (5.83)

These operators satisfy the algebraic relations

We remark that L will turn out to be related to a Hamiltonian just

as in the Veneziano model, and we conclude that there is no coupling between

the a- and b-operators. In a space-time picture of the hadrons this corresponds

to no spin-orbit coupling.

With the construction of L we know the structure of the propagator.

We now seek a vertex with the right commutation relations with the L 's. Such

i a vertex is found to be

V(k,z) = k-H(z) VQ(k,z) (5.85)

where V is the Veneziano model vertex.

An N-point function can be constructed in the following way

(5-86) -62-

This amplitude can be shown to have cyclic symmetry. The physical states

\$> coupling at the poles is seen to satisfy

(Ln - | «no)l

G | 0 (5.88) r

Hence, there are two infinite sets of gauge conditions for this model. This

is hopeful, since we have introduced a new set of ghost states in the Fock

space as compared to the Veneziano model.

When the Neveu-Schwarz model was first written down, it was written as

(5-8s"

This can be shown to be equal to (5*86). One usually says that (5«8°J is written in the ^..-space and (5.86) in the IPp-space indicating that the full

Pock spaces of the two amplitudes are different. There are more states in the JT.-space than in the y2-space. These extra states are, of course, decoupled in the amplitudes. The dual properties are easier to investigate in the 7*-space, while the properties of the physical states are more transparent in the 72-space. This is typical for gauge theories. We are really considering the amplitude in two different gauges, and different gauges display different properties more clearly.

The spectrum of the Neveu-Schwarz model is much richer than the one of the Veneziano model. The spacing between trajectories is 1/2. There is further in the theory a new conserved quantum number which can be thought of as a G-parity. Hence one can separate a ir-trajeetory and a p -trajectory. A remarkable thing is that the lowest state on thep-trajectory, -63-

which is a tachyon and hence unphysical, is decoupled from the theory. There 2 is, however, still a tachyon in the theory since m^j* -1. One can summarize

the spectrum by saying that the experimentally seen mesons that can occur in

TTW-scattering are included in the model but the trajectories with normal-parity couplings have intercepts that are 1/2 unit too high, while the trajectories with abnormal-parity couplings are correctly located.

The four-point function agrees with the Lovelace-Shapiro form (5.11*).

Even though the spectrum of the Neveu-Schwarz model is rather different from the spectrum of the Veneziano model, the algebraic structure of the two models is very similar and it is not surprising that Goddard and Thorn and 57) Brower could extend their proofs of no ghosts to the Neveu-Schwarz model also. In the construction of physical states one now has two sets of transverse states, which is natural with the introduction of the b-operators. These states are the only states with positive norm if d=10. The model is ghost free if d $10. The close algebraic relationship between the two models is further strengthened, when one considers the projection operator from the full Fock space onto the transverse subspace. This was done in paper V. This operator is

^°"H (5.90) wherethe contour encircles the origin twice, since JC-H may have integral or half-integral eigenvalues. «C and H are defined as in the Veneziano model.

The calculation is performed in the Kg-space. For the critical dimension it can be shown that

ENS "= ^o -H = (V1> + f, (D-nVL-nDn> n=1 S (EG-G E) (5.91) r=1/2 r r r r -6k-

with L and G defined in (5.81) and (5.83) and n r

D 1 z k H(z) 2 n " / Ife *" k^(7T < " ' h (k'H(z))U-P(Z))- ) (5.92)

E f 1/2 n - ffc '* l^(7)p/2 !l {k-H(Z)(k.P(z))- } (5.93)

As in the other case the projection operator is essentially constructed

from gauge operators and it is now an easy task to check that the operator acts as the identity operator in the residues.

It is also possible to construct unitarity corrections for this model.

In paper VI the single planar loop was calculated. It can be done with the same technique as was used for the Veneziano model. One starts with the expression from the tree theorem and the further calculations follow step by step the calculations in the first model. The results differ from a calculation with the naive sewing procedure by two inverse powers of a function, which is the corresponding partition function for the system.

Once again we stress that the algebraic procedures in these two models are almost identical, while certain results differ considerably. This is for example the case with the critical dimension and the spectrum. This strengthens our belief that the algebraic properties that we deal with are very general and could survive in a future, hopefully correct model for strong interaction. These properties are also the ones that are physically the most reasonable, as we have seen and it makes it really worthwhile to focus the attention on these matters. The unphysical properties of the models, such as the critical dimensions and the wrong mass spectra are very

model dependent. Therefore one might hope that a more realistic model will be free of these pathologies.

One could continue the investigation of the Neveu-Schwarz model to construct also the Pomeron and to check its properties etc. However, there is another aspect of this model that is more interesting and important. The -65-

model allows us to introduce fermions '\ which will be dealt with in the next

section. We conclude this section by noting that the Pomeron from the naive 68)

sewing procedure has been constructed , and if one extrapolates the know-

ledge from the planar loop one can conclude that the Pomeron trajectory is the

same as for the Veneziano model.

5.7 Fermions in dual models

For the two models described one could think of the propagator as the inverse of a generalized Klein-Gordon operator. It is then natural to investigate if there is a similar generalization of the Dirac operator. Ramond found this generalization and this led to an amplitude for the scattering of mesons on a fermion. To describe the model, let us introduce the field

n Y I d/ z" ] (5.9U) •* n=1 where yv and y are the ordinaru Dirac Y-matrices and d and d are anticommuting annihilation and creation operators that commute with the a's. They are analogous to the bfs in the meson sector.

r = V> J tVi _ _ W« p»v = 0,1,...,d-1

n,m = 0,1,2, (5.95)

With this field we can construct gauge operators just as for the Neveu-Schwarz model

a Ln - Ln • L/ (5.96)

= - i'ffe (5.97)

P(Z) -66-

These operators satisfy the algebraic relations I

(5'99)

(5.100) [v %] •

The generalized Dirac operator leads to a propagator (F - ) , where

the ground state fermion has mass m. For the emission of a scalar meson a

vertex can be found with the right transformation properties under the gau-

ge transformations. This is important, as we have seen, in order to have

no-ghosts. The scattering amplitude for two ground state fermions and N

ground state mesons corresponding to the diagram 12 3 K-1 N lit I ' I I I N+1 -0 1 J . J J. I. ....

Fig. 5.16 can now be written as

(2),

(5.102) F - ii

(8) U. (1) - r5 : (5.1O3)

where IV is defined to be

+ z dn -dn n 1 * (5.10U) -67-

and u and u are ordinary spinors. The upper index 2 for the vertex indicates

that the amplitude is written in the ?*p-space, or rather this form is the one

that connects to the Neveu-Schwarz model in the J^-space. A corresponding

formula for the amplitude in the yi -space is

^ (5.105)

(5.106)

and L is a redefined fermion -L such that o o

L=F2 = L - 4 o o o 1c (5.107)

The physical feraion states |> that couple in the residues of the poles satisfy in both descriptions

n > 0 (5.108)

= 0 m > 0 (5-109)

(po- > - o (5.110)

To obtain the vertex for emission of a ground state fermion one can dualize

Fig. 5.16 into 1 2 N-3 Nr2 N-1 i I 1 I I ' !

N+1

Fig. 5.17 -68-

This means that one has to find a vertex operator such that the expression

for the diagram in Fig. 5.17 agrees with the one for Fig. 5-16. In Fig 5.17

there are also vertices for emission of mesons from internal mesons. These

vertices are assumed to be the ones of the Neveu-Sehwarz model. We are

considering the amplitudes in the ¥ -picture since the duality properties

are exhibited most clearly in this picture. It is a very complicated

task to construct this vertex, but it was finally done by Corrigan and

Olive . It is a very complicated object. Let us denote the vertex

V_,(z). Then it can be written as

(5-111)

where

) >„ (s.m>

m=o n=1

<»•»»>

The (juestion arises whether ths right physical mesons couple to this vertex. In order to check this one can introduce a physical state projection operator in the residues of the meson propagator adjacent to the fermion vertex. To be able to check the effect of the projection operator we have to understand the gauge properties of the fermion vertex. This problem was analysed in paper IX.

To understand these gauge properties we define linear combinations of -69-

Ramond fermionic and Neveu-Schwarz mesonic gauge operators:

dx p<*> (5.116)

Ife •<*> (5-117) where r encircles the origin and one and r the origin tut not one. If (x) is expanded in a power series in x then F. and G. are seen to "be linear combinations of the ordinary gauge operators. The operators in (5-116) and (5.117) satisfy the anticommutation relations

(5-118)

(5-119)

where C_ and CQ are c-numbers

L, is defined in a similar way to (5.11 6) and (5.117) . From (5.120) it is seen that

(5.121)

69) Corrigan and Olive have shown that one gauge operator is given by

Lo ~ L-1 = L(1- -) (5.122) x

With the knowledge of (5.121) a good candidate for another gauge operator is then

\l K - 1,' (5.123) -70-

This is in fact a true gauge operator and the most general one is given by

(5.12U) where f(x) is a function analytic in a domain including the points 0 and 1 with the exception of possible poles at the origin.

The most general Ward identity satisfied by Vp(1) is then

f yo|\5 % - ^- d)] = 0

Consider the case when f(x) = 1. Then

(5'126)

We see that an incident G_ , is partly transmitted to F_ "s and partly re flected back as G 's, where n and r are non-negative. This shows a great similarity with the gauge conditions of the Pomeron vertex.

The vertex Vp( 1) was derived in the 7.-picture. With the help of

(5.125) and with the choices of (5.126) and (5.127) one can show that

=^-^11 Vp(D u(k)

(5-128) where |R,1> , |R,2> , |ws,1> and |lJS,2> denote Ramond and Neveu-Schwarz states in the Tr\- and J^-picture respectively.

Eq.. (5.128) shows that if m=0 one has to be careful to choose the picture in which the amplitude is non-singular. This can be shown to be the

^-picture.

With this knowledge, one can now insert the Neveu-Schwarz projection -71-

operator into a residue adjacent to the fermion vertex and show that it acts as

a unit operator provided the ferraion mass m is taken to be zero. This shows

that the Ramond- and Heveu-Schwarz-models are compatible with each other.

This result opens the way for the construction of multi-fermion amplitudes.

We know the vertex and the spectrum of states that is propagated and can in

principle calculate the amplitudes. However, the vertex Vp is rather complica-

ted. This could mean that the gauge in which we are performing the calculations 70) is not the most suitable one. We can now calculate the 4-fermion amplitude ,

but the calculation of other amplitudes looks very complicated and it is cer-

tainly worthwhile to try to find another gauge where the calculations are simp-

lified. This is a subject which is presently pursued.

5.8 The string interpretation of dual models

Even if the Veneziano model emerged as an S-matrix theory, it would still be interesting to investigate if there is some space-time description of the hadrons hidden in the model. Shortly after the construction of the N-point 71) amplitudes, Nielsen, Nambu and Susskind , independently suggested that the dual hadrons behave as strings, i.e. one-dimensional objects. We can interpret the harmonic creation operators a *^ as operators that create excitations, phonons, on the string. A direct problem then is to understand the time components that create excitations with the wrong norm. This problem took a long time to resolve. Meanwhile the string concept was used in a non-rigorous way. One can think of a string as being built up by point-particles like a one-dimensional crystal. The data on deep inelastic electron-proton scattering is consistent 72) with the proton being built up by point-like constituents . Feynman calls 37) these partons . We can regard the string as the limit of infinitely many partons in a finite one-dimensional extension. The scattering of N hadrons can then be viewed as scattering among these partons, where the partons, however, must "remember" that they are sitting on a string. One can try to eva- -72-

luate this amplitude by using complicated planar Feynman diagrams (Fig. 5-18)

Fig. 5.18 where each line shows a parton propagating. The diagrams must be complicated (we are really taking the limit of infinitely complicated), since there are many partons participating and it must be planar since by cutting the diagram at a certain point in time should leave us with a string of partons.

If the partons are assumed to be scalar and their propagators to be Gaussian, which they would be if the momentum flow through them is small, then one can prove that one gets the Veneziano amplitude by summing all these diagrams properly. No one has proven that it is legitimate to use Gaussian propagators, hence we take this procedure rather as an algorithm to evaluate the

N-point function. Though non-rigorous these ideas have given some insight into the model. Paper X is devoted to a study with this type of spring picture, but before we discuss this paper let us describe the further development of the string idea.

In the early treatments of strings one started directly with an equation of motion for the string. In this way one could not understand the sub-

sidiary conditions, the Virasoro conditions (5«55)» which had to be used to

show that the ghost states are decoupled. A fundamental step was taken by

Nambu23^ who suggested that the action of the relativistic string should

be proportional to the area of the "world sheet" swept out by the string in

space-time. This is in analogy with a point particle whose action is pro-

portional to the length of its "world-line". There remains an arbitrariness

in the parametrization of the area which makes it possible to choose an or-

thonormal parametrization. These orthonormality conditions can be shown -73-

to be equivalent to the Virasoro conditions. The problem was very carefully

examined by Goddard, Goldstone, Rebbi and Thorn23 ) and we give a brief account

of their work.

Consider a two-dimensional surface X^(O,T) in Minkowski space. This is

the surface swept out by the string. Here a is a parameter along the string and

T a time-like or light-like parameter. We take the classical action of the

free string to be

/" dx /* in X (5.129)

We have chosen the parameters such that the ends of the string correspond to o=0 and a=7T and the initial and final configurations of the string correspond to fixed values of T , T- and i . The action (5.129) is invariant under all reparametrizations that respect the boundary values. This provide for invariance under a large set of transformations.. In particular we can choose a parametrization such that

If'IT"0 (5.130)

(|^) + (|^) = 0 (5.131)

This choice linearizes the equations of motion which then become

2 2 (g^JT ~ •jto?)xM(a>T) = ° (5-132)

If we take the boundary condition into account the solutions can be written in terms of normal modes as

v P y inT x (a,r) = qQ + a r + i f — cos no e (5-133) n=—» n*o -7k-

where q V and a V are all constants of motion. If we further specialize to o n a pararaetrization where

x+ = p+t (5.13*0

(5.135)

PU = Si da Jf (5.136)

one can show that

ao+ = P+ (5-137)

— 1 ft 1 °° a = —q:

a = (0,ai,a2,0) (5.138)

We see that only a 1, q 1, a amd q are independent variables (i refers to a transverse direction).

With these results one can show that the string has a leading Regge trajectory with slope 1/2 in our units and intercept zero. Furthermore it is easy to see that the ends of the string always move with the velocity of light, which is true in all parametrizations. The spin of the string is only built up by the angular momentum and stretching the string makes it more massive and hence increases the spin.

We have so far treated the string as a classical object. In order to quantize it one can introduce Poisson brackets among the independent variables (P.T.O.) -75-

{a \ a J} = - in 6 61J n m n,-m

(5.139) it >= 0

' ao+) =

From these the algebra of the dependent variables follows

. aj} = ima J n' m m+n

We note that the choice (5.13^) is a non-covariant one. However, one can con-

struct the generators of the Lorentz group and check that these o"bey the right

algebra so the physical results will be covariant.

In constructing the quantum theory of the string we rely on the correspon-

dence principle. That means, that we regard the dynamical variables as opera-

tors whose equal-T commutators are obtained by the rule

i { Poisson bracket } "*" [commutator] (5.1^1)

We can perform the quantization by letting the a V and qV become operators. A definite problem is then if we shall assign commutation relations to all a v n and q or only to the real degrees of freedom, i.e. shall we apply the constraints before or after quantization.

Let us start by quantizing only the real degrees of freedom, that is we

apply (5.1U1) to (5.139). We regard a , n > 0, as annihilation operators

and consequently a =a_ as creation operators. -76-

There is now an ambiguity in the ordering of operators. We define the quan-

tum mechanical operators as the normal ordered operators. This leads to a

possible c-number difference between the classical Xo and the quantum

mechanical one

(5.1*2) classical q.m.

In the classical case one can show that the mass of the string is given by

2 (5.1*3) M classical n= I

In the quantum mechanical case the substitution (5.1*2) gives

2 M (5.1**)

We denote by |0,k} the ground state with momentum k. The first excited state a? |0,k} has M "1-ct . On the other hand it has only transverse degrees of freedom, so if the theory is to be Lorentz invariant this state must be massless, i.e. o «1. Hence we see that the quantized string has the leading trajectory

«(s) - 1 +£p* (5.1*5) as has the Veneziano model.

To check the Lorentz covariance one has to examine the algebra of the

Lorentz generators. This has to be done with great care since these operators are bilinear in the a's. One has to ensure that the operators are

Hermitian. When checking the algebra one finds that it only closes when

ao"1 and d* 26 (5.1*6)

This means that the string with this type of quantization can only be con- -77-

sistent in 26 space-time dimensions (25+1) and with unit intercept for the

leading trajectory.

If we instead quantize all a's and q's then we shall impose the constraints

only for matrix elements between physical states. One can show that this im-

plies that the physical string states have to satisfy

Ln Iphys > = 0 n* 0 (5-iVf)

(LQ-ao) | phys > = 0 (5-1U8)

where the L 's turn out to "be the Virasoro operators (5«3U). This means that

the physical string states satisfy exactly the same conditions as the physical

states in the Veneziano model. It is not a priori clear that these conditions

exclude ghost states, but it was proven "by Brower and Goddard and Thorn

that this is the case if a = 1 and 1 < d < 26 or o < 1 and 1 < d < 25. For

the other quantization it is clear that there are only positive-norm states

as there are only transverse creation operators.

We see hence that in the covariant gauge (we refer to a certain para-

metrization as a gauge) one can relax the condition d=26, but only at the ex-

pense of introducing new states into the theory that have no classical analogue.

It is intriguing that the results from the two different quantizations do not

agree. It is difficult to reject d < 26 in the covariant gauge at this stage.

Eventually we want an interacting string picture and then d=26 might be more

obvious in all gauges. We recall in this context that the Veneziano model

seems to be unitary only for d=26.

From the free string picture we can hence understand why there are only

transverse degrees of freedom for d=26 and a =1 and hence no negative-norm

states for d ^ 26, but one can still not understand from it why the Veneziano model seems to be consistent only for d=26. To do this one probably needs a mo-

del for interacting strings. Such a model has been constructed by Gervais -78-

and Sakita^ ' and by Mandelstam^' using functional integration methods but these investigations are still not brought to a fully satisfactory conclusion. The attempt to construct a second quantized field theory for strings have so far been unsuccessful. It is, however, an important problem, since it seems to be the simplest realistic extension of conventional field theories which only describe point particles. It is also important since it can give insight into the question how to construct new and better dual models.

There exists now also a model for a free string with the same mass spectrum as the Neveu-Schwarz model constructed by Iwasaki and Kikkawa

It is, however, a very complicated model and it has so far not yielded much of concrete results.

If we work in the "transverse gauge", the gauge choice that seems most natural, we have seen that the theory is only covariant if d=26. It would be nice to get a better physical understanding why this is so. In paper X we consider the propagation of a string. We represent its Green's function in two ways. Firstly we use the technique with complicated Feynman diagrams and secondly we write this Green's function as propagating in all possible states with the proper weight. We specialize to a string corresponding to the physical vacuum locally all along the string. This corresponds to complicated graphs like Fig. 5-19

b |p(e)=e> - - - - -

a Pig. 5.19

where one has to perform the appropriate summation over all ratios a/b to

get the Green's function. If we now use that the diagram in Fig. 5-19 equals -79-

the diagram in Fig. 5.20

|p(e)»o> |p(e)=o>

Fig. 5.20 and since we have vacuum all along the diagram, one is led to a self-consis tency relation

2 0|n>|2 exp(-m = o|n>| exp(-ni n RJ n n

(5.1^9)

O 1 where (TTO1) ) (5.150)

Here |n> denotes all the states of the string and the transformation (5.150) is called the Jacotoi imaginary transformation when it is applied to a parameter 77) of a 6-function .

If we specialize to the "Nambu-string" discussed above and consider it in the "transverse" gauge, we can solve the matrix elements in (5.1^9). We can

then compare with the Jacobi imaginary transformation on &1(0,T) and find that our self-consisting equation is essentially satisfied if

2 d-2 (5.151) mo =" where m is the mass of the ground state. We saw before that the spin-1 state with only transverse degrees of freedom has to be massless giving -80-

m2 = -h (5-152)

and hence 6=26.

We have here shovn that only when d=26 the string can propagate pro-

perly. We also remark that the Jacobi imaginary transformation plays an

important role in loop-diagrams. Here we can see it as a basic duality re-

quirement .

We should also remark in this context that the use of "complicated"

graphs can be justified from the more rigorous string picture.

The fact that the dimension d is intimately connected to the mass of

the ground state, led us in paper XI to investigate the quantum mechanical

zero-point fluctuations.of the string. It was shown that the energy due to

zero-point fluctuations is given by Ezero " <**>£ i i> h (5-153)

in the "transverse" gauge. E is the total energy of the string. This sum diverges. To be able to handle the sum we introduce a cut-off that respects the gauge symmetry and write H- fefef 'If' 'I' where f (x) is unity for x not too high and f -»> 0, when x •*• « fast enough for 'B^~M to be finite. If we now calculate the mass of the ground state we get

= ?? E - §&• (5-155)

The first term is non-covariant. To restore covariance we have to change the classical action that we started with i» a non-covwiant way. By per- -81-

forming the change

x°+ (1 +e) x° (5.156)

in the JTambu Lagrangian (5.129) one can show that one can find an e such that

the non-covariant term in (5.152) disappears and one is left with

that is, the same result as in (5.152). This calculation shows that the mass

shift is due to the zero-point fluctuations. It also shows that in order to

have a covariant quantum mechanical theory one has to start with a non-covari-

ant classical one.

The summation over n in the sum in (5.153) is related to the sum over all frequences that the excitations on the string can have. This is clearly strongly dependent on the boundary conditions on the string, so we can directly associate the critical dimension to how the ends of the string behave.

We can perform similar calculations for the "Neveu-Schwarz string" and the

"Ramond string". The argument is clearly heuristic since the extra fields in these models are anticommuting and the classical analogue is more difficult.

We can understand the Neveu-Schwarz and Ramond excitations (the excitations due to the b-operators and d-operators (5.79) and (5.95)) as excitations due to a generalized local spin-spin interaction along the string. Besides these excitations the strings also have the ordinary orbital excitations. One then gets for the contribution from the Neveu-Schwarz modes

E (d 2) ( zero,NS " ' Ji/2 -2"> *' If

While the orbital modes are such that the waves have a node at both ends of the string, the Neveu-Schwarz modes have a node at one end and a maximum at the other end. This explains why the summation is over half-integer n. -82-

In analogy with the calculations for the orbital modes we only keep the

covariant part of (5.158), and we can show that it gives an additional con-

tribution to the ground state mass

2 AmQ HS = - |g|r (5-159)

Adding this to the contribution from the orbital modes we get

m 2 „„ = - 4^r (5.160)

Once again we can use the transversality of the spin-1 state to con-

clude that ds»1O.

If we consider a corresponding "Ramond string", we assume that this is

a string almost identical to the "Neveu-Schwarz string" with the exception

that the total spin equals a half-integer. This leads to spin waves with nodes at both ends of the string and

E = (d 2) Zero,Ramond " J1 (- \) h ^ * <|> (5-160

This sum is seen to be equal to the contribution from the orbital modes

(5.15*0 but of opposite sign and we conclude that the ground state fermion

mass is zero.

We take notice of the fact that these results are rather strong. We have taken care of the orbital interaction and the spin-spin interaction, and it is probably difficult to change these and still keep the full gauge invariance. One can try a spin-orbit coupling in the Hamiltonian, but that would not alter the results above, since it is built up by commuting operators. One way to find a new model could be to introduce asymmetric boundary conditions. This could be achieved through a spontaneous breakdown or by putting different quarks at the ends, two alternatives which might turn out to be the same thing. An intriguing observation is that the -83-

mass squared of the Neveu Schwarz ground state is missing r—7 to reproduce

the pion mass squared (which we take to be zero), while the mass squared of the 3 Ramond ground state similarly is missing j—T to reproduce the nucleon mass

squared (which is very close to TTT) • One can imagine that a quark or anti-

quark adds —j—, to the mass squared if put on the string, and that we shall add

one quark and one antiquark to the "Neveu-Schwarz string" and three quarks to the "Ramond string". How this can be done we do not know yet. The word quark

is in this context used for something unknown. It should probably be interpre-

ted as some new kind of excitations on the string rather than as a particle.

It is most urgent to analyse these questions within the string picture to arrive at a more satisfactory model. If we can find new excitation modes which provide for the correct mass spectrum we may simultaneously find a way to incorporate the internal SU(3) symmetry that experiments bear witness to. We would then have a model which represents a most significant step forward in our understanding of the strong interactions. -8U-

6. Attempts to construct nev models

Since the advent of the Veneziano model there have been several attempts

to understand the underlying dynamics of duality and to find schemes to con-

struct new models. One of the basic features of the Veneziano amplitude

first noted *' was the Mobius symmetry of the amplitude, its invariance

under the SU(1,1) group. This invariance is quite difficult to understand

from a dynamical point of view. SU(1,1) is a subgroup of SL(2,C) and if the

invariance had been of the latter type then one could try to connect the in-

ternal invariance group to the invariance under the external Lorentz group.

Soon after the original Veneziano amplitude had been found, another dual

U-point function was constructed by Virasoro . It was later generalized

by Shapiro to an arbitrary number of external particles. The internal

symmetry group for this model is actually SI»(2,C). This observation made

it interesting to attempt formulation of models from a group theoretical

point of view since so much work has been done on SL(2,C) invariance. Pa-

per XII is such an investigation applying the technique of so-called homo-

geneous spaces.

A homogeneous space H of a topological group G has the following pro-

perties:

(i) it is a topological space on which the group G acts continuously,

i.e. let y be a point in H, then y* = g y, geG, is defined and

belongs to H and y •+ y' is a continuous transformation

(ii) the action is transitive, i.e. given any two points y and y_ in

H, then it is always possible to find a group element geG such

that y2 = gyr

One can show that there is a one-to-one correspondence between the

homogeneous spaces of G and the coset spaces of G.

Usually one constructs wave functions with the Minkowski space as the

carrier space, i.e. the wave functions are functions of the four-vector xv.

This is the homogeneous space corresponding to the coset space P/L, where -85-

P is the Poincare group and L the Lorentz group. One can, however, choose

other subgroups of P than the Lorentz group to construct homogeneous spaces.

All these choices are described in paper XII. By choosing smaller groups one

can construct homogeneous spaces with more parameters than xH. In paper XII

we chose the simplest nontrivial extension of the Minkowski space. It corres-

ponds to a space which is the topological product of the Minkowski space and

a space, which topologically is a sphere. This looks very similar to the Sha- 6k) piro-Virasoro model , since the integration domain for the Koba-Nielsen like

integration varables in this model is a sphere.

Since hadrons have a rich internal structure it is tempting to try to use

the non-Minkowski part of the space as a carrier of this structure. This could

be a natural way to describe particles with an internal structure. The homo-

geneous Lorentz group acts in a natural way on the homogeneous spaces. This

means that one can construct wave functions, which depend on the variables of

the homogeneous space and which transform in a covariant way when subjected to

Lorentz transformations. With the help of these wave functions one can construct

relativistically invariant scattering amplitudes by multiplying all the wave

functions of the external particles with a suitable kernel depending on all the

parameters and then integrating over all the parameters'°'. For the choice

of homogeneous space mentioned above one can show that a natural choice of amp-

litude for the scattering of N scalar particles is

s,...d-z J1 Iz.-z.l " -. -" 2 ,,., 1 n i

Z. o±. • ^ (6.2)

Z g. . = 0 0.. integer-valued (6.3) -86-

The integration extends over the whole complex plane.

/This amplitude can depend on the kinematical variables only through the

exponents a. Lorentz invariance together with (6.2) then implies that ex. . ij can only depend linearly on quantities like p.«p. and we write 1 o

(6.U)

The Shapiro-Virasoro model is now obtained apart from a scale factor by choosing

2 2 2 j d z

where z , zfe and z are any three of the integration variables.

Let us next consider the U-point function within this scheme. It can be given in closed form

r(i-a12/2+B1g/2) T

(6.6)

It is seen that this amplitude has poles for

ct12 - 2 + |B12| + 2H N=0,1,2,... (6.7)

and similarly for a and a , . Hence the amplitude has poles in all channels and does not satisfy planar duality. By choosing the ot..°'s and B..'s properly we can, however, fix the energyvalues at which the-string of poles -87-

start in each of the three channels. We can even let the poles in one channel

start arbitrarily high up in energy and recover a Beta-function as a limiting

case. This procedure seems, however, difficult to generalize to an N-point

function and we have not found a way to implement planar duality in general.

We have also factorized the amplitude (6.1) after having divided out the

Haar measure (6.5) using an operator formalism very similar to the one by which

the Shapiro-Virasoro model is factorized . We use an extra quantized space

dimension to carry the extra degree of freedom. The Fock space is hence built

up by two d-dimensional sets of harmonic oscillators (5.28) and the vertex is the

product of two ordinary Veneziano vertices, one for each set of oscillators. One

can then prove that this is a ghost free amplitude. However, this factorization

is rather restrictive, even if it allows for some asymmetry between the channels, and there might well be a better factorization that allows greater freedom. This line of research has not been explored further.

We remark that the scheme above includes in a natural way scattering amplitudes for particles with spin. However, the factorization of such amplitudes seems a very difficult problem. So far it does not seem to be worth the effort since the scalar amplitudes fail to be better than the dual amplitudes constructed within alternative schemes.

The string picture of dual models discussed in section 5-8 makes it seem more natural to interpret the SL(2,C) symmetry of the Shapiro-Virasoro model as due to the parameter invariance rather than due to Lorentz covariance. However, the attempts outlined above were the simplest non-trivial amplitudes that one can construct with the approach based on homogeneous spaces. In order to pur- sue this idea further we have investigated the nucleon form factors that one obtains by considering field theories on a homogeneous space. The results are given in paper XIII.

We have already noted that the method discussed here leads naturally to wave functions for free particles. Similarly we can perform a second quantiza- -88-

tion and construct free fields that depend on all the parameters of the

homogeneous space. We write a free field as t(x,z), where x denotes the

Minkowski parameters and z the additional parameters of the inner space.

We can then construct a Lagrangian density for this field as

•C (x, (6.8)

where (6.9)

If M are the generators of the Lorentz group S are given by the identity

M = i x 3 -ix 3 +S (6.10)

The S are linear in the derivatives of the parameters of the inner space.

We demand <£ to be linear in the derivatives of the Minkowski space parameters and in the parameters of the inner space in order to get an equation of motion, which is at most of second order in the derivatives. This is because we want to avoid all the problems with higher-order equations of motion leading, for example, to indefinite metric.

With this restriction we can construct the most general Hermitian La- grangian density. We can also generalize the functional variation of the action integral to give an equation of motion. We introduce the electromagnetic interaction through the usual minimal coupling technique.

3 -> 3 - i e A (x) (6.11) but to do so we must first bring the Lagrangian density into a form where it only contains the parameters of the Minkowski space. The obvious choice of a Lagrangian density on the Minkowski space is then

.C(x) = /dz£(x,z) (6.12) -89-

vhere we integrate over the parameters of the inner space. We then consider

electron-nucleon scattering where the nucleon Lagrangian corresponds to the one

above and the electron Lagrangian is the usual local one. The electromagnetic

interaction is introduced through the substitution (6.11). We consider the one-

photon exchange contribution and derive from this an expression for the nucleon

form factors. This was all done with a U-dimensional inner space, constituting a slight generalization of the case previously discussed. On this larger space we used the most general free fields.

The result of this investigation is that we get non-trivial form factors but they do not seem to represent observations in a satisfactory way. For one thing we find, that one cannot incorporate the inr-threshold and hence the analytic properti form factors are non-physical in this respect. This means that no matter how general free fields we use, there is an essential piece of the strong interaction missing. The result of this investigation is not encouraging in this respect.

Let us finally mention another attempt to find a scheme for the construc- 81 \ tion of new dual models. It was found by Nielsen that the duality transformation mentioned earlier could be interpreted in terms of an associative algebra.

Consider the dual vertex

a*b

Fig. 6.1 We define the state that the initial states a and b create as the product a-b.

Consider then the diagram

b c

a«b (a*b)»c

Fig. 6.2 -90-

We perform a duality transformation to get

Fig. 6.3

With this definition of a product we have found that

(a-b) *c = a- (b-c) (6.13)

This means that one can formally construct an associative algebra. (We

note in passing that the algebra is not commutative because of the cyclic symmetry of the vertex.) These algebras are mathematically well known and thoroughly investigated . This makes it interesting to try to deduce as much as possible about the vertex (in general the three-reggeon vertex)

from pure mathematical considerations. This could indicate the limits within which a dual model can be defined. An obvious difficulty is that the physical operators that appear may not have all the required properties to make the mathematical theory applicable. In order to see if the approach is meaningful we investigated in paper XIV the mathematical properties of the operator formalism of the Veneziano model. In most mathematical formulations of an algebra the operators are defined everywhere in a

Hilbert space or at least on a dense set. Our study of the Hilbert space properties of the operators in the Veneziano model was carried out before we knew that the physical states indeed span a Hilbert space and hence we did not use the ordinary inner product but had to define a suitable scalar product. However, this does not change our conclusion that the operators are quite badly defined as mathematical objects. For example, the ground state vertex is not defined anywhere within the Hilbe^t space. This is a reflection of the infinitely many degrees of freedom. This result does 1 -91-

not imply that the operations we have performed with the operator formalism in

section 5 are incorrect. To get a physical quantity we take always matrix ele-

ments and these are certainly well defined. It is only if we try to identify

the physical operators with elements of a "nice" mathematical operator algebra

that we can make mistakes. This result make this approach less attractive

since in most cases one cannot apply the strong mathematical theorems known

for associative algebras. We note, however, that with similar ideas Dethlefsen 81) and Nielsen proved that the ononl]y symmetries compatible with duality were

SU(N) where N can be any integer. -92-

7« Conclusions and speculations

The successes of dual models are indisputable. For the first time we

have models that lead to an S-matrix which seems to have all the desired

physical properties except for an additional pole corresponding to a par-

ticle with imaginary mass. Hence these models do not yet describe the

real world. It is a matter of faith to speculate about how far we are

from the correct theory describing the real world. In order to make further

progress we must understand clearly what is missing in the present picture.

It seems impossible to construct new and more realistic models by just add-

ing new degrees of freedom as was done going from the Veneziano model to

the Neveu-Schwarz-Ramond model. However, it may well be that the latter

model already provides for the right degrees of freedom. Schwarz has in a

recent publication ' pointed out that by reinterpreting some of the degrees

of freedom one has a model with critical dimension k, internal SU(3) symme-

try together with the right quark statistics. This model still has some

defects, however, since it involves a tachyon state. This can be under-

stood from the calculations of the energy due to zero-point fluctuations

which are not changed by the reinterpretation. The model also has an exact

rather than a broken internal SU(3) symmetry, since this symmetry is ob-

tained by multiplication by external factors. Introduction of a broken

SU(3) symmetry into dual models has been an outstandingly difficult problem

so far. There seems to be a strong correlation between the existence of

the tachyon state and the absence of the broken SU(3) symmetry. It would

be most helpful if one could prove that only a broken SU(3) symmetry can

completely remove the tachyon. Such an attempt might lead to a better understanding how to formulate the problem in the best way. We must extract those properties of the existing models that we believe are true also

in a correct model for strong interaction and try to incorporate them in a fully satisfactory theory.

It may be impossible to find the correct model directly on the basis -93-

of today's knowledge. Instead we ought to try to analyse the remaining diffi-

culties leading to non-physical consequences. For one thing one must under-

stand how one can escape the transversality of the states which requires the

p-meson to be massless. In gauge field theories one can give the vector meson

a mass while still keeping the symmetry by means of the so-called Higg's

trick°^'. This amounts to using a scalar massless particle in a suitable way

to implement the missing degree of freedom of the vector meson. One can try

a similar method for dual models but it turns out to be very complicated. This

is, however, a very important point to understand.

The restrictions on a dual model are so strong that a successful introduc-

tion of additional physical features might also resolve other difficulties within the present schemes. This could mean that there are no non-trivial intermediate stages left now if we are on the right track to find the correct model. It seems, for example, unlikely that one could introduce a shift in the vector meson mass without also introducing the broken SU(3) symmetry.

If the strong interactions are properly represented by a dual model this will certainly mean that for practical reasons we will never be able to calculate all details of the cross sections. Many of the observed dips and bumps will remain unaccounted for, since numerically it will be very difficult to calculate higher terms than single loop graphs, and more than 5 external particles will certainly lead to great numerical difficulties when checking theory versus experiment. It also seems unlikely that one will be able to calculate for example multiplicity distributions exactly. How can one then decide when one has found the right model? One must clearly have a fully consistent theory with (i) the right mass spectrum (as far as one knows experimentally) and (ii) the critical dimension k, which (iii) explains the internal. SU(3) structure and where (iv) the Born terms give a decent description of the strong processes. Unitarity corrections will, of course, change the mass spectrum.

The hope is that this change essentially consists of the introduction of an -9k-

imaginary part that puts the poles onto the second Riemann sheet while it

contributes little to the real part. This must be checked if possible.

The unitarity corrections will most likely be important at least locally in

the phase space. We need just remember that the Pomeron singularity comes

out as a unitarity correction.

As previously stated in a correct model the SU(3) symmetry must be

explained. We know that simple-minded quark models work quite well to de-

scribe the mass spectrum of elementary particles. Will a correct dual mo-

del explain what a quark is? This is, of course, impossible to know now.

We have for some time speculated about the hadron as being a string with 85)

something attached to the ends that carry the quark quantum numbers

This could be a gauge dependent picture. In another gauge this structure

might be washed out. This is, in that case, an explanation why free quarks

cannot be found in Nature. We hope that a correct introduction of inter-

nal quantum numbers will explain the successes of the quark concept.

The next few years will certainly be decisive as to the question

whether the dual model approach is the correct one to the strong interactions.

As a final question we ought to ask ourselves, we ask what will be the bene-

fit if the strong interaction one day is solved. If the space-time struc-

ture of the hadrons is known this will certainly lead to a knowledge how the

universe was created. In this context it will be very important philoso-

phically. The technological advantages are more difficult to foresee.

Those depend so much on the detailed structure of the strong interactions.

Since knowledge of the strong interactions most probably will radically change

our understanding of the submicroscopic phenomena, this could open up a new world to explore, the outcome of which is unforeseeable. Let us end by re- minding that the understanding of Planck's constant h opened up the way

from classical mechanics to quantum mechanics. Will an equally fertile

field open up when we understand the constant a1 that gives the scale for

strong interactions? -95-

Acknowledgement

I am deeply indebted to Prcf. Jan Nilsson, who introduced me into elemen-

tary particle physics and whose continuous support has been invaluable for me.

I have been fortuneate to collaborate with many scientists, whom I wish to

thank. Especially I want to thank Dr. David Olive, whose critical mind and

deep knowledge.in physics and mathematics have meant so much to me, and

Dr. Holger Bech-Nielsen, whose equally deep knowledge and physical intuition

have taught me the importance of letting the mind try intuitive ideas. Most

of the research that this thesis is based on was done at CERN. I am very grate-

ful to have had the opportunity to work there. The financial support from the

Swedish Atomic Research Council during my years in Sweden has been of great

help. Lastly I want to thank Mrs. Britta Bonde for her efficient typing of the manuscript. References: =j L- 1. E. Rutherford, Phil. Mag., 2J., 669 (1911) [ f 2. I. Curie-Joliot and P. Joliot, Compt. rend. 19^, 273 (1932) \ 3. Chadwick, Proc. Roy. Soc. (London) A136, 692 (1932) \ W. Heisenberg, Z. Physik 7J_, 1 (1932) \ 3. CD. Anderson, Phys. Rev. k3, k91 (1933) ;: k. 0. Chamberlain, E. Segri, C. Wiegand and T. Ypsilantis, jr Phys. Rev. _KK), 9^7 (1956) \ 5. H. Yukawa, Proc. Phys. Math. Soc. Japan JX, U8 (1935) | 6. C.M.G. Lattes, H. Muirhead, G.P.S. Occhialini and C.F. Powell, \ Nature Jj>2, 69k (19V7) f; ' 7. For a complete account of all detected particles, see Particle Data £ f Group, Reviews of Mod. Phys. j+5_ Wo 2 part II (1973) j 8. S. Sakata, Prog. Theor. Phys. j[6, 686 (1956) [ 9. M. Gell-Mann, Phys. Letters 8, 2"\k (196UJ ^ I? 10. G. Zweig, unpublished I.. 11. E.P. Wigner, Phys. Rev. U3, 252 (1933) and I Z. Physik 83, 253 (1933) I"; 12. H. Becquerel, Compt. Rend. 122 (1896) \-\ r i 13. E. Fermi, Z. Physik 88, 161 (193U) %\ 1U. T.D. Lee and C.N. Yang, Phys. Rev. JOU, 25^ (1956) | N. Bohr and F. Kalckar, Kgl. Danske Videnskap. Selskab, 15. N. Bohr, Nature J3J, 3^^ (1936) f Mat.-fys. Medd. _1_U, 10 (1937) 16. G. Breit and E.P. Wigner, Phys. Rev. J+9, 519, 6U2 (1936) 17. M. Gell-Mann, Report CTSL-20 (1961) Y. Ne'eman, Nucl. Phys. 26, 222 (1961) 18. See G.F. Chew, The Analytic S-matrix, W.A.Benjamin (1966) f 19» See for example J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields, Me Graw-Hill (196U) and references therein 20. E.C.G. Sudarshan and R.E. Marshak, Proc. Conf. of Newly Discovered Particles, Padua (1957) R.P. Feynman and M. Gell-Mann, Phys. Rev. _10£, 193 (1958 21. S. Vfeinberg, Fnys. Rev. Letters _1£, 126U (1967); 2J_, 1688 (1971) Abdus Salam in Nobel Symposium 8, ed. N. Svartholm Almqyist & Wiksell, 1968 22. H. Yukawa, Phys. Rev. 77., 219 (1950) 23. Y. Nambu, Lectures at the Copenhagen Summer Symposium (1970) P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl.Phys. B56, 109 (1973) - 97 - I \ 2k. C.N. Yang and R.L. Mills, Phys. Rev. ^6, 191 (195U) I 25. J.A. Wheeler, Phys. Rev. ^2, 1107 (1937) | W. Heisenberg, Z. Physik .120, 513 and 673 (19U3) \ 26. G.F. Chew, S-matrix Theory of Strong Interactions c W.A. Benjamin (1961) and ref. 18 [- 27. S. Mandelstam, Phys. Rev. 112, 13^ (1958) I 28. T. Regge, Nuovo Cim. jjt, 951 (1959) and JjJ, 9^7 (i960) I 29. G.F. Chew and F.E. Low, Phys. Rev. JM3, 16U0 (1959) [ C. Goebel, Phys. Rev. Letters 1, 337 (1958)

I'. 30. S.B. Treiman and C.N. Yang, Phys. Rev. Letters _8, 1U0 (1962) p; 31. R.T. Deck, Phys. Rev. Letters _13_, 169 (196U) I 32. K. Zalewski, talk presented at the 2nd Aix International I Conference on Elementary Particles, 1973 [; 33. R. Dolen, D. Horn and C. Schmid, Phys. Rev. _i66, 1768 (1968) \: 3k. E.L. Berger, Phys. Rev. 179, 1567 (1969) JL; 35- H.R. Rubinstein, Phys. Letters 33B, U77 (1970) H- 36. See for example T.D. Lee and C.N. Yang, Phys. Rev. 109, 1755 (1958) H] M. Peshkin, Phys. Rev. J23, 637 (1961) I | N. Byers and S. Fenster, Phys. Rev. Letters V\_t 52 (1963) j C. Zemach, Phys. Rev. 133B, 1201 (,196k) 37- R.P. Feynman, Phys. Rev. Letters 23, 1U15 (1969) 38. L.G. Ratner et al., Phys. Rev. Letters 27, 68 (1971) 39^ G. Jarlskog, private communication 1+0. M. Froissart, Phys. Rev. 123. 1053 (1961) 1*1. See for example Bateman Manuscript Project vol.1 Me Graw-Hill (1953) k2. G. Vene-iano, Nuovo Cim. 57A, 190 (1968) k3. British-Scandinavian Collaboration, CERN preprints (1973) kk. H.M. Chan and J. Paton, Nucl. Phys. B1£, 519 (1969) 1*5. C. Lovelace, Phys. Letters 28B, 265 (1968) J.A. Shapiro, Phys. Rev. JJ9, I3U5 (1969) k6. S.L. Adle- and R.F. Dashen, Current Algebras, W.A. Benjamin (1968) 1+7. Z. Koba and H.B. Nielsen, Nucl. Phys. B10., 633 (1969) U8. K. Bardakci and H. Ruegg, Phys. Rev. J_82, 188U (1969) k9. See for example M. Jacob, Brandeis lecture notes (1970) 50. S. Fubini, D. Gordon and G. Veneziano, Phys. Letters 29B, 679 (1969) Y. Nambu, Symmetries and Quark Models ed. by R. Chand, Gordon and Breach (1969), p. 269 51. M.A. Virasoro, Phys. Rev. D1., 2933 (1970) 52. R. Hagedorn, Nuovo Cim. j>6A, 1027 (1968) it. - 98 -

53. R- Carlitz, S. Frautschi and W. Nahm, Enrico Fermi Inst. Preprint EFI 72-56 (1972) 5U. S. Frautschi, Phys. Rev. D3, 2821 (1971) and ref. 52 55. E. Del Giudice , P. Di Vecchia and S. Fubini, Annals of Physics J0_, 378 (1972) 56. P. Goddard and C.B. Thorn, Phys. Letters ]tOB, 235 (1972) 57. R.C. Brower, Phys. Rev. D_6, 1655 (1972) 58. K. Bardakci, M.B. Halpern and J.A. Shapiro, Phys. Rev. 185, 1910 (1969) D. Amati, C. Bouchiat and J.L. Gervais, Nuovo Cim. Letters £, 399 (1969) 59- R.P. Feynman, Acta Phys. Pol. 2^, 697 (1963), Magic without Magic, ed. J. Klauder , Freeman (1972), p. 355 60. V.N. Popov and L.D. Fadeev NAL-THY-57 (1972), originally a Kiev Report No. ITP 67-36 (1967) 61. L. Caneschi, A. Schwimmer and G. Veneziano, Phys. Letters 30B, 351 (1969) D. Amati, M. Le Bellac and D. Olive, Nuovo Cim. 66_A, 831 (1970) D.J. Gross, A. Neveu, J. Scherk and J.H. Schwarz, Phys. Rev. D2, 697 (1970) L. Susskind, Nuovo Cim. 69A, 1+57 (1970) 62. C. Lovelace, Phys. Letters 3UB, 500 (1971) 63. E. Cremmer and J. Scherk, Nucl. Phys. B5J), 222 (1972) 6k. M.A. Virasoro, Phys. Rev. J77, 2309 (1969) J.A. Shapiro, Phys. Letters 33B, 361 (1970) E. Del Giudice and P. Di Vecchia, Nuovo Cim. _5A, 90 (1971) M. Yoshimura, Phys. Letters 3^B, 79 (1971) 65. D. Olive and J. Scherk, Phys. Letters kkB, 296 (1973) 66. A. Neveu and J.H. Schwarz, Nucl. Phys. B31_, 86 (1971) A. Neveu, J.H. Schwarz and C.B. Thorn, Phys. Letters 35B, 529 (1971) 67. P. Ramond, Phys. Rev. D3, 2U15 (1971) 68. P. Goddard and R.E. Waltz, Nucl. Phys. B3jt, 99 (1971) 69. E. Corrigan and D. Olive, Nuovo Cim. 11A, "Jk9 (1972) 70. E. Corrigan, P. Goddard, D. Olive and R.A. Smith, University of Durham preprint Oct. 1973 71. Y. Nambu see ref. 50 H. B. Nielsen, XV Int. Conf. on High Energy Physics, Kiev 1970 L. Susskind, Nuovo Cim. 69_A, ^57 (1970) 72. E.D. Bloom and F.J. Gilman, Phys. Rev. Letters 25., 11*10 (1970) 73. H.B. Nielsen see ref. 71 fk. J.L. Gervais and B. Sakita, Phys. Rev. Letters 30., 719 (1973) 75. S. Mandelstam, Berkeley preprint I, II and III, August 1973 76. Y. Iwasaki and K. Kikkawa, Phys. Rev. D8, kko (1973) 77. See E.T. Whittaker and G.N. Watson, Modern Analysis, Cambridge 1952, p. kjk 78. This observation has also been made by J.H. Schwarz, private communication I - 99 -

79. A. Kihlberg, Nuovo Cim. J53A 592 (1968) 80. The two last references in ref. 6k 81. H.B. Nielsen, unpublished J.M. Dethlevsen and H.B. Nielsen, NBI preprint, April 1972 82. See M.A. Naimark, Normed Rings, Wolters-Nordhoff (1970) 83. J.H. Schwarz, Caltech preprint, Sept. 1973 8U. P.W. Higgs, Phys. Letters JJ2, 132 (19^), Phys. Rev. Letters J3.» 508 (196U), Phys. Rev. J_U5., 1156 (1966) 85. L. Brink and H.B. Nielsen, to be published. See also M. Gell-Mann, concluding talk at the XVI International Conference on High Energy Physics, Chicago-Batavia (1972)